3.423 \(\int \frac {\tan ^{\frac {3}{2}}(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx\)
Optimal. Leaf size=237 \[ \frac {B (a-b) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )}-\frac {B (a-b) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )}+\frac {B (a+b) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}-\frac {B (a+b) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}+\frac {2 a^{3/2} B \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {b} d \left (a^2+b^2\right )} \]
[Out]
-1/2*(a-b)*B*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)/d*2^(1/2)-1/2*(a-b)*B*arctan(1+2^(1/2)*tan(d*x+c)^(
1/2))/(a^2+b^2)/d*2^(1/2)+1/4*(a+b)*B*ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)/d*2^(1/2)-1/4*(a+b)*
B*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)/d*2^(1/2)+2*a^(3/2)*B*arctan(b^(1/2)*tan(d*x+c)^(1/2)/a^
(1/2))/(a^2+b^2)/d/b^(1/2)
________________________________________________________________________________________
Rubi [A] time = 0.30, antiderivative size = 237, normalized size of antiderivative =
1.00, number of steps used = 15, number of rules used = 12, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) =
0.333, Rules used = {21, 3573, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205}
\[ \frac {2 a^{3/2} B \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {b} d \left (a^2+b^2\right )}+\frac {B (a-b) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )}-\frac {B (a-b) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )}+\frac {B (a+b) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}-\frac {B (a+b) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
[In]
Int[(Tan[c + d*x]^(3/2)*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]
[Out]
((a - b)*B*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - ((a - b)*B*ArcTan[1 + Sqrt[2]*Sqr
t[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) + (2*a^(3/2)*B*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[b
]*(a^2 + b^2)*d) + ((a + b)*B*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) -
((a + b)*B*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d)
Rule 21
Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])
Rule 63
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]
Rule 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 205
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rule 617
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 628
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Rule 1162
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Rule 1165
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Rule 1168
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[-(a*c)]
Rule 3534
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2/f, Subst[I
nt[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2,
0] && NeQ[c^2 + d^2, 0]
Rule 3573
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(3/2)/((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1
/(c^2 + d^2), Int[Simp[a^2*c - b^2*c + 2*a*b*d + (2*a*b*c - a^2*d + b^2*d)*Tan[e + f*x], x]/Sqrt[a + b*Tan[e +
f*x]], x], x] + Dist[(b*c - a*d)^2/(c^2 + d^2), Int[(1 + Tan[e + f*x]^2)/(Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan
[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2
, 0]
Rule 3634
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*
tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x]
/; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
Rubi steps
\begin {align*} \int \frac {\tan ^{\frac {3}{2}}(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx &=B \int \frac {\tan ^{\frac {3}{2}}(c+d x)}{a+b \tan (c+d x)} \, dx\\ &=\frac {B \int \frac {-a+b \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{a^2+b^2}+\frac {\left (a^2 B\right ) \int \frac {1+\tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{a^2+b^2}\\ &=\frac {(2 B) \operatorname {Subst}\left (\int \frac {-a+b x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}+\frac {\left (a^2 B\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d}\\ &=\frac {\left (2 a^2 B\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac {((a-b) B) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac {((a+b) B) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}\\ &=\frac {2 a^{3/2} B \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {b} \left (a^2+b^2\right ) d}-\frac {((a-b) B) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}-\frac {((a-b) B) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}+\frac {((a+b) B) \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}+\frac {((a+b) B) \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}\\ &=\frac {2 a^{3/2} B \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {b} \left (a^2+b^2\right ) d}+\frac {(a+b) B \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a+b) B \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {((a-b) B) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}+\frac {((a-b) B) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}\\ &=\frac {(a-b) B \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a-b) B \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}+\frac {2 a^{3/2} B \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {b} \left (a^2+b^2\right ) d}+\frac {(a+b) B \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a+b) B \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.29, size = 228, normalized size = 0.96 \[ \frac {B \left (3 a \left (8 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )+2 \sqrt {2} \sqrt {b} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-2 \sqrt {2} \sqrt {b} \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )+\sqrt {2} \sqrt {b} \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )-\sqrt {2} \sqrt {b} \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )\right )+8 b^{3/2} \tan ^{\frac {3}{2}}(c+d x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\tan ^2(c+d x)\right )\right )}{12 \sqrt {b} d \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
[In]
Integrate[(Tan[c + d*x]^(3/2)*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]
[Out]
(B*(3*a*(2*Sqrt[2]*Sqrt[b]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 2*Sqrt[2]*Sqrt[b]*ArcTan[1 + Sqrt[2]*Sqrt[
Tan[c + d*x]]] + 8*Sqrt[a]*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]] + Sqrt[2]*Sqrt[b]*Log[1 - Sqrt[2]*Sqrt
[Tan[c + d*x]] + Tan[c + d*x]] - Sqrt[2]*Sqrt[b]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]) + 8*b^(3/
2)*Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2]*Tan[c + d*x]^(3/2)))/(12*Sqrt[b]*(a^2 + b^2)*d)
________________________________________________________________________________________
fricas [B] time = 14.15, size = 8860, normalized size = 37.38 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tan(d*x+c)^(3/2)*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c))^2,x, algorithm="fricas")
[Out]
[-1/4*(4*sqrt(2)*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^5*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 - 2*(a^5*b +
2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*(B^4/((a^
4 + 2*a^2*b^2 + b^4)*d^4))^(3/4)*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^
2*b^6 + b^8)*d^4))*arctan(((B^6*a^8 + 2*B^6*a^6*b^2 - 2*B^6*a^2*b^6 - B^6*b^8)*d^4*sqrt(B^4/((a^4 + 2*a^2*b^2
+ b^4)*d^4))*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) +
sqrt(2)*((a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*d^7*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt((
B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + (B^2*a^6*b + 3*B^2
*a^4*b^3 + 3*B^2*a^2*b^5 + B^2*b^7)*d^5*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4
+ 4*a^2*b^6 + b^8)*d^4)))*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^
4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*sqrt(((B^4*a^6 - B^4*a^4*b^2 - B^4*a^2*
b^4 + B^4*b^6)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + sqrt(2)*((B^3*a^6*b - B^3*a^4*b^3 -
B^3*a^2*b^5 + B^3*b^7)*d^3*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + (B^5*a^5 - 2*B^5*a^3*b^2 + B
^5*a*b^4)*d*cos(d*x + c))*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4
/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(B^4/((a
^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (B^6*a^4 - 2*B^6*a^2*b^2 + B^6*b^4)*sin(d*x + c))/cos(d*x + c))*(B^4/((a^4
+ 2*a^2*b^2 + b^4)*d^4))^(3/4) + sqrt(2)*((B^3*a^11 + 3*B^3*a^9*b^2 + 2*B^3*a^7*b^4 - 2*B^3*a^5*b^6 - 3*B^3*a
^3*b^8 - B^3*a*b^10)*d^7*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^
8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + (B^5*a^8*b + 2*B^5*a^6*b^3 - 2*B^5*a^2*b^7 - B^5*b^9)*d^5
*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((B^2*a^
4 + 2*B^2*a^2*b^2 + B^2*b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*
a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4))/(B^
10*a^4 - 2*B^10*a^2*b^2 + B^10*b^4)) + 4*sqrt(2)*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^5*sqrt((B^2*a^4 + 2*B^2
*a^2*b^2 + B^2*b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B
^2*a^2*b^2 + B^2*b^4))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4)*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^
8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*arctan(-((B^6*a^8 + 2*B^6*a^6*b^2 - 2*B^6*a^2*b^6 - B^6*b^8
)*d^4*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a
^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - sqrt(2)*((a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*d^7*sqrt(B^4/((a^
4 + 2*a^2*b^2 + b^4)*d^4))*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6
+ b^8)*d^4)) + (B^2*a^6*b + 3*B^2*a^4*b^3 + 3*B^2*a^2*b^5 + B^2*b^7)*d^5*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b
^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 - 2*(a^5*b
+ 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*sqrt((
(B^4*a^6 - B^4*a^4*b^2 - B^4*a^2*b^4 + B^4*b^6)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - sqr
t(2)*((B^3*a^6*b - B^3*a^4*b^3 - B^3*a^2*b^5 + B^3*b^7)*d^3*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x +
c) + (B^5*a^5 - 2*B^5*a^3*b^2 + B^5*a*b^4)*d*cos(d*x + c))*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 - 2*(a^5*b
+ 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*sqrt(si
n(d*x + c)/cos(d*x + c))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (B^6*a^4 - 2*B^6*a^2*b^2 + B^6*b^4)*sin(d
*x + c))/cos(d*x + c))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4) - sqrt(2)*((B^3*a^11 + 3*B^3*a^9*b^2 + 2*B^3*
a^7*b^4 - 2*B^3*a^5*b^6 - 3*B^3*a^3*b^8 - B^3*a*b^10)*d^7*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt((B^4*a^
4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + (B^5*a^8*b + 2*B^5*a^6*b
^3 - 2*B^5*a^2*b^7 - B^5*b^9)*d^5*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a
^2*b^6 + b^8)*d^4)))*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^
4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(B^4/((a^4 +
2*a^2*b^2 + b^4)*d^4))^(3/4))/(B^10*a^4 - 2*B^10*a^2*b^2 + B^10*b^4)) - 2*B^5*a*sqrt(-a/b)*log(-(6*a*b*cos(d*x
+ c)*sin(d*x + c) - (a^2 - b^2)*cos(d*x + c)^2 - b^2 + 4*(a*b*cos(d*x + c)^2 - b^2*cos(d*x + c)*sin(d*x + c))
*sqrt(-a/b)*sqrt(sin(d*x + c)/cos(d*x + c)))/(2*a*b*cos(d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 + b
^2)) + sqrt(2)*(2*(B^2*a^3*b + B^2*a*b^3)*d^3*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + (B^4*a^2 + B^4*b^2)*d)
*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)
*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4)*log(((B^4*a^6 - B^4*a^4
*b^2 - B^4*a^2*b^4 + B^4*b^6)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + sqrt(2)*((B^3*a^6*b -
B^3*a^4*b^3 - B^3*a^2*b^5 + B^3*b^7)*d^3*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + (B^5*a^5 - 2*
B^5*a^3*b^2 + B^5*a*b^4)*d*cos(d*x + c))*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^
5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*sqrt(sin(d*x + c)/cos(d*x
+ c))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (B^6*a^4 - 2*B^6*a^2*b^2 + B^6*b^4)*sin(d*x + c))/cos(d*x +
c)) - sqrt(2)*(2*(B^2*a^3*b + B^2*a*b^3)*d^3*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + (B^4*a^2 + B^4*b^2)*d)
*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)
*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4)*log(((B^4*a^6 - B^4*a^4
*b^2 - B^4*a^2*b^4 + B^4*b^6)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - sqrt(2)*((B^3*a^6*b -
B^3*a^4*b^3 - B^3*a^2*b^5 + B^3*b^7)*d^3*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + (B^5*a^5 - 2*
B^5*a^3*b^2 + B^5*a*b^4)*d*cos(d*x + c))*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^
5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*sqrt(sin(d*x + c)/cos(d*x
+ c))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (B^6*a^4 - 2*B^6*a^2*b^2 + B^6*b^4)*sin(d*x + c))/cos(d*x +
c)))/((B^4*a^2 + B^4*b^2)*d), -1/4*(4*sqrt(2)*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^5*sqrt((B^2*a^4 + 2*B^2*a
^2*b^2 + B^2*b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2
*a^2*b^2 + B^2*b^4))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4)*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8
+ 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*arctan(((B^6*a^8 + 2*B^6*a^6*b^2 - 2*B^6*a^2*b^6 - B^6*b^8)*d
^4*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*
b^4 + 4*a^2*b^6 + b^8)*d^4)) + sqrt(2)*((a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*d^7*sqrt(B^4/((a^4 +
2*a^2*b^2 + b^4)*d^4))*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b
^8)*d^4)) + (B^2*a^6*b + 3*B^2*a^4*b^3 + 3*B^2*a^2*b^5 + B^2*b^7)*d^5*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)
/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 - 2*(a^5*b +
2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*sqrt(((B^
4*a^6 - B^4*a^4*b^2 - B^4*a^2*b^4 + B^4*b^6)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + sqrt(2
)*((B^3*a^6*b - B^3*a^4*b^3 - B^3*a^2*b^5 + B^3*b^7)*d^3*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c)
+ (B^5*a^5 - 2*B^5*a^3*b^2 + B^5*a*b^4)*d*cos(d*x + c))*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 - 2*(a^5*b + 2
*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*sqrt(sin(d
*x + c)/cos(d*x + c))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (B^6*a^4 - 2*B^6*a^2*b^2 + B^6*b^4)*sin(d*x
+ c))/cos(d*x + c))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4) + sqrt(2)*((B^3*a^11 + 3*B^3*a^9*b^2 + 2*B^3*a^7
*b^4 - 2*B^3*a^5*b^6 - 3*B^3*a^3*b^8 - B^3*a*b^10)*d^7*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt((B^4*a^4 -
2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + (B^5*a^8*b + 2*B^5*a^6*b^3
- 2*B^5*a^2*b^7 - B^5*b^9)*d^5*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*
b^6 + b^8)*d^4)))*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 +
2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(B^4/((a^4 + 2*a
^2*b^2 + b^4)*d^4))^(3/4))/(B^10*a^4 - 2*B^10*a^2*b^2 + B^10*b^4)) + 4*sqrt(2)*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 +
b^6)*d^5*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^
2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4)*sqrt((B^4*a^4 -
2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*arctan(-((B^6*a^8 + 2*B^6*a^6
*b^2 - 2*B^6*a^2*b^6 - B^6*b^8)*d^4*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^
4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) - sqrt(2)*((a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*
b^6 + a*b^8)*d^7*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^
6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)) + (B^2*a^6*b + 3*B^2*a^4*b^3 + 3*B^2*a^2*b^5 + B^2*b^7)*d^5*sqrt((B
^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((B^2*a^4 + 2*B^
2*a^2*b^2 + B^2*b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*
B^2*a^2*b^2 + B^2*b^4))*sqrt(((B^4*a^6 - B^4*a^4*b^2 - B^4*a^2*b^4 + B^4*b^6)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 +
b^4)*d^4))*cos(d*x + c) - sqrt(2)*((B^3*a^6*b - B^3*a^4*b^3 - B^3*a^2*b^5 + B^3*b^7)*d^3*sqrt(B^4/((a^4 + 2*a
^2*b^2 + b^4)*d^4))*cos(d*x + c) + (B^5*a^5 - 2*B^5*a^3*b^2 + B^5*a*b^4)*d*cos(d*x + c))*sqrt((B^2*a^4 + 2*B^2
*a^2*b^2 + B^2*b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B
^2*a^2*b^2 + B^2*b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) + (B^6*a^4 -
2*B^6*a^2*b^2 + B^6*b^4)*sin(d*x + c))/cos(d*x + c))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4) - sqrt(2)*((B^3
*a^11 + 3*B^3*a^9*b^2 + 2*B^3*a^7*b^4 - 2*B^3*a^5*b^6 - 3*B^3*a^3*b^8 - B^3*a*b^10)*d^7*sqrt(B^4/((a^4 + 2*a^2
*b^2 + b^4)*d^4))*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^
4)) + (B^5*a^8*b + 2*B^5*a^6*b^3 - 2*B^5*a^2*b^7 - B^5*b^9)*d^5*sqrt((B^4*a^4 - 2*B^4*a^2*b^2 + B^4*b^4)/((a^8
+ 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4)))*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 - 2*(a^5*b + 2*a^3*
b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*sqrt(sin(d*x +
c)/cos(d*x + c))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(3/4))/(B^10*a^4 - 2*B^10*a^2*b^2 + B^10*b^4)) - 8*B^5*a*
sqrt(a/b)*arctan(b*sqrt(a/b)*sqrt(sin(d*x + c)/cos(d*x + c))/a) + sqrt(2)*(2*(B^2*a^3*b + B^2*a*b^3)*d^3*sqrt(
B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + (B^4*a^2 + B^4*b^2)*d)*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 - 2*(a^5*b
+ 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*(B^4/(
(a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4)*log(((B^4*a^6 - B^4*a^4*b^2 - B^4*a^2*b^4 + B^4*b^6)*d^2*sqrt(B^4/((a^4 +
2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + sqrt(2)*((B^3*a^6*b - B^3*a^4*b^3 - B^3*a^2*b^5 + B^3*b^7)*d^3*sqrt(B^4/
((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + (B^5*a^5 - 2*B^5*a^3*b^2 + B^5*a*b^4)*d*cos(d*x + c))*sqrt((B^2*
a^4 + 2*B^2*a^2*b^2 + B^2*b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^
2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) +
(B^6*a^4 - 2*B^6*a^2*b^2 + B^6*b^4)*sin(d*x + c))/cos(d*x + c)) - sqrt(2)*(2*(B^2*a^3*b + B^2*a*b^3)*d^3*sqrt(
B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + (B^4*a^2 + B^4*b^2)*d)*sqrt((B^2*a^4 + 2*B^2*a^2*b^2 + B^2*b^4 - 2*(a^5*b
+ 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*(B^4/(
(a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4)*log(((B^4*a^6 - B^4*a^4*b^2 - B^4*a^2*b^4 + B^4*b^6)*d^2*sqrt(B^4/((a^4 +
2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) - sqrt(2)*((B^3*a^6*b - B^3*a^4*b^3 - B^3*a^2*b^5 + B^3*b^7)*d^3*sqrt(B^4/
((a^4 + 2*a^2*b^2 + b^4)*d^4))*cos(d*x + c) + (B^5*a^5 - 2*B^5*a^3*b^2 + B^5*a*b^4)*d*cos(d*x + c))*sqrt((B^2*
a^4 + 2*B^2*a^2*b^2 + B^2*b^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d^2*sqrt(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4)))/(B^
2*a^4 - 2*B^2*a^2*b^2 + B^2*b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(B^4/((a^4 + 2*a^2*b^2 + b^4)*d^4))^(1/4) +
(B^6*a^4 - 2*B^6*a^2*b^2 + B^6*b^4)*sin(d*x + c))/cos(d*x + c)))/((B^4*a^2 + B^4*b^2)*d)]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tan(d*x+c)^(3/2)*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c))^2,x, algorithm="giac")
[Out]
Timed out
________________________________________________________________________________________
maple [A] time = 0.34, size = 305, normalized size = 1.29 \[ \frac {2 a^{2} \arctan \left (\frac {\left (\sqrt {\tan }\left (d x +c \right )\right ) b}{\sqrt {a b}}\right ) B}{d \left (a^{2}+b^{2}\right ) \sqrt {a b}}-\frac {B \sqrt {2}\, \ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right ) a}{4 d \left (a^{2}+b^{2}\right )}-\frac {B \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) a}{2 d \left (a^{2}+b^{2}\right )}-\frac {B \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) a}{2 d \left (a^{2}+b^{2}\right )}+\frac {B \sqrt {2}\, \ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right ) b}{4 d \left (a^{2}+b^{2}\right )}+\frac {B \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) b}{2 d \left (a^{2}+b^{2}\right )}+\frac {B \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) b}{2 d \left (a^{2}+b^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tan(d*x+c)^(3/2)*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c))^2,x)
[Out]
2/d*a^2/(a^2+b^2)/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))*B-1/4/d/(a^2+b^2)*B*2^(1/2)*ln((1+2^(1/2)
*tan(d*x+c)^(1/2)+tan(d*x+c))/(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*a-1/2/d/(a^2+b^2)*B*2^(1/2)*arctan(1+2^
(1/2)*tan(d*x+c)^(1/2))*a-1/2/d/(a^2+b^2)*B*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a+1/4/d/(a^2+b^2)*B*2^
(1/2)*ln((1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*b+1/2/d/(a^2+b^2)*B*
2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*b+1/2/d/(a^2+b^2)*B*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*b
________________________________________________________________________________________
maxima [A] time = 0.83, size = 173, normalized size = 0.73 \[ \frac {\frac {8 \, B a^{2} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{2} + b^{2}\right )} \sqrt {a b}} - \frac {{\left (2 \, \sqrt {2} {\left (a - b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a - b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} {\left (a + b\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} {\left (a + b\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )} B}{a^{2} + b^{2}}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tan(d*x+c)^(3/2)*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c))^2,x, algorithm="maxima")
[Out]
1/4*(8*B*a^2*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^2 + b^2)*sqrt(a*b)) - (2*sqrt(2)*(a - b)*arctan(1/2*sq
rt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*(a - b)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c)
))) + sqrt(2)*(a + b)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) - sqrt(2)*(a + b)*log(-sqrt(2)*sqrt(t
an(d*x + c)) + tan(d*x + c) + 1))*B/(a^2 + b^2))/d
________________________________________________________________________________________
mupad [B] time = 32.87, size = 16878, normalized size = 71.22 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((tan(c + d*x)^(3/2)*(B*a + B*b*tan(c + d*x)))/(a + b*tan(c + d*x))^2,x)
[Out]
(log(((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)
^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) + (768*B*a^2*b^4*(a^2 + b^2))/d)*((
4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))
^(1/2))/4 + (64*B^2*a*b^2*tan(c + d*x)^(1/2)*(2*a^8 + 15*b^8 - 17*a^2*b^6 + 51*a^4*b^4 + 21*a^6*b^2))/(d^2*(a^
2 + b^2)^2))*((4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a*b^5*d^2)/(d^4*
(a^2 + b^2)^4))^(1/2))/4 + (32*B^3*a*b^3*(4*a^8 + b^8 - 77*a^2*b^6 + 47*a^4*b^4 + 33*a^6*b^2))/(d^3*(a^2 + b^2
)^3))*((4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a*b^5*d^2)/(d^4*(a^2 +
b^2)^4))^(1/2))/4 + (16*B^4*b^3*tan(c + d*x)^(1/2)*(a^10 - 2*b^10 - 4*a^2*b^8 - 27*a^4*b^6 + 15*a^6*b^4 + 9*a^
8*b^2))/(d^4*(a^2 + b^2)^4))*((4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*
a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (8*B^5*a^2*b^4*(a^6 + 10*b^6 + 27*a^2*b^4 + 10*a^4*b^2))/(d^5*(a^2
+ b^2)^4))*(((192*B^4*a^2*b^10*d^4 - 16*B^4*b^12*d^4 - 608*B^4*a^4*b^8*d^4 + 192*B^4*a^6*b^6*d^4 - 16*B^4*a^8*
b^4*d^4)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a*b^5*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4
*a^6*b^2*d^4))^(1/2))/4 + (log(((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*(-(4*(-B^4*b^4*d^4
*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) + (768*B
*a^2*b^4*(a^2 + b^2))/d)*(-(4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a*b
^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*B^2*a*b^2*tan(c + d*x)^(1/2)*(2*a^8 + 15*b^8 - 17*a^2*b^6 + 51*a^4
*b^4 + 21*a^6*b^2))/(d^2*(a^2 + b^2)^2))*(-(4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*
d^2 - 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (32*B^3*a*b^3*(4*a^8 + b^8 - 77*a^2*b^6 + 47*a^4*b^4 +
33*a^6*b^2))/(d^3*(a^2 + b^2)^3))*(-(4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 -
16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (16*B^4*b^3*tan(c + d*x)^(1/2)*(a^10 - 2*b^10 - 4*a^2*b^8 -
27*a^4*b^6 + 15*a^6*b^4 + 9*a^8*b^2))/(d^4*(a^2 + b^2)^4))*(-(4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2)
+ 16*B^2*a^3*b^3*d^2 - 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (8*B^5*a^2*b^4*(a^6 + 10*b^6 + 27*a^
2*b^4 + 10*a^4*b^2))/(d^5*(a^2 + b^2)^4))*(-((192*B^4*a^2*b^10*d^4 - 16*B^4*b^12*d^4 - 608*B^4*a^4*b^8*d^4 + 1
92*B^4*a^6*b^6*d^4 - 16*B^4*a^8*b^4*d^4)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a*b^5*d^2)/(a^8*d^4 + b^8*d^4 + 4
*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2))/4 - log((8*B^5*a^2*b^4*(a^6 + 10*b^6 + 27*a^2*b^4 + 10*a
^4*b^2))/(d^5*(a^2 + b^2)^4) - ((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-B^4*b^4*d^4*
(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) - (768*B*
a^2*b^4*(a^2 + b^2))/d)*((4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a*b^5
*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*B^2*a*b^2*tan(c + d*x)^(1/2)*(2*a^8 + 15*b^8 - 17*a^2*b^6 + 51*a^4*b
^4 + 21*a^6*b^2))/(d^2*(a^2 + b^2)^2))*((4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2
+ 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (32*B^3*a*b^3*(4*a^8 + b^8 - 77*a^2*b^6 + 47*a^4*b^4 + 33
*a^6*b^2))/(d^3*(a^2 + b^2)^3))*((4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B
^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (16*B^4*b^3*tan(c + d*x)^(1/2)*(a^10 - 2*b^10 - 4*a^2*b^8 - 27*a
^4*b^6 + 15*a^6*b^4 + 9*a^8*b^2))/(d^4*(a^2 + b^2)^4))*((4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16
*B^2*a^3*b^3*d^2 + 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4)*(((192*B^4*a^2*b^10*d^4 - 16*B^4*b^12*d^4
- 608*B^4*a^4*b^8*d^4 + 192*B^4*a^6*b^6*d^4 - 16*B^4*a^8*b^4*d^4)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a*b^5*d^
2)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2) - log((8*B^5*a^2*b^4*(a
^6 + 10*b^6 + 27*a^2*b^4 + 10*a^4*b^2))/(d^5*(a^2 + b^2)^4) - ((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(
a^2 + b^2)^2*(-(4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a*b^5*d^2)/(d^4
*(a^2 + b^2)^4))^(1/2) - (768*B*a^2*b^4*(a^2 + b^2))/d)*(-(4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) +
16*B^2*a^3*b^3*d^2 - 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*B^2*a*b^2*tan(c + d*x)^(1/2)*(2*a^8
+ 15*b^8 - 17*a^2*b^6 + 51*a^4*b^4 + 21*a^6*b^2))/(d^2*(a^2 + b^2)^2))*(-(4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*
b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (32*B^3*a*b^3*(4*a^8 +
b^8 - 77*a^2*b^6 + 47*a^4*b^4 + 33*a^6*b^2))/(d^3*(a^2 + b^2)^3))*(-(4*(-B^4*b^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2
)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (16*B^4*b^3*tan(c + d*x)^(1/2
)*(a^10 - 2*b^10 - 4*a^2*b^8 - 27*a^4*b^6 + 15*a^6*b^4 + 9*a^8*b^2))/(d^4*(a^2 + b^2)^4))*(-(4*(-B^4*b^4*d^4*(
a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a*b^5*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4)*(-((19
2*B^4*a^2*b^10*d^4 - 16*B^4*b^12*d^4 - 608*B^4*a^4*b^8*d^4 + 192*B^4*a^6*b^6*d^4 - 16*B^4*a^8*b^4*d^4)^(1/2) +
16*B^2*a^3*b^3*d^2 - 16*B^2*a*b^5*d^2)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^
2*d^4))^(1/2) + (log(((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-B^4*a^4*d^4*(a^4 + b^4
- 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) - (128*B*a^2*b^2*(5
*b^4 - a^4 + 4*a^2*b^2))/d)*((4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a
^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*B^2*a^3*b^2*tan(c + d*x)^(1/2)*(a^6 - 15*b^6 + 35*a^2*b^4 - 13*a
^4*b^2))/(d^2*(a^2 + b^2)^2))*((4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2
*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (32*B^3*a^5*b*(a^6 - 39*b^6 + 43*a^2*b^4 - 13*a^4*b^2))/(d^3*(a^2
+ b^2)^3))*((4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a^5*b*d^2)/(d^4*(a
^2 + b^2)^4))^(1/2))/4 - (16*B^4*a^4*b*tan(c + d*x)^(1/2)*(a^8 + 2*b^8 - 5*a^2*b^6 + 17*a^4*b^4 - 7*a^6*b^2))/
(d^4*(a^2 + b^2)^4))*((4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a^5*b*d^
2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (16*B^5*a^6*b^4*(a^2 - 3*b^2))/(d^5*(a^2 + b^2)^4))*(((192*B^4*a^6*b^6*d^4
- 16*B^4*a^4*b^8*d^4 - 16*B^4*a^12*d^4 - 608*B^4*a^8*b^4*d^4 + 192*B^4*a^10*b^2*d^4)^(1/2) - 16*B^2*a^3*b^3*d^
2 + 16*B^2*a^5*b*d^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2))/4 + (log((((
(((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*(-(4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2)
+ 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) - (128*B*a^2*b^2*(5*b^4 - a^4 + 4*a^2*b^2)
)/d)*(-(4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d^2)/(d^4*(a^2 +
b^2)^4))^(1/2))/4 + (64*B^2*a^3*b^2*tan(c + d*x)^(1/2)*(a^6 - 15*b^6 + 35*a^2*b^4 - 13*a^4*b^2))/(d^2*(a^2 + b
^2)^2))*(-(4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d^2)/(d^4*(a^2
+ b^2)^4))^(1/2))/4 + (32*B^3*a^5*b*(a^6 - 39*b^6 + 43*a^2*b^4 - 13*a^4*b^2))/(d^3*(a^2 + b^2)^3))*(-(4*(-B^4
*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))
/4 - (16*B^4*a^4*b*tan(c + d*x)^(1/2)*(a^8 + 2*b^8 - 5*a^2*b^6 + 17*a^4*b^4 - 7*a^6*b^2))/(d^4*(a^2 + b^2)^4))
*(-(4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)
^4))^(1/2))/4 + (16*B^5*a^6*b^4*(a^2 - 3*b^2))/(d^5*(a^2 + b^2)^4))*(-((192*B^4*a^6*b^6*d^4 - 16*B^4*a^4*b^8*d
^4 - 16*B^4*a^12*d^4 - 608*B^4*a^8*b^4*d^4 + 192*B^4*a^10*b^2*d^4)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d
^2)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))^(1/2))/4 - log((16*B^5*a^6*b^4*(a^2 -
3*b^2))/(d^5*(a^2 + b^2)^4) - ((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-B^4*a^4*d^4*
(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) + (128*B*
a^2*b^2*(5*b^4 - a^4 + 4*a^2*b^2))/d)*((4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2
+ 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*B^2*a^3*b^2*tan(c + d*x)^(1/2)*(a^6 - 15*b^6 + 35*a^2*
b^4 - 13*a^4*b^2))/(d^2*(a^2 + b^2)^2))*((4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^
2 + 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (32*B^3*a^5*b*(a^6 - 39*b^6 + 43*a^2*b^4 - 13*a^4*b^2))/
(d^3*(a^2 + b^2)^3))*((4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a^5*b*d^
2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (16*B^4*a^4*b*tan(c + d*x)^(1/2)*(a^8 + 2*b^8 - 5*a^2*b^6 + 17*a^4*b^4 - 7*
a^6*b^2))/(d^4*(a^2 + b^2)^4))*((4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^
2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4)*(((192*B^4*a^6*b^6*d^4 - 16*B^4*a^4*b^8*d^4 - 16*B^4*a^12*d^4 - 60
8*B^4*a^8*b^4*d^4 + 192*B^4*a^10*b^2*d^4)^(1/2) - 16*B^2*a^3*b^3*d^2 + 16*B^2*a^5*b*d^2)/(16*a^8*d^4 + 16*b^8*
d^4 + 64*a^2*b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2) - log((16*B^5*a^6*b^4*(a^2 - 3*b^2))/(d^5*(a^2
+ b^2)^4) - ((((((((128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*(-(4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2
*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2) + (128*B*a^2*b^2*(5*b^4 - a
^4 + 4*a^2*b^2))/d)*(-(4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d^
2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*B^2*a^3*b^2*tan(c + d*x)^(1/2)*(a^6 - 15*b^6 + 35*a^2*b^4 - 13*a^4*b^2)
)/(d^2*(a^2 + b^2)^2))*(-(4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b
*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (32*B^3*a^5*b*(a^6 - 39*b^6 + 43*a^2*b^4 - 13*a^4*b^2))/(d^3*(a^2 + b^2)
^3))*(-(4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d^2)/(d^4*(a^2 +
b^2)^4))^(1/2))/4 - (16*B^4*a^4*b*tan(c + d*x)^(1/2)*(a^8 + 2*b^8 - 5*a^2*b^6 + 17*a^4*b^4 - 7*a^6*b^2))/(d^4*
(a^2 + b^2)^4))*(-(4*(-B^4*a^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d^2)/(
d^4*(a^2 + b^2)^4))^(1/2))/4)*(-((192*B^4*a^6*b^6*d^4 - 16*B^4*a^4*b^8*d^4 - 16*B^4*a^12*d^4 - 608*B^4*a^8*b^4
*d^4 + 192*B^4*a^10*b^2*d^4)^(1/2) + 16*B^2*a^3*b^3*d^2 - 16*B^2*a^5*b*d^2)/(16*a^8*d^4 + 16*b^8*d^4 + 64*a^2*
b^6*d^4 + 96*a^4*b^4*d^4 + 64*a^6*b^2*d^4))^(1/2) - (atan(-((((((16*(8*B^3*a^7*b^7*d^2 - 78*B^3*a^5*b^9*d^2 +
60*B^3*a^9*b^5*d^2 - 24*B^3*a^11*b^3*d^2 + 2*B^3*a^13*b*d^2))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d
^5 + 4*a^6*b^2*d^5) - (((16*tan(c + d*x)^(1/2)*(20*B^2*a^5*b^10*d^2 - 60*B^2*a^3*b^12*d^2 + 168*B^2*a^7*b^8*d^
2 + 40*B^2*a^9*b^6*d^2 - 44*B^2*a^11*b^4*d^2 + 4*B^2*a^13*b^2*d^2))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4
*b^4*d^4 + 4*a^6*b^2*d^4) + (((16*(40*B*a^2*b^14*d^4 + 192*B*a^4*b^12*d^4 + 360*B*a^6*b^10*d^4 + 320*B*a^8*b^8
*d^4 + 120*B*a^10*b^6*d^4 - 8*B*a^14*b^2*d^4))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*
d^5) - (8*tan(c + d*x)^(1/2)*(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2)*(32*b^17*d^4 + 160*a^2*b^15*d^4 + 288*a
^4*b^13*d^4 + 160*a^6*b^11*d^4 - 160*a^8*b^9*d^4 - 288*a^10*b^7*d^4 - 160*a^12*b^5*d^4 - 32*a^14*b^3*d^4))/((-
(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^6*b^3*d^2
))^(1/2)*(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4)))*(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^
2*a^5*b^2))/(2*(-(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^
2 + 4*a^6*b^3*d^2))^(1/2)))*(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2))/(2*(-(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a
^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^6*b^3*d^2))^(1/2)))*(B^2*a^7 + 9*B^2*a^3*b^
4 - 6*B^2*a^5*b^2))/(2*(-(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^
4*b^5*d^2 + 4*a^6*b^3*d^2))^(1/2)) + (16*tan(c + d*x)^(1/2)*(B^4*a^12*b + 2*B^4*a^4*b^9 - 5*B^4*a^6*b^7 + 17*B
^4*a^8*b^5 - 7*B^4*a^10*b^3))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))*(B^2*a^7 +
9*B^2*a^3*b^4 - 6*B^2*a^5*b^2)*1i)/(2*(-(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2
*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^6*b^3*d^2))^(1/2)) - (((((16*(8*B^3*a^7*b^7*d^2 - 78*B^3*a^5*b^9*d^2 + 60*B^3*a
^9*b^5*d^2 - 24*B^3*a^11*b^3*d^2 + 2*B^3*a^13*b*d^2))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a
^6*b^2*d^5) + (((16*tan(c + d*x)^(1/2)*(20*B^2*a^5*b^10*d^2 - 60*B^2*a^3*b^12*d^2 + 168*B^2*a^7*b^8*d^2 + 40*B
^2*a^9*b^6*d^2 - 44*B^2*a^11*b^4*d^2 + 4*B^2*a^13*b^2*d^2))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4
+ 4*a^6*b^2*d^4) - (((16*(40*B*a^2*b^14*d^4 + 192*B*a^4*b^12*d^4 + 360*B*a^6*b^10*d^4 + 320*B*a^8*b^8*d^4 + 1
20*B*a^10*b^6*d^4 - 8*B*a^14*b^2*d^4))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) + (
8*tan(c + d*x)^(1/2)*(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2)*(32*b^17*d^4 + 160*a^2*b^15*d^4 + 288*a^4*b^13*
d^4 + 160*a^6*b^11*d^4 - 160*a^8*b^9*d^4 - 288*a^10*b^7*d^4 - 160*a^12*b^5*d^4 - 32*a^14*b^3*d^4))/((-(B^2*a^7
+ 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^6*b^3*d^2))^(1/2)
*(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4)))*(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^
2))/(2*(-(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^
6*b^3*d^2))^(1/2)))*(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2))/(2*(-(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2)*
(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^6*b^3*d^2))^(1/2)))*(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^
2*a^5*b^2))/(2*(-(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^
2 + 4*a^6*b^3*d^2))^(1/2)) - (16*tan(c + d*x)^(1/2)*(B^4*a^12*b + 2*B^4*a^4*b^9 - 5*B^4*a^6*b^7 + 17*B^4*a^8*b
^5 - 7*B^4*a^10*b^3))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))*(B^2*a^7 + 9*B^2*a^
3*b^4 - 6*B^2*a^5*b^2)*1i)/(2*(-(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2
+ 6*a^4*b^5*d^2 + 4*a^6*b^3*d^2))^(1/2)))/((((((16*(8*B^3*a^7*b^7*d^2 - 78*B^3*a^5*b^9*d^2 + 60*B^3*a^9*b^5*d
^2 - 24*B^3*a^11*b^3*d^2 + 2*B^3*a^13*b*d^2))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d
^5) - (((16*tan(c + d*x)^(1/2)*(20*B^2*a^5*b^10*d^2 - 60*B^2*a^3*b^12*d^2 + 168*B^2*a^7*b^8*d^2 + 40*B^2*a^9*b
^6*d^2 - 44*B^2*a^11*b^4*d^2 + 4*B^2*a^13*b^2*d^2))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6
*b^2*d^4) + (((16*(40*B*a^2*b^14*d^4 + 192*B*a^4*b^12*d^4 + 360*B*a^6*b^10*d^4 + 320*B*a^8*b^8*d^4 + 120*B*a^1
0*b^6*d^4 - 8*B*a^14*b^2*d^4))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) - (8*tan(c
+ d*x)^(1/2)*(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2)*(32*b^17*d^4 + 160*a^2*b^15*d^4 + 288*a^4*b^13*d^4 + 16
0*a^6*b^11*d^4 - 160*a^8*b^9*d^4 - 288*a^10*b^7*d^4 - 160*a^12*b^5*d^4 - 32*a^14*b^3*d^4))/((-(B^2*a^7 + 9*B^2
*a^3*b^4 - 6*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^6*b^3*d^2))^(1/2)*(a^8*d^
4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4)))*(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2))/(2*(
-(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^6*b^3*d^
2))^(1/2)))*(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2))/(2*(-(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2)*(b^9*d^2
+ a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^6*b^3*d^2))^(1/2)))*(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^
2))/(2*(-(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^
6*b^3*d^2))^(1/2)) + (16*tan(c + d*x)^(1/2)*(B^4*a^12*b + 2*B^4*a^4*b^9 - 5*B^4*a^6*b^7 + 17*B^4*a^8*b^5 - 7*B
^4*a^10*b^3))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))*(B^2*a^7 + 9*B^2*a^3*b^4 -
6*B^2*a^5*b^2))/(2*(-(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^
5*d^2 + 4*a^6*b^3*d^2))^(1/2)) - (32*(3*B^5*a^6*b^6 - B^5*a^8*b^4))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4
*b^4*d^5 + 4*a^6*b^2*d^5) + (((((16*(8*B^3*a^7*b^7*d^2 - 78*B^3*a^5*b^9*d^2 + 60*B^3*a^9*b^5*d^2 - 24*B^3*a^11
*b^3*d^2 + 2*B^3*a^13*b*d^2))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) + (((16*tan(
c + d*x)^(1/2)*(20*B^2*a^5*b^10*d^2 - 60*B^2*a^3*b^12*d^2 + 168*B^2*a^7*b^8*d^2 + 40*B^2*a^9*b^6*d^2 - 44*B^2*
a^11*b^4*d^2 + 4*B^2*a^13*b^2*d^2))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4) - (((1
6*(40*B*a^2*b^14*d^4 + 192*B*a^4*b^12*d^4 + 360*B*a^6*b^10*d^4 + 320*B*a^8*b^8*d^4 + 120*B*a^10*b^6*d^4 - 8*B*
a^14*b^2*d^4))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) + (8*tan(c + d*x)^(1/2)*(B^
2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2)*(32*b^17*d^4 + 160*a^2*b^15*d^4 + 288*a^4*b^13*d^4 + 160*a^6*b^11*d^4 -
160*a^8*b^9*d^4 - 288*a^10*b^7*d^4 - 160*a^12*b^5*d^4 - 32*a^14*b^3*d^4))/((-(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2
*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^6*b^3*d^2))^(1/2)*(a^8*d^4 + b^8*d^4 + 4*
a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4)))*(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2))/(2*(-(B^2*a^7 + 9*B^
2*a^3*b^4 - 6*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^6*b^3*d^2))^(1/2)))*(B^2
*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2))/(2*(-(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4
*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^6*b^3*d^2))^(1/2)))*(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2))/(2*(-(B^2*a^
7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^6*b^3*d^2))^(1/2
)) - (16*tan(c + d*x)^(1/2)*(B^4*a^12*b + 2*B^4*a^4*b^9 - 5*B^4*a^6*b^7 + 17*B^4*a^8*b^5 - 7*B^4*a^10*b^3))/(a
^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))*(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2))/
(2*(-(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^6*b^
3*d^2))^(1/2))))*(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2)*1i)/(-(B^2*a^7 + 9*B^2*a^3*b^4 - 6*B^2*a^5*b^2)*(b^
9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^6*b^3*d^2))^(1/2) - (atan(-((((16*tan(c + d*x)^(1/2)*(
2*B^4*b^13 + 4*B^4*a^2*b^11 + 27*B^4*a^4*b^9 - 15*B^4*a^6*b^7 - 9*B^4*a^8*b^5 - B^4*a^10*b^3))/(a^8*d^4 + b^8*
d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4) - (((8*(320*B^3*a^7*b^7*d^2 - 120*B^3*a^5*b^9*d^2 - 304*B
^3*a^3*b^11*d^2 + 148*B^3*a^9*b^5*d^2 + 16*B^3*a^11*b^3*d^2 + 4*B^3*a*b^13*d^2))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^
6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) + (((16*tan(c + d*x)^(1/2)*(52*B^2*a^3*b^12*d^2 + 128*B^2*a^5*b^10*d^2
+ 424*B^2*a^7*b^8*d^2 + 380*B^2*a^9*b^6*d^2 + 100*B^2*a^11*b^4*d^2 + 8*B^2*a^13*b^2*d^2 + 60*B^2*a*b^14*d^2))/
(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4) + (((8*(96*B*a^2*b^14*d^4 + 480*B*a^4*b^12
*d^4 + 960*B*a^6*b^10*d^4 + 960*B*a^8*b^8*d^4 + 480*B*a^10*b^6*d^4 + 96*B*a^12*b^4*d^4))/(a^8*d^5 + b^8*d^5 +
4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) - (8*tan(c + d*x)^(1/2)*(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*
b^2)*(32*b^17*d^4 + 160*a^2*b^15*d^4 + 288*a^4*b^13*d^4 + 160*a^6*b^11*d^4 - 160*a^8*b^9*d^4 - 288*a^10*b^7*d^
4 - 160*a^12*b^5*d^4 - 32*a^14*b^3*d^4))/((-(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 +
4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^6*b^3*d^2))^(1/2)*(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*
a^6*b^2*d^4)))*(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2))/(2*(-(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2)*(
b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^6*b^3*d^2))^(1/2)))*(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B
^2*a^5*b^2))/(2*(-(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5
*d^2 + 4*a^6*b^3*d^2))^(1/2)))*(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2))/(2*(-(B^2*a^7 + 25*B^2*a^3*b^4 + 1
0*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^6*b^3*d^2))^(1/2)))*(B^2*a^7 + 25*B^
2*a^3*b^4 + 10*B^2*a^5*b^2)*1i)/(2*(-(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*
b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^6*b^3*d^2))^(1/2)) + (((16*tan(c + d*x)^(1/2)*(2*B^4*b^13 + 4*B^4*a^2*b^11 + 27*
B^4*a^4*b^9 - 15*B^4*a^6*b^7 - 9*B^4*a^8*b^5 - B^4*a^10*b^3))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d
^4 + 4*a^6*b^2*d^4) + (((8*(320*B^3*a^7*b^7*d^2 - 120*B^3*a^5*b^9*d^2 - 304*B^3*a^3*b^11*d^2 + 148*B^3*a^9*b^5
*d^2 + 16*B^3*a^11*b^3*d^2 + 4*B^3*a*b^13*d^2))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2
*d^5) - (((16*tan(c + d*x)^(1/2)*(52*B^2*a^3*b^12*d^2 + 128*B^2*a^5*b^10*d^2 + 424*B^2*a^7*b^8*d^2 + 380*B^2*a
^9*b^6*d^2 + 100*B^2*a^11*b^4*d^2 + 8*B^2*a^13*b^2*d^2 + 60*B^2*a*b^14*d^2))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^
4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4) - (((8*(96*B*a^2*b^14*d^4 + 480*B*a^4*b^12*d^4 + 960*B*a^6*b^10*d^4 + 960*B
*a^8*b^8*d^4 + 480*B*a^10*b^6*d^4 + 96*B*a^12*b^4*d^4))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4
*a^6*b^2*d^5) + (8*tan(c + d*x)^(1/2)*(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2)*(32*b^17*d^4 + 160*a^2*b^15*
d^4 + 288*a^4*b^13*d^4 + 160*a^6*b^11*d^4 - 160*a^8*b^9*d^4 - 288*a^10*b^7*d^4 - 160*a^12*b^5*d^4 - 32*a^14*b^
3*d^4))/((-(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 +
4*a^6*b^3*d^2))^(1/2)*(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4)))*(B^2*a^7 + 25*B^2*
a^3*b^4 + 10*B^2*a^5*b^2))/(2*(-(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d
^2 + 6*a^4*b^5*d^2 + 4*a^6*b^3*d^2))^(1/2)))*(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2))/(2*(-(B^2*a^7 + 25*B
^2*a^3*b^4 + 10*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^6*b^3*d^2))^(1/2)))*(B
^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2))/(2*(-(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d
^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^6*b^3*d^2))^(1/2)))*(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2)*1i)/(
2*(-(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^6*b
^3*d^2))^(1/2)))/((16*(10*B^5*a^2*b^10 + 27*B^5*a^4*b^8 + 10*B^5*a^6*b^6 + B^5*a^8*b^4))/(a^8*d^5 + b^8*d^5 +
4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) - (((16*tan(c + d*x)^(1/2)*(2*B^4*b^13 + 4*B^4*a^2*b^11 + 27*B^
4*a^4*b^9 - 15*B^4*a^6*b^7 - 9*B^4*a^8*b^5 - B^4*a^10*b^3))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4
+ 4*a^6*b^2*d^4) - (((8*(320*B^3*a^7*b^7*d^2 - 120*B^3*a^5*b^9*d^2 - 304*B^3*a^3*b^11*d^2 + 148*B^3*a^9*b^5*d
^2 + 16*B^3*a^11*b^3*d^2 + 4*B^3*a*b^13*d^2))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d
^5) + (((16*tan(c + d*x)^(1/2)*(52*B^2*a^3*b^12*d^2 + 128*B^2*a^5*b^10*d^2 + 424*B^2*a^7*b^8*d^2 + 380*B^2*a^9
*b^6*d^2 + 100*B^2*a^11*b^4*d^2 + 8*B^2*a^13*b^2*d^2 + 60*B^2*a*b^14*d^2))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4
+ 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4) + (((8*(96*B*a^2*b^14*d^4 + 480*B*a^4*b^12*d^4 + 960*B*a^6*b^10*d^4 + 960*B*a
^8*b^8*d^4 + 480*B*a^10*b^6*d^4 + 96*B*a^12*b^4*d^4))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a
^6*b^2*d^5) - (8*tan(c + d*x)^(1/2)*(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2)*(32*b^17*d^4 + 160*a^2*b^15*d^
4 + 288*a^4*b^13*d^4 + 160*a^6*b^11*d^4 - 160*a^8*b^9*d^4 - 288*a^10*b^7*d^4 - 160*a^12*b^5*d^4 - 32*a^14*b^3*
d^4))/((-(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*
a^6*b^3*d^2))^(1/2)*(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4)))*(B^2*a^7 + 25*B^2*a^
3*b^4 + 10*B^2*a^5*b^2))/(2*(-(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2
+ 6*a^4*b^5*d^2 + 4*a^6*b^3*d^2))^(1/2)))*(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2))/(2*(-(B^2*a^7 + 25*B^2
*a^3*b^4 + 10*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^6*b^3*d^2))^(1/2)))*(B^2
*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2))/(2*(-(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2
+ 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^6*b^3*d^2))^(1/2)))*(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2))/(2*(-(
B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^6*b^3*d^
2))^(1/2)) + (((16*tan(c + d*x)^(1/2)*(2*B^4*b^13 + 4*B^4*a^2*b^11 + 27*B^4*a^4*b^9 - 15*B^4*a^6*b^7 - 9*B^4*a
^8*b^5 - B^4*a^10*b^3))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4) + (((8*(320*B^3*a^
7*b^7*d^2 - 120*B^3*a^5*b^9*d^2 - 304*B^3*a^3*b^11*d^2 + 148*B^3*a^9*b^5*d^2 + 16*B^3*a^11*b^3*d^2 + 4*B^3*a*b
^13*d^2))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) - (((16*tan(c + d*x)^(1/2)*(52*B
^2*a^3*b^12*d^2 + 128*B^2*a^5*b^10*d^2 + 424*B^2*a^7*b^8*d^2 + 380*B^2*a^9*b^6*d^2 + 100*B^2*a^11*b^4*d^2 + 8*
B^2*a^13*b^2*d^2 + 60*B^2*a*b^14*d^2))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4) - (
((8*(96*B*a^2*b^14*d^4 + 480*B*a^4*b^12*d^4 + 960*B*a^6*b^10*d^4 + 960*B*a^8*b^8*d^4 + 480*B*a^10*b^6*d^4 + 96
*B*a^12*b^4*d^4))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) + (8*tan(c + d*x)^(1/2)*
(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2)*(32*b^17*d^4 + 160*a^2*b^15*d^4 + 288*a^4*b^13*d^4 + 160*a^6*b^11*
d^4 - 160*a^8*b^9*d^4 - 288*a^10*b^7*d^4 - 160*a^12*b^5*d^4 - 32*a^14*b^3*d^4))/((-(B^2*a^7 + 25*B^2*a^3*b^4 +
10*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^6*b^3*d^2))^(1/2)*(a^8*d^4 + b^8*d
^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4)))*(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2))/(2*(-(B^2*a
^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^6*b^3*d^2))^(
1/2)))*(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2))/(2*(-(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2)*(b^9*d^2
+ a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^6*b^3*d^2))^(1/2)))*(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b
^2))/(2*(-(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4
*a^6*b^3*d^2))^(1/2)))*(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2))/(2*(-(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^
5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 + 6*a^4*b^5*d^2 + 4*a^6*b^3*d^2))^(1/2))))*(B^2*a^7 + 25*B^2*a^3*b
^4 + 10*B^2*a^5*b^2)*1i)/(-(B^2*a^7 + 25*B^2*a^3*b^4 + 10*B^2*a^5*b^2)*(b^9*d^2 + a^8*b*d^2 + 4*a^2*b^7*d^2 +
6*a^4*b^5*d^2 + 4*a^6*b^3*d^2))^(1/2)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ B \int \frac {\tan ^{\frac {3}{2}}{\left (c + d x \right )}}{a + b \tan {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tan(d*x+c)**(3/2)*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c))**2,x)
[Out]
B*Integral(tan(c + d*x)**(3/2)/(a + b*tan(c + d*x)), x)
________________________________________________________________________________________