3.406 \(\int \frac {\sqrt {\tan (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx\)
Optimal. Leaf size=391 \[ -\frac {\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}+\frac {\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}-\frac {(A b-a B) \sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {\left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}+\frac {\left (-\left (a^2 (A+B)\right )+2 a b (A-B)+b^2 (A+B)\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}-\frac {\left (a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} d \left (a^2+b^2\right )^2} \]
[Out]
1/2*(a^2*(A-B)-b^2*(A-B)+2*a*b*(A+B))*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)^2/d*2^(1/2)+1/2*(a^2*(A-B)
-b^2*(A-B)+2*a*b*(A+B))*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)^2/d*2^(1/2)-1/4*(2*a*b*(A-B)-a^2*(A+B)+b^
2*(A+B))*ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)^2/d*2^(1/2)+1/4*(2*a*b*(A-B)-a^2*(A+B)+b^2*(A+B))
*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)^2/d*2^(1/2)-(3*A*a^2*b-A*b^3-B*a^3+3*B*a*b^2)*arctan(b^(1
/2)*tan(d*x+c)^(1/2)/a^(1/2))/(a^2+b^2)^2/d/a^(1/2)/b^(1/2)-(A*b-B*a)*tan(d*x+c)^(1/2)/(a^2+b^2)/d/(a+b*tan(d*
x+c))
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Rubi [A] time = 0.82, antiderivative size = 391, normalized size of antiderivative =
1.00, number of steps used = 15, number of rules used = 12, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) =
0.364, Rules used = {3608, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205}
\[ -\frac {\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}+\frac {\left (a^2 (A-B)+2 a b (A+B)-b^2 (A-B)\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}-\frac {\left (3 a^2 A b+a^3 (-B)+3 a b^2 B-A b^3\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} d \left (a^2+b^2\right )^2}-\frac {(A b-a B) \sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {\left (a^2 (-(A+B))+2 a b (A-B)+b^2 (A+B)\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}+\frac {\left (a^2 (-(A+B))+2 a b (A-B)+b^2 (A+B)\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
[In]
Int[(Sqrt[Tan[c + d*x]]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]
[Out]
-(((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*
d)) + ((a^2*(A - B) - b^2*(A - B) + 2*a*b*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2
)^2*d) - ((3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*Sqrt[
b]*(a^2 + b^2)^2*d) - ((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c
+ d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) + ((2*a*b*(A - B) - a^2*(A + B) + b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c
+ d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - ((A*b - a*B)*Sqrt[Tan[c + d*x]])/((a^2 + b^2)*d*(a + b
*Tan[c + d*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 3608
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((A*b - a*B)*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n)/(
f*(m + 1)*(a^2 + b^2)), x] + Dist[1/(b*(m + 1)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f
*x])^(n - 1)*Simp[b*B*(b*c*(m + 1) + a*d*n) + A*b*(a*c*(m + 1) - b*d*n) - b*(A*(b*c - a*d) - B*(a*c + b*d))*(m
+ 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B},
x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && LtQ[0, n, 1] && (IntegerQ[
m] || IntegersQ[2*m, 2*n])
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]
Rule 3653
Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*
x])^n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 +
b^2), Int[((c + d*Tan[e + f*x])^n*(1 + Tan[e + f*x]^2))/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e,
f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !GtQ[n, 0] && !LeQ[n, -
1]
Rubi steps
\begin {align*} \int \frac {\sqrt {\tan (c+d x)} (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx &=-\frac {(A b-a B) \sqrt {\tan (c+d x)}}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {\int \frac {-\frac {1}{2} b (A b-a B)-b (a A+b B) \tan (c+d x)+\frac {1}{2} b (A b-a B) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{b \left (a^2+b^2\right )}\\ &=-\frac {(A b-a B) \sqrt {\tan (c+d x)}}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {\int \frac {-b \left (2 a A b-a^2 B+b^2 B\right )-b \left (a^2 A-A b^2+2 a b B\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{b \left (a^2+b^2\right )^2}-\frac {\left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) \int \frac {1+\tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{2 \left (a^2+b^2\right )^2}\\ &=-\frac {(A b-a B) \sqrt {\tan (c+d x)}}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {2 \operatorname {Subst}\left (\int \frac {-b \left (2 a A b-a^2 B+b^2 B\right )-b \left (a^2 A-A b^2+2 a b B\right ) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{b \left (a^2+b^2\right )^2 d}-\frac {\left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{2 \left (a^2+b^2\right )^2 d}\\ &=-\frac {(A b-a B) \sqrt {\tan (c+d x)}}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {\left (3 a^2 A b-A b^3-a^3 B+3 a b^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 )^2 d}+\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \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 )^2 d}+\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \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 )^2 d}\\ &=-\frac {\left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \left (a^2+b^2\right )^2 d}-\frac {(A b-a B) \sqrt {\tan (c+d x)}}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \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 )^2 d}+\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \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 )^2 d}-\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \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 )^2 d}-\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \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 )^2 d}\\ &=-\frac {\left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \left (a^2+b^2\right )^2 d}-\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {(A b-a B) \sqrt {\tan (c+d x)}}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \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 )^2 d}-\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \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 )^2 d}\\ &=-\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2 (A-B)-b^2 (A-B)+2 a b (A+B)\right ) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \left (a^2+b^2\right )^2 d}-\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (2 a b (A-B)-a^2 (A+B)+b^2 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {(A b-a B) \sqrt {\tan (c+d x)}}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [C] time = 1.43, size = 220, normalized size = 0.56 \[ \frac {\frac {\sqrt [4]{-1} a \left ((a+i b)^2 (B+i A) \tan ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+(a-i b)^2 (B-i A) \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )\right )}{a^2+b^2}+\frac {\sqrt {a} \left (a^3 B-3 a^2 A b-3 a b^2 B+A b^3\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {b} \left (a^2+b^2\right )}+\frac {b (A b-a B) \tan ^{\frac {3}{2}}(c+d x)}{a+b \tan (c+d x)}+(a B-A b) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
[In]
Integrate[(Sqrt[Tan[c + d*x]]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]
[Out]
((Sqrt[a]*(-3*a^2*A*b + A*b^3 + a^3*B - 3*a*b^2*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[b]*(a^2
+ b^2)) + ((-1)^(1/4)*a*((a + I*b)^2*(I*A + B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + (a - I*b)^2*((-I)*A +
B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]))/(a^2 + b^2) + (-(A*b) + a*B)*Sqrt[Tan[c + d*x]] + (b*(A*b - a*B)*T
an[c + d*x]^(3/2))/(a + b*Tan[c + d*x]))/(a*(a^2 + b^2)*d)
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fricas [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)^(1/2)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^2,x, algorithm="fricas")
[Out]
Timed out
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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)^(1/2)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^2,x, algorithm="giac")
[Out]
Timed out
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maple [B] time = 0.46, size = 1128, normalized size = 2.88 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tan(d*x+c)^(1/2)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^2,x)
[Out]
-3/d/(a^2+b^2)^2/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))*B*a*b^2+1/d/(a^2+b^2)^2*tan(d*x+c)^(1/2)/(
a+b*tan(d*x+c))*B*a*b^2+1/d/(a^2+b^2)^2*B*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a*b+1/d/(a^2+b^2)^2*B*2^(
1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a*b-1/d*a^2*b/(a^2+b^2)^2*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))*A-3/d*a^2
*b/(a^2+b^2)^2/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))*A+1/2/d/(a^2+b^2)^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*b+1/2/d/(a^2+b^2)^2*A*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*b+1/d/(a^2+b^2)^2*A*2^(1/2)*arct
an(1+2^(1/2)*tan(d*x+c)^(1/2))*a*b+1/d/(a^2+b^2)^2*A*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a*b+1/d*a^3/(
a^2+b^2)^2*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))*B+1/d*a^3/(a^2+b^2)^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)^2*A*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)+t
an(d*x+c)))*a^2-1/4/d/(a^2+b^2)^2*A*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^2-1/4/d/(a^2+b^2)^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^2+1/4/d/(a^2+b^2)^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^2-1/2/d/(a^2+b^2)^2*B*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^2+1/2/d/(a^
2+b^2)^2*B*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*b^2-1/2/d/(a^2+b^2)^2*B*2^(1/2)*arctan(-1+2^(1/2)*tan(d*
x+c)^(1/2))*a^2+1/2/d/(a^2+b^2)^2*B*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*b^2+1/2/d/(a^2+b^2)^2*A*2^(1/2
)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^2-1/2/d/(a^2+b^2)^2*A*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*b^2-1/
2/d/(a^2+b^2)^2*A*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*b^2+1/2/d/(a^2+b^2)^2*A*2^(1/2)*arctan(-1+2^(1/2
)*tan(d*x+c)^(1/2))*a^2-1/d/(a^2+b^2)^2*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))*A*b^3+1/d/(a^2+b^2)^2/(a*b)^(1/2)*ar
ctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))*A*b^3
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maxima [A] time = 0.71, size = 341, normalized size = 0.87 \[ \frac {\frac {4 \, {\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a b}} + \frac {4 \, {\left (B a - A b\right )} \sqrt {\tan \left (d x + c\right )}}{a^{3} + a b^{2} + {\left (a^{2} b + b^{3}\right )} \tan \left (d x + c\right )} + \frac {2 \, \sqrt {2} {\left ({\left (A - B\right )} a^{2} + 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left ({\left (A - B\right )} a^{2} + 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - \sqrt {2} {\left ({\left (A + B\right )} a^{2} - 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} {\left ({\left (A + B\right )} a^{2} - 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tan(d*x+c)^(1/2)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^2,x, algorithm="maxima")
[Out]
1/4*(4*(B*a^3 - 3*A*a^2*b - 3*B*a*b^2 + A*b^3)*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^4 + 2*a^2*b^2 + b^4)
*sqrt(a*b)) + 4*(B*a - A*b)*sqrt(tan(d*x + c))/(a^3 + a*b^2 + (a^2*b + b^3)*tan(d*x + c)) + (2*sqrt(2)*((A - B
)*a^2 + 2*(A + B)*a*b - (A - B)*b^2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*((A - B)
*a^2 + 2*(A + B)*a*b - (A - B)*b^2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c)))) - sqrt(2)*((A + B)*a
^2 - 2*(A - B)*a*b - (A + B)*b^2)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) + sqrt(2)*((A + B)*a^2 -
2*(A - B)*a*b - (A + B)*b^2)*log(-sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1))/(a^4 + 2*a^2*b^2 + b^4))/d
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mupad [B] time = 32.55, size = 17089, normalized size = 43.71 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((tan(c + d*x)^(1/2)*(A + B*tan(c + d*x)))/(a + b*tan(c + d*x))^2,x)
[Out]
(log((16*B^5*a*b^4*(a^2 - 3*b^2))/(d^5*(a^2 + b^2)^4) - (((((((((128*B*a*b^2*(5*b^4 - a^4 + 4*a^2*b^2))/d - 12
8*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a*b
^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))*((4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2
*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (64*B^2*a*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*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a*b^3
*d^2 + 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (32*B^3*a^2*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*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/
(d^4*(a^2 + b^2)^4))^(1/2))/4 + (16*B^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*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/
(d^4*(a^2 + b^2)^4))^(1/2))/4)*(((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^8*d^4 - 608*B^4*a^4*b^4*d^4
+ 192*B^4*a^6*b^2*d^4)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*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*b^4*(a^2 - 3*b^2))/(d^5*(a^2 + b^2)^4) - (((((((((128*B*a
*b^2*(5*b^4 - a^4 + 4*a^2*b^2))/d - 128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*(-(4*(-B^4*d^4*(a^4 +
b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))*(-(4*(-B^4*d^4*(
a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (64*B^2
*a*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*d^4*(a^4 +
b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (32*B^3*a^2*
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*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/
2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (16*B^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*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1
/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4)*(-((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8
*d^4 - 16*B^4*a^8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*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*b^4*(a^2 - 3*b
^2))/(d^5*(a^2 + b^2)^4) - (((((((((128*B*a*b^2*(5*b^4 - a^4 + 4*a^2*b^2))/d + 128*b^3*tan(c + d*x)^(1/2)*(a^2
- b^2)*(a^2 + b^2)^2*((4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d
^4*(a^2 + b^2)^4))^(1/2))*((4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2
)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*B^2*a*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*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4
*(a^2 + b^2)^4))^(1/2))/4 - (32*B^3*a^2*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*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4
- (16*B^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*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*B^2*a*b^3*d^2 + 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4
)*(((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^8*d^4 - 608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2)
- 16*B^2*a*b^3*d^2 + 16*B^2*a^3*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*b^4*(a^2 - 3*b^2))/(d^5*(a^2 + b^2)^4) - (((((((((128*B*a*b^2*(5*b^4 - a^4 + 4*a^
2*b^2))/d + 128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*(-(4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/
2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))*(-(4*(-B^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2
)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*B^2*a*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*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/
2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (32*B^3*a^2*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*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a*b^3*d^2 -
16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (16*B^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*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*B^2*a*b^3*d^2 -
16*B^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4)*(-((192*B^4*a^2*b^6*d^4 - 16*B^4*b^8*d^4 - 16*B^4*a^8*d^4 -
608*B^4*a^4*b^4*d^4 + 192*B^4*a^6*b^2*d^4)^(1/2) + 16*B^2*a*b^3*d^2 - 16*B^2*a^3*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(- (((((((((256*A*b^3*(2*a^4 - b^4 + a^2*b
^2))/d - 128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*((4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) +
16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))*((4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/
2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (64*A^2*a*b^2*tan(c + d*x)^(1/2)*(a^
6 + 17*b^6 - 29*a^2*b^4 + 19*a^4*b^2))/(d^2*(a^2 + b^2)^2))*((4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 1
6*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (32*A^3*a*b^2*(a^6 + 13*b^6 - 45*a^2*b^4 +
39*a^4*b^2))/(d^3*(a^2 + b^2)^3))*((4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*
a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (16*A^4*b^3*tan(c + d*x)^(1/2)*(9*a^6 - 3*b^6 + 3*a^2*b^4 - 17*a^4*
b^2))/(d^4*(a^2 + b^2)^4))*((4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^
2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (8*A^5*b^3*(9*a^4 - b^4))/(d^5*(a^2 + b^2)^4))*(((192*A^4*a^2*b^6*d^4 - 16*
A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^
3*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(- (((((((((256*A
*b^3*(2*a^4 - b^4 + a^2*b^2))/d - 128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*(-(4*(-A^4*d^4*(a^4 + b
^4 - 6*a^2*b^2)^2)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))*(-(4*(-A^4*d^4*(a^
4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (64*A^2*a
*b^2*tan(c + d*x)^(1/2)*(a^6 + 17*b^6 - 29*a^2*b^4 + 19*a^4*b^2))/(d^2*(a^2 + b^2)^2))*(-(4*(-A^4*d^4*(a^4 + b
^4 - 6*a^2*b^2)^2)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (32*A^3*a*b^2*
(a^6 + 13*b^6 - 45*a^2*b^4 + 39*a^4*b^2))/(d^3*(a^2 + b^2)^3))*(-(4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2)
- 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (16*A^4*b^3*tan(c + d*x)^(1/2)*(9*a^6
- 3*b^6 + 3*a^2*b^4 - 17*a^4*b^2))/(d^4*(a^2 + b^2)^4))*(-(4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*A
^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (8*A^5*b^3*(9*a^4 - b^4))/(d^5*(a^2 + b^2)^4)
)*(-((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2)
- 16*A^2*a*b^3*d^2 + 16*A^2*a^3*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(- (((((((((256*A*b^3*(2*a^4 - b^4 + a^2*b^2))/d + 128*b^3*tan(c + d*x)^(1/2)*(a^2 - b^2)*(a^2 +
b^2)^2*((4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^
4))^(1/2))*((4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b
^2)^4))^(1/2))/4 + (64*A^2*a*b^2*tan(c + d*x)^(1/2)*(a^6 + 17*b^6 - 29*a^2*b^4 + 19*a^4*b^2))/(d^2*(a^2 + b^2)
^2))*((4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4)
)^(1/2))/4 + (32*A^3*a*b^2*(a^6 + 13*b^6 - 45*a^2*b^4 + 39*a^4*b^2))/(d^3*(a^2 + b^2)^3))*((4*(-A^4*d^4*(a^4 +
b^4 - 6*a^2*b^2)^2)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (16*A^4*b^3*
tan(c + d*x)^(1/2)*(9*a^6 - 3*b^6 + 3*a^2*b^4 - 17*a^4*b^2))/(d^4*(a^2 + b^2)^4))*((4*(-A^4*d^4*(a^4 + b^4 - 6
*a^2*b^2)^2)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 - (8*A^5*b^3*(9*a^4 -
b^4))/(d^5*(a^2 + b^2)^4))*(((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 19
2*A^4*a^6*b^2*d^4)^(1/2) + 16*A^2*a*b^3*d^2 - 16*A^2*a^3*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(- (((((((((256*A*b^3*(2*a^4 - b^4 + a^2*b^2))/d + 128*b^3*tan(c +
d*x)^(1/2)*(a^2 - b^2)*(a^2 + b^2)^2*(-(4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A
^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))*(-(4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*A^2*a*b^3*d^2 +
16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (64*A^2*a*b^2*tan(c + d*x)^(1/2)*(a^6 + 17*b^6 - 29*a^2*b^4
+ 19*a^4*b^2))/(d^2*(a^2 + b^2)^2))*(-(4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A
^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^4))^(1/2))/4 + (32*A^3*a*b^2*(a^6 + 13*b^6 - 45*a^2*b^4 + 39*a^4*b^2))/(d^3*(a^
2 + b^2)^3))*(-(4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(d^4*(a^2
+ b^2)^4))^(1/2))/4 + (16*A^4*b^3*tan(c + d*x)^(1/2)*(9*a^6 - 3*b^6 + 3*a^2*b^4 - 17*a^4*b^2))/(d^4*(a^2 + b^2
)^4))*(-(4*(-A^4*d^4*(a^4 + b^4 - 6*a^2*b^2)^2)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*b*d^2)/(d^4*(a^2 + b^2)^
4))^(1/2))/4 - (8*A^5*b^3*(9*a^4 - b^4))/(d^5*(a^2 + b^2)^4))*(-((192*A^4*a^2*b^6*d^4 - 16*A^4*b^8*d^4 - 16*A^
4*a^8*d^4 - 608*A^4*a^4*b^4*d^4 + 192*A^4*a^6*b^2*d^4)^(1/2) - 16*A^2*a*b^3*d^2 + 16*A^2*a^3*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(((((((8*(160*A^3*a^7*b^4*d^2
- 24*A^3*a^5*b^6*d^2 - 128*A^3*a^3*b^8*d^2 + 4*A^3*a^9*b^2*d^2 + 52*A^3*a*b^10*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*A^2*a^3*b^10*d^2 - 88*A^2*a^5*b^8*d
^2 + 40*A^2*a^7*b^6*d^2 + 84*A^2*a^9*b^4*d^2 + 4*A^2*a^11*b^2*d^2 + 68*A^2*a*b^12*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*(320*A*a^6*b^9*d^4 - 96*A*a^2*b^13*d^4 - 32*A*b^15*d^4 +
480*A*a^8*b^7*d^4 + 288*A*a^10*b^5*d^4 + 64*A*a^12*b^3*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) - (4*tan(c + d*x)^(1/2)*(-4*(A^2*b^5 + 9*A^2*a^4*b - 6*A^2*a^2*b^3)*(a^9*d^2 + a*b^8*d^2 +
4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2)*(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))/((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)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*
b^2*d^2)))*(-4*(A^2*b^5 + 9*A^2*a^4*b - 6*A^2*a^2*b^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 +
4*a^7*b^2*d^2))^(1/2))/(4*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)))*(-4*(A^2*b^5
+ 9*A^2*a^4*b - 6*A^2*a^2*b^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2))/
(4*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)))*(-4*(A^2*b^5 + 9*A^2*a^4*b - 6*A^2*
a^2*b^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2))/(4*(a^9*d^2 + a*b^8*d^2
+ 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)) - (16*tan(c + d*x)^(1/2)*(3*A^4*b^9 - 3*A^4*a^2*b^7 + 17*A^
4*a^4*b^5 - 9*A^4*a^6*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))*(-4*(A^2*b^5
+ 9*A^2*a^4*b - 6*A^2*a^2*b^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2)*1i
)/(4*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)) - (((((8*(160*A^3*a^7*b^4*d^2 - 24
*A^3*a^5*b^6*d^2 - 128*A^3*a^3*b^8*d^2 + 4*A^3*a^9*b^2*d^2 + 52*A^3*a*b^10*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*A^2*a^3*b^10*d^2 - 88*A^2*a^5*b^8*d^2 +
40*A^2*a^7*b^6*d^2 + 84*A^2*a^9*b^4*d^2 + 4*A^2*a^11*b^2*d^2 + 68*A^2*a*b^12*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*(320*A*a^6*b^9*d^4 - 96*A*a^2*b^13*d^4 - 32*A*b^15*d^4 + 480*A
*a^8*b^7*d^4 + 288*A*a^10*b^5*d^4 + 64*A*a^12*b^3*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) + (4*tan(c + d*x)^(1/2)*(-4*(A^2*b^5 + 9*A^2*a^4*b - 6*A^2*a^2*b^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3
*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2)*(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))/((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)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d
^2)))*(-4*(A^2*b^5 + 9*A^2*a^4*b - 6*A^2*a^2*b^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7
*b^2*d^2))^(1/2))/(4*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)))*(-4*(A^2*b^5 + 9*
A^2*a^4*b - 6*A^2*a^2*b^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2))/(4*(a
^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)))*(-4*(A^2*b^5 + 9*A^2*a^4*b - 6*A^2*a^2*b
^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2))/(4*(a^9*d^2 + a*b^8*d^2 + 4*
a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)) + (16*tan(c + d*x)^(1/2)*(3*A^4*b^9 - 3*A^4*a^2*b^7 + 17*A^4*a^4
*b^5 - 9*A^4*a^6*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))*(-4*(A^2*b^5 + 9*A
^2*a^4*b - 6*A^2*a^2*b^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2)*1i)/(4*
(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)))/((((((8*(160*A^3*a^7*b^4*d^2 - 24*A^3*
a^5*b^6*d^2 - 128*A^3*a^3*b^8*d^2 + 4*A^3*a^9*b^2*d^2 + 52*A^3*a*b^10*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*A^2*a^3*b^10*d^2 - 88*A^2*a^5*b^8*d^2 + 40*A^
2*a^7*b^6*d^2 + 84*A^2*a^9*b^4*d^2 + 4*A^2*a^11*b^2*d^2 + 68*A^2*a*b^12*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*(320*A*a^6*b^9*d^4 - 96*A*a^2*b^13*d^4 - 32*A*b^15*d^4 + 480*A*a^8*
b^7*d^4 + 288*A*a^10*b^5*d^4 + 64*A*a^12*b^3*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) - (4*tan(c + d*x)^(1/2)*(-4*(A^2*b^5 + 9*A^2*a^4*b - 6*A^2*a^2*b^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*
d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2)*(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))/((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)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)))
*(-4*(A^2*b^5 + 9*A^2*a^4*b - 6*A^2*a^2*b^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*
d^2))^(1/2))/(4*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)))*(-4*(A^2*b^5 + 9*A^2*a
^4*b - 6*A^2*a^2*b^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2))/(4*(a^9*d^
2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)))*(-4*(A^2*b^5 + 9*A^2*a^4*b - 6*A^2*a^2*b^3)*(
a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2))/(4*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b
^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)) - (16*tan(c + d*x)^(1/2)*(3*A^4*b^9 - 3*A^4*a^2*b^7 + 17*A^4*a^4*b^5
- 9*A^4*a^6*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))*(-4*(A^2*b^5 + 9*A^2*a^
4*b - 6*A^2*a^2*b^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2))/(4*(a^9*d^2
+ a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)) - (16*(A^5*b^7 - 9*A^5*a^4*b^3))/(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*(160*A^3*a^7*b^4*d^2 - 24*A^3*a^5*b^6*d^2 - 128
*A^3*a^3*b^8*d^2 + 4*A^3*a^9*b^2*d^2 + 52*A^3*a*b^10*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*A^2*a^3*b^10*d^2 - 88*A^2*a^5*b^8*d^2 + 40*A^2*a^7*b^6*d^2 + 8
4*A^2*a^9*b^4*d^2 + 4*A^2*a^11*b^2*d^2 + 68*A^2*a*b^12*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*(320*A*a^6*b^9*d^4 - 96*A*a^2*b^13*d^4 - 32*A*b^15*d^4 + 480*A*a^8*b^7*d^4 + 288*A*a
^10*b^5*d^4 + 64*A*a^12*b^3*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) + (4*tan
(c + d*x)^(1/2)*(-4*(A^2*b^5 + 9*A^2*a^4*b - 6*A^2*a^2*b^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d
^2 + 4*a^7*b^2*d^2))^(1/2)*(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))/((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)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)))*(-4*(A^2*b^5 + 9
*A^2*a^4*b - 6*A^2*a^2*b^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2))/(4*(
a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)))*(-4*(A^2*b^5 + 9*A^2*a^4*b - 6*A^2*a^2*
b^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2))/(4*(a^9*d^2 + a*b^8*d^2 + 4
*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2)))*(-4*(A^2*b^5 + 9*A^2*a^4*b - 6*A^2*a^2*b^3)*(a^9*d^2 + a*b^8*d
^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2))/(4*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^
4*d^2 + 4*a^7*b^2*d^2)) + (16*tan(c + d*x)^(1/2)*(3*A^4*b^9 - 3*A^4*a^2*b^7 + 17*A^4*a^4*b^5 - 9*A^4*a^6*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))*(-4*(A^2*b^5 + 9*A^2*a^4*b - 6*A^2*a^2*b
^3)*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2))/(4*(a^9*d^2 + a*b^8*d^2 + 4*
a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))))*(-4*(A^2*b^5 + 9*A^2*a^4*b - 6*A^2*a^2*b^3)*(a^9*d^2 + a*b^8*d
^2 + 4*a^3*b^6*d^2 + 6*a^5*b^4*d^2 + 4*a^7*b^2*d^2))^(1/2)*1i)/(2*(a^9*d^2 + a*b^8*d^2 + 4*a^3*b^6*d^2 + 6*a^5
*b^4*d^2 + 4*a^7*b^2*d^2)) - (atan(((((16*tan(c + d*x)^(1/2)*(2*B^4*b^9 + B^4*a^8*b - 5*B^4*a^2*b^7 + 17*B^4*a
^4*b^5 - 7*B^4*a^6*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) + (((16*(8*B^3*a^
4*b^7*d^2 - 78*B^3*a^2*b^9*d^2 + 60*B^3*a^6*b^5*d^2 - 24*B^3*a^8*b^3*d^2 + 2*B^3*a^10*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*(40*B*a*b^14*d^4 + 192*B*a^3*b^12*d^4 + 360*B*a^
5*b^10*d^4 + 320*B*a^7*b^8*d^4 + 120*B*a^9*b^6*d^4 - 8*B*a^13*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) - (4*tan(c + d*x)^(1/2)*(-4*(B^2*a^5 + 9*B^2*a*b^4 - 6*B^2*a^3*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*b^17*d^4 + 160*a^2*b^15*d^4 + 288*a^4*b^1
3*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))/((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^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)))*(-4*(B^2*a^5 + 9*B^2*a*b^4 - 6*B^2*a^3*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))/(4*(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)) +
(16*tan(c + d*x)^(1/2)*(20*B^2*a^3*b^10*d^2 + 168*B^2*a^5*b^8*d^2 + 40*B^2*a^7*b^6*d^2 - 44*B^2*a^9*b^4*d^2 +
4*B^2*a^11*b^2*d^2 - 60*B^2*a*b^12*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))*
(-4*(B^2*a^5 + 9*B^2*a*b^4 - 6*B^2*a^3*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))/(4*(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)))*(-4*(B^2*a^5 + 9*B^2*a*
b^4 - 6*B^2*a^3*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))/(4*(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)))*(-4*(B^2*a^5 + 9*B^2*a*b^4 - 6*B^2*a^3*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)*1i)/(4*(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)) + (((16*tan(c + d*x)^(1/2)*(2*B^4*b^9 + B^4*a^8*b - 5*B^4*a^2*b^7 +
17*B^4*a^4*b^5 - 7*B^4*a^6*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) - (((16*
(8*B^3*a^4*b^7*d^2 - 78*B^3*a^2*b^9*d^2 + 60*B^3*a^6*b^5*d^2 - 24*B^3*a^8*b^3*d^2 + 2*B^3*a^10*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*(40*B*a*b^14*d^4 + 192*B*a^3*b^12*d^4 +
360*B*a^5*b^10*d^4 + 320*B*a^7*b^8*d^4 + 120*B*a^9*b^6*d^4 - 8*B*a^13*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) + (4*tan(c + d*x)^(1/2)*(-4*(B^2*a^5 + 9*B^2*a*b^4 - 6*B^2*a^3*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*b^17*d^4 + 160*a^2*b^15*d^4 + 28
8*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))/
((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^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)))*(-4*(B^2*a^5 + 9*B^2*a*b^4 - 6*B^2*a^3*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))/(4*(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)) - (16*tan(c + d*x)^(1/2)*(20*B^2*a^3*b^10*d^2 + 168*B^2*a^5*b^8*d^2 + 40*B^2*a^7*b^6*d^2 - 44*B^2*a^9*b
^4*d^2 + 4*B^2*a^11*b^2*d^2 - 60*B^2*a*b^12*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))*(-4*(B^2*a^5 + 9*B^2*a*b^4 - 6*B^2*a^3*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))/(4*(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)))*(-4*(B^2*a^5 +
9*B^2*a*b^4 - 6*B^2*a^3*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))/(4
*(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)))*(-4*(B^2*a^5 + 9*B^2*a*b^4 - 6*B^2*a^
3*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)*1i)/(4*(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)))/((32*(3*B^5*a*b^6 - B^5*a^3*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^9 + B^4*a^8*b - 5*B^4*a^2*b^7
+ 17*B^4*a^4*b^5 - 7*B^4*a^6*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) + (((1
6*(8*B^3*a^4*b^7*d^2 - 78*B^3*a^2*b^9*d^2 + 60*B^3*a^6*b^5*d^2 - 24*B^3*a^8*b^3*d^2 + 2*B^3*a^10*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*(40*B*a*b^14*d^4 + 192*B*a^3*b^12*d^4
+ 360*B*a^5*b^10*d^4 + 320*B*a^7*b^8*d^4 + 120*B*a^9*b^6*d^4 - 8*B*a^13*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) - (4*tan(c + d*x)^(1/2)*(-4*(B^2*a^5 + 9*B^2*a*b^4 - 6*B^2*a^3*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*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)
)/((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^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)))*(-4*(B^2*a^5 + 9*B^2*a*b^4 - 6*B^2*a^3*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))/(4*(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)) + (16*tan(c + d*x)^(1/2)*(20*B^2*a^3*b^10*d^2 + 168*B^2*a^5*b^8*d^2 + 40*B^2*a^7*b^6*d^2 - 44*B^2*a^9
*b^4*d^2 + 4*B^2*a^11*b^2*d^2 - 60*B^2*a*b^12*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))*(-4*(B^2*a^5 + 9*B^2*a*b^4 - 6*B^2*a^3*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))/(4*(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)))*(-4*(B^2*a^5
+ 9*B^2*a*b^4 - 6*B^2*a^3*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))/
(4*(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)))*(-4*(B^2*a^5 + 9*B^2*a*b^4 - 6*B^2*
a^3*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))/(4*(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)) + (((16*tan(c + d*x)^(1/2)*(2*B^4*b^9 + B^4*a^8*b - 5*B^4*a
^2*b^7 + 17*B^4*a^4*b^5 - 7*B^4*a^6*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)
- (((16*(8*B^3*a^4*b^7*d^2 - 78*B^3*a^2*b^9*d^2 + 60*B^3*a^6*b^5*d^2 - 24*B^3*a^8*b^3*d^2 + 2*B^3*a^10*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*(40*B*a*b^14*d^4 + 192*B*a^3*b^
12*d^4 + 360*B*a^5*b^10*d^4 + 320*B*a^7*b^8*d^4 + 120*B*a^9*b^6*d^4 - 8*B*a^13*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) + (4*tan(c + d*x)^(1/2)*(-4*(B^2*a^5 + 9*B^2*a*b^4 - 6*B^2*a^3*
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*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))/((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^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)))*(-4*(B^2*a^5 + 9*B^2*a*b^4 - 6*B^2*a^3*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))/(4*(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)) - (16*tan(c + d*x)^(1/2)*(20*B^2*a^3*b^10*d^2 + 168*B^2*a^5*b^8*d^2 + 40*B^2*a^7*b^6*d^2 - 44*B
^2*a^9*b^4*d^2 + 4*B^2*a^11*b^2*d^2 - 60*B^2*a*b^12*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))*(-4*(B^2*a^5 + 9*B^2*a*b^4 - 6*B^2*a^3*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))/(4*(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)))*(-4*(B
^2*a^5 + 9*B^2*a*b^4 - 6*B^2*a^3*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))/(4*(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)))*(-4*(B^2*a^5 + 9*B^2*a*b^4 -
6*B^2*a^3*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))/(4*(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))))*(-4*(B^2*a^5 + 9*B^2*a*b^4 - 6*B^2*a^3*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)*1i)/(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)) - (A*b*tan(c + d*x)^(1/2))/((a*d + b*d*tan(c + d*x))*(a^2 + b^2)) + (B*a
*tan(c + d*x)^(1/2))/((a*d + b*d*tan(c + d*x))*(a^2 + b^2))
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \sqrt {\tan {\left (c + d x \right )}}}{\left (a + b \tan {\left (c + d x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tan(d*x+c)**(1/2)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))**2,x)
[Out]
Integral((A + B*tan(c + d*x))*sqrt(tan(c + d*x))/(a + b*tan(c + d*x))**2, x)
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