3.369 \(\int \frac {\cot (c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx\)
Optimal. Leaf size=154 \[ -\frac {2 B \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {2 b^2 B}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {B \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d (a-i b)^{3/2}}+\frac {B \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d (a+i b)^{3/2}} \]
[Out]
-2*B*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/a^(3/2)/d+B*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/(a-I*b)
^(3/2)/d+B*arctanh((a+b*tan(d*x+c))^(1/2)/(a+I*b)^(1/2))/(a+I*b)^(3/2)/d+2*b^2*B/a/(a^2+b^2)/d/(a+b*tan(d*x+c)
)^(1/2)
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Rubi [A] time = 0.49, antiderivative size = 154, normalized size of antiderivative = 1.00,
number of steps used = 13, number of rules used = 8, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used
= {21, 3569, 3653, 3539, 3537, 63, 208, 3634} \[ \frac {2 b^2 B}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}-\frac {2 B \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {B \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d (a-i b)^{3/2}}+\frac {B \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d (a+i b)^{3/2}} \]
Antiderivative was successfully verified.
[In]
Int[(Cot[c + d*x]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2),x]
[Out]
(-2*B*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) + (B*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*
b]])/((a - I*b)^(3/2)*d) + (B*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(3/2)*d) + (2*b^2*B)
/(a*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])
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 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rule 3537
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c*
d)/f, Subst[Int[(a + (b*x)/d)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &&
NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
Rule 3539
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c
+ I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(c - I*d)/2, Int[(a + b*Tan[e + f*x]
)^m*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0]
&& NeQ[c^2 + d^2, 0] && !IntegerQ[m]
Rule 3569
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b^2*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d)), x] + D
ist[1/((m + 1)*(a^2 + b^2)*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c -
a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && I
ntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] &&
NeQ[a, 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]
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 {\cot (c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx &=B \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx\\ &=\frac {2 b^2 B}{a \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {(2 B) \int \frac {\cot (c+d x) \left (\frac {1}{2} \left (a^2+b^2\right )-\frac {1}{2} a b \tan (c+d x)+\frac {1}{2} b^2 \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{a \left (a^2+b^2\right )}\\ &=\frac {2 b^2 B}{a \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {B \int \frac {\cot (c+d x) \left (1+\tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{a}+\frac {(2 B) \int \frac {-\frac {a b}{2}-\frac {1}{2} a^2 \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{a \left (a^2+b^2\right )}\\ &=\frac {2 b^2 B}{a \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {B \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 (i a-b)}-\frac {B \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 (i a+b)}+\frac {B \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{a d}\\ &=\frac {2 b^2 B}{a \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}-\frac {B \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 (a-i b) d}-\frac {B \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 (a+i b) d}+\frac {(2 B) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{a b d}\\ &=-\frac {2 B \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {2 b^2 B}{a \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}-\frac {B \operatorname {Subst}\left (\int \frac {1}{-1+\frac {i a}{b}-\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{(i a-b) b d}+\frac {B \operatorname {Subst}\left (\int \frac {1}{-1-\frac {i a}{b}+\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{b (i a+b) d}\\ &=-\frac {2 B \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {B \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{3/2} d}+\frac {B \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{3/2} d}+\frac {2 b^2 B}{a \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 1.22, size = 166, normalized size = 1.08 \[ \frac {B \left (-\frac {2 \left (a^2+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {2 b^2}{\sqrt {a+b \tan (c+d x)}}+\frac {a (a+i b) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b}}+\frac {a (a-i b) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{\sqrt {a+i b}}\right )}{a d \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
[In]
Integrate[(Cot[c + d*x]*(a*B + b*B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^(5/2),x]
[Out]
(B*((-2*(a^2 + b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/Sqrt[a] + (a*(a + I*b)*ArcTanh[Sqrt[a + b*Tan[c
+ d*x]]/Sqrt[a - I*b]])/Sqrt[a - I*b] + (a*(a - I*b)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/Sqrt[a
+ I*b] + (2*b^2)/Sqrt[a + b*Tan[c + d*x]]))/(a*(a^2 + b^2)*d)
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fricas [B] time = 2.56, size = 13991, normalized size = 90.85 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c))^(5/2),x, algorithm="fricas")
[Out]
[1/4*(4*sqrt(2)*((a^12 + 3*a^10*b^2 + 2*a^8*b^4 - 2*a^6*b^6 - 3*a^4*b^8 - a^2*b^10)*d^5*cos(d*x + c)^2 + 2*(a^
11*b + 4*a^9*b^3 + 6*a^7*b^5 + 4*a^5*b^7 + a^3*b^9)*d^5*cos(d*x + c)*sin(d*x + c) + (a^10*b^2 + 4*a^8*b^4 + 6*
a^6*b^6 + 4*a^4*b^8 + a^2*b^10)*d^5)*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^
4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b
^4 + B^2*b^6))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4)*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4
*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*arctan(-((3*B^6*a^
12 + 14*B^6*a^10*b^2 + 25*B^6*a^8*b^4 + 20*B^6*a^6*b^6 + 5*B^6*a^4*b^8 - 2*B^6*a^2*b^10 - B^6*b^12)*d^4*sqrt(B
^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b
^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + (3*B^8*a^9 + 8*B^8*a^7*b^2 + 6*B^8*a^5*
b^4 - B^8*a*b^8)*d^2*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*
b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + sqrt(2)*((a^14 + 5*a^12*b^2 + 9*a^10*b^4 + 5*a^8*b^6 - 5*a^6*b^8
- 9*a^4*b^10 - 5*a^2*b^12 - b^14)*d^7*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*sqrt((9*B^4*a^4*b^2
- 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^
4)) + (B^2*a^11 + 5*B^2*a^9*b^2 + 10*B^2*a^7*b^4 + 10*B^2*a^5*b^6 + 5*B^2*a^3*b^8 + B^2*a*b^10)*d^5*sqrt((9*B^
4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 +
b^12)*d^4)))*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8
)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*sqrt(((9
*B^4*a^8 + 12*B^4*a^6*b^2 - 2*B^4*a^4*b^4 - 4*B^4*a^2*b^6 + B^4*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^
4 + b^6)*d^4))*cos(d*x + c) + sqrt(2)*((9*B^3*a^9 + 12*B^3*a^7*b^2 - 2*B^3*a^5*b^4 - 4*B^3*a^3*b^6 + B^3*a*b^8
)*d^3*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) + (9*B^5*a^6 - 15*B^5*a^4*b^2 + 7*B^5*a
^2*b^4 - B^5*b^6)*d*cos(d*x + c))*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 -
8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4
+ B^2*b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4
))^(1/4) + (9*B^6*a^5 - 6*B^6*a^3*b^2 + B^6*a*b^4)*cos(d*x + c) + (9*B^6*a^4*b - 6*B^6*a^2*b^3 + B^6*b^5)*sin(
d*x + c))/cos(d*x + c))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4) + sqrt(2)*((3*B^3*a^16 + 14*B^3*
a^14*b^2 + 22*B^3*a^12*b^4 + 6*B^3*a^10*b^6 - 20*B^3*a^8*b^8 - 22*B^3*a^6*b^10 - 6*B^3*a^4*b^12 + 2*B^3*a^2*b^
14 + B^3*b^16)*d^7*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B
^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + (3*B^5*a^13 +
14*B^5*a^11*b^2 + 25*B^5*a^9*b^4 + 20*B^5*a^7*b^6 + 5*B^5*a^5*b^8 - 2*B^5*a^3*b^10 - B^5*a*b^12)*d^5*sqrt((9*B
^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10
+ b^12)*d^4)))*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^
8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*sqrt((a
*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4))/(9*B^10*a
^4*b^2 - 6*B^10*a^2*b^4 + B^10*b^6)) + 4*sqrt(2)*((a^12 + 3*a^10*b^2 + 2*a^8*b^4 - 2*a^6*b^6 - 3*a^4*b^8 - a^2
*b^10)*d^5*cos(d*x + c)^2 + 2*(a^11*b + 4*a^9*b^3 + 6*a^7*b^5 + 4*a^5*b^7 + a^3*b^9)*d^5*cos(d*x + c)*sin(d*x
+ c) + (a^10*b^2 + 4*a^8*b^4 + 6*a^6*b^6 + 4*a^4*b^8 + a^2*b^10)*d^5)*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^
2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^
4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4)*sqrt((9*
B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10
+ b^12)*d^4))*arctan(((3*B^6*a^12 + 14*B^6*a^10*b^2 + 25*B^6*a^8*b^4 + 20*B^6*a^6*b^6 + 5*B^6*a^4*b^8 - 2*B^6
*a^2*b^10 - B^6*b^12)*d^4*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*
b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + (3*B^8*
a^9 + 8*B^8*a^7*b^2 + 6*B^8*a^5*b^4 - B^8*a*b^8)*d^2*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6
*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) - sqrt(2)*((a^14 + 5*a^12*b^2 + 9*
a^10*b^4 + 5*a^8*b^6 - 5*a^6*b^8 - 9*a^4*b^10 - 5*a^2*b^12 - b^14)*d^7*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4
+ b^6)*d^4))*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15
*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + (B^2*a^11 + 5*B^2*a^9*b^2 + 10*B^2*a^7*b^4 + 10*B^2*a^5*b^6 + 5*B^2*a^3*
b^8 + B^2*a*b^10)*d^5*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6
*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 -
6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*
B^2*a^2*b^4 + B^2*b^6))*sqrt(((9*B^4*a^8 + 12*B^4*a^6*b^2 - 2*B^4*a^4*b^4 - 4*B^4*a^2*b^6 + B^4*b^8)*d^2*sqrt(
B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) - sqrt(2)*((9*B^3*a^9 + 12*B^3*a^7*b^2 - 2*B^3*a^5
*b^4 - 4*B^3*a^3*b^6 + B^3*a*b^8)*d^3*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) + (9*B^
5*a^6 - 15*B^5*a^4*b^2 + 7*B^5*a^2*b^4 - B^5*b^6)*d*cos(d*x + c))*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^
4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))
/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(B^4/((a^6 +
3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(1/4) + (9*B^6*a^5 - 6*B^6*a^3*b^2 + B^6*a*b^4)*cos(d*x + c) + (9*B^6*a^4*b
- 6*B^6*a^2*b^3 + B^6*b^5)*sin(d*x + c))/cos(d*x + c))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4)
- sqrt(2)*((3*B^3*a^16 + 14*B^3*a^14*b^2 + 22*B^3*a^12*b^4 + 6*B^3*a^10*b^6 - 20*B^3*a^8*b^8 - 22*B^3*a^6*b^10
- 6*B^3*a^4*b^12 + 2*B^3*a^2*b^14 + B^3*b^16)*d^7*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*sqrt((9
*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^1
0 + b^12)*d^4)) + (3*B^5*a^13 + 14*B^5*a^11*b^2 + 25*B^5*a^9*b^4 + 20*B^5*a^7*b^6 + 5*B^5*a^5*b^8 - 2*B^5*a^3*
b^10 - B^5*a*b^12)*d^5*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^
6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9
- 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6
*B^2*a^2*b^4 + B^2*b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b
^4 + b^6)*d^4))^(3/4))/(9*B^10*a^4*b^2 - 6*B^10*a^2*b^4 + B^10*b^6)) + sqrt(2)*((B^4*a^6 - B^4*a^2*b^4)*d*cos(
d*x + c)^2 + 2*(B^4*a^5*b + B^4*a^3*b^3)*d*cos(d*x + c)*sin(d*x + c) + (B^4*a^4*b^2 + B^4*a^2*b^4)*d + ((B^2*a
^9 - 3*B^2*a^7*b^2 - B^2*a^5*b^4 + 3*B^2*a^3*b^6)*d^3*cos(d*x + c)^2 + 2*(B^2*a^8*b - 2*B^2*a^6*b^3 - 3*B^2*a^
4*b^5)*d^3*cos(d*x + c)*sin(d*x + c) + (B^2*a^7*b^2 - 2*B^2*a^5*b^4 - 3*B^2*a^3*b^6)*d^3)*sqrt(B^4/((a^6 + 3*a
^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 -
8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 +
B^2*b^6))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(1/4)*log(((9*B^4*a^8 + 12*B^4*a^6*b^2 - 2*B^4*a^4*
b^4 - 4*B^4*a^2*b^6 + B^4*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) + sqrt(2)*
((9*B^3*a^9 + 12*B^3*a^7*b^2 - 2*B^3*a^5*b^4 - 4*B^3*a^3*b^6 + B^3*a*b^8)*d^3*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a
^2*b^4 + b^6)*d^4))*cos(d*x + c) + (9*B^5*a^6 - 15*B^5*a^4*b^2 + 7*B^5*a^2*b^4 - B^5*b^6)*d*cos(d*x + c))*sqrt
((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a
^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*sqrt((a*cos(d*x + c) + b*s
in(d*x + c))/cos(d*x + c))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(1/4) + (9*B^6*a^5 - 6*B^6*a^3*b^2
+ B^6*a*b^4)*cos(d*x + c) + (9*B^6*a^4*b - 6*B^6*a^2*b^3 + B^6*b^5)*sin(d*x + c))/cos(d*x + c)) - sqrt(2)*((B^
4*a^6 - B^4*a^2*b^4)*d*cos(d*x + c)^2 + 2*(B^4*a^5*b + B^4*a^3*b^3)*d*cos(d*x + c)*sin(d*x + c) + (B^4*a^4*b^2
+ B^4*a^2*b^4)*d + ((B^2*a^9 - 3*B^2*a^7*b^2 - B^2*a^5*b^4 + 3*B^2*a^3*b^6)*d^3*cos(d*x + c)^2 + 2*(B^2*a^8*b
- 2*B^2*a^6*b^3 - 3*B^2*a^4*b^5)*d^3*cos(d*x + c)*sin(d*x + c) + (B^2*a^7*b^2 - 2*B^2*a^5*b^4 - 3*B^2*a^3*b^6
)*d^3)*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^
2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^
2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(1/4)*log(((9*B^4*a^8 +
12*B^4*a^6*b^2 - 2*B^4*a^4*b^4 - 4*B^4*a^2*b^6 + B^4*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^
4))*cos(d*x + c) - sqrt(2)*((9*B^3*a^9 + 12*B^3*a^7*b^2 - 2*B^3*a^5*b^4 - 4*B^3*a^3*b^6 + B^3*a*b^8)*d^3*sqrt(
B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) + (9*B^5*a^6 - 15*B^5*a^4*b^2 + 7*B^5*a^2*b^4 - B^
5*b^6)*d*cos(d*x + c))*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6
- 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))
*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(1/4) +
(9*B^6*a^5 - 6*B^6*a^3*b^2 + B^6*a*b^4)*cos(d*x + c) + (9*B^6*a^4*b - 6*B^6*a^2*b^3 + B^6*b^5)*sin(d*x + c))/c
os(d*x + c)) + 2*(B^5*a^2*b^2 + B^5*b^4 + (B^5*a^4 - B^5*b^4)*cos(d*x + c)^2 + 2*(B^5*a^3*b + B^5*a*b^3)*cos(d
*x + c)*sin(d*x + c))*sqrt(a)*log(-(8*a*b*cos(d*x + c)*sin(d*x + c) + (8*a^2 - b^2)*cos(d*x + c)^2 + b^2 - 4*(
2*a*cos(d*x + c)^2 + b*cos(d*x + c)*sin(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))
)/(cos(d*x + c)^2 - 1)) + 8*(B^5*a^2*b^2*cos(d*x + c)^2 + B^5*a*b^3*cos(d*x + c)*sin(d*x + c))*sqrt((a*cos(d*x
+ c) + b*sin(d*x + c))/cos(d*x + c)))/((B^4*a^6 - B^4*a^2*b^4)*d*cos(d*x + c)^2 + 2*(B^4*a^5*b + B^4*a^3*b^3)
*d*cos(d*x + c)*sin(d*x + c) + (B^4*a^4*b^2 + B^4*a^2*b^4)*d), 1/4*(4*sqrt(2)*((a^12 + 3*a^10*b^2 + 2*a^8*b^4
- 2*a^6*b^6 - 3*a^4*b^8 - a^2*b^10)*d^5*cos(d*x + c)^2 + 2*(a^11*b + 4*a^9*b^3 + 6*a^7*b^5 + 4*a^5*b^7 + a^3*b
^9)*d^5*cos(d*x + c)*sin(d*x + c) + (a^10*b^2 + 4*a^8*b^4 + 6*a^6*b^6 + 4*a^4*b^8 + a^2*b^10)*d^5)*sqrt((B^2*a
^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*
a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^
4 + b^6)*d^4))^(3/4)*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*
b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*arctan(-((3*B^6*a^12 + 14*B^6*a^10*b^2 + 25*B^6*a^8*b^4 + 20*B^6*a
^6*b^6 + 5*B^6*a^4*b^8 - 2*B^6*a^2*b^10 - B^6*b^12)*d^4*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*sq
rt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^
2*b^10 + b^12)*d^4)) + (3*B^8*a^9 + 8*B^8*a^7*b^2 + 6*B^8*a^5*b^4 - B^8*a*b^8)*d^2*sqrt((9*B^4*a^4*b^2 - 6*B^4
*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + sq
rt(2)*((a^14 + 5*a^12*b^2 + 9*a^10*b^4 + 5*a^8*b^6 - 5*a^6*b^8 - 9*a^4*b^10 - 5*a^2*b^12 - b^14)*d^7*sqrt(B^4/
((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2
+ 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + (B^2*a^11 + 5*B^2*a^9*b^2 + 10*B^2*a^7*b^4
+ 10*B^2*a^5*b^6 + 5*B^2*a^3*b^8 + B^2*a*b^10)*d^5*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*
a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*
B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b
^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*sqrt(((9*B^4*a^8 + 12*B^4*a^6*b^2 - 2*B^4*a^4*b^4 - 4*B^
4*a^2*b^6 + B^4*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) + sqrt(2)*((9*B^3*a^
9 + 12*B^3*a^7*b^2 - 2*B^3*a^5*b^4 - 4*B^3*a^3*b^6 + B^3*a*b^8)*d^3*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b
^6)*d^4))*cos(d*x + c) + (9*B^5*a^6 - 15*B^5*a^4*b^2 + 7*B^5*a^2*b^4 - B^5*b^6)*d*cos(d*x + c))*sqrt((B^2*a^6
+ 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4
*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c
))/cos(d*x + c))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(1/4) + (9*B^6*a^5 - 6*B^6*a^3*b^2 + B^6*a*b^
4)*cos(d*x + c) + (9*B^6*a^4*b - 6*B^6*a^2*b^3 + B^6*b^5)*sin(d*x + c))/cos(d*x + c))*(B^4/((a^6 + 3*a^4*b^2 +
3*a^2*b^4 + b^6)*d^4))^(3/4) + sqrt(2)*((3*B^3*a^16 + 14*B^3*a^14*b^2 + 22*B^3*a^12*b^4 + 6*B^3*a^10*b^6 - 20
*B^3*a^8*b^8 - 22*B^3*a^6*b^10 - 6*B^3*a^4*b^12 + 2*B^3*a^2*b^14 + B^3*b^16)*d^7*sqrt(B^4/((a^6 + 3*a^4*b^2 +
3*a^2*b^4 + b^6)*d^4))*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^
6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + (3*B^5*a^13 + 14*B^5*a^11*b^2 + 25*B^5*a^9*b^4 + 20*B^5*a^7*b^
6 + 5*B^5*a^5*b^8 - 2*B^5*a^3*b^10 - B^5*a*b^12)*d^5*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6
*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3
*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 +
b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(B
^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4))/(9*B^10*a^4*b^2 - 6*B^10*a^2*b^4 + B^10*b^6)) + 4*sqrt(2)
*((a^12 + 3*a^10*b^2 + 2*a^8*b^4 - 2*a^6*b^6 - 3*a^4*b^8 - a^2*b^10)*d^5*cos(d*x + c)^2 + 2*(a^11*b + 4*a^9*b^
3 + 6*a^7*b^5 + 4*a^5*b^7 + a^3*b^9)*d^5*cos(d*x + c)*sin(d*x + c) + (a^10*b^2 + 4*a^8*b^4 + 6*a^6*b^6 + 4*a^4
*b^8 + a^2*b^10)*d^5)*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 -
3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*
(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4)*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 +
6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*arctan(((3*B^6*a^12 + 14*B^6*a^10
*b^2 + 25*B^6*a^8*b^4 + 20*B^6*a^6*b^6 + 5*B^6*a^4*b^8 - 2*B^6*a^2*b^10 - B^6*b^12)*d^4*sqrt(B^4/((a^6 + 3*a^4
*b^2 + 3*a^2*b^4 + b^6)*d^4))*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4
+ 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + (3*B^8*a^9 + 8*B^8*a^7*b^2 + 6*B^8*a^5*b^4 - B^8*a*b^8)
*d^2*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8
+ 6*a^2*b^10 + b^12)*d^4)) - sqrt(2)*((a^14 + 5*a^12*b^2 + 9*a^10*b^4 + 5*a^8*b^6 - 5*a^6*b^8 - 9*a^4*b^10 -
5*a^2*b^12 - b^14)*d^7*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4
+ B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + (B^2*a^11
+ 5*B^2*a^9*b^2 + 10*B^2*a^7*b^4 + 10*B^2*a^5*b^6 + 5*B^2*a^3*b^8 + B^2*a*b^10)*d^5*sqrt((9*B^4*a^4*b^2 - 6*B^
4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))*sq
rt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/(
(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*sqrt(((9*B^4*a^8 + 12*B^
4*a^6*b^2 - 2*B^4*a^4*b^4 - 4*B^4*a^2*b^6 + B^4*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*c
os(d*x + c) - sqrt(2)*((9*B^3*a^9 + 12*B^3*a^7*b^2 - 2*B^3*a^5*b^4 - 4*B^3*a^3*b^6 + B^3*a*b^8)*d^3*sqrt(B^4/(
(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) + (9*B^5*a^6 - 15*B^5*a^4*b^2 + 7*B^5*a^2*b^4 - B^5*b^6
)*d*cos(d*x + c))*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a
*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*sqrt
((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(1/4) + (9*B^
6*a^5 - 6*B^6*a^3*b^2 + B^6*a*b^4)*cos(d*x + c) + (9*B^6*a^4*b - 6*B^6*a^2*b^3 + B^6*b^5)*sin(d*x + c))/cos(d*
x + c))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4) - sqrt(2)*((3*B^3*a^16 + 14*B^3*a^14*b^2 + 22*B^
3*a^12*b^4 + 6*B^3*a^10*b^6 - 20*B^3*a^8*b^8 - 22*B^3*a^6*b^10 - 6*B^3*a^4*b^12 + 2*B^3*a^2*b^14 + B^3*b^16)*d
^7*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*sqrt((9*B^4*a^4*b^2 - 6*B^4*a^2*b^4 + B^4*b^6)/((a^12 +
6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + (3*B^5*a^13 + 14*B^5*a^11*b^2
+ 25*B^5*a^9*b^4 + 20*B^5*a^7*b^6 + 5*B^5*a^5*b^8 - 2*B^5*a^3*b^10 - B^5*a*b^12)*d^5*sqrt((9*B^4*a^4*b^2 - 6*B
^4*a^2*b^4 + B^4*b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))*s
qrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/
((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*sqrt((a*cos(d*x + c) +
b*sin(d*x + c))/cos(d*x + c))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4))/(9*B^10*a^4*b^2 - 6*B^10*
a^2*b^4 + B^10*b^6)) + sqrt(2)*((B^4*a^6 - B^4*a^2*b^4)*d*cos(d*x + c)^2 + 2*(B^4*a^5*b + B^4*a^3*b^3)*d*cos(d
*x + c)*sin(d*x + c) + (B^4*a^4*b^2 + B^4*a^2*b^4)*d + ((B^2*a^9 - 3*B^2*a^7*b^2 - B^2*a^5*b^4 + 3*B^2*a^3*b^6
)*d^3*cos(d*x + c)^2 + 2*(B^2*a^8*b - 2*B^2*a^6*b^3 - 3*B^2*a^4*b^5)*d^3*cos(d*x + c)*sin(d*x + c) + (B^2*a^7*
b^2 - 2*B^2*a^5*b^4 - 3*B^2*a^3*b^6)*d^3)*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))*sqrt((B^2*a^6 +
3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*
b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 +
b^6)*d^4))^(1/4)*log(((9*B^4*a^8 + 12*B^4*a^6*b^2 - 2*B^4*a^4*b^4 - 4*B^4*a^2*b^6 + B^4*b^8)*d^2*sqrt(B^4/((a^
6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) + sqrt(2)*((9*B^3*a^9 + 12*B^3*a^7*b^2 - 2*B^3*a^5*b^4 - 4
*B^3*a^3*b^6 + B^3*a*b^8)*d^3*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) + (9*B^5*a^6 -
15*B^5*a^4*b^2 + 7*B^5*a^2*b^4 - B^5*b^6)*d*cos(d*x + c))*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*
b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*
a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(B^4/((a^6 + 3*a^4*b^
2 + 3*a^2*b^4 + b^6)*d^4))^(1/4) + (9*B^6*a^5 - 6*B^6*a^3*b^2 + B^6*a*b^4)*cos(d*x + c) + (9*B^6*a^4*b - 6*B^6
*a^2*b^3 + B^6*b^5)*sin(d*x + c))/cos(d*x + c)) - sqrt(2)*((B^4*a^6 - B^4*a^2*b^4)*d*cos(d*x + c)^2 + 2*(B^4*a
^5*b + B^4*a^3*b^3)*d*cos(d*x + c)*sin(d*x + c) + (B^4*a^4*b^2 + B^4*a^2*b^4)*d + ((B^2*a^9 - 3*B^2*a^7*b^2 -
B^2*a^5*b^4 + 3*B^2*a^3*b^6)*d^3*cos(d*x + c)^2 + 2*(B^2*a^8*b - 2*B^2*a^6*b^3 - 3*B^2*a^4*b^5)*d^3*cos(d*x +
c)*sin(d*x + c) + (B^2*a^7*b^2 - 2*B^2*a^5*b^4 - 3*B^2*a^3*b^6)*d^3)*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 +
b^6)*d^4)))*sqrt((B^2*a^6 + 3*B^2*a^4*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*
d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*(B^4/((a^6
+ 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(1/4)*log(((9*B^4*a^8 + 12*B^4*a^6*b^2 - 2*B^4*a^4*b^4 - 4*B^4*a^2*b^6 +
B^4*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) - sqrt(2)*((9*B^3*a^9 + 12*B^3*
a^7*b^2 - 2*B^3*a^5*b^4 - 4*B^3*a^3*b^6 + B^3*a*b^8)*d^3*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))*c
os(d*x + c) + (9*B^5*a^6 - 15*B^5*a^4*b^2 + 7*B^5*a^2*b^4 - B^5*b^6)*d*cos(d*x + c))*sqrt((B^2*a^6 + 3*B^2*a^4
*b^2 + 3*B^2*a^2*b^4 + B^2*b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(B^4/((a^6 + 3*a^4*b^2 + 3*a^
2*b^4 + b^6)*d^4)))/(9*B^2*a^4*b^2 - 6*B^2*a^2*b^4 + B^2*b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x
+ c))*(B^4/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(1/4) + (9*B^6*a^5 - 6*B^6*a^3*b^2 + B^6*a*b^4)*cos(d*x
+ c) + (9*B^6*a^4*b - 6*B^6*a^2*b^3 + B^6*b^5)*sin(d*x + c))/cos(d*x + c)) + 8*(B^5*a^2*b^2 + B^5*b^4 + (B^5*a
^4 - B^5*b^4)*cos(d*x + c)^2 + 2*(B^5*a^3*b + B^5*a*b^3)*cos(d*x + c)*sin(d*x + c))*sqrt(-a)*arctan(sqrt(-a)*s
qrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))/a) + 8*(B^5*a^2*b^2*cos(d*x + c)^2 + B^5*a*b^3*cos(d*x + c
)*sin(d*x + c))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c)))/((B^4*a^6 - B^4*a^2*b^4)*d*cos(d*x + c)^
2 + 2*(B^4*a^5*b + B^4*a^3*b^3)*d*cos(d*x + c)*sin(d*x + c) + (B^4*a^4*b^2 + B^4*a^2*b^4)*d)]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c))^(5/2),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Warning, need to choose a
branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [d]=[-4,
-97]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueWarning, need t
o choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assumi
ng [d]=[69,80]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2p
oly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen
& e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m
& i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur &
l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argu
ment Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/
r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e
,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,
const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l)
Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument
Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sy
m(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,con
st index_m & i,const vecteur & l) Error: Bad Argument ValueWarning, integration of abs or sign assumes constan
t sign by intervals (correct if the argument is real):Check [abs(t_nostep^2-1)]Discontinuities at zeroes of t_
nostep^2-1 were not checkedEvaluation time: 80.75Done
________________________________________________________________________________________
maple [C] time = 2.60, size = 39359, normalized size = 255.58 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(d*x+c)*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c))^(5/2),x)
[Out]
result too large to display
________________________________________________________________________________________
maxima [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(cot(d*x+c)*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c))^(5/2),x, algorithm="maxima")
[Out]
Timed out
________________________________________________________________________________________
mupad [B] time = 11.76, size = 7172, normalized size = 46.57 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((cot(c + d*x)*(B*a + B*b*tan(c + d*x)))/(a + b*tan(c + d*x))^(5/2),x)
[Out]
(log(((((96*B^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*a^4*b^2*d^4)^(1/2) + 4*B^2*a^3*d^2 - 12*B^2*a*b^2*d^2)/
(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4))^(1/2)*(((((((96*B^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^
4*a^4*b^2*d^4)^(1/2) + 4*B^2*a^3*d^2 - 12*B^2*a*b^2*d^2)/(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4))^
(1/2)*(512*B*a^8*b^28*d^8 - ((((96*B^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*a^4*b^2*d^4)^(1/2) + 4*B^2*a^3*d
^2 - 12*B^2*a*b^2*d^2)/(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(
512*a^9*b^28*d^9 + 5376*a^11*b^26*d^9 + 25344*a^13*b^24*d^9 + 70656*a^15*b^22*d^9 + 129024*a^17*b^20*d^9 + 161
280*a^19*b^18*d^9 + 139776*a^21*b^16*d^9 + 82944*a^23*b^14*d^9 + 32256*a^25*b^12*d^9 + 7424*a^27*b^10*d^9 + 76
8*a^29*b^8*d^9))/4 + 5248*B*a^10*b^26*d^8 + 23936*B*a^12*b^24*d^8 + 64000*B*a^14*b^22*d^8 + 111104*B*a^16*b^20
*d^8 + 130816*B*a^18*b^18*d^8 + 105728*B*a^20*b^16*d^8 + 57856*B*a^22*b^14*d^8 + 20480*B*a^24*b^12*d^8 + 4224*
B*a^26*b^10*d^8 + 384*B*a^28*b^8*d^8))/4 + (a + b*tan(c + d*x))^(1/2)*(256*B^2*a^8*b^26*d^7 + 1472*B^2*a^10*b^
24*d^7 + 3712*B^2*a^12*b^22*d^7 + 6272*B^2*a^14*b^20*d^7 + 9856*B^2*a^16*b^18*d^7 + 14336*B^2*a^18*b^16*d^7 +
15232*B^2*a^20*b^14*d^7 + 10112*B^2*a^22*b^12*d^7 + 3712*B^2*a^24*b^10*d^7 + 576*B^2*a^26*b^8*d^7))*(((96*B^4*
a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*a^4*b^2*d^4)^(1/2) + 4*B^2*a^3*d^2 - 12*B^2*a*b^2*d^2)/(a^6*d^4 + b^6*d
^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4))^(1/2))/4 - 128*B^3*a^7*b^26*d^6 - 128*B^3*a^9*b^24*d^6 + 2592*B^3*a^11*b^
22*d^6 + 10976*B^3*a^13*b^20*d^6 + 20384*B^3*a^15*b^18*d^6 + 20832*B^3*a^17*b^16*d^6 + 11872*B^3*a^19*b^14*d^6
+ 3232*B^3*a^21*b^12*d^6 + 96*B^3*a^23*b^10*d^6 - 96*B^3*a^25*b^8*d^6))/4 - (a + b*tan(c + d*x))^(1/2)*(1120*
B^4*a^15*b^16*d^5 - 352*B^4*a^9*b^22*d^5 - 672*B^4*a^11*b^20*d^5 - 224*B^4*a^13*b^18*d^5 - 64*B^4*a^7*b^24*d^5
+ 2016*B^4*a^17*b^14*d^5 + 1568*B^4*a^19*b^12*d^5 + 608*B^4*a^21*b^10*d^5 + 96*B^4*a^23*b^8*d^5))*(((96*B^4*a
^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*a^4*b^2*d^4)^(1/2) + 4*B^2*a^3*d^2 - 12*B^2*a*b^2*d^2)/(a^6*d^4 + b^6*d^
4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4))^(1/2))/4 + (log(((-((96*B^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*a^4*b^2
*d^4)^(1/2) - 4*B^2*a^3*d^2 + 12*B^2*a*b^2*d^2)/(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4))^(1/2)*(((
((-((96*B^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*a^4*b^2*d^4)^(1/2) - 4*B^2*a^3*d^2 + 12*B^2*a*b^2*d^2)/(a^6
*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4))^(1/2)*(512*B*a^8*b^28*d^8 - ((-((96*B^4*a^2*b^4*d^4 - 16*B^4*
b^6*d^4 - 144*B^4*a^4*b^2*d^4)^(1/2) - 4*B^2*a^3*d^2 + 12*B^2*a*b^2*d^2)/(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 +
3*a^4*b^2*d^4))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(512*a^9*b^28*d^9 + 5376*a^11*b^26*d^9 + 25344*a^13*b^24*d^9
+ 70656*a^15*b^22*d^9 + 129024*a^17*b^20*d^9 + 161280*a^19*b^18*d^9 + 139776*a^21*b^16*d^9 + 82944*a^23*b^14*d
^9 + 32256*a^25*b^12*d^9 + 7424*a^27*b^10*d^9 + 768*a^29*b^8*d^9))/4 + 5248*B*a^10*b^26*d^8 + 23936*B*a^12*b^2
4*d^8 + 64000*B*a^14*b^22*d^8 + 111104*B*a^16*b^20*d^8 + 130816*B*a^18*b^18*d^8 + 105728*B*a^20*b^16*d^8 + 578
56*B*a^22*b^14*d^8 + 20480*B*a^24*b^12*d^8 + 4224*B*a^26*b^10*d^8 + 384*B*a^28*b^8*d^8))/4 + (a + b*tan(c + d*
x))^(1/2)*(256*B^2*a^8*b^26*d^7 + 1472*B^2*a^10*b^24*d^7 + 3712*B^2*a^12*b^22*d^7 + 6272*B^2*a^14*b^20*d^7 + 9
856*B^2*a^16*b^18*d^7 + 14336*B^2*a^18*b^16*d^7 + 15232*B^2*a^20*b^14*d^7 + 10112*B^2*a^22*b^12*d^7 + 3712*B^2
*a^24*b^10*d^7 + 576*B^2*a^26*b^8*d^7))*(-((96*B^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*a^4*b^2*d^4)^(1/2) -
4*B^2*a^3*d^2 + 12*B^2*a*b^2*d^2)/(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4))^(1/2))/4 - 128*B^3*a^7
*b^26*d^6 - 128*B^3*a^9*b^24*d^6 + 2592*B^3*a^11*b^22*d^6 + 10976*B^3*a^13*b^20*d^6 + 20384*B^3*a^15*b^18*d^6
+ 20832*B^3*a^17*b^16*d^6 + 11872*B^3*a^19*b^14*d^6 + 3232*B^3*a^21*b^12*d^6 + 96*B^3*a^23*b^10*d^6 - 96*B^3*a
^25*b^8*d^6))/4 - (a + b*tan(c + d*x))^(1/2)*(1120*B^4*a^15*b^16*d^5 - 352*B^4*a^9*b^22*d^5 - 672*B^4*a^11*b^2
0*d^5 - 224*B^4*a^13*b^18*d^5 - 64*B^4*a^7*b^24*d^5 + 2016*B^4*a^17*b^14*d^5 + 1568*B^4*a^19*b^12*d^5 + 608*B^
4*a^21*b^10*d^5 + 96*B^4*a^23*b^8*d^5))*(-((96*B^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*a^4*b^2*d^4)^(1/2) -
4*B^2*a^3*d^2 + 12*B^2*a*b^2*d^2)/(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4))^(1/2))/4 - log((((96*B
^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*a^4*b^2*d^4)^(1/2) + 4*B^2*a^3*d^2 - 12*B^2*a*b^2*d^2)/(16*a^6*d^4 +
16*b^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^2*d^4))^(1/2)*(((((96*B^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*a^4*b
^2*d^4)^(1/2) + 4*B^2*a^3*d^2 - 12*B^2*a*b^2*d^2)/(16*a^6*d^4 + 16*b^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^2*d^4))
^(1/2)*((((96*B^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*a^4*b^2*d^4)^(1/2) + 4*B^2*a^3*d^2 - 12*B^2*a*b^2*d^2
)/(16*a^6*d^4 + 16*b^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^2*d^4))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(512*a^9*b^28*
d^9 + 5376*a^11*b^26*d^9 + 25344*a^13*b^24*d^9 + 70656*a^15*b^22*d^9 + 129024*a^17*b^20*d^9 + 161280*a^19*b^18
*d^9 + 139776*a^21*b^16*d^9 + 82944*a^23*b^14*d^9 + 32256*a^25*b^12*d^9 + 7424*a^27*b^10*d^9 + 768*a^29*b^8*d^
9) + 512*B*a^8*b^28*d^8 + 5248*B*a^10*b^26*d^8 + 23936*B*a^12*b^24*d^8 + 64000*B*a^14*b^22*d^8 + 111104*B*a^16
*b^20*d^8 + 130816*B*a^18*b^18*d^8 + 105728*B*a^20*b^16*d^8 + 57856*B*a^22*b^14*d^8 + 20480*B*a^24*b^12*d^8 +
4224*B*a^26*b^10*d^8 + 384*B*a^28*b^8*d^8) - (a + b*tan(c + d*x))^(1/2)*(256*B^2*a^8*b^26*d^7 + 1472*B^2*a^10*
b^24*d^7 + 3712*B^2*a^12*b^22*d^7 + 6272*B^2*a^14*b^20*d^7 + 9856*B^2*a^16*b^18*d^7 + 14336*B^2*a^18*b^16*d^7
+ 15232*B^2*a^20*b^14*d^7 + 10112*B^2*a^22*b^12*d^7 + 3712*B^2*a^24*b^10*d^7 + 576*B^2*a^26*b^8*d^7))*(((96*B^
4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*a^4*b^2*d^4)^(1/2) + 4*B^2*a^3*d^2 - 12*B^2*a*b^2*d^2)/(16*a^6*d^4 +
16*b^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^2*d^4))^(1/2) - 128*B^3*a^7*b^26*d^6 - 128*B^3*a^9*b^24*d^6 + 2592*B^3*
a^11*b^22*d^6 + 10976*B^3*a^13*b^20*d^6 + 20384*B^3*a^15*b^18*d^6 + 20832*B^3*a^17*b^16*d^6 + 11872*B^3*a^19*b
^14*d^6 + 3232*B^3*a^21*b^12*d^6 + 96*B^3*a^23*b^10*d^6 - 96*B^3*a^25*b^8*d^6) + (a + b*tan(c + d*x))^(1/2)*(1
120*B^4*a^15*b^16*d^5 - 352*B^4*a^9*b^22*d^5 - 672*B^4*a^11*b^20*d^5 - 224*B^4*a^13*b^18*d^5 - 64*B^4*a^7*b^24
*d^5 + 2016*B^4*a^17*b^14*d^5 + 1568*B^4*a^19*b^12*d^5 + 608*B^4*a^21*b^10*d^5 + 96*B^4*a^23*b^8*d^5))*(((96*B
^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*a^4*b^2*d^4)^(1/2) + 4*B^2*a^3*d^2 - 12*B^2*a*b^2*d^2)/(16*a^6*d^4 +
16*b^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^2*d^4))^(1/2) - log((-((96*B^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*
a^4*b^2*d^4)^(1/2) - 4*B^2*a^3*d^2 + 12*B^2*a*b^2*d^2)/(16*a^6*d^4 + 16*b^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^2*
d^4))^(1/2)*(((-((96*B^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^4*a^4*b^2*d^4)^(1/2) - 4*B^2*a^3*d^2 + 12*B^2*a*
b^2*d^2)/(16*a^6*d^4 + 16*b^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^2*d^4))^(1/2)*((-((96*B^4*a^2*b^4*d^4 - 16*B^4*b
^6*d^4 - 144*B^4*a^4*b^2*d^4)^(1/2) - 4*B^2*a^3*d^2 + 12*B^2*a*b^2*d^2)/(16*a^6*d^4 + 16*b^6*d^4 + 48*a^2*b^4*
d^4 + 48*a^4*b^2*d^4))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(512*a^9*b^28*d^9 + 5376*a^11*b^26*d^9 + 25344*a^13*b^
24*d^9 + 70656*a^15*b^22*d^9 + 129024*a^17*b^20*d^9 + 161280*a^19*b^18*d^9 + 139776*a^21*b^16*d^9 + 82944*a^23
*b^14*d^9 + 32256*a^25*b^12*d^9 + 7424*a^27*b^10*d^9 + 768*a^29*b^8*d^9) + 512*B*a^8*b^28*d^8 + 5248*B*a^10*b^
26*d^8 + 23936*B*a^12*b^24*d^8 + 64000*B*a^14*b^22*d^8 + 111104*B*a^16*b^20*d^8 + 130816*B*a^18*b^18*d^8 + 105
728*B*a^20*b^16*d^8 + 57856*B*a^22*b^14*d^8 + 20480*B*a^24*b^12*d^8 + 4224*B*a^26*b^10*d^8 + 384*B*a^28*b^8*d^
8) - (a + b*tan(c + d*x))^(1/2)*(256*B^2*a^8*b^26*d^7 + 1472*B^2*a^10*b^24*d^7 + 3712*B^2*a^12*b^22*d^7 + 6272
*B^2*a^14*b^20*d^7 + 9856*B^2*a^16*b^18*d^7 + 14336*B^2*a^18*b^16*d^7 + 15232*B^2*a^20*b^14*d^7 + 10112*B^2*a^
22*b^12*d^7 + 3712*B^2*a^24*b^10*d^7 + 576*B^2*a^26*b^8*d^7))*(-((96*B^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*B^
4*a^4*b^2*d^4)^(1/2) - 4*B^2*a^3*d^2 + 12*B^2*a*b^2*d^2)/(16*a^6*d^4 + 16*b^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*b^
2*d^4))^(1/2) - 128*B^3*a^7*b^26*d^6 - 128*B^3*a^9*b^24*d^6 + 2592*B^3*a^11*b^22*d^6 + 10976*B^3*a^13*b^20*d^6
+ 20384*B^3*a^15*b^18*d^6 + 20832*B^3*a^17*b^16*d^6 + 11872*B^3*a^19*b^14*d^6 + 3232*B^3*a^21*b^12*d^6 + 96*B
^3*a^23*b^10*d^6 - 96*B^3*a^25*b^8*d^6) + (a + b*tan(c + d*x))^(1/2)*(1120*B^4*a^15*b^16*d^5 - 352*B^4*a^9*b^2
2*d^5 - 672*B^4*a^11*b^20*d^5 - 224*B^4*a^13*b^18*d^5 - 64*B^4*a^7*b^24*d^5 + 2016*B^4*a^17*b^14*d^5 + 1568*B^
4*a^19*b^12*d^5 + 608*B^4*a^21*b^10*d^5 + 96*B^4*a^23*b^8*d^5))*(-((96*B^4*a^2*b^4*d^4 - 16*B^4*b^6*d^4 - 144*
B^4*a^4*b^2*d^4)^(1/2) - 4*B^2*a^3*d^2 + 12*B^2*a*b^2*d^2)/(16*a^6*d^4 + 16*b^6*d^4 + 48*a^2*b^4*d^4 + 48*a^4*
b^2*d^4))^(1/2) + (B*atan((B^5*a^3*b^28*d^4*(a + b*tan(c + d*x))^(1/2)*1024i)/((a^3)^(1/2)*(1024*B^5*a^2*b^28*
d^4 + 10240*B^5*a^4*b^26*d^4 + 47616*B^5*a^6*b^24*d^4 + 133184*B^5*a^8*b^22*d^4 + 244160*B^5*a^10*b^20*d^4 + 3
02400*B^5*a^12*b^18*d^4 + 253120*B^5*a^14*b^16*d^4 + 139456*B^5*a^16*b^14*d^4 + 47424*B^5*a^18*b^12*d^4 + 8640
*B^5*a^20*b^10*d^4 + 576*B^5*a^22*b^8*d^4)) + (B^5*a^5*b^26*d^4*(a + b*tan(c + d*x))^(1/2)*10240i)/((a^3)^(1/2
)*(1024*B^5*a^2*b^28*d^4 + 10240*B^5*a^4*b^26*d^4 + 47616*B^5*a^6*b^24*d^4 + 133184*B^5*a^8*b^22*d^4 + 244160*
B^5*a^10*b^20*d^4 + 302400*B^5*a^12*b^18*d^4 + 253120*B^5*a^14*b^16*d^4 + 139456*B^5*a^16*b^14*d^4 + 47424*B^5
*a^18*b^12*d^4 + 8640*B^5*a^20*b^10*d^4 + 576*B^5*a^22*b^8*d^4)) + (B^5*a^7*b^24*d^4*(a + b*tan(c + d*x))^(1/2
)*47616i)/((a^3)^(1/2)*(1024*B^5*a^2*b^28*d^4 + 10240*B^5*a^4*b^26*d^4 + 47616*B^5*a^6*b^24*d^4 + 133184*B^5*a
^8*b^22*d^4 + 244160*B^5*a^10*b^20*d^4 + 302400*B^5*a^12*b^18*d^4 + 253120*B^5*a^14*b^16*d^4 + 139456*B^5*a^16
*b^14*d^4 + 47424*B^5*a^18*b^12*d^4 + 8640*B^5*a^20*b^10*d^4 + 576*B^5*a^22*b^8*d^4)) + (B^5*a^9*b^22*d^4*(a +
b*tan(c + d*x))^(1/2)*133184i)/((a^3)^(1/2)*(1024*B^5*a^2*b^28*d^4 + 10240*B^5*a^4*b^26*d^4 + 47616*B^5*a^6*b
^24*d^4 + 133184*B^5*a^8*b^22*d^4 + 244160*B^5*a^10*b^20*d^4 + 302400*B^5*a^12*b^18*d^4 + 253120*B^5*a^14*b^16
*d^4 + 139456*B^5*a^16*b^14*d^4 + 47424*B^5*a^18*b^12*d^4 + 8640*B^5*a^20*b^10*d^4 + 576*B^5*a^22*b^8*d^4)) +
(B^5*a^11*b^20*d^4*(a + b*tan(c + d*x))^(1/2)*244160i)/((a^3)^(1/2)*(1024*B^5*a^2*b^28*d^4 + 10240*B^5*a^4*b^2
6*d^4 + 47616*B^5*a^6*b^24*d^4 + 133184*B^5*a^8*b^22*d^4 + 244160*B^5*a^10*b^20*d^4 + 302400*B^5*a^12*b^18*d^4
+ 253120*B^5*a^14*b^16*d^4 + 139456*B^5*a^16*b^14*d^4 + 47424*B^5*a^18*b^12*d^4 + 8640*B^5*a^20*b^10*d^4 + 57
6*B^5*a^22*b^8*d^4)) + (B^5*a^13*b^18*d^4*(a + b*tan(c + d*x))^(1/2)*302400i)/((a^3)^(1/2)*(1024*B^5*a^2*b^28*
d^4 + 10240*B^5*a^4*b^26*d^4 + 47616*B^5*a^6*b^24*d^4 + 133184*B^5*a^8*b^22*d^4 + 244160*B^5*a^10*b^20*d^4 + 3
02400*B^5*a^12*b^18*d^4 + 253120*B^5*a^14*b^16*d^4 + 139456*B^5*a^16*b^14*d^4 + 47424*B^5*a^18*b^12*d^4 + 8640
*B^5*a^20*b^10*d^4 + 576*B^5*a^22*b^8*d^4)) + (B^5*a^15*b^16*d^4*(a + b*tan(c + d*x))^(1/2)*253120i)/((a^3)^(1
/2)*(1024*B^5*a^2*b^28*d^4 + 10240*B^5*a^4*b^26*d^4 + 47616*B^5*a^6*b^24*d^4 + 133184*B^5*a^8*b^22*d^4 + 24416
0*B^5*a^10*b^20*d^4 + 302400*B^5*a^12*b^18*d^4 + 253120*B^5*a^14*b^16*d^4 + 139456*B^5*a^16*b^14*d^4 + 47424*B
^5*a^18*b^12*d^4 + 8640*B^5*a^20*b^10*d^4 + 576*B^5*a^22*b^8*d^4)) + (B^5*a^17*b^14*d^4*(a + b*tan(c + d*x))^(
1/2)*139456i)/((a^3)^(1/2)*(1024*B^5*a^2*b^28*d^4 + 10240*B^5*a^4*b^26*d^4 + 47616*B^5*a^6*b^24*d^4 + 133184*B
^5*a^8*b^22*d^4 + 244160*B^5*a^10*b^20*d^4 + 302400*B^5*a^12*b^18*d^4 + 253120*B^5*a^14*b^16*d^4 + 139456*B^5*
a^16*b^14*d^4 + 47424*B^5*a^18*b^12*d^4 + 8640*B^5*a^20*b^10*d^4 + 576*B^5*a^22*b^8*d^4)) + (B^5*a^19*b^12*d^4
*(a + b*tan(c + d*x))^(1/2)*47424i)/((a^3)^(1/2)*(1024*B^5*a^2*b^28*d^4 + 10240*B^5*a^4*b^26*d^4 + 47616*B^5*a
^6*b^24*d^4 + 133184*B^5*a^8*b^22*d^4 + 244160*B^5*a^10*b^20*d^4 + 302400*B^5*a^12*b^18*d^4 + 253120*B^5*a^14*
b^16*d^4 + 139456*B^5*a^16*b^14*d^4 + 47424*B^5*a^18*b^12*d^4 + 8640*B^5*a^20*b^10*d^4 + 576*B^5*a^22*b^8*d^4)
) + (B^5*a^21*b^10*d^4*(a + b*tan(c + d*x))^(1/2)*8640i)/((a^3)^(1/2)*(1024*B^5*a^2*b^28*d^4 + 10240*B^5*a^4*b
^26*d^4 + 47616*B^5*a^6*b^24*d^4 + 133184*B^5*a^8*b^22*d^4 + 244160*B^5*a^10*b^20*d^4 + 302400*B^5*a^12*b^18*d
^4 + 253120*B^5*a^14*b^16*d^4 + 139456*B^5*a^16*b^14*d^4 + 47424*B^5*a^18*b^12*d^4 + 8640*B^5*a^20*b^10*d^4 +
576*B^5*a^22*b^8*d^4)) + (B^5*a^23*b^8*d^4*(a + b*tan(c + d*x))^(1/2)*576i)/((a^3)^(1/2)*(1024*B^5*a^2*b^28*d^
4 + 10240*B^5*a^4*b^26*d^4 + 47616*B^5*a^6*b^24*d^4 + 133184*B^5*a^8*b^22*d^4 + 244160*B^5*a^10*b^20*d^4 + 302
400*B^5*a^12*b^18*d^4 + 253120*B^5*a^14*b^16*d^4 + 139456*B^5*a^16*b^14*d^4 + 47424*B^5*a^18*b^12*d^4 + 8640*B
^5*a^20*b^10*d^4 + 576*B^5*a^22*b^8*d^4)))*2i)/(d*(a^3)^(1/2)) + (2*B*b^2)/(d*(a*b^2 + a^3)*(a + b*tan(c + d*x
))^(1/2))
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ B \int \frac {\cot {\left (c + d x \right )}}{a \sqrt {a + b \tan {\left (c + d x \right )}} + b \sqrt {a + b \tan {\left (c + d x \right )}} \tan {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)*(a*B+b*B*tan(d*x+c))/(a+b*tan(d*x+c))**(5/2),x)
[Out]
B*Integral(cot(c + d*x)/(a*sqrt(a + b*tan(c + d*x)) + b*sqrt(a + b*tan(c + d*x))*tan(c + d*x)), x)
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