\(\int \cot (c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx\) [29]

Optimal result

Integrand size = 32, antiderivative size = 142 \[ \int \cot (c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=8 a^4 (i A+B) x+\frac {a^4 (7 A-8 i B) \log (\cos (c+d x))}{d}+\frac {a^4 A \log (\sin (c+d x))}{d}+\frac {i a B (a+i a \tan (c+d x))^3}{3 d}-\frac {(A-2 i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac {(3 A-4 i B) \left (a^4+i a^4 \tan (c+d x)\right )}{d} \]

[Out]

8*a^4*(I*A+B)*x+a^4*(7*A-8*I*B)*ln(cos(d*x+c))/d+a^4*A*ln(sin(d*x+c))/d+1/3*I*a*B*(a+I*a*tan(d*x+c))^3/d-1/2*(
A-2*I*B)*(a^2+I*a^2*tan(d*x+c))^2/d-(3*A-4*I*B)*(a^4+I*a^4*tan(d*x+c))/d

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3675, 3670, 3556, 3612} \[ \int \cot (c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {(3 A-4 i B) \left (a^4+i a^4 \tan (c+d x)\right )}{d}+\frac {a^4 (7 A-8 i B) \log (\cos (c+d x))}{d}+8 a^4 x (B+i A)+\frac {a^4 A \log (\sin (c+d x))}{d}-\frac {(A-2 i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}+\frac {i a B (a+i a \tan (c+d x))^3}{3 d} \]

[In]

Int[Cot[c + d*x]*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]

[Out]

8*a^4*(I*A + B)*x + (a^4*(7*A - (8*I)*B)*Log[Cos[c + d*x]])/d + (a^4*A*Log[Sin[c + d*x]])/d + ((I/3)*a*B*(a +
I*a*Tan[c + d*x])^3)/d - ((A - (2*I)*B)*(a^2 + I*a^2*Tan[c + d*x])^2)/(2*d) - ((3*A - (4*I)*B)*(a^4 + I*a^4*Ta
n[c + d*x]))/d

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3612

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*c +
b*d)*(x/(a^2 + b^2)), x] + Dist[(b*c - a*d)/(a^2 + b^2), Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[a*c + b*d, 0]

Rule 3670

Int[(((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]))/((a_.) + (b_.)*tan[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Dist[B*(d/b), Int[Tan[e + f*x], x], x] + Dist[1/b, Int[Simp[A*b*c + (A*b*d + B*(
b*c - a*d))*Tan[e + f*x], x]/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a
*d, 0]

Rule 3675

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*B*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*
(m + n))), x] + Dist[1/(d*(m + n)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n*Simp[a*A*d*(m + n)
+ B*(a*c*(m - 1) - b*d*(n + 1)) - (B*(b*c - a*d)*(m - 1) - d*(A*b + a*B)*(m + n))*Tan[e + f*x], x], x], x] /;
FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && GtQ[m, 1] &&  !LtQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {i a B (a+i a \tan (c+d x))^3}{3 d}+\frac {1}{3} \int \cot (c+d x) (a+i a \tan (c+d x))^3 (3 a A+3 a (i A+2 B) \tan (c+d x)) \, dx \\ & = \frac {i a B (a+i a \tan (c+d x))^3}{3 d}-\frac {(A-2 i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}+\frac {1}{6} \int \cot (c+d x) (a+i a \tan (c+d x))^2 \left (6 a^2 A+6 a^2 (3 i A+4 B) \tan (c+d x)\right ) \, dx \\ & = \frac {i a B (a+i a \tan (c+d x))^3}{3 d}-\frac {(A-2 i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac {(3 A-4 i B) \left (a^4+i a^4 \tan (c+d x)\right )}{d}+\frac {1}{6} \int \cot (c+d x) (a+i a \tan (c+d x)) \left (6 a^3 A+6 a^3 (7 i A+8 B) \tan (c+d x)\right ) \, dx \\ & = \frac {i a B (a+i a \tan (c+d x))^3}{3 d}-\frac {(A-2 i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac {(3 A-4 i B) \left (a^4+i a^4 \tan (c+d x)\right )}{d}+\frac {1}{6} \int \cot (c+d x) \left (6 a^4 A+48 a^4 (i A+B) \tan (c+d x)\right ) \, dx-\left (a^4 (7 A-8 i B)\right ) \int \tan (c+d x) \, dx \\ & = 8 a^4 (i A+B) x+\frac {a^4 (7 A-8 i B) \log (\cos (c+d x))}{d}+\frac {i a B (a+i a \tan (c+d x))^3}{3 d}-\frac {(A-2 i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac {(3 A-4 i B) \left (a^4+i a^4 \tan (c+d x)\right )}{d}+\left (a^4 A\right ) \int \cot (c+d x) \, dx \\ & = 8 a^4 (i A+B) x+\frac {a^4 (7 A-8 i B) \log (\cos (c+d x))}{d}+\frac {a^4 A \log (\sin (c+d x))}{d}+\frac {i a B (a+i a \tan (c+d x))^3}{3 d}-\frac {(A-2 i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac {(3 A-4 i B) \left (a^4+i a^4 \tan (c+d x)\right )}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.21 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.63 \[ \int \cot (c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {a^4 \left (6 A \log (\tan (c+d x))-8 (A-i B) (1+6 \log (i+\tan (c+d x)))+(-24 i A-42 B) \tan (c+d x)+3 (A-4 i B) \tan ^2(c+d x)+2 B \tan ^3(c+d x)\right )}{6 d} \]

[In]

Integrate[Cot[c + d*x]*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]

[Out]

(a^4*(6*A*Log[Tan[c + d*x]] - 8*(A - I*B)*(1 + 6*Log[I + Tan[c + d*x]]) + ((-24*I)*A - 42*B)*Tan[c + d*x] + 3*
(A - (4*I)*B)*Tan[c + d*x]^2 + 2*B*Tan[c + d*x]^3))/(6*d)

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.77

method	result	size
parallelrisch	\(-\frac {a^{4} \left (-48 i A x d +12 i B \left (\tan ^{2}\left (d x +c \right )\right )-2 B \left (\tan ^{3}\left (d x +c \right )\right )+24 i A \tan \left (d x +c \right )-3 A \left (\tan ^{2}\left (d x +c \right )\right )-24 i B \ln \left (\sec ^{2}\left (d x +c \right )\right )-48 B d x +24 A \ln \left (\sec ^{2}\left (d x +c \right )\right )-6 A \ln \left (\tan \left (d x +c \right )\right )+42 B \tan \left (d x +c \right )\right )}{6 d}\)	\(109\)
derivativedivides	\(\frac {a^{4} \left (\frac {\left (8 i B -8 A \right ) \ln \left (\cot ^{2}\left (d x +c \right )+1\right )}{2}+\left (-8 i A -8 B \right ) \left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (d x +c \right )\right )\right )+\left (-8 i B +7 A \right ) \ln \left (\cot \left (d x +c \right )\right )-\frac {4 i A +7 B}{\cot \left (d x +c \right )}-\frac {4 i B -A}{2 \cot \left (d x +c \right )^{2}}+\frac {B}{3 \cot \left (d x +c \right )^{3}}\right )}{d}\)	\(115\)
default	\(\frac {a^{4} \left (\frac {\left (8 i B -8 A \right ) \ln \left (\cot ^{2}\left (d x +c \right )+1\right )}{2}+\left (-8 i A -8 B \right ) \left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (d x +c \right )\right )\right )+\left (-8 i B +7 A \right ) \ln \left (\cot \left (d x +c \right )\right )-\frac {4 i A +7 B}{\cot \left (d x +c \right )}-\frac {4 i B -A}{2 \cot \left (d x +c \right )^{2}}+\frac {B}{3 \cot \left (d x +c \right )^{3}}\right )}{d}\)	\(115\)
norman	\(\left (8 i A \,a^{4}+8 B \,a^{4}\right ) x -\frac {\left (4 i A \,a^{4}+7 B \,a^{4}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (-4 i B \,a^{4}+A \,a^{4}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {B \,a^{4} \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}+\frac {A \,a^{4} \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {4 \left (-i B \,a^{4}+A \,a^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\)	\(130\)
risch	\(-\frac {16 a^{4} B c}{d}-\frac {16 i a^{4} A c}{d}-\frac {2 i a^{4} \left (15 i A \,{\mathrm e}^{4 i \left (d x +c \right )}+36 B \,{\mathrm e}^{4 i \left (d x +c \right )}+27 i A \,{\mathrm e}^{2 i \left (d x +c \right )}+54 B \,{\mathrm e}^{2 i \left (d x +c \right )}+12 i A +22 B \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}-\frac {8 i a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) B}{d}+\frac {7 a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) A}{d}+\frac {A \,a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\)	\(166\)

[In]

int(cot(d*x+c)*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-1/6*a^4*(-48*I*A*x*d+12*I*B*tan(d*x+c)^2-2*B*tan(d*x+c)^3+24*I*A*tan(d*x+c)-3*A*tan(d*x+c)^2-24*I*B*ln(sec(d*
x+c)^2)-48*B*d*x+24*A*ln(sec(d*x+c)^2)-6*A*ln(tan(d*x+c))+42*B*tan(d*x+c))/d

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (122) = 244\).

Time = 0.25 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.73 \[ \int \cot (c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {6 \, {\left (5 \, A - 12 i \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 54 \, {\left (A - 2 i \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 4 \, {\left (6 \, A - 11 i \, B\right )} a^{4} + 3 \, {\left ({\left (7 \, A - 8 i \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, {\left (7 \, A - 8 i \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, {\left (7 \, A - 8 i \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (7 \, A - 8 i \, B\right )} a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 3 \, {\left (A a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, A a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, A a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + A a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}{3 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

[In]

integrate(cot(d*x+c)*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/3*(6*(5*A - 12*I*B)*a^4*e^(4*I*d*x + 4*I*c) + 54*(A - 2*I*B)*a^4*e^(2*I*d*x + 2*I*c) + 4*(6*A - 11*I*B)*a^4
+ 3*((7*A - 8*I*B)*a^4*e^(6*I*d*x + 6*I*c) + 3*(7*A - 8*I*B)*a^4*e^(4*I*d*x + 4*I*c) + 3*(7*A - 8*I*B)*a^4*e^(
2*I*d*x + 2*I*c) + (7*A - 8*I*B)*a^4)*log(e^(2*I*d*x + 2*I*c) + 1) + 3*(A*a^4*e^(6*I*d*x + 6*I*c) + 3*A*a^4*e^
(4*I*d*x + 4*I*c) + 3*A*a^4*e^(2*I*d*x + 2*I*c) + A*a^4)*log(e^(2*I*d*x + 2*I*c) - 1))/(d*e^(6*I*d*x + 6*I*c)
+ 3*d*e^(4*I*d*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) + d)

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (121) = 242\).

Time = 1.85 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.04 \[ \int \cot (c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {A a^{4} \log {\left (\frac {- 3 A a^{4} + 4 i B a^{4}}{3 A a^{4} e^{2 i c} - 4 i B a^{4} e^{2 i c}} + e^{2 i d x} \right )}}{d} + \frac {a^{4} \cdot \left (7 A - 8 i B\right ) \log {\left (e^{2 i d x} + \frac {- 4 A a^{4} + 4 i B a^{4} + a^{4} \cdot \left (7 A - 8 i B\right )}{3 A a^{4} e^{2 i c} - 4 i B a^{4} e^{2 i c}} \right )}}{d} + \frac {24 A a^{4} - 44 i B a^{4} + \left (54 A a^{4} e^{2 i c} - 108 i B a^{4} e^{2 i c}\right ) e^{2 i d x} + \left (30 A a^{4} e^{4 i c} - 72 i B a^{4} e^{4 i c}\right ) e^{4 i d x}}{3 d e^{6 i c} e^{6 i d x} + 9 d e^{4 i c} e^{4 i d x} + 9 d e^{2 i c} e^{2 i d x} + 3 d} \]

[In]

integrate(cot(d*x+c)*(a+I*a*tan(d*x+c))**4*(A+B*tan(d*x+c)),x)

[Out]

A*a**4*log((-3*A*a**4 + 4*I*B*a**4)/(3*A*a**4*exp(2*I*c) - 4*I*B*a**4*exp(2*I*c)) + exp(2*I*d*x))/d + a**4*(7*
A - 8*I*B)*log(exp(2*I*d*x) + (-4*A*a**4 + 4*I*B*a**4 + a**4*(7*A - 8*I*B))/(3*A*a**4*exp(2*I*c) - 4*I*B*a**4*
exp(2*I*c)))/d + (24*A*a**4 - 44*I*B*a**4 + (54*A*a**4*exp(2*I*c) - 108*I*B*a**4*exp(2*I*c))*exp(2*I*d*x) + (3
0*A*a**4*exp(4*I*c) - 72*I*B*a**4*exp(4*I*c))*exp(4*I*d*x))/(3*d*exp(6*I*c)*exp(6*I*d*x) + 9*d*exp(4*I*c)*exp(
4*I*d*x) + 9*d*exp(2*I*c)*exp(2*I*d*x) + 3*d)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.75 \[ \int \cot (c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {2 \, B a^{4} \tan \left (d x + c\right )^{3} + 3 \, {\left (A - 4 i \, B\right )} a^{4} \tan \left (d x + c\right )^{2} - 48 \, {\left (d x + c\right )} {\left (-i \, A - B\right )} a^{4} - 24 \, {\left (A - i \, B\right )} a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 6 \, A a^{4} \log \left (\tan \left (d x + c\right )\right ) - 6 \, {\left (4 i \, A + 7 \, B\right )} a^{4} \tan \left (d x + c\right )}{6 \, d} \]

[In]

integrate(cot(d*x+c)*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/6*(2*B*a^4*tan(d*x + c)^3 + 3*(A - 4*I*B)*a^4*tan(d*x + c)^2 - 48*(d*x + c)*(-I*A - B)*a^4 - 24*(A - I*B)*a^
4*log(tan(d*x + c)^2 + 1) + 6*A*a^4*log(tan(d*x + c)) - 6*(4*I*A + 7*B)*a^4*tan(d*x + c))/d

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 332 vs. \(2 (122) = 244\).

Time = 1.23 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.34 \[ \int \cot (c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {6 \, A a^{4} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 6 \, {\left (7 \, A a^{4} - 8 i \, B a^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) - 96 \, {\left (A a^{4} - i \, B a^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) + 6 \, {\left (7 \, A a^{4} - 8 i \, B a^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right ) - \frac {77 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 88 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 48 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 84 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 243 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 312 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 96 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 184 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 243 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 312 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 48 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 84 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 77 \, A a^{4} + 88 i \, B a^{4}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]

[In]

integrate(cot(d*x+c)*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

1/6*(6*A*a^4*log(tan(1/2*d*x + 1/2*c)) + 6*(7*A*a^4 - 8*I*B*a^4)*log(tan(1/2*d*x + 1/2*c) + 1) - 96*(A*a^4 - I
*B*a^4)*log(tan(1/2*d*x + 1/2*c) + I) + 6*(7*A*a^4 - 8*I*B*a^4)*log(tan(1/2*d*x + 1/2*c) - 1) - (77*A*a^4*tan(
1/2*d*x + 1/2*c)^6 - 88*I*B*a^4*tan(1/2*d*x + 1/2*c)^6 - 48*I*A*a^4*tan(1/2*d*x + 1/2*c)^5 - 84*B*a^4*tan(1/2*
d*x + 1/2*c)^5 - 243*A*a^4*tan(1/2*d*x + 1/2*c)^4 + 312*I*B*a^4*tan(1/2*d*x + 1/2*c)^4 + 96*I*A*a^4*tan(1/2*d*
x + 1/2*c)^3 + 184*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 243*A*a^4*tan(1/2*d*x + 1/2*c)^2 - 312*I*B*a^4*tan(1/2*d*x +
1/2*c)^2 - 48*I*A*a^4*tan(1/2*d*x + 1/2*c) - 84*B*a^4*tan(1/2*d*x + 1/2*c) - 77*A*a^4 + 88*I*B*a^4)/(tan(1/2*
d*x + 1/2*c)^2 - 1)^3)/d

Mupad [B] (verification not implemented)

Time = 7.24 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.94 \[ \int \cot (c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {A\,a^4\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d}-\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (a^4\,\left (A-B\,1{}\mathrm {i}\right )\,3{}\mathrm {i}+B\,a^4+a^4\,\left (3\,B+A\,1{}\mathrm {i}\right )\right )}{d}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {B\,a^4\,1{}\mathrm {i}}{2}+\frac {a^4\,\left (3\,B+A\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}\right )}{d}-\frac {8\,a^4\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )}{d}+\frac {B\,a^4\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3\,d} \]

[In]

int(cot(c + d*x)*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^4,x)

[Out]

(A*a^4*log(tan(c + d*x)))/d - (tan(c + d*x)*(a^4*(A - B*1i)*3i + B*a^4 + a^4*(A*1i + 3*B)))/d - (tan(c + d*x)^
2*((B*a^4*1i)/2 + (a^4*(A*1i + 3*B)*1i)/2))/d - (8*a^4*log(tan(c + d*x) + 1i)*(A - B*1i))/d + (B*a^4*tan(c + d
*x)^3)/(3*d)