\(\int \cot ^2(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx\) [30]

Optimal result

Integrand size = 34, antiderivative size = 144 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-8 a^4 (A-i B) x+\frac {a^4 (4 i A+7 B) \log (\cos (c+d x))}{d}+\frac {a^4 (4 i A+B) \log (\sin (c+d x))}{d}-\frac {a A \cot (c+d x) (a+i a \tan (c+d x))^3}{d}+\frac {(2 i A-B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac {3 B \left (a^4+i a^4 \tan (c+d x)\right )}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {3674, 3675, 3670, 3556, 3612} \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {a^4 (B+4 i A) \log (\sin (c+d x))}{d}+\frac {a^4 (7 B+4 i A) \log (\cos (c+d x))}{d}-8 a^4 x (A-i B)-\frac {3 B \left (a^4+i a^4 \tan (c+d x)\right )}{d}+\frac {(-B+2 i A) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac {a A \cot (c+d x) (a+i a \tan (c+d x))^3}{d} \]

[In]

Int[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]

[Out]

-8*a^4*(A - I*B)*x + (a^4*((4*I)*A + 7*B)*Log[Cos[c + d*x]])/d + (a^4*((4*I)*A + B)*Log[Sin[c + d*x]])/d - (a*
A*Cot[c + d*x]*(a + I*a*Tan[c + d*x])^3)/d + (((2*I)*A - B)*(a^2 + I*a^2*Tan[c + d*x])^2)/(2*d) - (3*B*(a^4 +
I*a^4*Tan[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 3674

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^2)*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x]
)^(n + 1)/(d*f*(b*c + a*d)*(n + 1))), x] - Dist[a/(d*(b*c + a*d)*(n + 1)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c
+ d*Tan[e + f*x])^(n + 1)*Simp[A*b*d*(m - n - 2) - B*(b*c*(m - 1) + a*d*(n + 1)) + (a*A*d*(m + n) - B*(a*c*(m
- 1) + b*d*(n + 1)))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && E
qQ[a^2 + b^2, 0] && GtQ[m, 1] && LtQ[n, -1]

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 {a A \cot (c+d x) (a+i a \tan (c+d x))^3}{d}+\int \cot (c+d x) (a+i a \tan (c+d x))^3 (a (4 i A+B)+a (2 A+i B) \tan (c+d x)) \, dx \\ & = -\frac {a A \cot (c+d x) (a+i a \tan (c+d x))^3}{d}+\frac {(2 i A-B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}+\frac {1}{2} \int \cot (c+d x) (a+i a \tan (c+d x))^2 \left (2 a^2 (4 i A+B)+6 i a^2 B \tan (c+d x)\right ) \, dx \\ & = -\frac {a A \cot (c+d x) (a+i a \tan (c+d x))^3}{d}+\frac {(2 i A-B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac {3 B \left (a^4+i a^4 \tan (c+d x)\right )}{d}+\frac {1}{2} \int \cot (c+d x) (a+i a \tan (c+d x)) \left (2 a^3 (4 i A+B)-2 a^3 (4 A-7 i B) \tan (c+d x)\right ) \, dx \\ & = -\frac {a A \cot (c+d x) (a+i a \tan (c+d x))^3}{d}+\frac {(2 i A-B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac {3 B \left (a^4+i a^4 \tan (c+d x)\right )}{d}+\frac {1}{2} \int \cot (c+d x) \left (2 a^4 (4 i A+B)-16 a^4 (A-i B) \tan (c+d x)\right ) \, dx-\left (a^4 (4 i A+7 B)\right ) \int \tan (c+d x) \, dx \\ & = -8 a^4 (A-i B) x+\frac {a^4 (4 i A+7 B) \log (\cos (c+d x))}{d}-\frac {a A \cot (c+d x) (a+i a \tan (c+d x))^3}{d}+\frac {(2 i A-B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac {3 B \left (a^4+i a^4 \tan (c+d x)\right )}{d}+\left (a^4 (4 i A+B)\right ) \int \cot (c+d x) \, dx \\ & = -8 a^4 (A-i B) x+\frac {a^4 (4 i A+7 B) \log (\cos (c+d x))}{d}+\frac {a^4 (4 i A+B) \log (\sin (c+d x))}{d}-\frac {a A \cot (c+d x) (a+i a \tan (c+d x))^3}{d}+\frac {(2 i A-B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac {3 B \left (a^4+i a^4 \tan (c+d x)\right )}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.37 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.84 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=a^4 \left (-\frac {A \cot (c+d x)}{d}+\frac {4 i A \log (\tan (c+d x))}{d}+\frac {B \log (\tan (c+d x))}{d}-\frac {8 i A \log (i+\tan (c+d x))}{d}-\frac {8 B \log (i+\tan (c+d x))}{d}+\frac {A \tan (c+d x)}{d}-\frac {4 i B \tan (c+d x)}{d}+\frac {B \tan ^2(c+d x)}{2 d}\right ) \]

[In]

Integrate[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]

[Out]

a^4*(-((A*Cot[c + d*x])/d) + ((4*I)*A*Log[Tan[c + d*x]])/d + (B*Log[Tan[c + d*x]])/d - ((8*I)*A*Log[I + Tan[c
+ d*x]])/d - (8*B*Log[I + Tan[c + d*x]])/d + (A*Tan[c + d*x])/d - ((4*I)*B*Tan[c + d*x])/d + (B*Tan[c + d*x]^2
)/(2*d))

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.63

method	result	size
parallelrisch	\(\frac {4 a^{4} \left (\left (-i A -B \right ) \ln \left (\sec ^{2}\left (d x +c \right )\right )+\left (i A +\frac {B}{4}\right ) \ln \left (\tan \left (d x +c \right )\right )+\frac {B \left (\tan ^{2}\left (d x +c \right )\right )}{8}+\left (-i B +\frac {A}{4}\right ) \tan \left (d x +c \right )-\frac {A \cot \left (d x +c \right )}{4}+2 x d \left (i B -A \right )\right )}{d}\)	\(91\)
derivativedivides	\(\frac {a^{4} \left (-A \cot \left (d x +c \right )+\frac {\left (-8 i A -8 B \right ) \ln \left (\cot ^{2}\left (d x +c \right )+1\right )}{2}+\left (-8 i B +8 A \right ) \left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (d x +c \right )\right )\right )+\left (4 i A +7 B \right ) \ln \left (\cot \left (d x +c \right )\right )-\frac {4 i B -A}{\cot \left (d x +c \right )}+\frac {B}{2 \cot \left (d x +c \right )^{2}}\right )}{d}\)	\(106\)
default	\(\frac {a^{4} \left (-A \cot \left (d x +c \right )+\frac {\left (-8 i A -8 B \right ) \ln \left (\cot ^{2}\left (d x +c \right )+1\right )}{2}+\left (-8 i B +8 A \right ) \left (\frac {\pi }{2}-\operatorname {arccot}\left (\cot \left (d x +c \right )\right )\right )+\left (4 i A +7 B \right ) \ln \left (\cot \left (d x +c \right )\right )-\frac {4 i B -A}{\cot \left (d x +c \right )}+\frac {B}{2 \cot \left (d x +c \right )^{2}}\right )}{d}\)	\(106\)
norman	\(\frac {\left (8 i B \,a^{4}-8 A \,a^{4}\right ) x \tan \left (d x +c \right )+\frac {\left (-4 i B \,a^{4}+A \,a^{4}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{d}-\frac {A \,a^{4}}{d}+\frac {B \,a^{4} \left (\tan ^{3}\left (d x +c \right )\right )}{2 d}}{\tan \left (d x +c \right )}+\frac {\left (4 i A \,a^{4}+B \,a^{4}\right ) \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {4 \left (i A \,a^{4}+B \,a^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\)	\(138\)
risch	\(-\frac {16 i a^{4} B c}{d}+\frac {16 a^{4} A c}{d}+\frac {2 a^{4} \left (5 B \,{\mathrm e}^{4 i \left (d x +c \right )}-2 i A \,{\mathrm e}^{2 i \left (d x +c \right )}-B \,{\mathrm e}^{2 i \left (d x +c \right )}-2 i A -4 B \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}+\frac {7 a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) B}{d}+\frac {4 i a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) A}{d}+\frac {a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B}{d}+\frac {4 i a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) A}{d}\)	\(187\)

[In]

int(cot(d*x+c)^2*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

4*a^4*((-I*A-B)*ln(sec(d*x+c)^2)+(I*A+1/4*B)*ln(tan(d*x+c))+1/8*B*tan(d*x+c)^2+(-I*B+1/4*A)*tan(d*x+c)-1/4*A*c
ot(d*x+c)+2*x*d*(-A+I*B))/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. 254 vs. \(2 (126) = 252\).

Time = 0.27 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.76 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {10 \, B a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, {\left (2 i \, A + B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 4 \, {\left (i \, A + 2 \, B\right )} a^{4} + {\left ({\left (4 i \, A + 7 \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + {\left (4 i \, A + 7 \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (-4 i \, A - 7 \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-4 i \, A - 7 \, B\right )} a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + {\left ({\left (4 i \, A + B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + {\left (4 i \, A + B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (-4 i \, A - B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-4 i \, A - B\right )} a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}{d e^{\left (6 i \, d x + 6 i \, c\right )} + d e^{\left (4 i \, d x + 4 i \, c\right )} - d e^{\left (2 i \, d x + 2 i \, c\right )} - d} \]

[In]

integrate(cot(d*x+c)^2*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

(10*B*a^4*e^(4*I*d*x + 4*I*c) - 2*(2*I*A + B)*a^4*e^(2*I*d*x + 2*I*c) - 4*(I*A + 2*B)*a^4 + ((4*I*A + 7*B)*a^4
*e^(6*I*d*x + 6*I*c) + (4*I*A + 7*B)*a^4*e^(4*I*d*x + 4*I*c) + (-4*I*A - 7*B)*a^4*e^(2*I*d*x + 2*I*c) + (-4*I*
A - 7*B)*a^4)*log(e^(2*I*d*x + 2*I*c) + 1) + ((4*I*A + B)*a^4*e^(6*I*d*x + 6*I*c) + (4*I*A + B)*a^4*e^(4*I*d*x
+ 4*I*c) + (-4*I*A - B)*a^4*e^(2*I*d*x + 2*I*c) + (-4*I*A - B)*a^4)*log(e^(2*I*d*x + 2*I*c) - 1))/(d*e^(6*I*d
*x + 6*I*c) + d*e^(4*I*d*x + 4*I*c) - 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. 260 vs. \(2 (124) = 248\).

Time = 1.30 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.81 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {i a^{4} \cdot \left (4 A - 7 i B\right ) \log {\left (e^{2 i d x} + \frac {\left (- 4 i A a^{4} - 4 B a^{4} + i a^{4} \cdot \left (4 A - 7 i B\right )\right ) e^{- 2 i c}}{3 B a^{4}} \right )}}{d} + \frac {i a^{4} \cdot \left (4 A - i B\right ) \log {\left (e^{2 i d x} + \frac {\left (- 4 i A a^{4} - 4 B a^{4} + i a^{4} \cdot \left (4 A - i B\right )\right ) e^{- 2 i c}}{3 B a^{4}} \right )}}{d} + \frac {- 4 i A a^{4} + 10 B a^{4} e^{4 i c} e^{4 i d x} - 8 B a^{4} + \left (- 4 i A a^{4} e^{2 i c} - 2 B a^{4} e^{2 i c}\right ) e^{2 i d x}}{d e^{6 i c} e^{6 i d x} + d e^{4 i c} e^{4 i d x} - d e^{2 i c} e^{2 i d x} - d} \]

[In]

integrate(cot(d*x+c)**2*(a+I*a*tan(d*x+c))**4*(A+B*tan(d*x+c)),x)

[Out]

I*a**4*(4*A - 7*I*B)*log(exp(2*I*d*x) + (-4*I*A*a**4 - 4*B*a**4 + I*a**4*(4*A - 7*I*B))*exp(-2*I*c)/(3*B*a**4)
)/d + I*a**4*(4*A - I*B)*log(exp(2*I*d*x) + (-4*I*A*a**4 - 4*B*a**4 + I*a**4*(4*A - I*B))*exp(-2*I*c)/(3*B*a**
4))/d + (-4*I*A*a**4 + 10*B*a**4*exp(4*I*c)*exp(4*I*d*x) - 8*B*a**4 + (-4*I*A*a**4*exp(2*I*c) - 2*B*a**4*exp(2
*I*c))*exp(2*I*d*x))/(d*exp(6*I*c)*exp(6*I*d*x) + d*exp(4*I*c)*exp(4*I*d*x) - d*exp(2*I*c)*exp(2*I*d*x) - d)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.71 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {B a^{4} \tan \left (d x + c\right )^{2} - 16 \, {\left (d x + c\right )} {\left (A - i \, B\right )} a^{4} - 8 \, {\left (i \, A + B\right )} a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \, {\left (4 i \, A + B\right )} a^{4} \log \left (\tan \left (d x + c\right )\right ) + 2 \, {\left (A - 4 i \, B\right )} a^{4} \tan \left (d x + c\right ) - \frac {2 \, A a^{4}}{\tan \left (d x + c\right )}}{2 \, d} \]

[In]

integrate(cot(d*x+c)^2*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/2*(B*a^4*tan(d*x + c)^2 - 16*(d*x + c)*(A - I*B)*a^4 - 8*(I*A + B)*a^4*log(tan(d*x + c)^2 + 1) + 2*(4*I*A +
B)*a^4*log(tan(d*x + c)) + 2*(A - 4*I*B)*a^4*tan(d*x + c) - 2*A*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. 336 vs. \(2 (126) = 252\).

Time = 0.79 (sec) , antiderivative size = 336, normalized size of antiderivative = 2.33 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, {\left (4 i \, A a^{4} + 7 \, B a^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) - 32 \, {\left (i \, A a^{4} + B a^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) - 2 \, {\left (-4 i \, A a^{4} - 7 \, B a^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right ) - 2 \, {\left (-4 i \, A a^{4} - B a^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - \frac {8 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A a^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - \frac {12 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 21 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 4 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 16 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 i \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 46 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 16 i \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 i \, A a^{4} + 21 \, B a^{4}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \]

[In]

integrate(cot(d*x+c)^2*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

1/2*(A*a^4*tan(1/2*d*x + 1/2*c) + 2*(4*I*A*a^4 + 7*B*a^4)*log(tan(1/2*d*x + 1/2*c) + 1) - 32*(I*A*a^4 + B*a^4)
*log(tan(1/2*d*x + 1/2*c) + I) - 2*(-4*I*A*a^4 - 7*B*a^4)*log(tan(1/2*d*x + 1/2*c) - 1) - 2*(-4*I*A*a^4 - B*a^
4)*log(tan(1/2*d*x + 1/2*c)) - (8*I*A*a^4*tan(1/2*d*x + 1/2*c) + 2*B*a^4*tan(1/2*d*x + 1/2*c) + A*a^4)/tan(1/2
*d*x + 1/2*c) - (12*I*A*a^4*tan(1/2*d*x + 1/2*c)^4 + 21*B*a^4*tan(1/2*d*x + 1/2*c)^4 + 4*A*a^4*tan(1/2*d*x + 1
/2*c)^3 - 16*I*B*a^4*tan(1/2*d*x + 1/2*c)^3 - 24*I*A*a^4*tan(1/2*d*x + 1/2*c)^2 - 46*B*a^4*tan(1/2*d*x + 1/2*c
)^2 - 4*A*a^4*tan(1/2*d*x + 1/2*c) + 16*I*B*a^4*tan(1/2*d*x + 1/2*c) + 12*I*A*a^4 + 21*B*a^4)/(tan(1/2*d*x + 1
/2*c)^2 - 1)^2)/d

Mupad [B] (verification not implemented)

Time = 8.38 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.76 \[ \int \cot ^2(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {B\,a^4\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2\,d}+\frac {a^4\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (B+A\,4{}\mathrm {i}\right )}{d}-\frac {8\,a^4\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )}{d}-\frac {A\,a^4\,\mathrm {cot}\left (c+d\,x\right )}{d}-\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (B\,a^4\,1{}\mathrm {i}+a^4\,\left (3\,B+A\,1{}\mathrm {i}\right )\,1{}\mathrm {i}\right )}{d} \]

[In]

int(cot(c + d*x)^2*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^4,x)

[Out]

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