3.1.33 \(\int \cot ^6(c+d x) (a+i a \tan (c+d x))^3 \, dx\) [33]

Optimal. Leaf size=126 \[ -4 a^3 x-\frac {4 a^3 \cot (c+d x)}{d}+\frac {2 i a^3 \cot ^2(c+d x)}{d}+\frac {4 a^3 \cot ^3(c+d x)}{3 d}-\frac {11 i a^3 \cot ^4(c+d x)}{20 d}+\frac {4 i a^3 \log (\sin (c+d x))}{d}-\frac {\cot ^5(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{5 d} \]

[Out]

-4*a^3*x-4*a^3*cot(d*x+c)/d+2*I*a^3*cot(d*x+c)^2/d+4/3*a^3*cot(d*x+c)^3/d-11/20*I*a^3*cot(d*x+c)^4/d+4*I*a^3*l
n(sin(d*x+c))/d-1/5*cot(d*x+c)^5*(a^3+I*a^3*tan(d*x+c))/d

________________________________________________________________________________________

Rubi [A]
time = 0.15, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3634, 3672, 3610, 3612, 3556} \begin {gather*} -\frac {11 i a^3 \cot ^4(c+d x)}{20 d}+\frac {4 a^3 \cot ^3(c+d x)}{3 d}+\frac {2 i a^3 \cot ^2(c+d x)}{d}-\frac {4 a^3 \cot (c+d x)}{d}+\frac {4 i a^3 \log (\sin (c+d x))}{d}-\frac {\cot ^5(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{5 d}-4 a^3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^6*(a + I*a*Tan[c + d*x])^3,x]

[Out]

-4*a^3*x - (4*a^3*Cot[c + d*x])/d + ((2*I)*a^3*Cot[c + d*x]^2)/d + (4*a^3*Cot[c + d*x]^3)/(3*d) - (((11*I)/20)
*a^3*Cot[c + d*x]^4)/d + ((4*I)*a^3*Log[Sin[c + d*x]])/d - (Cot[c + d*x]^5*(a^3 + I*a^3*Tan[c + d*x]))/(5*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 3610

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

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 3634

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(-a^2)*(b*c - a*d)*(a + b*Tan[e + f*x])^(m - 2)*((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 - 2)*(c + d*Tan[e + f*x])^(n + 1)*Simp[b*(b*c*(
m - 2) - a*d*(m - 2*n - 4)) + (a*b*c*(m - 2) + b^2*d*(n + 1) - a^2*d*(m + n - 1))*Tan[e + f*x], x], x], x] /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && Lt
Q[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])

Rule 3672

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

Rubi steps

\begin {align*} \int \cot ^6(c+d x) (a+i a \tan (c+d x))^3 \, dx &=-\frac {\cot ^5(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{5 d}-\frac {1}{5} \int \cot ^5(c+d x) (a+i a \tan (c+d x)) \left (-11 i a^2+9 a^2 \tan (c+d x)\right ) \, dx\\ &=-\frac {11 i a^3 \cot ^4(c+d x)}{20 d}-\frac {\cot ^5(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{5 d}-\frac {1}{5} \int \cot ^4(c+d x) \left (20 a^3+20 i a^3 \tan (c+d x)\right ) \, dx\\ &=\frac {4 a^3 \cot ^3(c+d x)}{3 d}-\frac {11 i a^3 \cot ^4(c+d x)}{20 d}-\frac {\cot ^5(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{5 d}-\frac {1}{5} \int \cot ^3(c+d x) \left (20 i a^3-20 a^3 \tan (c+d x)\right ) \, dx\\ &=\frac {2 i a^3 \cot ^2(c+d x)}{d}+\frac {4 a^3 \cot ^3(c+d x)}{3 d}-\frac {11 i a^3 \cot ^4(c+d x)}{20 d}-\frac {\cot ^5(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{5 d}-\frac {1}{5} \int \cot ^2(c+d x) \left (-20 a^3-20 i a^3 \tan (c+d x)\right ) \, dx\\ &=-\frac {4 a^3 \cot (c+d x)}{d}+\frac {2 i a^3 \cot ^2(c+d x)}{d}+\frac {4 a^3 \cot ^3(c+d x)}{3 d}-\frac {11 i a^3 \cot ^4(c+d x)}{20 d}-\frac {\cot ^5(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{5 d}-\frac {1}{5} \int \cot (c+d x) \left (-20 i a^3+20 a^3 \tan (c+d x)\right ) \, dx\\ &=-4 a^3 x-\frac {4 a^3 \cot (c+d x)}{d}+\frac {2 i a^3 \cot ^2(c+d x)}{d}+\frac {4 a^3 \cot ^3(c+d x)}{3 d}-\frac {11 i a^3 \cot ^4(c+d x)}{20 d}-\frac {\cot ^5(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{5 d}+\left (4 i a^3\right ) \int \cot (c+d x) \, dx\\ &=-4 a^3 x-\frac {4 a^3 \cot (c+d x)}{d}+\frac {2 i a^3 \cot ^2(c+d x)}{d}+\frac {4 a^3 \cot ^3(c+d x)}{3 d}-\frac {11 i a^3 \cot ^4(c+d x)}{20 d}+\frac {4 i a^3 \log (\sin (c+d x))}{d}-\frac {\cot ^5(c+d x) \left (a^3+i a^3 \tan (c+d x)\right )}{5 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(359\) vs. \(2(126)=252\).
time = 1.65, size = 359, normalized size = 2.85 \begin {gather*} \frac {a^3 \csc (c) \csc ^5(c+d x) (\cos (3 d x)+i \sin (3 d x)) \left (-225 i \cos (2 c+d x)+600 d x \cos (2 c+d x)-105 i \cos (2 c+3 d x)+300 d x \cos (2 c+3 d x)+105 i \cos (4 c+3 d x)-300 d x \cos (4 c+3 d x)-60 d x \cos (4 c+5 d x)+60 d x \cos (6 c+5 d x)-75 \cos (d x) \left (-3 i+8 d x-2 i \log \left (\sin ^2(c+d x)\right )\right )-150 i \cos (2 c+d x) \log \left (\sin ^2(c+d x)\right )-75 i \cos (2 c+3 d x) \log \left (\sin ^2(c+d x)\right )+75 i \cos (4 c+3 d x) \log \left (\sin ^2(c+d x)\right )+15 i \cos (4 c+5 d x) \log \left (\sin ^2(c+d x)\right )-15 i \cos (6 c+5 d x) \log \left (\sin ^2(c+d x)\right )+470 \sin (d x)+960 \text {ArcTan}(\tan (4 c+d x)) \sin (c) \sin ^5(c+d x)+360 \sin (2 c+d x)-280 \sin (2 c+3 d x)-135 \sin (4 c+3 d x)+83 \sin (4 c+5 d x)\right )}{240 d (\cos (d x)+i \sin (d x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^6*(a + I*a*Tan[c + d*x])^3,x]

[Out]

(a^3*Csc[c]*Csc[c + d*x]^5*(Cos[3*d*x] + I*Sin[3*d*x])*((-225*I)*Cos[2*c + d*x] + 600*d*x*Cos[2*c + d*x] - (10
5*I)*Cos[2*c + 3*d*x] + 300*d*x*Cos[2*c + 3*d*x] + (105*I)*Cos[4*c + 3*d*x] - 300*d*x*Cos[4*c + 3*d*x] - 60*d*
x*Cos[4*c + 5*d*x] + 60*d*x*Cos[6*c + 5*d*x] - 75*Cos[d*x]*(-3*I + 8*d*x - (2*I)*Log[Sin[c + d*x]^2]) - (150*I
)*Cos[2*c + d*x]*Log[Sin[c + d*x]^2] - (75*I)*Cos[2*c + 3*d*x]*Log[Sin[c + d*x]^2] + (75*I)*Cos[4*c + 3*d*x]*L
og[Sin[c + d*x]^2] + (15*I)*Cos[4*c + 5*d*x]*Log[Sin[c + d*x]^2] - (15*I)*Cos[6*c + 5*d*x]*Log[Sin[c + d*x]^2]
+ 470*Sin[d*x] + 960*ArcTan[Tan[4*c + d*x]]*Sin[c]*Sin[c + d*x]^5 + 360*Sin[2*c + d*x] - 280*Sin[2*c + 3*d*x]
- 135*Sin[4*c + 3*d*x] + 83*Sin[4*c + 5*d*x]))/(240*d*(Cos[d*x] + I*Sin[d*x])^3)

________________________________________________________________________________________

Maple [A]
time = 0.20, size = 132, normalized size = 1.05

method	result	size
risch	\(\frac {8 a^{3} c}{d}-\frac {2 i a^{3} \left (240 \,{\mathrm e}^{8 i \left (d x +c \right )}-585 \,{\mathrm e}^{6 i \left (d x +c \right )}+695 \,{\mathrm e}^{4 i \left (d x +c \right )}-385 \,{\mathrm e}^{2 i \left (d x +c \right )}+83\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+\frac {4 i a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\)	\(100\)
derivativedivides	\(\frac {-i a^{3} \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )-3 a^{3} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+3 i a^{3} \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\)	\(132\)
default	\(\frac {-i a^{3} \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )-3 a^{3} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+3 i a^{3} \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\)	\(132\)
norman	\(\frac {-\frac {a^{3}}{5 d}-4 a^{3} x \left (\tan ^{5}\left (d x +c \right )\right )+\frac {4 a^{3} \left (\tan ^{2}\left (d x +c \right )\right )}{3 d}-\frac {4 a^{3} \left (\tan ^{4}\left (d x +c \right )\right )}{d}-\frac {3 i a^{3} \tan \left (d x +c \right )}{4 d}+\frac {2 i a^{3} \left (\tan ^{3}\left (d x +c \right )\right )}{d}}{\tan \left (d x +c \right )^{5}}+\frac {4 i a^{3} \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {2 i a^{3} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\)	\(134\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^6*(a+I*a*tan(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-I*a^3*(-1/2*cot(d*x+c)^2-ln(sin(d*x+c)))-3*a^3*(-1/3*cot(d*x+c)^3+cot(d*x+c)+d*x+c)+3*I*a^3*(-1/4*cot(d*
x+c)^4+1/2*cot(d*x+c)^2+ln(sin(d*x+c)))+a^3*(-1/5*cot(d*x+c)^5+1/3*cot(d*x+c)^3-cot(d*x+c)-d*x-c))

________________________________________________________________________________________

Maxima [A]
time = 0.59, size = 109, normalized size = 0.87 \begin {gather*} -\frac {240 \, {\left (d x + c\right )} a^{3} + 120 i \, a^{3} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 240 i \, a^{3} \log \left (\tan \left (d x + c\right )\right ) + \frac {240 \, a^{3} \tan \left (d x + c\right )^{4} - 120 i \, a^{3} \tan \left (d x + c\right )^{3} - 80 \, a^{3} \tan \left (d x + c\right )^{2} + 45 i \, a^{3} \tan \left (d x + c\right ) + 12 \, a^{3}}{\tan \left (d x + c\right )^{5}}}{60 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6*(a+I*a*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/60*(240*(d*x + c)*a^3 + 120*I*a^3*log(tan(d*x + c)^2 + 1) - 240*I*a^3*log(tan(d*x + c)) + (240*a^3*tan(d*x
+ c)^4 - 120*I*a^3*tan(d*x + c)^3 - 80*a^3*tan(d*x + c)^2 + 45*I*a^3*tan(d*x + c) + 12*a^3)/tan(d*x + c)^5)/d

________________________________________________________________________________________

Fricas [A]
time = 0.44, size = 219, normalized size = 1.74 \begin {gather*} -\frac {2 \, {\left (240 i \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} - 585 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} + 695 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 385 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 83 i \, a^{3} + 30 \, {\left (-i \, a^{3} e^{\left (10 i \, d x + 10 i \, c\right )} + 5 i \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} - 10 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} + 10 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 5 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{3}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )\right )}}{15 \, {\left (d e^{\left (10 i \, d x + 10 i \, c\right )} - 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} - 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6*(a+I*a*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

-2/15*(240*I*a^3*e^(8*I*d*x + 8*I*c) - 585*I*a^3*e^(6*I*d*x + 6*I*c) + 695*I*a^3*e^(4*I*d*x + 4*I*c) - 385*I*a
^3*e^(2*I*d*x + 2*I*c) + 83*I*a^3 + 30*(-I*a^3*e^(10*I*d*x + 10*I*c) + 5*I*a^3*e^(8*I*d*x + 8*I*c) - 10*I*a^3*
e^(6*I*d*x + 6*I*c) + 10*I*a^3*e^(4*I*d*x + 4*I*c) - 5*I*a^3*e^(2*I*d*x + 2*I*c) + I*a^3)*log(e^(2*I*d*x + 2*I
*c) - 1))/(d*e^(10*I*d*x + 10*I*c) - 5*d*e^(8*I*d*x + 8*I*c) + 10*d*e^(6*I*d*x + 6*I*c) - 10*d*e^(4*I*d*x + 4*
I*c) + 5*d*e^(2*I*d*x + 2*I*c) - d)

________________________________________________________________________________________

Sympy [A]
time = 0.36, size = 218, normalized size = 1.73 \begin {gather*} \frac {4 i a^{3} \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {- 480 i a^{3} e^{8 i c} e^{8 i d x} + 1170 i a^{3} e^{6 i c} e^{6 i d x} - 1390 i a^{3} e^{4 i c} e^{4 i d x} + 770 i a^{3} e^{2 i c} e^{2 i d x} - 166 i a^{3}}{15 d e^{10 i c} e^{10 i d x} - 75 d e^{8 i c} e^{8 i d x} + 150 d e^{6 i c} e^{6 i d x} - 150 d e^{4 i c} e^{4 i d x} + 75 d e^{2 i c} e^{2 i d x} - 15 d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**6*(a+I*a*tan(d*x+c))**3,x)

[Out]

4*I*a**3*log(exp(2*I*d*x) - exp(-2*I*c))/d + (-480*I*a**3*exp(8*I*c)*exp(8*I*d*x) + 1170*I*a**3*exp(6*I*c)*exp
(6*I*d*x) - 1390*I*a**3*exp(4*I*c)*exp(4*I*d*x) + 770*I*a**3*exp(2*I*c)*exp(2*I*d*x) - 166*I*a**3)/(15*d*exp(1
0*I*c)*exp(10*I*d*x) - 75*d*exp(8*I*c)*exp(8*I*d*x) + 150*d*exp(6*I*c)*exp(6*I*d*x) - 150*d*exp(4*I*c)*exp(4*I
*d*x) + 75*d*exp(2*I*c)*exp(2*I*d*x) - 15*d)

________________________________________________________________________________________

Giac [A]
time = 1.61, size = 212, normalized size = 1.68 \begin {gather*} \frac {6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 45 i \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 190 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 660 i \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 7680 i \, a^{3} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) + 3840 i \, a^{3} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 2460 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {-8768 i \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2460 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 660 i \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 190 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 45 i \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{960 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6*(a+I*a*tan(d*x+c))^3,x, algorithm="giac")

[Out]

1/960*(6*a^3*tan(1/2*d*x + 1/2*c)^5 - 45*I*a^3*tan(1/2*d*x + 1/2*c)^4 - 190*a^3*tan(1/2*d*x + 1/2*c)^3 + 660*I
*a^3*tan(1/2*d*x + 1/2*c)^2 - 7680*I*a^3*log(tan(1/2*d*x + 1/2*c) + I) + 3840*I*a^3*log(tan(1/2*d*x + 1/2*c))
+ 2460*a^3*tan(1/2*d*x + 1/2*c) + (-8768*I*a^3*tan(1/2*d*x + 1/2*c)^5 - 2460*a^3*tan(1/2*d*x + 1/2*c)^4 + 660*
I*a^3*tan(1/2*d*x + 1/2*c)^3 + 190*a^3*tan(1/2*d*x + 1/2*c)^2 - 45*I*a^3*tan(1/2*d*x + 1/2*c) - 6*a^3)/tan(1/2
*d*x + 1/2*c)^5)/d

________________________________________________________________________________________

Mupad [B]
time = 4.11, size = 92, normalized size = 0.73 \begin {gather*} -\frac {8\,a^3\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{d}-\frac {4\,a^3\,{\mathrm {tan}\left (c+d\,x\right )}^4-a^3\,{\mathrm {tan}\left (c+d\,x\right )}^3\,2{}\mathrm {i}-\frac {4\,a^3\,{\mathrm {tan}\left (c+d\,x\right )}^2}{3}+\frac {a^3\,\mathrm {tan}\left (c+d\,x\right )\,3{}\mathrm {i}}{4}+\frac {a^3}{5}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^6*(a + a*tan(c + d*x)*1i)^3,x)

[Out]

- (8*a^3*atan(2*tan(c + d*x) + 1i))/d - ((a^3*tan(c + d*x)*3i)/4 + a^3/5 - (4*a^3*tan(c + d*x)^2)/3 - a^3*tan(
c + d*x)^3*2i + 4*a^3*tan(c + d*x)^4)/(d*tan(c + d*x)^5)

________________________________________________________________________________________