3.1.43 \(\int \cot ^6(c+d x) (a+i a \tan (c+d x))^4 \, dx\) [43]

Optimal. Leaf size=142 \[ -8 a^4 x-\frac {8 a^4 \cot (c+d x)}{d}+\frac {4 i a^4 \cot ^2(c+d x)}{d}+\frac {23 a^4 \cot ^3(c+d x)}{15 d}+\frac {8 i a^4 \log (\sin (c+d x))}{d}-\frac {\cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{5 d}-\frac {3 i \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{5 d} \]

[Out]

-8*a^4*x-8*a^4*cot(d*x+c)/d+4*I*a^4*cot(d*x+c)^2/d+23/15*a^4*cot(d*x+c)^3/d+8*I*a^4*ln(sin(d*x+c))/d-1/5*cot(d
*x+c)^5*(a^2+I*a^2*tan(d*x+c))^2/d-3/5*I*cot(d*x+c)^4*(a^4+I*a^4*tan(d*x+c))/d

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Rubi [A]
time = 0.21, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3634, 3674, 3672, 3610, 3612, 3556} \begin {gather*} \frac {23 a^4 \cot ^3(c+d x)}{15 d}+\frac {4 i a^4 \cot ^2(c+d x)}{d}-\frac {8 a^4 \cot (c+d x)}{d}+\frac {8 i a^4 \log (\sin (c+d x))}{d}-\frac {3 i \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{5 d}-8 a^4 x-\frac {\cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^6*(a + I*a*Tan[c + d*x])^4,x]

[Out]

-8*a^4*x - (8*a^4*Cot[c + d*x])/d + ((4*I)*a^4*Cot[c + d*x]^2)/d + (23*a^4*Cot[c + d*x]^3)/(15*d) + ((8*I)*a^4
*Log[Sin[c + d*x]])/d - (Cot[c + d*x]^5*(a^2 + I*a^2*Tan[c + d*x])^2)/(5*d) - (((3*I)/5)*Cot[c + d*x]^4*(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 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]

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]

Rubi steps

\begin {align*} \int \cot ^6(c+d x) (a+i a \tan (c+d x))^4 \, dx &=-\frac {\cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{5 d}-\frac {1}{5} \int \cot ^5(c+d x) (a+i a \tan (c+d x))^2 \left (-12 i a^2+8 a^2 \tan (c+d x)\right ) \, dx\\ &=-\frac {\cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{5 d}-\frac {3 i \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{5 d}-\frac {1}{20} \int \cot ^4(c+d x) (a+i a \tan (c+d x)) \left (92 a^3+68 i a^3 \tan (c+d x)\right ) \, dx\\ &=\frac {23 a^4 \cot ^3(c+d x)}{15 d}-\frac {\cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{5 d}-\frac {3 i \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{5 d}-\frac {1}{20} \int \cot ^3(c+d x) \left (160 i a^4-160 a^4 \tan (c+d x)\right ) \, dx\\ &=\frac {4 i a^4 \cot ^2(c+d x)}{d}+\frac {23 a^4 \cot ^3(c+d x)}{15 d}-\frac {\cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{5 d}-\frac {3 i \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{5 d}-\frac {1}{20} \int \cot ^2(c+d x) \left (-160 a^4-160 i a^4 \tan (c+d x)\right ) \, dx\\ &=-\frac {8 a^4 \cot (c+d x)}{d}+\frac {4 i a^4 \cot ^2(c+d x)}{d}+\frac {23 a^4 \cot ^3(c+d x)}{15 d}-\frac {\cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{5 d}-\frac {3 i \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{5 d}-\frac {1}{20} \int \cot (c+d x) \left (-160 i a^4+160 a^4 \tan (c+d x)\right ) \, dx\\ &=-8 a^4 x-\frac {8 a^4 \cot (c+d x)}{d}+\frac {4 i a^4 \cot ^2(c+d x)}{d}+\frac {23 a^4 \cot ^3(c+d x)}{15 d}-\frac {\cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{5 d}-\frac {3 i \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{5 d}+\left (8 i a^4\right ) \int \cot (c+d x) \, dx\\ &=-8 a^4 x-\frac {8 a^4 \cot (c+d x)}{d}+\frac {4 i a^4 \cot ^2(c+d x)}{d}+\frac {23 a^4 \cot ^3(c+d x)}{15 d}+\frac {8 i a^4 \log (\sin (c+d x))}{d}-\frac {\cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{5 d}-\frac {3 i \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{5 d}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(359\) vs. \(2(142)=284\).
time = 2.83, size = 359, normalized size = 2.53 \begin {gather*} \frac {a^4 \csc (c) \csc ^5(c+d x) (\cos (4 d x)+i \sin (4 d x)) \left (-210 i \cos (2 c+d x)+600 d x \cos (2 c+d x)-90 i \cos (2 c+3 d x)+300 d x \cos (2 c+3 d x)+90 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)-30 \cos (d x) \left (-7 i+20 d x-5 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 )+445 \sin (d x)+960 \text {ArcTan}(\tan (5 c+d x)) \sin (c) \sin ^5(c+d x)+345 \sin (2 c+d x)-275 \sin (2 c+3 d x)-120 \sin (4 c+3 d x)+79 \sin (4 c+5 d x)\right )}{120 d (\cos (d x)+i \sin (d x))^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^6*(a + I*a*Tan[c + d*x])^4,x]

[Out]

(a^4*Csc[c]*Csc[c + d*x]^5*(Cos[4*d*x] + I*Sin[4*d*x])*((-210*I)*Cos[2*c + d*x] + 600*d*x*Cos[2*c + d*x] - (90
*I)*Cos[2*c + 3*d*x] + 300*d*x*Cos[2*c + 3*d*x] + (90*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] - 30*Cos[d*x]*(-7*I + 20*d*x - (5*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]*Lo
g[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]
+ 445*Sin[d*x] + 960*ArcTan[Tan[5*c + d*x]]*Sin[c]*Sin[c + d*x]^5 + 345*Sin[2*c + d*x] - 275*Sin[2*c + 3*d*x]
- 120*Sin[4*c + 3*d*x] + 79*Sin[4*c + 5*d*x]))/(120*d*(Cos[d*x] + I*Sin[d*x])^4)

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Maple [A]
time = 0.21, size = 152, normalized size = 1.07

method	result	size
risch	\(\frac {16 a^{4} c}{d}-\frac {4 i a^{4} \left (210 \,{\mathrm e}^{8 i \left (d x +c \right )}-555 \,{\mathrm e}^{6 i \left (d x +c \right )}+655 \,{\mathrm e}^{4 i \left (d x +c \right )}-365 \,{\mathrm e}^{2 i \left (d x +c \right )}+79\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+\frac {8 i a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\)	\(100\)
norman	\(\frac {-\frac {a^{4}}{5 d}-8 a^{4} x \left (\tan ^{5}\left (d x +c \right )\right )+\frac {7 a^{4} \left (\tan ^{2}\left (d x +c \right )\right )}{3 d}-\frac {8 a^{4} \left (\tan ^{4}\left (d x +c \right )\right )}{d}-\frac {i a^{4} \tan \left (d x +c \right )}{d}+\frac {4 i a^{4} \left (\tan ^{3}\left (d x +c \right )\right )}{d}}{\tan \left (d x +c \right )^{5}}+\frac {8 i a^{4} \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {4 i a^{4} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\)	\(134\)
derivativedivides	\(\frac {a^{4} \left (-\cot \left (d x +c \right )-d x -c \right )-4 i a^{4} \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )-6 a^{4} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+4 i a^{4} \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+a^{4} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\)	\(152\)
default	\(\frac {a^{4} \left (-\cot \left (d x +c \right )-d x -c \right )-4 i a^{4} \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )-6 a^{4} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+4 i a^{4} \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+a^{4} \left (-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\)	\(152\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^6*(a+I*a*tan(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(a^4*(-cot(d*x+c)-d*x-c)-4*I*a^4*(-1/2*cot(d*x+c)^2-ln(sin(d*x+c)))-6*a^4*(-1/3*cot(d*x+c)^3+cot(d*x+c)+d*
x+c)+4*I*a^4*(-1/4*cot(d*x+c)^4+1/2*cot(d*x+c)^2+ln(sin(d*x+c)))+a^4*(-1/5*cot(d*x+c)^5+1/3*cot(d*x+c)^3-cot(d
*x+c)-d*x-c))

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Maxima [A]
time = 0.49, size = 109, normalized size = 0.77 \begin {gather*} -\frac {120 \, {\left (d x + c\right )} a^{4} + 60 i \, a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 120 i \, a^{4} \log \left (\tan \left (d x + c\right )\right ) + \frac {120 \, a^{4} \tan \left (d x + c\right )^{4} - 60 i \, a^{4} \tan \left (d x + c\right )^{3} - 35 \, a^{4} \tan \left (d x + c\right )^{2} + 15 i \, a^{4} \tan \left (d x + c\right ) + 3 \, a^{4}}{\tan \left (d x + c\right )^{5}}}{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))^4,x, algorithm="maxima")

[Out]

-1/15*(120*(d*x + c)*a^4 + 60*I*a^4*log(tan(d*x + c)^2 + 1) - 120*I*a^4*log(tan(d*x + c)) + (120*a^4*tan(d*x +
c)^4 - 60*I*a^4*tan(d*x + c)^3 - 35*a^4*tan(d*x + c)^2 + 15*I*a^4*tan(d*x + c) + 3*a^4)/tan(d*x + c)^5)/d

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Fricas [A]
time = 0.37, size = 219, normalized size = 1.54 \begin {gather*} -\frac {4 \, {\left (210 i \, a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} - 555 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 655 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 365 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 79 i \, a^{4} + 30 \, {\left (-i \, a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} + 5 i \, a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} - 10 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 10 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 5 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{4}\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))^4,x, algorithm="fricas")

[Out]

-4/15*(210*I*a^4*e^(8*I*d*x + 8*I*c) - 555*I*a^4*e^(6*I*d*x + 6*I*c) + 655*I*a^4*e^(4*I*d*x + 4*I*c) - 365*I*a
^4*e^(2*I*d*x + 2*I*c) + 79*I*a^4 + 30*(-I*a^4*e^(10*I*d*x + 10*I*c) + 5*I*a^4*e^(8*I*d*x + 8*I*c) - 10*I*a^4*
e^(6*I*d*x + 6*I*c) + 10*I*a^4*e^(4*I*d*x + 4*I*c) - 5*I*a^4*e^(2*I*d*x + 2*I*c) + I*a^4)*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)

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Sympy [A]
time = 0.41, size = 218, normalized size = 1.54 \begin {gather*} \frac {8 i a^{4} \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {- 840 i a^{4} e^{8 i c} e^{8 i d x} + 2220 i a^{4} e^{6 i c} e^{6 i d x} - 2620 i a^{4} e^{4 i c} e^{4 i d x} + 1460 i a^{4} e^{2 i c} e^{2 i d x} - 316 i a^{4}}{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))**4,x)

[Out]

8*I*a**4*log(exp(2*I*d*x) - exp(-2*I*c))/d + (-840*I*a**4*exp(8*I*c)*exp(8*I*d*x) + 2220*I*a**4*exp(6*I*c)*exp
(6*I*d*x) - 2620*I*a**4*exp(4*I*c)*exp(4*I*d*x) + 1460*I*a**4*exp(2*I*c)*exp(2*I*d*x) - 316*I*a**4)/(15*d*exp(
10*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)

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Giac [A]
time = 0.85, size = 212, normalized size = 1.49 \begin {gather*} \frac {3 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 30 i \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 155 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 600 i \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 7680 i \, a^{4} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) + 3840 i \, a^{4} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 2370 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {-8768 i \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2370 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 600 i \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 155 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 30 i \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{480 \, 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))^4,x, algorithm="giac")

[Out]

1/480*(3*a^4*tan(1/2*d*x + 1/2*c)^5 - 30*I*a^4*tan(1/2*d*x + 1/2*c)^4 - 155*a^4*tan(1/2*d*x + 1/2*c)^3 + 600*I
*a^4*tan(1/2*d*x + 1/2*c)^2 - 7680*I*a^4*log(tan(1/2*d*x + 1/2*c) + I) + 3840*I*a^4*log(tan(1/2*d*x + 1/2*c))
+ 2370*a^4*tan(1/2*d*x + 1/2*c) + (-8768*I*a^4*tan(1/2*d*x + 1/2*c)^5 - 2370*a^4*tan(1/2*d*x + 1/2*c)^4 + 600*
I*a^4*tan(1/2*d*x + 1/2*c)^3 + 155*a^4*tan(1/2*d*x + 1/2*c)^2 - 30*I*a^4*tan(1/2*d*x + 1/2*c) - 3*a^4)/tan(1/2
*d*x + 1/2*c)^5)/d

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Mupad [B]
time = 4.27, size = 92, normalized size = 0.65 \begin {gather*} -\frac {16\,a^4\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{d}-\frac {8\,a^4\,{\mathrm {tan}\left (c+d\,x\right )}^4-a^4\,{\mathrm {tan}\left (c+d\,x\right )}^3\,4{}\mathrm {i}-\frac {7\,a^4\,{\mathrm {tan}\left (c+d\,x\right )}^2}{3}+a^4\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}+\frac {a^4}{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)^4,x)

[Out]

- (16*a^4*atan(2*tan(c + d*x) + 1i))/d - (a^4*tan(c + d*x)*1i + a^4/5 - (7*a^4*tan(c + d*x)^2)/3 - a^4*tan(c +
d*x)^3*4i + 8*a^4*tan(c + d*x)^4)/(d*tan(c + d*x)^5)

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