3.43 \(\int \cot ^6(c+d x) (a+i a \tan (c+d x))^4 \, dx\)

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

[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

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Rubi [A]  time = 0.291448, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3553, 3593, 3591, 3529, 3531, 3475} \[ \frac{23 a^4 \cot ^3(c+d x)}{15 d}+\frac{4 i a^4 \cot ^2(c+d x)}{d}-\frac{8 a^4 \cot (c+d x)}{d}+\frac{8 i a^4 \log (\sin (c+d x))}{d}-\frac{\cot ^5(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{5 d}-\frac{3 i \cot ^4(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{5 d}-8 a^4 x \]

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 3553

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Si
mp[(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] /; Fr
eeQ[{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] && LtQ[
n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])

Rule 3593

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] && EqQ
[a^2 + b^2, 0] && GtQ[m, 1] && LtQ[n, -1]

Rule 3591

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 3529

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 3531

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 3475

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

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*}

Mathematica [B]  time = 3.46985, size = 359, normalized size = 2.53 \[ \frac{a^4 \csc (c) \csc ^5(c+d x) (\cos (4 d x)+i \sin (4 d x)) \left (345 \sin (2 c+d x)-275 \sin (2 c+3 d x)-120 \sin (4 c+3 d x)+79 \sin (4 c+5 d x)+600 d x \cos (2 c+d x)-210 i \cos (2 c+d x)+300 d x \cos (2 c+3 d x)-90 i \cos (2 c+3 d x)-300 d x \cos (4 c+3 d x)+90 i \cos (4 c+3 d x)-60 d x \cos (4 c+5 d x)+60 d x \cos (6 c+5 d x)+960 \sin (c) \sin ^5(c+d x) \tan ^{-1}(\tan (5 c+d x))-30 \cos (d x) \left (-5 i \log \left (\sin ^2(c+d x)\right )+20 d x-7 i\right )-150 i \cos (2 c+d x) \log \left (\sin ^2(c+d x)\right )-75 i \cos (2 c+3 d x) \log \left (\sin ^2(c+d x)\right )+75 i \cos (4 c+3 d x) \log \left (\sin ^2(c+d x)\right )+15 i \cos (4 c+5 d x) \log \left (\sin ^2(c+d x)\right )-15 i \cos (6 c+5 d x) \log \left (\sin ^2(c+d x)\right )+445 \sin (d x)\right )}{120 d (\cos (d x)+i \sin (d x))^4} \]

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 = 0.056, size = 113, normalized size = 0.8 \begin{align*} -8\,{a}^{4}x-8\,{\frac{{a}^{4}\cot \left ( dx+c \right ) }{d}}-8\,{\frac{{a}^{4}c}{d}}+{\frac{4\,i{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{d}}-{\frac{i{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{d}}+{\frac{7\,{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{8\,i{a}^{4}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5\,d}} \end{align*}

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)

[Out]

-8*a^4*x-8*a^4*cot(d*x+c)/d-8/d*a^4*c+4*I*a^4*cot(d*x+c)^2/d-I/d*a^4*cot(d*x+c)^4+7/3*a^4*cot(d*x+c)^3/d+8*I*a
^4*ln(sin(d*x+c))/d-1/5/d*a^4*cot(d*x+c)^5

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Maxima [A]  time = 1.57646, size = 147, normalized size = 1.04 \begin{align*} -\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{align*}

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 = 2.31537, size = 666, normalized size = 4.69 \begin{align*} \frac{-840 i \, a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 2220 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 2620 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 1460 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 316 i \, a^{4} +{\left (120 i \, a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} - 600 i \, a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 1200 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 1200 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 600 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 120 i \, a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\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{align*}

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]

1/15*(-840*I*a^4*e^(8*I*d*x + 8*I*c) + 2220*I*a^4*e^(6*I*d*x + 6*I*c) - 2620*I*a^4*e^(4*I*d*x + 4*I*c) + 1460*
I*a^4*e^(2*I*d*x + 2*I*c) - 316*I*a^4 + (120*I*a^4*e^(10*I*d*x + 10*I*c) - 600*I*a^4*e^(8*I*d*x + 8*I*c) + 120
0*I*a^4*e^(6*I*d*x + 6*I*c) - 1200*I*a^4*e^(4*I*d*x + 4*I*c) + 600*I*a^4*e^(2*I*d*x + 2*I*c) - 120*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 = 5.93892, size = 226, normalized size = 1.59 \begin{align*} \frac{8 i a^{4} \log{\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac{- \frac{56 i a^{4} e^{- 2 i c} e^{8 i d x}}{d} + \frac{148 i a^{4} e^{- 4 i c} e^{6 i d x}}{d} - \frac{524 i a^{4} e^{- 6 i c} e^{4 i d x}}{3 d} + \frac{292 i a^{4} e^{- 8 i c} e^{2 i d x}}{3 d} - \frac{316 i a^{4} e^{- 10 i c}}{15 d}}{e^{10 i d x} - 5 e^{- 2 i c} e^{8 i d x} + 10 e^{- 4 i c} e^{6 i d x} - 10 e^{- 6 i c} e^{4 i d x} + 5 e^{- 8 i c} e^{2 i d x} - e^{- 10 i c}} \end{align*}

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 + (-56*I*a**4*exp(-2*I*c)*exp(8*I*d*x)/d + 148*I*a**4*exp(-4*I*c)*e
xp(6*I*d*x)/d - 524*I*a**4*exp(-6*I*c)*exp(4*I*d*x)/(3*d) + 292*I*a**4*exp(-8*I*c)*exp(2*I*d*x)/(3*d) - 316*I*
a**4*exp(-10*I*c)/(15*d))/(exp(10*I*d*x) - 5*exp(-2*I*c)*exp(8*I*d*x) + 10*exp(-4*I*c)*exp(6*I*d*x) - 10*exp(-
6*I*c)*exp(4*I*d*x) + 5*exp(-8*I*c)*exp(2*I*d*x) - exp(-10*I*c))

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Giac [A]  time = 1.53522, size = 288, normalized size = 2.03 \begin{align*} \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 ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \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{align*}

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(abs(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)/ta
n(1/2*d*x + 1/2*c)^5)/d