3.42 \(\int \cot ^5(c+d x) (a+i a \tan (c+d x))^4 \, dx\)

Optimal. Leaf size=134 \[ \frac{4 i a^4 \cot (c+d x)}{d}+\frac{8 a^4 \log (\sin (c+d x))}{d}+\frac{\cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}+8 i a^4 x-\frac{i a \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}-\frac{\cot ^4(c+d x) (a+i a \tan (c+d x))^4}{4 d} \]

[Out]

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

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Rubi [A]  time = 0.20305, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {3548, 3545, 3542, 3531, 3475} \[ \frac{4 i a^4 \cot (c+d x)}{d}+\frac{8 a^4 \log (\sin (c+d x))}{d}+\frac{\cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}+8 i a^4 x-\frac{i a \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}-\frac{\cot ^4(c+d x) (a+i a \tan (c+d x))^4}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])^4,x]

[Out]

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

Rule 3548

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

Rule 3545

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(a*b*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m - 1)*(a*c - b*d)), x] + Dist[(2*a^2)/(
a*c - b*d), Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1), 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] && EqQ[m + n, 0] && GtQ[m, 1/2]

Rule 3542

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

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 ^5(c+d x) (a+i a \tan (c+d x))^4 \, dx &=-\frac{\cot ^4(c+d x) (a+i a \tan (c+d x))^4}{4 d}+i \int \cot ^4(c+d x) (a+i a \tan (c+d x))^4 \, dx\\ &=-\frac{i a \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}-\frac{\cot ^4(c+d x) (a+i a \tan (c+d x))^4}{4 d}-(2 a) \int \cot ^3(c+d x) (a+i a \tan (c+d x))^3 \, dx\\ &=-\frac{i a \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}-\frac{\cot ^4(c+d x) (a+i a \tan (c+d x))^4}{4 d}+\frac{\cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}-\left (4 i a^2\right ) \int \cot ^2(c+d x) (a+i a \tan (c+d x))^2 \, dx\\ &=\frac{4 i a^4 \cot (c+d x)}{d}-\frac{i a \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}-\frac{\cot ^4(c+d x) (a+i a \tan (c+d x))^4}{4 d}+\frac{\cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}-\left (4 i a^2\right ) \int \cot (c+d x) \left (2 i a^2-2 a^2 \tan (c+d x)\right ) \, dx\\ &=8 i a^4 x+\frac{4 i a^4 \cot (c+d x)}{d}-\frac{i a \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}-\frac{\cot ^4(c+d x) (a+i a \tan (c+d x))^4}{4 d}+\frac{\cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}+\left (8 a^4\right ) \int \cot (c+d x) \, dx\\ &=8 i a^4 x+\frac{4 i a^4 \cot (c+d x)}{d}+\frac{8 a^4 \log (\sin (c+d x))}{d}-\frac{i a \cot ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}-\frac{\cot ^4(c+d x) (a+i a \tan (c+d x))^4}{4 d}+\frac{\cot ^2(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}\\ \end{align*}

Mathematica [A]  time = 1.06595, size = 245, normalized size = 1.83 \[ \frac{a^4 \csc (c) \csc ^4(c+d x) \left (36 i d x \sin (c)+24 i d x \sin (c+2 d x)+12 \sin (c+2 d x)-24 i d x \sin (3 c+2 d x)-12 \sin (3 c+2 d x)-6 i d x \sin (3 c+4 d x)+6 i d x \sin (5 c+4 d x)+38 i \cos (c+2 d x)+18 i \cos (3 c+2 d x)-14 i \cos (3 c+4 d x)+18 \sin (c) \log \left (\sin ^2(c+d x)\right )+12 \sin (c+2 d x) \log \left (\sin ^2(c+d x)\right )-12 \sin (3 c+2 d x) \log \left (\sin ^2(c+d x)\right )-3 \sin (3 c+4 d x) \log \left (\sin ^2(c+d x)\right )+3 \sin (5 c+4 d x) \log \left (\sin ^2(c+d x)\right )+21 \sin (c)-42 i \cos (c)\right )}{12 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^5*(a + I*a*Tan[c + d*x])^4,x]

[Out]

(a^4*Csc[c]*Csc[c + d*x]^4*((-42*I)*Cos[c] + (38*I)*Cos[c + 2*d*x] + (18*I)*Cos[3*c + 2*d*x] - (14*I)*Cos[3*c
+ 4*d*x] + 21*Sin[c] + (36*I)*d*x*Sin[c] + 18*Log[Sin[c + d*x]^2]*Sin[c] + 12*Sin[c + 2*d*x] + (24*I)*d*x*Sin[
c + 2*d*x] + 12*Log[Sin[c + d*x]^2]*Sin[c + 2*d*x] - 12*Sin[3*c + 2*d*x] - (24*I)*d*x*Sin[3*c + 2*d*x] - 12*Lo
g[Sin[c + d*x]^2]*Sin[3*c + 2*d*x] - (6*I)*d*x*Sin[3*c + 4*d*x] - 3*Log[Sin[c + d*x]^2]*Sin[3*c + 4*d*x] + (6*
I)*d*x*Sin[5*c + 4*d*x] + 3*Log[Sin[c + d*x]^2]*Sin[5*c + 4*d*x]))/(12*d)

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Maple [A]  time = 0.056, size = 98, normalized size = 0.7 \begin{align*} 8\,{\frac{{a}^{4}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}+8\,i{a}^{4}x+{\frac{8\,i\cot \left ( dx+c \right ){a}^{4}}{d}}+{\frac{8\,i{a}^{4}c}{d}}+{\frac{7\,{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{{\frac{4\,i}{3}}{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{d}}-{\frac{{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{4\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^5*(a+I*a*tan(d*x+c))^4,x)

[Out]

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

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Maxima [A]  time = 1.57589, size = 130, normalized size = 0.97 \begin{align*} -\frac{-96 i \,{\left (d x + c\right )} a^{4} + 48 \, a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 96 \, a^{4} \log \left (\tan \left (d x + c\right )\right ) + \frac{-96 i \, a^{4} \tan \left (d x + c\right )^{3} - 42 \, a^{4} \tan \left (d x + c\right )^{2} + 16 i \, a^{4} \tan \left (d x + c\right ) + 3 \, a^{4}}{\tan \left (d x + c\right )^{4}}}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^5*(a+I*a*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

-1/12*(-96*I*(d*x + c)*a^4 + 48*a^4*log(tan(d*x + c)^2 + 1) - 96*a^4*log(tan(d*x + c)) + (-96*I*a^4*tan(d*x +
c)^3 - 42*a^4*tan(d*x + c)^2 + 16*I*a^4*tan(d*x + c) + 3*a^4)/tan(d*x + c)^4)/d

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Fricas [A]  time = 2.15734, size = 482, normalized size = 3.6 \begin{align*} -\frac{4 \,{\left (30 \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 63 \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 50 \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 14 \, a^{4} - 6 \,{\left (a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} - 4 \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 4 \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )\right )}}{3 \,{\left (d e^{\left (8 i \, d x + 8 i \, c\right )} - 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} - 4 \, 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)^5*(a+I*a*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

-4/3*(30*a^4*e^(6*I*d*x + 6*I*c) - 63*a^4*e^(4*I*d*x + 4*I*c) + 50*a^4*e^(2*I*d*x + 2*I*c) - 14*a^4 - 6*(a^4*e
^(8*I*d*x + 8*I*c) - 4*a^4*e^(6*I*d*x + 6*I*c) + 6*a^4*e^(4*I*d*x + 4*I*c) - 4*a^4*e^(2*I*d*x + 2*I*c) + a^4)*
log(e^(2*I*d*x + 2*I*c) - 1))/(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e^(4*I*d*x + 4*I*c) - 4*d
*e^(2*I*d*x + 2*I*c) + d)

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Sympy [A]  time = 5.85144, size = 175, normalized size = 1.31 \begin{align*} \frac{8 a^{4} \log{\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac{- \frac{40 a^{4} e^{- 2 i c} e^{6 i d x}}{d} + \frac{84 a^{4} e^{- 4 i c} e^{4 i d x}}{d} - \frac{200 a^{4} e^{- 6 i c} e^{2 i d x}}{3 d} + \frac{56 a^{4} e^{- 8 i c}}{3 d}}{e^{8 i d x} - 4 e^{- 2 i c} e^{6 i d x} + 6 e^{- 4 i c} e^{4 i d x} - 4 e^{- 6 i c} e^{2 i d x} + e^{- 8 i c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**5*(a+I*a*tan(d*x+c))**4,x)

[Out]

8*a**4*log(exp(2*I*d*x) - exp(-2*I*c))/d + (-40*a**4*exp(-2*I*c)*exp(6*I*d*x)/d + 84*a**4*exp(-4*I*c)*exp(4*I*
d*x)/d - 200*a**4*exp(-6*I*c)*exp(2*I*d*x)/(3*d) + 56*a**4*exp(-8*I*c)/(3*d))/(exp(8*I*d*x) - 4*exp(-2*I*c)*ex
p(6*I*d*x) + 6*exp(-4*I*c)*exp(4*I*d*x) - 4*exp(-6*I*c)*exp(2*I*d*x) + exp(-8*I*c))

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Giac [A]  time = 1.61977, size = 244, normalized size = 1.82 \begin{align*} -\frac{3 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 32 i \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 180 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 3072 \, a^{4} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + i\right ) - 1536 \, a^{4} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 864 i \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{3200 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 864 i \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 180 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 32 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 )^{4}}}{192 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^5*(a+I*a*tan(d*x+c))^4,x, algorithm="giac")

[Out]

-1/192*(3*a^4*tan(1/2*d*x + 1/2*c)^4 - 32*I*a^4*tan(1/2*d*x + 1/2*c)^3 - 180*a^4*tan(1/2*d*x + 1/2*c)^2 + 3072
*a^4*log(tan(1/2*d*x + 1/2*c) + I) - 1536*a^4*log(abs(tan(1/2*d*x + 1/2*c))) + 864*I*a^4*tan(1/2*d*x + 1/2*c)
+ (3200*a^4*tan(1/2*d*x + 1/2*c)^4 - 864*I*a^4*tan(1/2*d*x + 1/2*c)^3 - 180*a^4*tan(1/2*d*x + 1/2*c)^2 + 32*I*
a^4*tan(1/2*d*x + 1/2*c) + 3*a^4)/tan(1/2*d*x + 1/2*c)^4)/d