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

Optimal. Leaf size=100 \[ -\frac{a \cot ^5(c+d x)}{5 d}-\frac{i a \cot ^4(c+d x)}{4 d}+\frac{a \cot ^3(c+d x)}{3 d}+\frac{i a \cot ^2(c+d x)}{2 d}-\frac{a \cot (c+d x)}{d}+\frac{i a \log (\sin (c+d x))}{d}-a x \]

[Out]

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

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Rubi [A]  time = 0.131811, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {3529, 3531, 3475} \[ -\frac{a \cot ^5(c+d x)}{5 d}-\frac{i a \cot ^4(c+d x)}{4 d}+\frac{a \cot ^3(c+d x)}{3 d}+\frac{i a \cot ^2(c+d x)}{2 d}-\frac{a \cot (c+d x)}{d}+\frac{i a \log (\sin (c+d x))}{d}-a x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [C]  time = 0.358327, size = 84, normalized size = 0.84 \[ -\frac{a \cot ^5(c+d x) \, _2F_1\left (-\frac{5}{2},1;-\frac{3}{2};-\tan ^2(c+d x)\right )}{5 d}+\frac{i a \left (-\cot ^4(c+d x)+2 \cot ^2(c+d x)+4 \log (\tan (c+d x))+4 \log (\cos (c+d x))\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*Cot[c + d*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, -Tan[c + d*x]^2])/(5*d) + ((I/4)*a*(2*Cot[c + d*x]^2 - Cot
[c + d*x]^4 + 4*Log[Cos[c + d*x]] + 4*Log[Tan[c + d*x]]))/d

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Maple [A]  time = 0.043, size = 97, normalized size = 1. \begin{align*}{\frac{-{\frac{i}{4}}a \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{d}}+{\frac{{\frac{i}{2}}a \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{d}}+{\frac{ia\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{a \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5\,d}}+{\frac{a \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{\cot \left ( dx+c \right ) a}{d}}-ax-{\frac{ac}{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)),x)

[Out]

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

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Maxima [A]  time = 2.19778, size = 126, normalized size = 1.26 \begin{align*} -\frac{60 \,{\left (d x + c\right )} a + 30 i \, a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 60 i \, a \log \left (\tan \left (d x + c\right )\right ) + \frac{60 \, a \tan \left (d x + c\right )^{4} - 30 i \, a \tan \left (d x + c\right )^{3} - 20 \, a \tan \left (d x + c\right )^{2} + 15 i \, a \tan \left (d x + c\right ) + 12 \, a}{\tan \left (d x + c\right )^{5}}}{60 \, 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)),x, algorithm="maxima")

[Out]

-1/60*(60*(d*x + c)*a + 30*I*a*log(tan(d*x + c)^2 + 1) - 60*I*a*log(tan(d*x + c)) + (60*a*tan(d*x + c)^4 - 30*
I*a*tan(d*x + c)^3 - 20*a*tan(d*x + c)^2 + 15*I*a*tan(d*x + c) + 12*a)/tan(d*x + c)^5)/d

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Fricas [B]  time = 2.22204, size = 622, normalized size = 6.22 \begin{align*} \frac{-150 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} + 300 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} - 400 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 200 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} +{\left (15 i \, a e^{\left (10 i \, d x + 10 i \, c\right )} - 75 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} + 150 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} - 150 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 75 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} - 15 i \, a\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right ) - 46 i \, a}{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)),x, algorithm="fricas")

[Out]

1/15*(-150*I*a*e^(8*I*d*x + 8*I*c) + 300*I*a*e^(6*I*d*x + 6*I*c) - 400*I*a*e^(4*I*d*x + 4*I*c) + 200*I*a*e^(2*
I*d*x + 2*I*c) + (15*I*a*e^(10*I*d*x + 10*I*c) - 75*I*a*e^(8*I*d*x + 8*I*c) + 150*I*a*e^(6*I*d*x + 6*I*c) - 15
0*I*a*e^(4*I*d*x + 4*I*c) + 75*I*a*e^(2*I*d*x + 2*I*c) - 15*I*a)*log(e^(2*I*d*x + 2*I*c) - 1) - 46*I*a)/(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 [B]  time = 8.60432, size = 214, normalized size = 2.14 \begin{align*} \frac{i a \log{\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac{- \frac{10 i a e^{- 2 i c} e^{8 i d x}}{d} + \frac{20 i a e^{- 4 i c} e^{6 i d x}}{d} - \frac{80 i a e^{- 6 i c} e^{4 i d x}}{3 d} + \frac{40 i a e^{- 8 i c} e^{2 i d x}}{3 d} - \frac{46 i a 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)),x)

[Out]

I*a*log(exp(2*I*d*x) - exp(-2*I*c))/d + (-10*I*a*exp(-2*I*c)*exp(8*I*d*x)/d + 20*I*a*exp(-4*I*c)*exp(6*I*d*x)/
d - 80*I*a*exp(-6*I*c)*exp(4*I*d*x)/(3*d) + 40*I*a*exp(-8*I*c)*exp(2*I*d*x)/(3*d) - 46*I*a*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 [B]  time = 1.47314, size = 252, normalized size = 2.52 \begin{align*} \frac{6 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 15 i \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 70 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 180 i \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1920 i \, a \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + i\right ) + 960 i \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 660 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{-2192 i \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 660 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 180 i \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 70 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 i \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 6 \, a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}}}{960 \, 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)),x, algorithm="giac")

[Out]

1/960*(6*a*tan(1/2*d*x + 1/2*c)^5 - 15*I*a*tan(1/2*d*x + 1/2*c)^4 - 70*a*tan(1/2*d*x + 1/2*c)^3 + 180*I*a*tan(
1/2*d*x + 1/2*c)^2 - 1920*I*a*log(tan(1/2*d*x + 1/2*c) + I) + 960*I*a*log(abs(tan(1/2*d*x + 1/2*c))) + 660*a*t
an(1/2*d*x + 1/2*c) + (-2192*I*a*tan(1/2*d*x + 1/2*c)^5 - 660*a*tan(1/2*d*x + 1/2*c)^4 + 180*I*a*tan(1/2*d*x +
1/2*c)^3 + 70*a*tan(1/2*d*x + 1/2*c)^2 - 15*I*a*tan(1/2*d*x + 1/2*c) - 6*a)/tan(1/2*d*x + 1/2*c)^5)/d