3.112 \(\int \frac{\coth ^6(x)}{i+\text{csch}(x)} \, dx\)

Optimal. Leaf size=43 \[ -i x-\frac{3}{8} \tanh ^{-1}(\cosh (x))+\frac{1}{12} \coth ^3(x) (-3 \text{csch}(x)+4 i)+\frac{1}{8} \coth (x) (-3 \text{csch}(x)+8 i) \]

[Out]

(-I)*x - (3*ArcTanh[Cosh[x]])/8 + (Coth[x]^3*(4*I - 3*Csch[x]))/12 + (Coth[x]*(8*I - 3*Csch[x]))/8

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Rubi [A]  time = 0.0749438, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3888, 3881, 3770} \[ -i x-\frac{3}{8} \tanh ^{-1}(\cosh (x))+\frac{1}{12} \coth ^3(x) (-3 \text{csch}(x)+4 i)+\frac{1}{8} \coth (x) (-3 \text{csch}(x)+8 i) \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]^6/(I + Csch[x]),x]

[Out]

(-I)*x - (3*ArcTanh[Cosh[x]])/8 + (Coth[x]^3*(4*I - 3*Csch[x]))/12 + (Coth[x]*(8*I - 3*Csch[x]))/8

Rule 3888

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Dist[a^(2*n
)/e^(2*n), Int[(e*Cot[c + d*x])^(m + 2*n)/(-a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && E
qQ[a^2 - b^2, 0] && ILtQ[n, 0]

Rule 3881

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[(e*(e*Cot[
c + d*x])^(m - 1)*(a*m + b*(m - 1)*Csc[c + d*x]))/(d*m*(m - 1)), x] - Dist[e^2/m, Int[(e*Cot[c + d*x])^(m - 2)
*(a*m + b*(m - 1)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[m, 1]

Rule 3770

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

Rubi steps

\begin{align*} \int \frac{\coth ^6(x)}{i+\text{csch}(x)} \, dx &=\int \coth ^4(x) (-i+\text{csch}(x)) \, dx\\ &=\frac{1}{12} \coth ^3(x) (4 i-3 \text{csch}(x))+\frac{1}{4} \int \coth ^2(x) (-4 i+3 \text{csch}(x)) \, dx\\ &=\frac{1}{12} \coth ^3(x) (4 i-3 \text{csch}(x))+\frac{1}{8} \coth (x) (8 i-3 \text{csch}(x))+\frac{1}{8} \int (-8 i+3 \text{csch}(x)) \, dx\\ &=-i x+\frac{1}{12} \coth ^3(x) (4 i-3 \text{csch}(x))+\frac{1}{8} \coth (x) (8 i-3 \text{csch}(x))+\frac{3}{8} \int \text{csch}(x) \, dx\\ &=-i x-\frac{3}{8} \tanh ^{-1}(\cosh (x))+\frac{1}{12} \coth ^3(x) (4 i-3 \text{csch}(x))+\frac{1}{8} \coth (x) (8 i-3 \text{csch}(x))\\ \end{align*}

Mathematica [B]  time = 0.0405122, size = 129, normalized size = 3. \[ -i x+\frac{2}{3} i \tanh \left (\frac{x}{2}\right )+\frac{2}{3} i \coth \left (\frac{x}{2}\right )-\frac{1}{64} \text{csch}^4\left (\frac{x}{2}\right )-\frac{5}{32} \text{csch}^2\left (\frac{x}{2}\right )+\frac{1}{64} \text{sech}^4\left (\frac{x}{2}\right )-\frac{5}{32} \text{sech}^2\left (\frac{x}{2}\right )+\frac{3}{8} \log \left (\tanh \left (\frac{x}{2}\right )\right )+\frac{1}{24} i \coth \left (\frac{x}{2}\right ) \text{csch}^2\left (\frac{x}{2}\right )-\frac{1}{24} i \tanh \left (\frac{x}{2}\right ) \text{sech}^2\left (\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]^6/(I + Csch[x]),x]

[Out]

(-I)*x + ((2*I)/3)*Coth[x/2] - (5*Csch[x/2]^2)/32 + (I/24)*Coth[x/2]*Csch[x/2]^2 - Csch[x/2]^4/64 + (3*Log[Tan
h[x/2]])/8 - (5*Sech[x/2]^2)/32 + Sech[x/2]^4/64 + ((2*I)/3)*Tanh[x/2] - (I/24)*Sech[x/2]^2*Tanh[x/2]

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Maple [B]  time = 0.092, size = 95, normalized size = 2.2 \begin{align*}{\frac{5\,i}{8}}\tanh \left ({\frac{x}{2}} \right ) +{\frac{1}{64} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{4}}+{\frac{i}{24}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}+{\frac{1}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}-i\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -{\frac{1}{64} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-4}}+{{\frac{5\,i}{8}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+{{\frac{i}{24}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-3}}-{\frac{1}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}+{\frac{3}{8}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }+i\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^6/(I+csch(x)),x)

[Out]

5/8*I*tanh(1/2*x)+1/64*tanh(1/2*x)^4+1/24*I*tanh(1/2*x)^3+1/8*tanh(1/2*x)^2-I*ln(tanh(1/2*x)+1)-1/64/tanh(1/2*
x)^4+5/8*I/tanh(1/2*x)+1/24*I/tanh(1/2*x)^3-1/8/tanh(1/2*x)^2+3/8*ln(tanh(1/2*x))+I*ln(tanh(1/2*x)-1)

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Maxima [B]  time = 1.03415, size = 130, normalized size = 3.02 \begin{align*} -i \, x + \frac{15 \, e^{\left (-x\right )} + 80 i \, e^{\left (-2 \, x\right )} + 9 \, e^{\left (-3 \, x\right )} - 96 i \, e^{\left (-4 \, x\right )} + 9 \, e^{\left (-5 \, x\right )} + 48 i \, e^{\left (-6 \, x\right )} + 15 \, e^{\left (-7 \, x\right )} - 32 i}{12 \,{\left (4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1\right )}} - \frac{3}{8} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac{3}{8} \, \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^6/(I+csch(x)),x, algorithm="maxima")

[Out]

-I*x + 1/12*(15*e^(-x) + 80*I*e^(-2*x) + 9*e^(-3*x) - 96*I*e^(-4*x) + 9*e^(-5*x) + 48*I*e^(-6*x) + 15*e^(-7*x)
- 32*I)/(4*e^(-2*x) - 6*e^(-4*x) + 4*e^(-6*x) - e^(-8*x) - 1) - 3/8*log(e^(-x) + 1) + 3/8*log(e^(-x) - 1)

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Fricas [B]  time = 1.79844, size = 478, normalized size = 11.12 \begin{align*} \frac{-24 i \, x e^{\left (8 \, x\right )} +{\left (96 i \, x + 96 i\right )} e^{\left (6 \, x\right )} +{\left (-144 i \, x - 192 i\right )} e^{\left (4 \, x\right )} +{\left (96 i \, x + 160 i\right )} e^{\left (2 \, x\right )} - 9 \,{\left (e^{\left (8 \, x\right )} - 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} - 4 \, e^{\left (2 \, x\right )} + 1\right )} \log \left (e^{x} + 1\right ) + 9 \,{\left (e^{\left (8 \, x\right )} - 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} - 4 \, e^{\left (2 \, x\right )} + 1\right )} \log \left (e^{x} - 1\right ) - 24 i \, x - 30 \, e^{\left (7 \, x\right )} - 18 \, e^{\left (5 \, x\right )} - 18 \, e^{\left (3 \, x\right )} - 30 \, e^{x} - 64 i}{24 \,{\left (e^{\left (8 \, x\right )} - 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} - 4 \, e^{\left (2 \, x\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^6/(I+csch(x)),x, algorithm="fricas")

[Out]

1/24*(-24*I*x*e^(8*x) + (96*I*x + 96*I)*e^(6*x) + (-144*I*x - 192*I)*e^(4*x) + (96*I*x + 160*I)*e^(2*x) - 9*(e
^(8*x) - 4*e^(6*x) + 6*e^(4*x) - 4*e^(2*x) + 1)*log(e^x + 1) + 9*(e^(8*x) - 4*e^(6*x) + 6*e^(4*x) - 4*e^(2*x)
+ 1)*log(e^x - 1) - 24*I*x - 30*e^(7*x) - 18*e^(5*x) - 18*e^(3*x) - 30*e^x - 64*I)/(e^(8*x) - 4*e^(6*x) + 6*e^
(4*x) - 4*e^(2*x) + 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)**6/(I+csch(x)),x)

[Out]

Timed out

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Giac [B]  time = 1.15593, size = 104, normalized size = 2.42 \begin{align*} -\frac{15 \, e^{\left (7 \, x\right )} - 48 i \, e^{\left (6 \, x\right )} + 9 \, e^{\left (5 \, x\right )} + 96 i \, e^{\left (4 \, x\right )} + 9 \, e^{\left (3 \, x\right )} - 80 i \, e^{\left (2 \, x\right )} + 15 \, e^{x} + 32 i}{12 \,{\left (i \, e^{\left (2 \, x\right )} - i\right )}^{4}} - i \, \log \left (-i \, e^{x}\right ) - \frac{3}{8} \, \log \left (e^{x} + 1\right ) + \frac{3}{8} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^6/(I+csch(x)),x, algorithm="giac")

[Out]

-1/12*(15*e^(7*x) - 48*I*e^(6*x) + 9*e^(5*x) + 96*I*e^(4*x) + 9*e^(3*x) - 80*I*e^(2*x) + 15*e^x + 32*I)/(I*e^(
2*x) - I)^4 - I*log(-I*e^x) - 3/8*log(e^x + 1) + 3/8*log(abs(e^x - 1))