3.68 \(\int \frac{\text{csch}^3(x)}{i+\text{csch}(x)} \, dx\)

Optimal. Leaf size=26 \[ -\coth (x)+i \tanh ^{-1}(\cosh (x))-\frac{i \coth (x)}{\text{csch}(x)+i} \]

[Out]

I*ArcTanh[Cosh[x]] - Coth[x] - (I*Coth[x])/(I + Csch[x])

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Rubi [A]  time = 0.0836923, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {3790, 3789, 3770, 3794} \[ -\coth (x)+i \tanh ^{-1}(\cosh (x))-\frac{i \coth (x)}{\text{csch}(x)+i} \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]^3/(I + Csch[x]),x]

[Out]

I*ArcTanh[Cosh[x]] - Coth[x] - (I*Coth[x])/(I + Csch[x])

Rule 3790

Int[csc[(e_.) + (f_.)*(x_)]^3/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[Cot[e + f*x]/(b*f), x
] - Dist[a/b, Int[Csc[e + f*x]^2/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x]

Rule 3789

Int[csc[(e_.) + (f_.)*(x_)]^2/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[Csc[e + f*x],
x], x] - Dist[a/b, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x]

Rule 3770

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

Rule 3794

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> -Simp[Cot[e + f*x]/(f*(b + a*
Csc[e + f*x])), x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

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

Mathematica [B]  time = 0.108823, size = 70, normalized size = 2.69 \[ -\frac{1}{2} \tanh \left (\frac{x}{2}\right )-\frac{1}{2} \coth \left (\frac{x}{2}\right )-i \log \left (\sinh \left (\frac{x}{2}\right )\right )+i \log \left (\cosh \left (\frac{x}{2}\right )\right )-\frac{2 \sinh \left (\frac{x}{2}\right )}{\cosh \left (\frac{x}{2}\right )+i \sinh \left (\frac{x}{2}\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]^3/(I + Csch[x]),x]

[Out]

-Coth[x/2]/2 + I*Log[Cosh[x/2]] - I*Log[Sinh[x/2]] - (2*Sinh[x/2])/(Cosh[x/2] + I*Sinh[x/2]) - Tanh[x/2]/2

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Maple [A]  time = 0.018, size = 35, normalized size = 1.4 \begin{align*} -{\frac{1}{2}\tanh \left ({\frac{x}{2}} \right ) }-2\, \left ( \tanh \left ( x/2 \right ) -i \right ) ^{-1}-{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-i\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)^3/(I+csch(x)),x)

[Out]

-1/2*tanh(1/2*x)-2/(tanh(1/2*x)-I)-1/2/tanh(1/2*x)-I*ln(tanh(1/2*x))

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Maxima [B]  time = 1.02521, size = 74, normalized size = 2.85 \begin{align*} -\frac{8 \,{\left (e^{\left (-x\right )} - i \, e^{\left (-2 \, x\right )} + 2 i\right )}}{4 \, e^{\left (-x\right )} - 4 i \, e^{\left (-2 \, x\right )} - 4 \, e^{\left (-3 \, x\right )} + 4 i} + i \, \log \left (e^{\left (-x\right )} + 1\right ) - i \, \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^3/(I+csch(x)),x, algorithm="maxima")

[Out]

-8*(e^(-x) - I*e^(-2*x) + 2*I)/(4*e^(-x) - 4*I*e^(-2*x) - 4*e^(-3*x) + 4*I) + I*log(e^(-x) + 1) - I*log(e^(-x)
- 1)

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Fricas [B]  time = 1.64018, size = 216, normalized size = 8.31 \begin{align*} \frac{{\left (i \, e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} - i \, e^{x} - 1\right )} \log \left (e^{x} + 1\right ) +{\left (-i \, e^{\left (3 \, x\right )} - e^{\left (2 \, x\right )} + i \, e^{x} + 1\right )} \log \left (e^{x} - 1\right ) - 2 i \, e^{\left (2 \, x\right )} - 2 \, e^{x} + 4 i}{e^{\left (3 \, x\right )} - i \, e^{\left (2 \, x\right )} - e^{x} + i} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^3/(I+csch(x)),x, algorithm="fricas")

[Out]

((I*e^(3*x) + e^(2*x) - I*e^x - 1)*log(e^x + 1) + (-I*e^(3*x) - e^(2*x) + I*e^x + 1)*log(e^x - 1) - 2*I*e^(2*x
) - 2*e^x + 4*I)/(e^(3*x) - I*e^(2*x) - e^x + I)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}^{3}{\left (x \right )}}{\operatorname{csch}{\left (x \right )} + i}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)**3/(I+csch(x)),x)

[Out]

Integral(csch(x)**3/(csch(x) + I), x)

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Giac [B]  time = 1.21046, size = 62, normalized size = 2.38 \begin{align*} \frac{2 \,{\left (e^{\left (2 \, x\right )} - i \, e^{x} - 2\right )}}{i \, e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} - i \, e^{x} - 1} + i \, \log \left (e^{x} + 1\right ) - i \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^3/(I+csch(x)),x, algorithm="giac")

[Out]

2*(e^(2*x) - I*e^x - 2)/(I*e^(3*x) + e^(2*x) - I*e^x - 1) + I*log(e^x + 1) - I*log(abs(e^x - 1))