\(\int \frac {\text {csch}^3(x)}{1+\coth (x)} \, dx\) [95]

Optimal result

Integrand size = 11, antiderivative size = 8 \[ \int \frac {\text {csch}^3(x)}{1+\coth (x)} \, dx=\text {arctanh}(\cosh (x))-\text {csch}(x) \]

[Out]

arctanh(cosh(x))-csch(x)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3582, 3855} \[ \int \frac {\text {csch}^3(x)}{1+\coth (x)} \, dx=\text {arctanh}(\cosh (x))-\text {csch}(x) \]

[In]

Int[Csch[x]^3/(1 + Coth[x]),x]

[Out]

ArcTanh[Cosh[x]] - Csch[x]

Rule 3582

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

Rule 3855

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

Rubi steps \begin{align*} \text {integral}& = -\text {csch}(x)-\int \text {csch}(x) \, dx \\ & = \text {arctanh}(\cosh (x))-\text {csch}(x) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(21\) vs. \(2(8)=16\).

Time = 0.32 (sec) , antiderivative size = 21, normalized size of antiderivative = 2.62 \[ \int \frac {\text {csch}^3(x)}{1+\coth (x)} \, dx=-\text {csch}(x)+\log \left (\cosh \left (\frac {x}{2}\right )\right )-\log \left (\sinh \left (\frac {x}{2}\right )\right ) \]

[In]

Integrate[Csch[x]^3/(1 + Coth[x]),x]

[Out]

-Csch[x] + Log[Cosh[x/2]] - Log[Sinh[x/2]]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(22\) vs. \(2(8)=16\).

Time = 0.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 2.88

method	result	size
default	\(\frac {\tanh \left (\frac {x}{2}\right )}{2}-\frac {1}{2 \tanh \left (\frac {x}{2}\right )}-\ln \left (\tanh \left (\frac {x}{2}\right )\right )\)	\(23\)
risch	\(-\frac {2 \,{\mathrm e}^{x}}{{\mathrm e}^{2 x}-1}+\ln \left ({\mathrm e}^{x}+1\right )-\ln \left ({\mathrm e}^{x}-1\right )\)	\(26\)

[In]

int(csch(x)^3/(1+coth(x)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (8) = 16\).

Time = 0.25 (sec) , antiderivative size = 77, normalized size of antiderivative = 9.62 \[ \int \frac {\text {csch}^3(x)}{1+\coth (x)} \, dx=\frac {{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) - 2 \, \cosh \left (x\right ) - 2 \, \sinh \left (x\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1} \]

[In]

integrate(csch(x)^3/(1+coth(x)),x, algorithm="fricas")

[Out]

((cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*log(cosh(x) + sinh(x) + 1) - (cosh(x)^2 + 2*cosh(x)*sinh(x) +
sinh(x)^2 - 1)*log(cosh(x) + sinh(x) - 1) - 2*cosh(x) - 2*sinh(x))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2
- 1)

Sympy [F]

\[ \int \frac {\text {csch}^3(x)}{1+\coth (x)} \, dx=\int \frac {\operatorname {csch}^{3}{\left (x \right )}}{\coth {\left (x \right )} + 1}\, dx \]

[In]

integrate(csch(x)**3/(1+coth(x)),x)

[Out]

Integral(csch(x)**3/(coth(x) + 1), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (8) = 16\).

Time = 0.18 (sec) , antiderivative size = 31, normalized size of antiderivative = 3.88 \[ \int \frac {\text {csch}^3(x)}{1+\coth (x)} \, dx=\frac {2 \, e^{\left (-x\right )}}{e^{\left (-2 \, x\right )} - 1} + \log \left (e^{\left (-x\right )} + 1\right ) - \log \left (e^{\left (-x\right )} - 1\right ) \]

[In]

integrate(csch(x)^3/(1+coth(x)),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (8) = 16\).

Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 3.25 \[ \int \frac {\text {csch}^3(x)}{1+\coth (x)} \, dx=-\frac {2 \, e^{x}}{e^{\left (2 \, x\right )} - 1} + \log \left (e^{x} + 1\right ) - \log \left ({\left | e^{x} - 1 \right |}\right ) \]

[In]

integrate(csch(x)^3/(1+coth(x)),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 3.62 \[ \int \frac {\text {csch}^3(x)}{1+\coth (x)} \, dx=\ln \left (2\,{\mathrm {e}}^x+2\right )-\ln \left (2\,{\mathrm {e}}^x-2\right )-\frac {2\,{\mathrm {e}}^x}{{\mathrm {e}}^{2\,x}-1} \]

[In]

int(1/(sinh(x)^3*(coth(x) + 1)),x)

[Out]

log(2*exp(x) + 2) - log(2*exp(x) - 2) - (2*exp(x))/(exp(2*x) - 1)