\(\int \frac {\sinh ^3(x)}{(1-\cosh (x))^3} \, dx\) [151]

Optimal result

Integrand size = 13, antiderivative size = 20 \[ \int \frac {\sinh ^3(x)}{(1-\cosh (x))^3} \, dx=-\frac {2}{1-\cosh (x)}-\log (1-\cosh (x)) \]

[Out]

-2/(1-cosh(x))-ln(1-cosh(x))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2746, 45} \[ \int \frac {\sinh ^3(x)}{(1-\cosh (x))^3} \, dx=-\frac {2}{1-\cosh (x)}-\log (1-\cosh (x)) \]

[In]

Int[Sinh[x]^3/(1 - Cosh[x])^3,x]

[Out]

-2/(1 - Cosh[x]) - Log[1 - Cosh[x]]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2746

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1-x}{(1+x)^2} \, dx,x,-\cosh (x)\right ) \\ & = \text {Subst}\left (\int \left (\frac {1}{-1-x}+\frac {2}{(1+x)^2}\right ) \, dx,x,-\cosh (x)\right ) \\ & = -\frac {2}{1-\cosh (x)}-\log (1-\cosh (x)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {\sinh ^3(x)}{(1-\cosh (x))^3} \, dx=\coth ^2\left (\frac {x}{2}\right )-2 \log \left (\cosh \left (\frac {x}{2}\right )\right )-2 \log \left (\tanh \left (\frac {x}{2}\right )\right ) \]

[In]

Integrate[Sinh[x]^3/(1 - Cosh[x])^3,x]

[Out]

Coth[x/2]^2 - 2*Log[Cosh[x/2]] - 2*Log[Tanh[x/2]]

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85

method	result	size
derivativedivides	\(\frac {2}{\cosh \left (x \right )-1}-\ln \left (\cosh \left (x \right )-1\right )\)	\(17\)
default	\(\frac {2}{\cosh \left (x \right )-1}-\ln \left (\cosh \left (x \right )-1\right )\)	\(17\)
risch	\(x +\frac {4 \,{\mathrm e}^{x}}{\left ({\mathrm e}^{x}-1\right )^{2}}-2 \ln \left ({\mathrm e}^{x}-1\right )\)	\(20\)

[In]

int(sinh(x)^3/(1-cosh(x))^3,x,method=_RETURNVERBOSE)

[Out]

2/(cosh(x)-1)-ln(cosh(x)-1)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (18) = 36\).

Time = 0.25 (sec) , antiderivative size = 90, normalized size of antiderivative = 4.50 \[ \int \frac {\sinh ^3(x)}{(1-\cosh (x))^3} \, dx=\frac {x \cosh \left (x\right )^{2} + x \sinh \left (x\right )^{2} - 2 \, {\left (x - 2\right )} \cosh \left (x\right ) - 2 \, {\left (\cosh \left (x\right )^{2} + 2 \, {\left (\cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 2 \, {\left (x \cosh \left (x\right ) - x + 2\right )} \sinh \left (x\right ) + x}{\cosh \left (x\right )^{2} + 2 \, {\left (\cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) + 1} \]

[In]

integrate(sinh(x)^3/(1-cosh(x))^3,x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (14) = 28\).

Time = 0.26 (sec) , antiderivative size = 126, normalized size of antiderivative = 6.30 \[ \int \frac {\sinh ^3(x)}{(1-\cosh (x))^3} \, dx=- \frac {2 \log {\left (\cosh {\left (x \right )} - 1 \right )} \cosh ^{2}{\left (x \right )}}{2 \cosh ^{2}{\left (x \right )} - 4 \cosh {\left (x \right )} + 2} + \frac {4 \log {\left (\cosh {\left (x \right )} - 1 \right )} \cosh {\left (x \right )}}{2 \cosh ^{2}{\left (x \right )} - 4 \cosh {\left (x \right )} + 2} - \frac {2 \log {\left (\cosh {\left (x \right )} - 1 \right )}}{2 \cosh ^{2}{\left (x \right )} - 4 \cosh {\left (x \right )} + 2} + \frac {\sinh ^{2}{\left (x \right )}}{2 \cosh ^{2}{\left (x \right )} - 4 \cosh {\left (x \right )} + 2} + \frac {2 \cosh {\left (x \right )}}{2 \cosh ^{2}{\left (x \right )} - 4 \cosh {\left (x \right )} + 2} - \frac {2}{2 \cosh ^{2}{\left (x \right )} - 4 \cosh {\left (x \right )} + 2} \]

[In]

integrate(sinh(x)**3/(1-cosh(x))**3,x)

[Out]

-2*log(cosh(x) - 1)*cosh(x)**2/(2*cosh(x)**2 - 4*cosh(x) + 2) + 4*log(cosh(x) - 1)*cosh(x)/(2*cosh(x)**2 - 4*c
osh(x) + 2) - 2*log(cosh(x) - 1)/(2*cosh(x)**2 - 4*cosh(x) + 2) + sinh(x)**2/(2*cosh(x)**2 - 4*cosh(x) + 2) +
2*cosh(x)/(2*cosh(x)**2 - 4*cosh(x) + 2) - 2/(2*cosh(x)**2 - 4*cosh(x) + 2)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.75 \[ \int \frac {\sinh ^3(x)}{(1-\cosh (x))^3} \, dx=-x - \frac {4 \, e^{\left (-x\right )}}{2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} - 1} - 2 \, \log \left (e^{\left (-x\right )} - 1\right ) \]

[In]

integrate(sinh(x)^3/(1-cosh(x))^3,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\sinh ^3(x)}{(1-\cosh (x))^3} \, dx=x + \frac {4 \, e^{x}}{{\left (e^{x} - 1\right )}^{2}} - 2 \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]

[In]

integrate(sinh(x)^3/(1-cosh(x))^3,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.67 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {\sinh ^3(x)}{(1-\cosh (x))^3} \, dx=\frac {2}{\mathrm {cosh}\left (x\right )-1}-\ln \left (\mathrm {cosh}\left (x\right )-1\right ) \]

[In]

int(-sinh(x)^3/(cosh(x) - 1)^3,x)

[Out]

2/(cosh(x) - 1) - log(cosh(x) - 1)