\(\int \sqrt {a \text {csch}^4(x)} \, dx\) [45]

Optimal result

Integrand size = 10, antiderivative size = 16 \[ \int \sqrt {a \text {csch}^4(x)} \, dx=-\cosh (x) \sqrt {a \text {csch}^4(x)} \sinh (x) \]

[Out]

-cosh(x)*sinh(x)*(a*csch(x)^4)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4208, 3852, 8} \[ \int \sqrt {a \text {csch}^4(x)} \, dx=\sinh (x) (-\cosh (x)) \sqrt {a \text {csch}^4(x)} \]

[In]

Int[Sqrt[a*Csch[x]^4],x]

[Out]

-(Cosh[x]*Sqrt[a*Csch[x]^4]*Sinh[x])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 4208

Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[b^IntPart[p]*((b*(c*Sec[e + f*x])^n)^
FracPart[p]/(c*Sec[e + f*x])^(n*FracPart[p])), Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p},
x] &&  !IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {a \text {csch}^4(x)} \sinh ^2(x)\right ) \int \text {csch}^2(x) \, dx \\ & = -\left (\left (i \sqrt {a \text {csch}^4(x)} \sinh ^2(x)\right ) \text {Subst}(\int 1 \, dx,x,-i \coth (x))\right ) \\ & = -\cosh (x) \sqrt {a \text {csch}^4(x)} \sinh (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \sqrt {a \text {csch}^4(x)} \, dx=-\cosh (x) \sqrt {a \text {csch}^4(x)} \sinh (x) \]

[In]

Integrate[Sqrt[a*Csch[x]^4],x]

[Out]

-(Cosh[x]*Sqrt[a*Csch[x]^4]*Sinh[x])

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.81

method	result	size
risch	\(-2 \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}\, {\mathrm e}^{-2 x} \left ({\mathrm e}^{2 x}-1\right )\)	\(29\)

[In]

int((a*csch(x)^4)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2*(a*exp(4*x)/(exp(2*x)-1)^4)^(1/2)*exp(-2*x)*(exp(2*x)-1)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (14) = 28\).

Time = 0.26 (sec) , antiderivative size = 81, normalized size of antiderivative = 5.06 \[ \int \sqrt {a \text {csch}^4(x)} \, dx=-\frac {2 \, \sqrt {\frac {a}{e^{\left (8 \, x\right )} - 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} - 4 \, e^{\left (2 \, x\right )} + 1}} {\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (2 \, x\right )}}{2 \, \cosh \left (x\right ) e^{\left (2 \, x\right )} \sinh \left (x\right ) + e^{\left (2 \, x\right )} \sinh \left (x\right )^{2} + {\left (\cosh \left (x\right )^{2} - 1\right )} e^{\left (2 \, x\right )}} \]

[In]

integrate((a*csch(x)^4)^(1/2),x, algorithm="fricas")

[Out]

-2*sqrt(a/(e^(8*x) - 4*e^(6*x) + 6*e^(4*x) - 4*e^(2*x) + 1))*(e^(4*x) - 2*e^(2*x) + 1)*e^(2*x)/(2*cosh(x)*e^(2
*x)*sinh(x) + e^(2*x)*sinh(x)^2 + (cosh(x)^2 - 1)*e^(2*x))

Sympy [F]

\[ \int \sqrt {a \text {csch}^4(x)} \, dx=\int \sqrt {a \operatorname {csch}^{4}{\left (x \right )}}\, dx \]

[In]

integrate((a*csch(x)**4)**(1/2),x)

[Out]

Integral(sqrt(a*csch(x)**4), x)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int \sqrt {a \text {csch}^4(x)} \, dx=\frac {2 \, \sqrt {a}}{e^{\left (-2 \, x\right )} - 1} \]

[In]

integrate((a*csch(x)^4)^(1/2),x, algorithm="maxima")

[Out]

2*sqrt(a)/(e^(-2*x) - 1)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int \sqrt {a \text {csch}^4(x)} \, dx=-\frac {2 \, \sqrt {a}}{e^{\left (2 \, x\right )} - 1} \]

[In]

integrate((a*csch(x)^4)^(1/2),x, algorithm="giac")

[Out]

-2*sqrt(a)/(e^(2*x) - 1)

Mupad [B] (verification not implemented)

Time = 2.14 (sec) , antiderivative size = 71, normalized size of antiderivative = 4.44 \[ \int \sqrt {a \text {csch}^4(x)} \, dx=-\frac {\sqrt {a}\,\sqrt {\frac {1}{{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (3\,{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^{6\,x}+\frac {{\mathrm {e}}^{8\,x}}{2}+\frac {1}{2}\right )}{\left ({\mathrm {e}}^{2\,x}-1\right )\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )} \]

[In]

int((a/sinh(x)^4)^(1/2),x)

[Out]

-(a^(1/2)*(1/(exp(-x)/2 - exp(x)/2)^4)^(1/2)*(3*exp(4*x) - 2*exp(2*x) - 2*exp(6*x) + exp(8*x)/2 + 1/2))/((exp(
2*x) - 1)*(exp(2*x) - 2*exp(4*x) + exp(6*x)))