\(\int (-\text {csch}^2(x))^{3/2} \, dx\) [23]

Optimal result

Integrand size = 10, antiderivative size = 24 \[ \int \left (-\text {csch}^2(x)\right )^{3/2} \, dx=\frac {1}{2} \arcsin (\coth (x))+\frac {1}{2} \coth (x) \sqrt {-\text {csch}^2(x)} \]

[Out]

1/2*arcsin(coth(x))+1/2*coth(x)*(-csch(x)^2)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 24, 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 = {4207, 201, 222} \[ \int \left (-\text {csch}^2(x)\right )^{3/2} \, dx=\frac {1}{2} \arcsin (\coth (x))+\frac {1}{2} \coth (x) \sqrt {-\text {csch}^2(x)} \]

[In]

Int[(-Csch[x]^2)^(3/2),x]

[Out]

ArcSin[Coth[x]]/2 + (Coth[x]*Sqrt[-Csch[x]^2])/2

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 4207

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[b*(ff/
f), Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(51\) vs. \(2(24)=48\).

Time = 0.04 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.12 \[ \int \left (-\text {csch}^2(x)\right )^{3/2} \, dx=\frac {1}{8} \sqrt {-\text {csch}^2(x)} \left (\text {csch}^2\left (\frac {x}{2}\right )-4 \log \left (\cosh \left (\frac {x}{2}\right )\right )+4 \log \left (\sinh \left (\frac {x}{2}\right )\right )+\text {sech}^2\left (\frac {x}{2}\right )\right ) \sinh (x) \]

[In]

Integrate[(-Csch[x]^2)^(3/2),x]

[Out]

(Sqrt[-Csch[x]^2]*(Csch[x/2]^2 - 4*Log[Cosh[x/2]] + 4*Log[Sinh[x/2]] + Sech[x/2]^2)*Sinh[x])/8

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(98\) vs. \(2(18)=36\).

Time = 0.16 (sec) , antiderivative size = 99, normalized size of antiderivative = 4.12

method	result	size
risch	\(\frac {\sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}{{\mathrm e}^{2 x}-1}-\frac {\sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, {\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right ) \ln \left ({\mathrm e}^{x}+1\right )}{2}+\frac {\sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, {\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right ) \ln \left ({\mathrm e}^{x}-1\right )}{2}\)	\(99\)

[In]

int((-csch(x)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/(exp(2*x)-1)*(-1/(exp(2*x)-1)^2*exp(2*x))^(1/2)*(1+exp(2*x))-1/2*(-1/(exp(2*x)-1)^2*exp(2*x))^(1/2)*exp(-x)*
(exp(2*x)-1)*ln(exp(x)+1)+1/2*(-1/(exp(2*x)-1)^2*exp(2*x))^(1/2)*exp(-x)*(exp(2*x)-1)*ln(exp(x)-1)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.79 \[ \int \left (-\text {csch}^2(x)\right )^{3/2} \, dx=\frac {{\left (-i \, e^{\left (4 \, x\right )} + 2 i \, e^{\left (2 \, x\right )} - i\right )} \log \left (e^{x} + 1\right ) + {\left (i \, e^{\left (4 \, x\right )} - 2 i \, e^{\left (2 \, x\right )} + i\right )} \log \left (e^{x} - 1\right ) + 2 i \, e^{\left (3 \, x\right )} + 2 i \, e^{x}}{2 \, {\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right )}} \]

[In]

integrate((-csch(x)^2)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \left (-\text {csch}^2(x)\right )^{3/2} \, dx=\int \left (- \operatorname {csch}^{2}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \]

[In]

integrate((-csch(x)**2)**(3/2),x)

[Out]

Integral((-csch(x)**2)**(3/2), x)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.04 \[ \int \left (-\text {csch}^2(x)\right )^{3/2} \, dx=\frac {i \, e^{\left (-x\right )} + i \, e^{\left (-3 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} + \frac {1}{2} i \, \log \left (e^{\left (-x\right )} + 1\right ) - \frac {1}{2} i \, \log \left (e^{\left (-x\right )} - 1\right ) \]

[In]

integrate((-csch(x)^2)^(3/2),x, algorithm="maxima")

[Out]

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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.38 \[ \int \left (-\text {csch}^2(x)\right )^{3/2} \, dx=-\frac {1}{4} \, {\left (\frac {4 i \, {\left (e^{\left (-x\right )} + e^{x}\right )}}{{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4} - i \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) + i \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right )\right )} \mathrm {sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right ) \]

[In]

integrate((-csch(x)^2)^(3/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \left (-\text {csch}^2(x)\right )^{3/2} \, dx=\int {\left (-\frac {1}{{\mathrm {sinh}\left (x\right )}^2}\right )}^{3/2} \,d x \]

[In]

int((-1/sinh(x)^2)^(3/2),x)

[Out]

int((-1/sinh(x)^2)^(3/2), x)