3.27 \(\int \frac {1}{(-\text {csch}^2(x))^{5/2}} \, dx\)

Optimal. Leaf size=49 \[ \frac {8 \coth (x)}{15 \sqrt {-\text {csch}^2(x)}}+\frac {4 \coth (x)}{15 \left (-\text {csch}^2(x)\right )^{3/2}}+\frac {\coth (x)}{5 \left (-\text {csch}^2(x)\right )^{5/2}} \]

[Out]

1/5*coth(x)/(-csch(x)^2)^(5/2)+4/15*coth(x)/(-csch(x)^2)^(3/2)+8/15*coth(x)/(-csch(x)^2)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4122, 192, 191} \[ \frac {8 \coth (x)}{15 \sqrt {-\text {csch}^2(x)}}+\frac {4 \coth (x)}{15 \left (-\text {csch}^2(x)\right )^{3/2}}+\frac {\coth (x)}{5 \left (-\text {csch}^2(x)\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(-Csch[x]^2)^(-5/2),x]

[Out]

Coth[x]/(5*(-Csch[x]^2)^(5/2)) + (4*Coth[x])/(15*(-Csch[x]^2)^(3/2)) + (8*Coth[x])/(15*Sqrt[-Csch[x]^2])

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 0] && NeQ[p, -1]

Rule 4122

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*} \int \frac {1}{\left (-\text {csch}^2(x)\right )^{5/2}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right )^{7/2}} \, dx,x,\coth (x)\right )\\ &=\frac {\coth (x)}{5 \left (-\text {csch}^2(x)\right )^{5/2}}+\frac {4}{5} \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right )^{5/2}} \, dx,x,\coth (x)\right )\\ &=\frac {\coth (x)}{5 \left (-\text {csch}^2(x)\right )^{5/2}}+\frac {4 \coth (x)}{15 \left (-\text {csch}^2(x)\right )^{3/2}}+\frac {8}{15} \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right )^{3/2}} \, dx,x,\coth (x)\right )\\ &=\frac {\coth (x)}{5 \left (-\text {csch}^2(x)\right )^{5/2}}+\frac {4 \coth (x)}{15 \left (-\text {csch}^2(x)\right )^{3/2}}+\frac {8 \coth (x)}{15 \sqrt {-\text {csch}^2(x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 33, normalized size = 0.67 \[ \frac {(150 \cosh (x)-25 \cosh (3 x)+3 \cosh (5 x)) \text {csch}(x)}{240 \sqrt {-\text {csch}^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-Csch[x]^2)^(-5/2),x]

[Out]

((150*Cosh[x] - 25*Cosh[3*x] + 3*Cosh[5*x])*Csch[x])/(240*Sqrt[-Csch[x]^2])

________________________________________________________________________________________

fricas [C]  time = 1.45, size = 38, normalized size = 0.78 \[ \frac {1}{480} \, {\left (-3 i \, e^{\left (10 \, x\right )} + 25 i \, e^{\left (8 \, x\right )} - 150 i \, e^{\left (6 \, x\right )} - 150 i \, e^{\left (4 \, x\right )} + 25 i \, e^{\left (2 \, x\right )} - 3 i\right )} e^{\left (-5 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-csch(x)^2)^(5/2),x, algorithm="fricas")

[Out]

1/480*(-3*I*e^(10*x) + 25*I*e^(8*x) - 150*I*e^(6*x) - 150*I*e^(4*x) + 25*I*e^(2*x) - 3*I)*e^(-5*x)

________________________________________________________________________________________

giac [C]  time = 0.17, size = 64, normalized size = 1.31 \[ \frac {i \, {\left (150 \, e^{\left (4 \, x\right )} - 25 \, e^{\left (2 \, x\right )} + 3\right )} e^{\left (-5 \, x\right )}}{480 \, \mathrm {sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right )} + \frac {i \, {\left (3 \, e^{\left (5 \, x\right )} - 25 \, e^{\left (3 \, x\right )} + 150 \, e^{x}\right )}}{480 \, \mathrm {sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-csch(x)^2)^(5/2),x, algorithm="giac")

[Out]

1/480*I*(150*e^(4*x) - 25*e^(2*x) + 3)*e^(-5*x)/sgn(-e^(3*x) + e^x) + 1/480*I*(3*e^(5*x) - 25*e^(3*x) + 150*e^
x)/sgn(-e^(3*x) + e^x)

________________________________________________________________________________________

maple [B]  time = 0.18, size = 178, normalized size = 3.63 \[ \frac {{\mathrm e}^{6 x}}{160 \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}-\frac {5 \,{\mathrm e}^{4 x}}{96 \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}+\frac {5 \,{\mathrm e}^{2 x}}{16 \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 x}-1\right )}+\frac {5}{16 \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}-\frac {5 \,{\mathrm e}^{-2 x}}{96 \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}+\frac {{\mathrm e}^{-4 x}}{160 \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-csch(x)^2)^(5/2),x)

[Out]

1/160*exp(6*x)/(exp(2*x)-1)/(-exp(2*x)/(exp(2*x)-1)^2)^(1/2)-5/96*exp(4*x)/(exp(2*x)-1)/(-exp(2*x)/(exp(2*x)-1
)^2)^(1/2)+5/16/(-exp(2*x)/(exp(2*x)-1)^2)^(1/2)/(exp(2*x)-1)*exp(2*x)+5/16/(exp(2*x)-1)/(-exp(2*x)/(exp(2*x)-
1)^2)^(1/2)-5/96*exp(-2*x)/(exp(2*x)-1)/(-exp(2*x)/(exp(2*x)-1)^2)^(1/2)+1/160*exp(-4*x)/(exp(2*x)-1)/(-exp(2*
x)/(exp(2*x)-1)^2)^(1/2)

________________________________________________________________________________________

maxima [C]  time = 0.46, size = 35, normalized size = 0.71 \[ \frac {1}{160} i \, e^{\left (5 \, x\right )} - \frac {5}{96} i \, e^{\left (3 \, x\right )} + \frac {5}{16} i \, e^{\left (-x\right )} - \frac {5}{96} i \, e^{\left (-3 \, x\right )} + \frac {1}{160} i \, e^{\left (-5 \, x\right )} + \frac {5}{16} i \, e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-csch(x)^2)^(5/2),x, algorithm="maxima")

[Out]

1/160*I*e^(5*x) - 5/96*I*e^(3*x) + 5/16*I*e^(-x) - 5/96*I*e^(-3*x) + 1/160*I*e^(-5*x) + 5/16*I*e^x

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (-\frac {1}{{\mathrm {sinh}\relax (x)}^2}\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-1/sinh(x)^2)^(5/2),x)

[Out]

int(1/(-1/sinh(x)^2)^(5/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (- \operatorname {csch}^{2}{\relax (x )}\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-csch(x)**2)**(5/2),x)

[Out]

Integral((-csch(x)**2)**(-5/2), x)

________________________________________________________________________________________