3.26 \(\int \frac {1}{(-\text {csch}^2(x))^{3/2}} \, dx\)

Optimal. Leaf size=33 \[ \frac {2 \coth (x)}{3 \sqrt {-\text {csch}^2(x)}}+\frac {\coth (x)}{3 \left (-\text {csch}^2(x)\right )^{3/2}} \]

[Out]

1/3*coth(x)/(-csch(x)^2)^(3/2)+2/3*coth(x)/(-csch(x)^2)^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4122, 192, 191} \[ \frac {2 \coth (x)}{3 \sqrt {-\text {csch}^2(x)}}+\frac {\coth (x)}{3 \left (-\text {csch}^2(x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Coth[x]/(3*(-Csch[x]^2)^(3/2)) + (2*Coth[x])/(3*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 )^{3/2}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right )^{5/2}} \, dx,x,\coth (x)\right )\\ &=\frac {\coth (x)}{3 \left (-\text {csch}^2(x)\right )^{3/2}}+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right )^{3/2}} \, dx,x,\coth (x)\right )\\ &=\frac {\coth (x)}{3 \left (-\text {csch}^2(x)\right )^{3/2}}+\frac {2 \coth (x)}{3 \sqrt {-\text {csch}^2(x)}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 0.82 \[ \frac {9 \coth (x)-\cosh (3 x) \text {csch}(x)}{12 \sqrt {-\text {csch}^2(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9*Coth[x] - Cosh[3*x]*Csch[x])/(12*Sqrt[-Csch[x]^2])

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fricas [C]  time = 2.47, size = 26, normalized size = 0.79 \[ \frac {1}{24} \, {\left (i \, e^{\left (6 \, x\right )} - 9 i \, e^{\left (4 \, x\right )} - 9 i \, e^{\left (2 \, x\right )} + i\right )} e^{\left (-3 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-csch(x)^2)^(3/2),x, algorithm="fricas")

[Out]

1/24*(I*e^(6*x) - 9*I*e^(4*x) - 9*I*e^(2*x) + I)*e^(-3*x)

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giac [C]  time = 0.16, size = 50, normalized size = 1.52 \[ \frac {i \, {\left (9 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-3 \, x\right )}}{24 \, \mathrm {sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right )} - \frac {i \, {\left (e^{\left (3 \, x\right )} - 9 \, e^{x}\right )}}{24 \, \mathrm {sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-csch(x)^2)^(3/2),x, algorithm="giac")

[Out]

1/24*I*(9*e^(2*x) - 1)*e^(-3*x)/sgn(-e^(3*x) + e^x) - 1/24*I*(e^(3*x) - 9*e^x)/sgn(-e^(3*x) + e^x)

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maple [B]  time = 0.18, size = 118, normalized size = 3.58 \[ -\frac {{\mathrm e}^{4 x}}{24 \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}+\frac {3 \,{\mathrm e}^{2 x}}{8 \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 x}-1\right )}+\frac {3}{8 \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}-\frac {{\mathrm e}^{-2 x}}{24 \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*exp(4*x)/(exp(2*x)-1)/(-exp(2*x)/(exp(2*x)-1)^2)^(1/2)+3/8/(-exp(2*x)/(exp(2*x)-1)^2)^(1/2)/(exp(2*x)-1)
*exp(2*x)+3/8/(exp(2*x)-1)/(-exp(2*x)/(exp(2*x)-1)^2)^(1/2)-1/24*exp(-2*x)/(exp(2*x)-1)/(-exp(2*x)/(exp(2*x)-1
)^2)^(1/2)

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maxima [C]  time = 0.44, size = 23, normalized size = 0.70 \[ -\frac {1}{24} i \, e^{\left (3 \, x\right )} + \frac {3}{8} i \, e^{\left (-x\right )} - \frac {1}{24} i \, e^{\left (-3 \, x\right )} + \frac {3}{8} i \, e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-csch(x)^2)^(3/2),x, algorithm="maxima")

[Out]

-1/24*I*e^(3*x) + 3/8*I*e^(-x) - 1/24*I*e^(-3*x) + 3/8*I*e^x

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{{\left (-\frac {1}{{\mathrm {sinh}\relax (x)}^2}\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(-1/sinh(x)^2)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (- \operatorname {csch}^{2}{\relax (x )}\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-csch(x)**2)**(3/2),x)

[Out]

Integral((-csch(x)**2)**(-3/2), x)

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