\(\int \frac {1}{(-\text {csch}^2(x))^{3/2}} \, dx\) [26]

Optimal result

Integrand size = 10, antiderivative size = 33 \[ \int \frac {1}{\left (-\text {csch}^2(x)\right )^{3/2}} \, dx=\frac {\coth (x)}{3 \left (-\text {csch}^2(x)\right )^{3/2}}+\frac {2 \coth (x)}{3 \sqrt {-\text {csch}^2(x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 33, 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, 198, 197} \[ \int \frac {1}{\left (-\text {csch}^2(x)\right )^{3/2}} \, dx=\frac {2 \coth (x)}{3 \sqrt {-\text {csch}^2(x)}}+\frac {\coth (x)}{3 \left (-\text {csch}^2(x)\right )^{3/2}} \]

[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 197

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 198

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 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 \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} \text {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*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (-\text {csch}^2(x)\right )^{3/2}} \, dx=\frac {9 \coth (x)-\cosh (3 x) \text {csch}(x)}{12 \sqrt {-\text {csch}^2(x)}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(117\) vs. \(2(25)=50\).

Time = 0.15 (sec) , antiderivative size = 118, normalized size of antiderivative = 3.58

method	result	size
risch	\(-\frac {{\mathrm e}^{4 x}}{24 \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}+\frac {3 \,{\mathrm e}^{2 x}}{8 \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 x}-1\right )}+\frac {3}{8 \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}-\frac {{\mathrm e}^{-2 x}}{24 \left ({\mathrm e}^{2 x}-1\right ) \sqrt {-\frac {{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}\)	\(118\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\left (-\text {csch}^2(x)\right )^{3/2}} \, dx=\frac {1}{24} \, {\left (i \, e^{\left (6 \, x\right )} - 9 i \, e^{\left (4 \, x\right )} - 9 i \, e^{\left (2 \, x\right )} + i\right )} e^{\left (-3 \, x\right )} \]

[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)

Sympy [F]

\[ \int \frac {1}{\left (-\text {csch}^2(x)\right )^{3/2}} \, dx=\int \frac {1}{\left (- \operatorname {csch}^{2}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(1/(-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 = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\left (-\text {csch}^2(x)\right )^{3/2}} \, dx=-\frac {1}{24} i \, e^{\left (3 \, x\right )} + \frac {3}{8} i \, e^{\left (-x\right )} - \frac {1}{24} i \, e^{\left (-3 \, x\right )} + \frac {3}{8} i \, e^{x} \]

[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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.15 \[ \int \frac {1}{\left (-\text {csch}^2(x)\right )^{3/2}} \, dx=\frac {{\left (9 i \, e^{\left (2 \, x\right )} - i\right )} e^{\left (-3 \, x\right )} - i \, e^{\left (3 \, x\right )} + 9 i \, e^{x}}{24 \, \mathrm {sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right )} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

int(1/(-1/sinh(x)^2)^(3/2), x)