\(\int \frac {1}{(a \sinh ^2(x))^{3/2}} \, dx\) [144]

Optimal result

Integrand size = 10, antiderivative size = 42 \[ \int \frac {1}{\left (a \sinh ^2(x)\right )^{3/2}} \, dx=-\frac {\coth (x)}{2 a \sqrt {a \sinh ^2(x)}}+\frac {\text {arctanh}(\cosh (x)) \sinh (x)}{2 a \sqrt {a \sinh ^2(x)}} \]

[Out]

-1/2*coth(x)/a/(a*sinh(x)^2)^(1/2)+1/2*arctanh(cosh(x))*sinh(x)/a/(a*sinh(x)^2)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 42, 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 = {3283, 3286, 3855} \[ \int \frac {1}{\left (a \sinh ^2(x)\right )^{3/2}} \, dx=\frac {\sinh (x) \text {arctanh}(\cosh (x))}{2 a \sqrt {a \sinh ^2(x)}}-\frac {\coth (x)}{2 a \sqrt {a \sinh ^2(x)}} \]

[In]

Int[(a*Sinh[x]^2)^(-3/2),x]

[Out]

-1/2*Coth[x]/(a*Sqrt[a*Sinh[x]^2]) + (ArcTanh[Cosh[x]]*Sinh[x])/(2*a*Sqrt[a*Sinh[x]^2])

Rule 3283

Int[((b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[Cot[e + f*x]*((b*Sin[e + f*x]^2)^(p + 1)/(b*f*(2
*p + 1))), x] + Dist[2*((p + 1)/(b*(2*p + 1))), Int[(b*Sin[e + f*x]^2)^(p + 1), x], x] /; FreeQ[{b, e, f}, x]
&&  !IntegerQ[p] && LtQ[p, -1]

Rule 3286

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[(b*ff^n)^IntPart[p]*((b*Sin[e + f*x]^n)^FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p])), Int[ActivateTrig[u]
*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {\coth (x)}{2 a \sqrt {a \sinh ^2(x)}}-\frac {\int \frac {1}{\sqrt {a \sinh ^2(x)}} \, dx}{2 a} \\ & = -\frac {\coth (x)}{2 a \sqrt {a \sinh ^2(x)}}-\frac {\sinh (x) \int \text {csch}(x) \, dx}{2 a \sqrt {a \sinh ^2(x)}} \\ & = -\frac {\coth (x)}{2 a \sqrt {a \sinh ^2(x)}}+\frac {\text {arctanh}(\cosh (x)) \sinh (x)}{2 a \sqrt {a \sinh ^2(x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\left (a \sinh ^2(x)\right )^{3/2}} \, dx=-\frac {\left (\text {csch}^2\left (\frac {x}{2}\right )-4 \log \left (\cosh \left (\frac {x}{2}\right )\right )+4 \log \left (\sinh \left (\frac {x}{2}\right )\right )+\text {sech}^2\left (\frac {x}{2}\right )\right ) \sinh ^3(x)}{8 \left (a \sinh ^2(x)\right )^{3/2}} \]

[In]

Integrate[(a*Sinh[x]^2)^(-3/2),x]

[Out]

-1/8*((Csch[x/2]^2 - 4*Log[Cosh[x/2]] + 4*Log[Sinh[x/2]] + Sech[x/2]^2)*Sinh[x]^3)/(a*Sinh[x]^2)^(3/2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(70\) vs. \(2(34)=68\).

Time = 0.74 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.69

method	result	size
default	\(-\frac {\sqrt {a \cosh \left (x \right )^{2}}\, \left (-\ln \left (\frac {2 \sqrt {a}\, \sqrt {a \cosh \left (x \right )^{2}}+2 a}{\sinh \left (x \right )}\right ) a \sinh \left (x \right )^{2}+\sqrt {a}\, \sqrt {a \cosh \left (x \right )^{2}}\right )}{2 a^{\frac {5}{2}} \sinh \left (x \right ) \cosh \left (x \right ) \sqrt {a \sinh \left (x \right )^{2}}}\)	\(71\)
risch	\(-\frac {1+{\mathrm e}^{2 x}}{a \left ({\mathrm e}^{2 x}-1\right ) \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-2 x}}}+\frac {\left ({\mathrm e}^{2 x}-1\right ) {\mathrm e}^{-x} \ln \left ({\mathrm e}^{x}+1\right )}{2 a \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-2 x}}}-\frac {\left ({\mathrm e}^{2 x}-1\right ) {\mathrm e}^{-x} \ln \left ({\mathrm e}^{x}-1\right )}{2 a \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-2 x}}}\)	\(109\)

[In]

int(1/(a*sinh(x)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/2/a^(5/2)/sinh(x)*(a*cosh(x)^2)^(1/2)*(-ln(2*(a^(1/2)*(a*cosh(x)^2)^(1/2)+a)/sinh(x))*a*sinh(x)^2+a^(1/2)*(
a*cosh(x)^2)^(1/2))/cosh(x)/(a*sinh(x)^2)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 327 vs. \(2 (34) = 68\).

Time = 0.30 (sec) , antiderivative size = 327, normalized size of antiderivative = 7.79 \[ \int \frac {1}{\left (a \sinh ^2(x)\right )^{3/2}} \, dx=\frac {{\left (6 \, \cosh \left (x\right ) e^{x} \sinh \left (x\right )^{2} + 2 \, e^{x} \sinh \left (x\right )^{3} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} e^{x} \sinh \left (x\right ) + 2 \, {\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} e^{x} - {\left (4 \, \cosh \left (x\right ) e^{x} \sinh \left (x\right )^{3} + e^{x} \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} e^{x} \sinh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} e^{x} \sinh \left (x\right ) + {\left (\cosh \left (x\right )^{4} - 2 \, \cosh \left (x\right )^{2} + 1\right )} e^{x}\right )} \log \left (\frac {\cosh \left (x\right ) + \sinh \left (x\right ) + 1}{\cosh \left (x\right ) + \sinh \left (x\right ) - 1}\right )\right )} \sqrt {a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a} e^{\left (-x\right )}}{2 \, {\left (a^{2} \cosh \left (x\right )^{4} - {\left (a^{2} e^{\left (2 \, x\right )} - a^{2}\right )} \sinh \left (x\right )^{4} - 2 \, a^{2} \cosh \left (x\right )^{2} - 4 \, {\left (a^{2} \cosh \left (x\right ) e^{\left (2 \, x\right )} - a^{2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 2 \, {\left (3 \, a^{2} \cosh \left (x\right )^{2} - a^{2} - {\left (3 \, a^{2} \cosh \left (x\right )^{2} - a^{2}\right )} e^{\left (2 \, x\right )}\right )} \sinh \left (x\right )^{2} + a^{2} - {\left (a^{2} \cosh \left (x\right )^{4} - 2 \, a^{2} \cosh \left (x\right )^{2} + a^{2}\right )} e^{\left (2 \, x\right )} + 4 \, {\left (a^{2} \cosh \left (x\right )^{3} - a^{2} \cosh \left (x\right ) - {\left (a^{2} \cosh \left (x\right )^{3} - a^{2} \cosh \left (x\right )\right )} e^{\left (2 \, x\right )}\right )} \sinh \left (x\right )\right )}} \]

[In]

integrate(1/(a*sinh(x)^2)^(3/2),x, algorithm="fricas")

[Out]

1/2*(6*cosh(x)*e^x*sinh(x)^2 + 2*e^x*sinh(x)^3 + 2*(3*cosh(x)^2 + 1)*e^x*sinh(x) + 2*(cosh(x)^3 + cosh(x))*e^x
- (4*cosh(x)*e^x*sinh(x)^3 + e^x*sinh(x)^4 + 2*(3*cosh(x)^2 - 1)*e^x*sinh(x)^2 + 4*(cosh(x)^3 - cosh(x))*e^x*
sinh(x) + (cosh(x)^4 - 2*cosh(x)^2 + 1)*e^x)*log((cosh(x) + sinh(x) + 1)/(cosh(x) + sinh(x) - 1)))*sqrt(a*e^(4
*x) - 2*a*e^(2*x) + a)*e^(-x)/(a^2*cosh(x)^4 - (a^2*e^(2*x) - a^2)*sinh(x)^4 - 2*a^2*cosh(x)^2 - 4*(a^2*cosh(x
)*e^(2*x) - a^2*cosh(x))*sinh(x)^3 + 2*(3*a^2*cosh(x)^2 - a^2 - (3*a^2*cosh(x)^2 - a^2)*e^(2*x))*sinh(x)^2 + a
^2 - (a^2*cosh(x)^4 - 2*a^2*cosh(x)^2 + a^2)*e^(2*x) + 4*(a^2*cosh(x)^3 - a^2*cosh(x) - (a^2*cosh(x)^3 - a^2*c
osh(x))*e^(2*x))*sinh(x))

Sympy [F]

\[ \int \frac {1}{\left (a \sinh ^2(x)\right )^{3/2}} \, dx=\int \frac {1}{\left (a \sinh ^{2}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(1/(a*sinh(x)**2)**(3/2),x)

[Out]

Integral((a*sinh(x)**2)**(-3/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.48 \[ \int \frac {1}{\left (a \sinh ^2(x)\right )^{3/2}} \, dx=-\frac {e^{\left (-x\right )} + e^{\left (-3 \, x\right )}}{2 \, a^{\frac {3}{2}} e^{\left (-2 \, x\right )} - a^{\frac {3}{2}} e^{\left (-4 \, x\right )} - a^{\frac {3}{2}}} - \frac {\log \left (e^{\left (-x\right )} + 1\right )}{2 \, a^{\frac {3}{2}}} + \frac {\log \left (e^{\left (-x\right )} - 1\right )}{2 \, a^{\frac {3}{2}}} \]

[In]

integrate(1/(a*sinh(x)^2)^(3/2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a \sinh ^2(x)\right )^{3/2}} \, dx=-\frac {e^{\left (-x\right )} + e^{x}}{{\left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4\right )} a^{\frac {3}{2}} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )} \]

[In]

integrate(1/(a*sinh(x)^2)^(3/2),x, algorithm="giac")

[Out]

-(e^(-x) + e^x)/(((e^(-x) + e^x)^2 - 4)*a^(3/2)*sgn(e^(3*x) - e^x))

Mupad [F(-1)]

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

[In]

int(1/(a*sinh(x)^2)^(3/2),x)

[Out]

int(1/(a*sinh(x)^2)^(3/2), x)