\(\int \frac {1}{\sqrt {a \sinh ^2(x)}} \, dx\) [143]

Optimal result

Integrand size = 10, antiderivative size = 17 \[ \int \frac {1}{\sqrt {a \sinh ^2(x)}} \, dx=-\frac {\text {arctanh}(\cosh (x)) \sinh (x)}{\sqrt {a \sinh ^2(x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3286, 3855} \[ \int \frac {1}{\sqrt {a \sinh ^2(x)}} \, dx=-\frac {\sinh (x) \text {arctanh}(\cosh (x))}{\sqrt {a \sinh ^2(x)}} \]

[In]

Int[1/Sqrt[a*Sinh[x]^2],x]

[Out]

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

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 {\sinh (x) \int \text {csch}(x) \, dx}{\sqrt {a \sinh ^2(x)}} \\ & = -\frac {\text {arctanh}(\cosh (x)) \sinh (x)}{\sqrt {a \sinh ^2(x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.76 \[ \int \frac {1}{\sqrt {a \sinh ^2(x)}} \, dx=\frac {\left (-\log \left (\cosh \left (\frac {x}{2}\right )\right )+\log \left (\sinh \left (\frac {x}{2}\right )\right )\right ) \sinh (x)}{\sqrt {a \sinh ^2(x)}} \]

[In]

Integrate[1/Sqrt[a*Sinh[x]^2],x]

[Out]

((-Log[Cosh[x/2]] + Log[Sinh[x/2]])*Sinh[x])/Sqrt[a*Sinh[x]^2]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(48\) vs. \(2(15)=30\).

Time = 0.91 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.88

method	result	size
default	\(-\frac {\sinh \left (x \right ) \sqrt {a \cosh \left (x \right )^{2}}\, \ln \left (\frac {2 \sqrt {a}\, \sqrt {a \cosh \left (x \right )^{2}}+2 a}{\sinh \left (x \right )}\right )}{\sqrt {a}\, \cosh \left (x \right ) \sqrt {a \sinh \left (x \right )^{2}}}\)	\(49\)
risch	\(-\frac {{\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right ) \ln \left ({\mathrm e}^{x}+1\right )}{\sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-2 x}}}+\frac {{\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right ) \ln \left ({\mathrm e}^{x}-1\right )}{\sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-2 x}}}\)	\(67\)

[In]

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

[Out]

-sinh(x)*(a*cosh(x)^2)^(1/2)/a^(1/2)*ln(2*(a^(1/2)*(a*cosh(x)^2)^(1/2)+a)/sinh(x))/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. 63 vs. \(2 (15) = 30\).

Time = 0.32 (sec) , antiderivative size = 110, normalized size of antiderivative = 6.47 \[ \int \frac {1}{\sqrt {a \sinh ^2(x)}} \, dx=\left [\frac {\sqrt {a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a} \log \left (\frac {\cosh \left (x\right ) + \sinh \left (x\right ) - 1}{\cosh \left (x\right ) + \sinh \left (x\right ) + 1}\right )}{a e^{\left (2 \, x\right )} - a}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a} \sqrt {-a}}{a \cosh \left (x\right ) e^{\left (2 \, x\right )} - a \cosh \left (x\right ) + {\left (a e^{\left (2 \, x\right )} - a\right )} \sinh \left (x\right )}\right )}{a}\right ] \]

[In]

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

[Out]

[sqrt(a*e^(4*x) - 2*a*e^(2*x) + a)*log((cosh(x) + sinh(x) - 1)/(cosh(x) + sinh(x) + 1))/(a*e^(2*x) - a), 2*sqr
t(-a)*arctan(sqrt(a*e^(4*x) - 2*a*e^(2*x) + a)*sqrt(-a)/(a*cosh(x)*e^(2*x) - a*cosh(x) + (a*e^(2*x) - a)*sinh(
x)))/a]

Sympy [F]

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

[In]

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

[Out]

Integral(1/sqrt(a*sinh(x)**2), x)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.41 \[ \int \frac {1}{\sqrt {a \sinh ^2(x)}} \, dx=\frac {\log \left (e^{\left (-x\right )} + 1\right )}{\sqrt {a}} - \frac {\log \left (e^{\left (-x\right )} - 1\right )}{\sqrt {a}} \]

[In]

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

[Out]

log(e^(-x) + 1)/sqrt(a) - log(e^(-x) - 1)/sqrt(a)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.06 \[ \int \frac {1}{\sqrt {a \sinh ^2(x)}} \, dx=0 \]

[In]

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

[Out]

0

Mupad [F(-1)]

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

[In]

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

[Out]

int(1/(a*sinh(x)^2)^(1/2), x)