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

Optimal. Leaf size=17 \[ -\frac {\sinh (x) \tanh ^{-1}(\cosh (x))}{\sqrt {a \sinh ^2(x)}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3207, 3770} \[ -\frac {\sinh (x) \tanh ^{-1}(\cosh (x))}{\sqrt {a \sinh ^2(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3207

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 3770

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {a \sinh ^2(x)}} \, dx &=\frac {\sinh (x) \int \text {csch}(x) \, dx}{\sqrt {a \sinh ^2(x)}}\\ &=-\frac {\tanh ^{-1}(\cosh (x)) \sinh (x)}{\sqrt {a \sinh ^2(x)}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 1.18 \[ \frac {\sinh (x) \log \left (\tanh \left (\frac {x}{2}\right )\right )}{\sqrt {a \sinh ^2(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Log[Tanh[x/2]]*Sinh[x])/Sqrt[a*Sinh[x]^2]

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fricas [B]  time = 0.80, size = 110, normalized size = 6.47 \[ \left [\frac {\sqrt {a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a} \log \left (\frac {\cosh \relax (x) + \sinh \relax (x) - 1}{\cosh \relax (x) + \sinh \relax (x) + 1}\right )}{a e^{\left (2 \, x\right )} - a}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a} \sqrt {-a}}{a \cosh \relax (x) e^{\left (2 \, x\right )} - a \cosh \relax (x) + {\left (a e^{\left (2 \, x\right )} - a\right )} \sinh \relax (x)}\right )}{a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[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]

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giac [A]  time = 0.23, size = 1, normalized size = 0.06 \[ 0 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

0

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maple [B]  time = 0.08, size = 49, normalized size = 2.88 \[ -\frac {\sinh \relax (x ) \sqrt {a \left (\cosh ^{2}\relax (x )\right )}\, \ln \left (\frac {2 \sqrt {a}\, \sqrt {a \left (\cosh ^{2}\relax (x )\right )}+2 a}{\sinh \relax (x )}\right )}{\sqrt {a}\, \cosh \relax (x ) \sqrt {a \left (\sinh ^{2}\relax (x )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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maxima [A]  time = 0.42, size = 24, normalized size = 1.41 \[ \frac {\log \left (e^{\left (-x\right )} + 1\right )}{\sqrt {a}} - \frac {\log \left (e^{\left (-x\right )} - 1\right )}{\sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {1}{\sqrt {a\,{\mathrm {sinh}\relax (x)}^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(a*sinh(x)^2)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \sinh ^{2}{\relax (x )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a*sinh(x)**2), x)

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