3.149 \(\int \frac {1}{\sqrt {a \sinh ^3(x)}} \, dx\)

Optimal. Leaf size=60 \[ -\frac {2 \sinh (x) \cosh (x)}{\sqrt {a \sinh ^3(x)}}+\frac {2 i \sinh ^2(x) E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{\sqrt {i \sinh (x)} \sqrt {a \sinh ^3(x)}} \]

[Out]

-2*cosh(x)*sinh(x)/(a*sinh(x)^3)^(1/2)+2*I*(sin(1/4*Pi+1/2*I*x)^2)^(1/2)/sin(1/4*Pi+1/2*I*x)*EllipticE(cos(1/4
*Pi+1/2*I*x),2^(1/2))*sinh(x)^2/(I*sinh(x))^(1/2)/(a*sinh(x)^3)^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3207, 2636, 2640, 2639} \[ -\frac {2 \sinh (x) \cosh (x)}{\sqrt {a \sinh ^3(x)}}+\frac {2 i \sinh ^2(x) E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{\sqrt {i \sinh (x)} \sqrt {a \sinh ^3(x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a*Sinh[x]^3],x]

[Out]

(-2*Cosh[x]*Sinh[x])/Sqrt[a*Sinh[x]^3] + ((2*I)*EllipticE[Pi/4 - (I/2)*x, 2]*Sinh[x]^2)/(Sqrt[I*Sinh[x]]*Sqrt[
a*Sinh[x]^3])

Rule 2636

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1))/(b*d*(n +
1)), x] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[2*n]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rule 2640

Int[Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[b*Sin[c + d*x]]/Sqrt[Sin[c + d*x]], Int[Sqrt[Si
n[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

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
]])

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {a \sinh ^3(x)}} \, dx &=\frac {\sinh ^{\frac {3}{2}}(x) \int \frac {1}{\sinh ^{\frac {3}{2}}(x)} \, dx}{\sqrt {a \sinh ^3(x)}}\\ &=-\frac {2 \cosh (x) \sinh (x)}{\sqrt {a \sinh ^3(x)}}+\frac {\sinh ^{\frac {3}{2}}(x) \int \sqrt {\sinh (x)} \, dx}{\sqrt {a \sinh ^3(x)}}\\ &=-\frac {2 \cosh (x) \sinh (x)}{\sqrt {a \sinh ^3(x)}}+\frac {\sinh ^2(x) \int \sqrt {i \sinh (x)} \, dx}{\sqrt {i \sinh (x)} \sqrt {a \sinh ^3(x)}}\\ &=-\frac {2 \cosh (x) \sinh (x)}{\sqrt {a \sinh ^3(x)}}+\frac {2 i E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sinh ^2(x)}{\sqrt {i \sinh (x)} \sqrt {a \sinh ^3(x)}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 42, normalized size = 0.70 \[ -\frac {2 \sinh (x) \left (\cosh (x)-\sqrt {i \sinh (x)} E\left (\left .\frac {1}{4} (\pi -2 i x)\right |2\right )\right )}{\sqrt {a \sinh ^3(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a*Sinh[x]^3],x]

[Out]

(-2*(Cosh[x] - EllipticE[(Pi - (2*I)*x)/4, 2]*Sqrt[I*Sinh[x]])*Sinh[x])/Sqrt[a*Sinh[x]^3]

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fricas [F]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \sinh \relax (x)^{3}}}{a \sinh \relax (x)^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(a*sinh(x)^3)/(a*sinh(x)^3), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \sinh \relax (x)^{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(a*sinh(x)^3), x)

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \left (\sinh ^{3}\relax (x )\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \sinh \relax (x)^{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(a*sinh(x)^3), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {a\,{\mathrm {sinh}\relax (x)}^3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(a*sinh(x)^3)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \sinh ^{3}{\relax (x )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a*sinh(x)**3), x)

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