3.2.31 \(\int \frac {1}{\sqrt {a \cosh ^3(x)}} \, dx\) [131]

Optimal. Leaf size=46 \[ \frac {2 i \cosh ^{\frac {3}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right )}{\sqrt {a \cosh ^3(x)}}+\frac {2 \cosh (x) \sinh (x)}{\sqrt {a \cosh ^3(x)}} \]

[Out]

2*I*cosh(x)^(3/2)*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticE(I*sinh(1/2*x),2^(1/2))/(a*cosh(x)^3)^(1/2)+2*cos
h(x)*sinh(x)/(a*cosh(x)^3)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3286, 2716, 2719} \begin {gather*} \frac {2 \sinh (x) \cosh (x)}{\sqrt {a \cosh ^3(x)}}+\frac {2 i \cosh ^{\frac {3}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right )}{\sqrt {a \cosh ^3(x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2716

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 2719

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

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
]])

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 36, normalized size = 0.78 \begin {gather*} \frac {2 \cosh (x) \left (i \sqrt {\cosh (x)} E\left (\left .\frac {i x}{2}\right |2\right )+\sinh (x)\right )}{\sqrt {a \cosh ^3(x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 1.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {a \left (\cosh ^{3}\left (x \right )\right )}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(a*cosh(x)^3), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.11, size = 93, normalized size = 2.02 \begin {gather*} \frac {2 \, {\left ({\left (\sqrt {2} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} \sinh \left (x\right )^{2} + \sqrt {2}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) + 2 \, \sqrt {a \cosh \left (x\right )} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}\right )}}{a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2} + a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*((sqrt(2)*cosh(x)^2 + 2*sqrt(2)*cosh(x)*sinh(x) + sqrt(2)*sinh(x)^2 + sqrt(2))*sqrt(a)*weierstrassZeta(-4, 0
, weierstrassPInverse(-4, 0, cosh(x) + sinh(x))) + 2*sqrt(a*cosh(x))*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^
2))/(a*cosh(x)^2 + 2*a*cosh(x)*sinh(x) + a*sinh(x)^2 + a)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a \cosh ^{3}{\left (x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a*cosh(x)**3), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(a*cosh(x)^3), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {a\,{\mathrm {cosh}\left (x\right )}^3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(a*cosh(x)^3)^(1/2), x)

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