3.20 \(\int \frac {1}{(a \cosh (x))^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2636, 2640, 2639} \[ \frac {2 \sinh (x)}{a \sqrt {a \cosh (x)}}+\frac {2 i E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \cosh (x)}}{a^2 \sqrt {\cosh (x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 34, normalized size = 0.74 \[ \frac {2 \cosh (x) \left (\sinh (x)+i \sqrt {\cosh (x)} E\left (\left .\frac {i x}{2}\right |2\right )\right )}{(a \cosh (x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \cosh \relax (x)}}{a^{2} \cosh \relax (x)^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(a*cosh(x))/(a^2*cosh(x)^2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \cosh \relax (x)\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*cosh(x))^(-3/2), x)

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maple [B]  time = 0.37, size = 159, normalized size = 3.46 \[ -\frac {\sqrt {2 a \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+a \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \left (\sqrt {2}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )-1}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \EllipticF \left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )-2 \sqrt {2}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )-1}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \EllipticE \left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )-4 \left (\sinh ^{2}\left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )\right )}{a \sqrt {a \left (2 \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \sinh \left (\frac {x}{2}\right ) \sqrt {a \left (2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )-1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a*(2*a*sinh(1/2*x)^4+a*sinh(1/2*x)^2)^(1/2)*(2^(1/2)*(-2*sinh(1/2*x)^2-1)^(1/2)*(-sinh(1/2*x)^2)^(1/2)*Elli
pticF(2^(1/2)*cosh(1/2*x),1/2*2^(1/2))-2*2^(1/2)*(-2*sinh(1/2*x)^2-1)^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticE(2
^(1/2)*cosh(1/2*x),1/2*2^(1/2))-4*sinh(1/2*x)^2*cosh(1/2*x))/(a*(2*sinh(1/2*x)^4+sinh(1/2*x)^2))^(1/2)/sinh(1/
2*x)/(a*(2*cosh(1/2*x)^2-1))^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \cosh \relax (x)\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*cosh(x))^(-3/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (a\,\mathrm {cosh}\relax (x)\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(a*cosh(x))^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \cosh {\relax (x )}\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*cosh(x))**(-3/2), x)

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