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

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Rubi [A]  time = 0.0253687, antiderivative size = 46, normalized size of antiderivative = 1., 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 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 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]

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*}

Mathematica [A]  time = 0.0216983, 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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Maple [B]  time = 0.052, size = 159, normalized size = 3.5 \begin{align*} -{\frac{1}{a}\sqrt{2\, \left ( \sinh \left ( x/2 \right ) \right ) ^{4}a+ \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2}a} \left ( \sqrt{2}\sqrt{-2\, \left ( \sinh \left ( x/2 \right ) \right ) ^{2}-1}\sqrt{- \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2}}{\it EllipticF} \left ( \cosh \left ({\frac{x}{2}} \right ) \sqrt{2},{\frac{\sqrt{2}}{2}} \right ) -2\,\sqrt{2}\sqrt{-2\, \left ( \sinh \left ( x/2 \right ) \right ) ^{2}-1}\sqrt{- \left ( \sinh \left ( x/2 \right ) \right ) ^{2}}{\it EllipticE} \left ( \cosh \left ( x/2 \right ) \sqrt{2},1/2\,\sqrt{2} \right ) -4\, \left ( \sinh \left ( x/2 \right ) \right ) ^{2}\cosh \left ( x/2 \right ) \right ){\frac{1}{\sqrt{a \left ( 2\, \left ( \sinh \left ( x/2 \right ) \right ) ^{4}+ \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2} \right ) }}} \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{a \left ( 2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}-1 \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a*(2*sinh(1/2*x)^4*a+sinh(1/2*x)^2*a)^(1/2)*(2^(1/2)*(-2*sinh(1/2*x)^2-1)^(1/2)*(-sinh(1/2*x)^2)^(1/2)*Elli
pticF(cosh(1/2*x)*2^(1/2),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(c
osh(1/2*x)*2^(1/2),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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \cosh \left (x\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}

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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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a \cosh \left (x\right )}}{a^{2} \cosh \left (x\right )^{2}}, x\right ) \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \cosh \left (x\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}

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)