3.17 \(\int (a \cosh (x))^{3/2} \, dx\)

Optimal. Leaf size=48 \[ \frac{2}{3} a \sinh (x) \sqrt{a \cosh (x)}-\frac{2 i a^2 \sqrt{\cosh (x)} \text{EllipticF}\left (\frac{i x}{2},2\right )}{3 \sqrt{a \cosh (x)}} \]

[Out]

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

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Rubi [A]  time = 0.0243887, antiderivative size = 48, 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 = {2635, 2642, 2641} \[ \frac{2}{3} a \sinh (x) \sqrt{a \cosh (x)}-\frac{2 i a^2 \sqrt{\cosh (x)} F\left (\left .\frac{i x}{2}\right |2\right )}{3 \sqrt{a \cosh (x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2635

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

Rule 2642

Int[1/Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[c + d*x]]/Sqrt[b*Sin[c + d*x]], Int[1/Sqr
t[Sin[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

Rule 2641

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

Rubi steps

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

Mathematica [C]  time = 0.0553999, size = 57, normalized size = 1.19 \[ \frac{2}{3} (a \cosh (x))^{3/2} \left (\text{sech}^2(x) \sqrt{\sinh (2 x)+\cosh (2 x)+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\cosh (2 x)-\sinh (2 x)\right )+\tanh (x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a*Cosh[x])^(3/2)*(Hypergeometric2F1[1/4, 1/2, 5/4, -Cosh[2*x] - Sinh[2*x]]*Sech[x]^2*Sqrt[1 + Cosh[2*x] +
Sinh[2*x]] + Tanh[x]))/3

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Maple [B]  time = 0.041, size = 130, normalized size = 2.7 \begin{align*}{\frac{{a}^{2}}{3}\sqrt{a \left ( 2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}-1 \right ) \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2}} \left ( 8\, \left ( \sinh \left ( x/2 \right ) \right ) ^{4}\cosh \left ( x/2 \right ) +\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 ) +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((a*cosh(x))^(3/2),x)

[Out]

1/3*(a*(2*cosh(1/2*x)^2-1)*sinh(1/2*x)^2)^(1/2)*a^2*(8*sinh(1/2*x)^4*cosh(1/2*x)+2^(1/2)*(-2*sinh(1/2*x)^2-1)^
(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticF(cosh(1/2*x)*2^(1/2),1/2*2^(1/2))+4*sinh(1/2*x)^2*cosh(1/2*x))/(a*(2*sin
h(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 \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((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 (\sqrt{a \cosh \left (x\right )} a \cosh \left (x\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(a*cosh(x))*a*cosh(x), 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((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 \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((a*cosh(x))^(3/2),x, algorithm="giac")

[Out]

integrate((a*cosh(x))^(3/2), x)