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

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

[Out]

-2/3*I*a^2*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticF(I*sinh(1/2*x),2^(1/2))*cosh(x)^(1/2)/(a*cosh(x))^(1/2)+
2/3*a*sinh(x)*(a*cosh(x))^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 48, 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 = {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 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]

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]

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*}

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Mathematica [C]  time = 0.06, 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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fricas [F]  time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a \cosh \relax (x)} a \cosh \relax (x), x\right ) \]

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

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)

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maple [B]  time = 0.38, size = 130, normalized size = 2.71 \[ \frac {\sqrt {a \left (2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, a^{2} \left (8 \left (\sinh ^{4}\left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )+\sqrt {2}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )-1}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \EllipticF \left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )+4 \left (\sinh ^{2}\left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )\right )}{3 \sqrt {a \left (2 \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \sinh \left (\frac {x}{2}\right ) \sqrt {a \left (2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )-1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a*cosh(x))^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*cosh(x))**(3/2), x)

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