3.16 \(\int (a \cosh (x))^{5/2} \, dx\)

Optimal. Leaf size=48 \[ \frac {2}{5} a \sinh (x) (a \cosh (x))^{3/2}-\frac {6 i a^2 E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \cosh (x)}}{5 \sqrt {\cosh (x)}} \]

[Out]

2/5*a*(a*cosh(x))^(3/2)*sinh(x)-6/5*I*a^2*(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)/cosh(x)^(1/2)

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Rubi [A]  time = 0.03, 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, 2640, 2639} \[ \frac {2}{5} a \sinh (x) (a \cosh (x))^{3/2}-\frac {6 i a^2 E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \cosh (x)}}{5 \sqrt {\cosh (x)}} \]

Antiderivative was successfully verified.

[In]

Int[(a*Cosh[x])^(5/2),x]

[Out]

(((-6*I)/5)*a^2*Sqrt[a*Cosh[x]]*EllipticE[(I/2)*x, 2])/Sqrt[Cosh[x]] + (2*a*(a*Cosh[x])^(3/2)*Sinh[x])/5

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 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 (a \cosh (x))^{5/2} \, dx &=\frac {2}{5} a (a \cosh (x))^{3/2} \sinh (x)+\frac {1}{5} \left (3 a^2\right ) \int \sqrt {a \cosh (x)} \, dx\\ &=\frac {2}{5} a (a \cosh (x))^{3/2} \sinh (x)+\frac {\left (3 a^2 \sqrt {a \cosh (x)}\right ) \int \sqrt {\cosh (x)} \, dx}{5 \sqrt {\cosh (x)}}\\ &=-\frac {6 i a^2 \sqrt {a \cosh (x)} E\left (\left .\frac {i x}{2}\right |2\right )}{5 \sqrt {\cosh (x)}}+\frac {2}{5} a (a \cosh (x))^{3/2} \sinh (x)\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 41, normalized size = 0.85 \[ \frac {2 (a \cosh (x))^{5/2} \left (\sinh (x) \cosh ^{\frac {3}{2}}(x)-3 i E\left (\left .\frac {i x}{2}\right |2\right )\right )}{5 \cosh ^{\frac {5}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Cosh[x])^(5/2),x]

[Out]

(2*(a*Cosh[x])^(5/2)*((-3*I)*EllipticE[(I/2)*x, 2] + Cosh[x]^(3/2)*Sinh[x]))/(5*Cosh[x]^(5/2))

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fricas [F]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a \cosh \relax (x)} a^{2} \cosh \relax (x)^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cosh(x))^(5/2),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 \[ \int \left (a \cosh \relax (x)\right )^{\frac {5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cosh(x))^(5/2),x, algorithm="giac")

[Out]

integrate((a*cosh(x))^(5/2), x)

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maple [B]  time = 0.37, size = 184, normalized size = 3.83 \[ \frac {\sqrt {a \left (2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, a^{3} \left (16 \left (\sinh ^{6}\left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )+16 \left (\sinh ^{4}\left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )+3 \sqrt {2}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )-1}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \EllipticF \left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )-6 \sqrt {2}\, \sqrt {-2 \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )-1}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \EllipticE \left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )+4 \left (\sinh ^{2}\left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )\right )}{5 \sqrt {a \left (2 \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \sinh \left (\frac {x}{2}\right ) \sqrt {a \left (2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )-1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*cosh(x))^(5/2),x)

[Out]

1/5*(a*(2*cosh(1/2*x)^2-1)*sinh(1/2*x)^2)^(1/2)*a^3*(16*sinh(1/2*x)^6*cosh(1/2*x)+16*sinh(1/2*x)^4*cosh(1/2*x)
+3*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))-6*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 \[ \int \left (a \cosh \relax (x)\right )^{\frac {5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cosh(x))^(5/2),x, algorithm="maxima")

[Out]

integrate((a*cosh(x))^(5/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (a\,\mathrm {cosh}\relax (x)\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*cosh(x))^(5/2),x)

[Out]

int((a*cosh(x))^(5/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cosh(x))**(5/2),x)

[Out]

Timed out

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