3.15 \(\int (a \cosh (x))^{7/2} \, dx\)

Optimal. Leaf size=65 \[ -\frac{10 i a^4 \sqrt{\cosh (x)} \text{EllipticF}\left (\frac{i x}{2},2\right )}{21 \sqrt{a \cosh (x)}}+\frac{10}{21} a^3 \sinh (x) \sqrt{a \cosh (x)}+\frac{2}{7} a \sinh (x) (a \cosh (x))^{5/2} \]

[Out]

(((-10*I)/21)*a^4*Sqrt[Cosh[x]]*EllipticF[(I/2)*x, 2])/Sqrt[a*Cosh[x]] + (10*a^3*Sqrt[a*Cosh[x]]*Sinh[x])/21 +
(2*a*(a*Cosh[x])^(5/2)*Sinh[x])/7

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Rubi [A]  time = 0.0374672, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2635, 2642, 2641} \[ \frac{10}{21} a^3 \sinh (x) \sqrt{a \cosh (x)}-\frac{10 i a^4 \sqrt{\cosh (x)} F\left (\left .\frac{i x}{2}\right |2\right )}{21 \sqrt{a \cosh (x)}}+\frac{2}{7} a \sinh (x) (a \cosh (x))^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[(a*Cosh[x])^(7/2),x]

[Out]

(((-10*I)/21)*a^4*Sqrt[Cosh[x]]*EllipticF[(I/2)*x, 2])/Sqrt[a*Cosh[x]] + (10*a^3*Sqrt[a*Cosh[x]]*Sinh[x])/21 +
(2*a*(a*Cosh[x])^(5/2)*Sinh[x])/7

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))^{7/2} \, dx &=\frac{2}{7} a (a \cosh (x))^{5/2} \sinh (x)+\frac{1}{7} \left (5 a^2\right ) \int (a \cosh (x))^{3/2} \, dx\\ &=\frac{10}{21} a^3 \sqrt{a \cosh (x)} \sinh (x)+\frac{2}{7} a (a \cosh (x))^{5/2} \sinh (x)+\frac{1}{21} \left (5 a^4\right ) \int \frac{1}{\sqrt{a \cosh (x)}} \, dx\\ &=\frac{10}{21} a^3 \sqrt{a \cosh (x)} \sinh (x)+\frac{2}{7} a (a \cosh (x))^{5/2} \sinh (x)+\frac{\left (5 a^4 \sqrt{\cosh (x)}\right ) \int \frac{1}{\sqrt{\cosh (x)}} \, dx}{21 \sqrt{a \cosh (x)}}\\ &=-\frac{10 i a^4 \sqrt{\cosh (x)} F\left (\left .\frac{i x}{2}\right |2\right )}{21 \sqrt{a \cosh (x)}}+\frac{10}{21} a^3 \sqrt{a \cosh (x)} \sinh (x)+\frac{2}{7} a (a \cosh (x))^{5/2} \sinh (x)\\ \end{align*}

Mathematica [A]  time = 0.0477263, size = 53, normalized size = 0.82 \[ \frac{a^3 \sqrt{a \cosh (x)} \left ((23 \sinh (x)+3 \sinh (3 x)) \sqrt{\cosh (x)}-20 i \text{EllipticF}\left (\frac{i x}{2},2\right )\right )}{42 \sqrt{\cosh (x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Cosh[x])^(7/2),x]

[Out]

(a^3*Sqrt[a*Cosh[x]]*((-20*I)*EllipticF[(I/2)*x, 2] + Sqrt[Cosh[x]]*(23*Sinh[x] + 3*Sinh[3*x])))/(42*Sqrt[Cosh
[x]])

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Maple [B]  time = 0.078, size = 145, normalized size = 2.2 \begin{align*}{\frac{{a}^{4}}{21}\sqrt{a \left ( 2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}-1 \right ) \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2}} \left ( 96\, \left ( \cosh \left ( x/2 \right ) \right ) ^{9}-240\, \left ( \cosh \left ( x/2 \right ) \right ) ^{7}+256\, \left ( \cosh \left ( x/2 \right ) \right ) ^{5}+5\,\sqrt{2}\sqrt{-2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}+1}\sqrt{- \left ( \sinh \left ( x/2 \right ) \right ) ^{2}}{\it EllipticF} \left ( \cosh \left ( x/2 \right ) \sqrt{2},1/2\,\sqrt{2} \right ) -144\, \left ( \cosh \left ( x/2 \right ) \right ) ^{3}+32\,\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))^(7/2),x)

[Out]

1/21*(a*(2*cosh(1/2*x)^2-1)*sinh(1/2*x)^2)^(1/2)*a^4*(96*cosh(1/2*x)^9-240*cosh(1/2*x)^7+256*cosh(1/2*x)^5+5*2
^(1/2)*(-2*cosh(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))-144*cosh(1
/2*x)^3+32*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 \left (a \cosh \left (x\right )\right )^{\frac{7}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cosh(x))^(7/2),x, algorithm="maxima")

[Out]

integrate((a*cosh(x))^(7/2), 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^{3} \cosh \left (x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cosh(x))^(7/2),x, algorithm="fricas")

[Out]

integral(sqrt(a*cosh(x))*a^3*cosh(x)^3, 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))**(7/2),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{7}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cosh(x))^(7/2),x, algorithm="giac")

[Out]

integrate((a*cosh(x))^(7/2), x)