3.22 \(\int \frac{1}{(a \cosh (x))^{7/2}} \, dx\)

Optimal. Leaf size=67 \[ \frac{6 \sinh (x)}{5 a^3 \sqrt{a \cosh (x)}}+\frac{6 i E\left (\left .\frac{i x}{2}\right |2\right ) \sqrt{a \cosh (x)}}{5 a^4 \sqrt{\cosh (x)}}+\frac{2 \sinh (x)}{5 a (a \cosh (x))^{5/2}} \]

[Out]

(((6*I)/5)*Sqrt[a*Cosh[x]]*EllipticE[(I/2)*x, 2])/(a^4*Sqrt[Cosh[x]]) + (2*Sinh[x])/(5*a*(a*Cosh[x])^(5/2)) +
(6*Sinh[x])/(5*a^3*Sqrt[a*Cosh[x]])

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Rubi [A]  time = 0.0387202, antiderivative size = 67, 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 = {2636, 2640, 2639} \[ \frac{6 \sinh (x)}{5 a^3 \sqrt{a \cosh (x)}}+\frac{6 i E\left (\left .\frac{i x}{2}\right |2\right ) \sqrt{a \cosh (x)}}{5 a^4 \sqrt{\cosh (x)}}+\frac{2 \sinh (x)}{5 a (a \cosh (x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((6*I)/5)*Sqrt[a*Cosh[x]]*EllipticE[(I/2)*x, 2])/(a^4*Sqrt[Cosh[x]]) + (2*Sinh[x])/(5*a*(a*Cosh[x])^(5/2)) +
(6*Sinh[x])/(5*a^3*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))^{7/2}} \, dx &=\frac{2 \sinh (x)}{5 a (a \cosh (x))^{5/2}}+\frac{3 \int \frac{1}{(a \cosh (x))^{3/2}} \, dx}{5 a^2}\\ &=\frac{2 \sinh (x)}{5 a (a \cosh (x))^{5/2}}+\frac{6 \sinh (x)}{5 a^3 \sqrt{a \cosh (x)}}-\frac{3 \int \sqrt{a \cosh (x)} \, dx}{5 a^4}\\ &=\frac{2 \sinh (x)}{5 a (a \cosh (x))^{5/2}}+\frac{6 \sinh (x)}{5 a^3 \sqrt{a \cosh (x)}}-\frac{\left (3 \sqrt{a \cosh (x)}\right ) \int \sqrt{\cosh (x)} \, dx}{5 a^4 \sqrt{\cosh (x)}}\\ &=\frac{6 i \sqrt{a \cosh (x)} E\left (\left .\frac{i x}{2}\right |2\right )}{5 a^4 \sqrt{\cosh (x)}}+\frac{2 \sinh (x)}{5 a (a \cosh (x))^{5/2}}+\frac{6 \sinh (x)}{5 a^3 \sqrt{a \cosh (x)}}\\ \end{align*}

Mathematica [A]  time = 0.0446776, size = 43, normalized size = 0.64 \[ \frac{2 \left (\tanh (x)+3 i \cosh ^{\frac{3}{2}}(x) E\left (\left .\frac{i x}{2}\right |2\right )+3 \sinh (x) \cosh (x)\right )}{5 a^2 (a \cosh (x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.082, size = 254, normalized size = 3.8 \begin{align*} 2\,{\frac{\sqrt{a \left ( 2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}-1 \right ) \left ( \sinh \left ( x/2 \right ) \right ) ^{2}}}{{a}^{3}\sinh \left ( x/2 \right ) \sqrt{a \left ( 2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}-1 \right ) }} \left ( 1/20\,{\frac{\cosh \left ( x/2 \right ) \sqrt{a \left ( 2\, \left ( \sinh \left ( x/2 \right ) \right ) ^{4}+ \left ( \sinh \left ( x/2 \right ) \right ) ^{2} \right ) }}{a \left ( \left ( \cosh \left ( x/2 \right ) \right ) ^{2}-1/2 \right ) ^{3}}}+6/5\,{\frac{ \left ( \sinh \left ( x/2 \right ) \right ) ^{2}\cosh \left ( x/2 \right ) }{\sqrt{a \left ( 2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}-1 \right ) \left ( \sinh \left ( x/2 \right ) \right ) ^{2}}}}+3/10\,{\frac{\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 ) }{\sqrt{a \left ( 2\, \left ( \sinh \left ( x/2 \right ) \right ) ^{4}+ \left ( \sinh \left ( x/2 \right ) \right ) ^{2} \right ) }}}-3/5\,{\frac{\sqrt{2}\sqrt{-2\, \left ( \cosh \left ( x/2 \right ) \right ) ^{2}+1}\sqrt{- \left ( \sinh \left ( x/2 \right ) \right ) ^{2}} \left ({\it EllipticF} \left ( \cosh \left ( x/2 \right ) \sqrt{2},1/2\,\sqrt{2} \right ) -{\it EllipticE} \left ( \cosh \left ( x/2 \right ) \sqrt{2},1/2\,\sqrt{2} \right ) \right ) }{\sqrt{a \left ( 2\, \left ( \sinh \left ( x/2 \right ) \right ) ^{4}+ \left ( \sinh \left ( x/2 \right ) \right ) ^{2} \right ) }}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*cosh(x))^(7/2),x)

[Out]

2*(a*(2*cosh(1/2*x)^2-1)*sinh(1/2*x)^2)^(1/2)/a^3*(1/20*cosh(1/2*x)/a*(a*(2*sinh(1/2*x)^4+sinh(1/2*x)^2))^(1/2
)/(cosh(1/2*x)^2-1/2)^3+6/5*sinh(1/2*x)^2*cosh(1/2*x)/(a*(2*cosh(1/2*x)^2-1)*sinh(1/2*x)^2)^(1/2)+3/10*2^(1/2)
*(-2*cosh(1/2*x)^2+1)^(1/2)*(-sinh(1/2*x)^2)^(1/2)/(a*(2*sinh(1/2*x)^4+sinh(1/2*x)^2))^(1/2)*EllipticF(cosh(1/
2*x)*2^(1/2),1/2*2^(1/2))-3/5*2^(1/2)*(-2*cosh(1/2*x)^2+1)^(1/2)*(-sinh(1/2*x)^2)^(1/2)/(a*(2*sinh(1/2*x)^4+si
nh(1/2*x)^2))^(1/2)*(EllipticF(cosh(1/2*x)*2^(1/2),1/2*2^(1/2))-EllipticE(cosh(1/2*x)*2^(1/2),1/2*2^(1/2))))/s
inh(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{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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 (\frac{\sqrt{a \cosh \left (x\right )}}{a^{4} \cosh \left (x\right )^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(x))^(7/2),x, algorithm="fricas")

[Out]

integral(sqrt(a*cosh(x))/(a^4*cosh(x)^4), 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))**(7/2),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{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(x))^(7/2),x, algorithm="giac")

[Out]

integrate((a*cosh(x))^(-7/2), x)