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

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

[Out]

2/5*sinh(x)/a/(a*cosh(x))^(5/2)+6/5*sinh(x)/a^3/(a*cosh(x))^(1/2)+6/5*I*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*Elli
pticE(I*sinh(1/2*x),2^(1/2))*(a*cosh(x))^(1/2)/a^4/cosh(x)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 67, normalized size of antiderivative = 1.00, 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 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 \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*}

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Mathematica [A]  time = 0.05, 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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fricas [F]  time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \cosh \relax (x)}}{a^{4} \cosh \relax (x)^{4}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \cosh \relax (x)\right )^{\frac {7}{2}}}\,{d x} \]

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)

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maple [B]  time = 0.60, size = 254, normalized size = 3.79 \[ \frac {2 \sqrt {a \left (2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \left (\frac {\cosh \left (\frac {x}{2}\right ) \sqrt {a \left (2 \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\sinh ^{2}\left (\frac {x}{2}\right )\right )}}{20 a \left (\cosh ^{2}\left (\frac {x}{2}\right )-\frac {1}{2}\right )^{3}}+\frac {6 \left (\sinh ^{2}\left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )}{5 \sqrt {a \left (2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}}+\frac {3 \sqrt {2}\, \sqrt {-2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )+1}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \EllipticF \left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )}{10 \sqrt {a \left (2 \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\sinh ^{2}\left (\frac {x}{2}\right )\right )}}-\frac {3 \sqrt {2}\, \sqrt {-2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )+1}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \left (\EllipticF \left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )-\EllipticE \left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )\right )}{5 \sqrt {a \left (2 \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\sinh ^{2}\left (\frac {x}{2}\right )\right )}}\right )}{a^{3} \sinh \left (\frac {x}{2}\right ) \sqrt {a \left (2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )-1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(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(2^(1/2)
*cosh(1/2*x),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(2^(1/2)*cosh(1/2*x),1/2*2^(1/2))-EllipticE(2^(1/2)*cosh(1/2*x),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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \cosh \relax (x)\right )^{\frac {7}{2}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a\,\mathrm {cosh}\relax (x)\right )}^{7/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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