3.1.22 \(\int \frac {1}{(a \cosh (x))^{7/2}} \, dx\) [22]

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

[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 = 0.03, 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 = {2716, 2721, 2719} \begin {gather*} \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}} \end {gather*}

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 2716

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 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 2721

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(b*Sin[c + d*x])^n/Sin[c + d*x]^n, Int[Sin[c + d*x]
^n, x], x] /; FreeQ[{b, c, d}, x] && LtQ[-1, n, 1] && IntegerQ[2*n]

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 = 0.03, size = 43, normalized size = 0.64 \begin {gather*} \frac {2 \left (3 i \cosh ^{\frac {3}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right )+3 \cosh (x) \sinh (x)+\tanh (x)\right )}{5 a^2 (a \cosh (x))^{3/2}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(253\) vs. \(2(68)=136\).
time = 1.36, size = 254, normalized size = 3.79

method	result	size
default	\(\frac {2 \sqrt {a \left (2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \left (\frac {\cosh \left (\frac {x}{2}\right ) \sqrt {a \left (2 \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\sinh ^{2}\left (\frac {x}{2}\right )\right )}}{20 a \left (\cosh ^{2}\left (\frac {x}{2}\right )-\frac {1}{2}\right )^{3}}+\frac {6 \left (\sinh ^{2}\left (\frac {x}{2}\right )\right ) \cosh \left (\frac {x}{2}\right )}{5 \sqrt {a \left (2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )-1\right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}}+\frac {3 \sqrt {2}\, \sqrt {-2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )+1}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \EllipticF \left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )}{10 \sqrt {a \left (2 \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\sinh ^{2}\left (\frac {x}{2}\right )\right )}}-\frac {3 \sqrt {2}\, \sqrt {-2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )+1}\, \sqrt {-\left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}\, \left (\EllipticF \left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )-\EllipticE \left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )\right )}{5 \sqrt {a \left (2 \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\sinh ^{2}\left (\frac {x}{2}\right )\right )}}\right )}{a^{3} \sinh \left (\frac {x}{2}\right ) \sqrt {a \left (2 \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )-1\right )}}\)	\(254\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.10, size = 422, normalized size = 6.30 \begin {gather*} \frac {2 \, {\left (3 \, {\left (\sqrt {2} \cosh \left (x\right )^{6} + 6 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sqrt {2} \sinh \left (x\right )^{6} + 3 \, {\left (5 \, \sqrt {2} \cosh \left (x\right )^{2} + \sqrt {2}\right )} \sinh \left (x\right )^{4} + 3 \, \sqrt {2} \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \sqrt {2} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, \sqrt {2} \cosh \left (x\right )^{4} + 6 \, \sqrt {2} \cosh \left (x\right )^{2} + \sqrt {2}\right )} \sinh \left (x\right )^{2} + 3 \, \sqrt {2} \cosh \left (x\right )^{2} + 6 \, {\left (\sqrt {2} \cosh \left (x\right )^{5} + 2 \, \sqrt {2} \cosh \left (x\right )^{3} + \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right ) + \sqrt {2}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) + 2 \, {\left (3 \, \cosh \left (x\right )^{6} + 18 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + 3 \, \sinh \left (x\right )^{6} + {\left (45 \, \cosh \left (x\right )^{2} + 8\right )} \sinh \left (x\right )^{4} + 8 \, \cosh \left (x\right )^{4} + 4 \, {\left (15 \, \cosh \left (x\right )^{3} + 8 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + {\left (45 \, \cosh \left (x\right )^{4} + 48 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} + 2 \, {\left (9 \, \cosh \left (x\right )^{5} + 16 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {a \cosh \left (x\right )}\right )}}{5 \, {\left (a^{4} \cosh \left (x\right )^{6} + 6 \, a^{4} \cosh \left (x\right ) \sinh \left (x\right )^{5} + a^{4} \sinh \left (x\right )^{6} + 3 \, a^{4} \cosh \left (x\right )^{4} + 3 \, a^{4} \cosh \left (x\right )^{2} + 3 \, {\left (5 \, a^{4} \cosh \left (x\right )^{2} + a^{4}\right )} \sinh \left (x\right )^{4} + a^{4} + 4 \, {\left (5 \, a^{4} \cosh \left (x\right )^{3} + 3 \, a^{4} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, a^{4} \cosh \left (x\right )^{4} + 6 \, a^{4} \cosh \left (x\right )^{2} + a^{4}\right )} \sinh \left (x\right )^{2} + 6 \, {\left (a^{4} \cosh \left (x\right )^{5} + 2 \, a^{4} \cosh \left (x\right )^{3} + a^{4} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(3*(sqrt(2)*cosh(x)^6 + 6*sqrt(2)*cosh(x)*sinh(x)^5 + sqrt(2)*sinh(x)^6 + 3*(5*sqrt(2)*cosh(x)^2 + sqrt(2)
)*sinh(x)^4 + 3*sqrt(2)*cosh(x)^4 + 4*(5*sqrt(2)*cosh(x)^3 + 3*sqrt(2)*cosh(x))*sinh(x)^3 + 3*(5*sqrt(2)*cosh(
x)^4 + 6*sqrt(2)*cosh(x)^2 + sqrt(2))*sinh(x)^2 + 3*sqrt(2)*cosh(x)^2 + 6*(sqrt(2)*cosh(x)^5 + 2*sqrt(2)*cosh(
x)^3 + sqrt(2)*cosh(x))*sinh(x) + sqrt(2))*sqrt(a)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cosh(x) +
sinh(x))) + 2*(3*cosh(x)^6 + 18*cosh(x)*sinh(x)^5 + 3*sinh(x)^6 + (45*cosh(x)^2 + 8)*sinh(x)^4 + 8*cosh(x)^4
+ 4*(15*cosh(x)^3 + 8*cosh(x))*sinh(x)^3 + (45*cosh(x)^4 + 48*cosh(x)^2 + 1)*sinh(x)^2 + cosh(x)^2 + 2*(9*cosh
(x)^5 + 16*cosh(x)^3 + cosh(x))*sinh(x))*sqrt(a*cosh(x)))/(a^4*cosh(x)^6 + 6*a^4*cosh(x)*sinh(x)^5 + a^4*sinh(
x)^6 + 3*a^4*cosh(x)^4 + 3*a^4*cosh(x)^2 + 3*(5*a^4*cosh(x)^2 + a^4)*sinh(x)^4 + a^4 + 4*(5*a^4*cosh(x)^3 + 3*
a^4*cosh(x))*sinh(x)^3 + 3*(5*a^4*cosh(x)^4 + 6*a^4*cosh(x)^2 + a^4)*sinh(x)^2 + 6*(a^4*cosh(x)^5 + 2*a^4*cosh
(x)^3 + a^4*cosh(x))*sinh(x))

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(x))**(7/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3880 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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