3.2.32 \(\int \frac {1}{(a \cosh ^3(x))^{3/2}} \, dx\) [132]

Optimal. Leaf size=75 \[ -\frac {10 i \cosh ^{\frac {3}{2}}(x) F\left (\left .\frac {i x}{2}\right |2\right )}{21 a \sqrt {a \cosh ^3(x)}}+\frac {10 \sinh (x)}{21 a \sqrt {a \cosh ^3(x)}}+\frac {2 \text {sech}(x) \tanh (x)}{7 a \sqrt {a \cosh ^3(x)}} \]

[Out]

-10/21*I*cosh(x)^(3/2)*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticF(I*sinh(1/2*x),2^(1/2))/a/(a*cosh(x)^3)^(1/2
)+10/21*sinh(x)/a/(a*cosh(x)^3)^(1/2)+2/7*sech(x)*tanh(x)/a/(a*cosh(x)^3)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3286, 2716, 2720} \begin {gather*} \frac {10 \sinh (x)}{21 a \sqrt {a \cosh ^3(x)}}-\frac {10 i \cosh ^{\frac {3}{2}}(x) F\left (\left .\frac {i x}{2}\right |2\right )}{21 a \sqrt {a \cosh ^3(x)}}+\frac {2 \tanh (x) \text {sech}(x)}{7 a \sqrt {a \cosh ^3(x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*Cosh[x]^3)^(-3/2),x]

[Out]

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

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 2720

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

Rule 3286

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[(b*ff^n)^IntPart[p]*((b*Sin[e + f*x]^n)^FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p])), Int[ActivateTrig[u]
*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rubi steps

\begin {align*} \int \frac {1}{\left (a \cosh ^3(x)\right )^{3/2}} \, dx &=\frac {\cosh ^{\frac {3}{2}}(x) \int \frac {1}{\cosh ^{\frac {9}{2}}(x)} \, dx}{a \sqrt {a \cosh ^3(x)}}\\ &=\frac {2 \text {sech}(x) \tanh (x)}{7 a \sqrt {a \cosh ^3(x)}}+\frac {\left (5 \cosh ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\cosh ^{\frac {5}{2}}(x)} \, dx}{7 a \sqrt {a \cosh ^3(x)}}\\ &=\frac {10 \sinh (x)}{21 a \sqrt {a \cosh ^3(x)}}+\frac {2 \text {sech}(x) \tanh (x)}{7 a \sqrt {a \cosh ^3(x)}}+\frac {\left (5 \cosh ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sqrt {\cosh (x)}} \, dx}{21 a \sqrt {a \cosh ^3(x)}}\\ &=-\frac {10 i \cosh ^{\frac {3}{2}}(x) F\left (\left .\frac {i x}{2}\right |2\right )}{21 a \sqrt {a \cosh ^3(x)}}+\frac {10 \sinh (x)}{21 a \sqrt {a \cosh ^3(x)}}+\frac {2 \text {sech}(x) \tanh (x)}{7 a \sqrt {a \cosh ^3(x)}}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 48, normalized size = 0.64 \begin {gather*} \frac {2 \cosh ^2(x) \left (-5 i \cosh ^{\frac {5}{2}}(x) F\left (\left .\frac {i x}{2}\right |2\right )+5 \cosh (x) \sinh (x)+3 \tanh (x)\right )}{21 \left (a \cosh ^3(x)\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*Cosh[x]^3)^(-3/2),x]

[Out]

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

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Maple [F]
time = 0.95, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a \left (\cosh ^{3}\left (x \right )\right )\right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*cosh(x)^3)^(3/2),x)

[Out]

int(1/(a*cosh(x)^3)^(3/2),x)

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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)^3)^(3/2),x, algorithm="maxima")

[Out]

integrate((a*cosh(x)^3)^(-3/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.16, size = 629, normalized size = 8.39 \begin {gather*} \frac {2 \, {\left (5 \, {\left (\sqrt {2} \cosh \left (x\right )^{8} + 8 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sqrt {2} \sinh \left (x\right )^{8} + 4 \, {\left (7 \, \sqrt {2} \cosh \left (x\right )^{2} + \sqrt {2}\right )} \sinh \left (x\right )^{6} + 4 \, \sqrt {2} \cosh \left (x\right )^{6} + 8 \, {\left (7 \, \sqrt {2} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (35 \, \sqrt {2} \cosh \left (x\right )^{4} + 30 \, \sqrt {2} \cosh \left (x\right )^{2} + 3 \, \sqrt {2}\right )} \sinh \left (x\right )^{4} + 6 \, \sqrt {2} \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \sqrt {2} \cosh \left (x\right )^{5} + 10 \, \sqrt {2} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \sqrt {2} \cosh \left (x\right )^{6} + 15 \, \sqrt {2} \cosh \left (x\right )^{4} + 9 \, \sqrt {2} \cosh \left (x\right )^{2} + \sqrt {2}\right )} \sinh \left (x\right )^{2} + 4 \, \sqrt {2} \cosh \left (x\right )^{2} + 8 \, {\left (\sqrt {2} \cosh \left (x\right )^{7} + 3 \, \sqrt {2} \cosh \left (x\right )^{5} + 3 \, \sqrt {2} \cosh \left (x\right )^{3} + \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right ) + \sqrt {2}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + 2 \, {\left (5 \, \cosh \left (x\right )^{7} + 35 \, \cosh \left (x\right ) \sinh \left (x\right )^{6} + 5 \, \sinh \left (x\right )^{7} + {\left (105 \, \cosh \left (x\right )^{2} + 17\right )} \sinh \left (x\right )^{5} + 17 \, \cosh \left (x\right )^{5} + 5 \, {\left (35 \, \cosh \left (x\right )^{3} + 17 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{4} + {\left (175 \, \cosh \left (x\right )^{4} + 170 \, \cosh \left (x\right )^{2} - 17\right )} \sinh \left (x\right )^{3} - 17 \, \cosh \left (x\right )^{3} + {\left (105 \, \cosh \left (x\right )^{5} + 170 \, \cosh \left (x\right )^{3} - 51 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + {\left (35 \, \cosh \left (x\right )^{6} + 85 \, \cosh \left (x\right )^{4} - 51 \, \cosh \left (x\right )^{2} - 5\right )} \sinh \left (x\right ) - 5 \, \cosh \left (x\right )\right )} \sqrt {a \cosh \left (x\right )}\right )}}{21 \, {\left (a^{2} \cosh \left (x\right )^{8} + 8 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right )^{7} + a^{2} \sinh \left (x\right )^{8} + 4 \, a^{2} \cosh \left (x\right )^{6} + 4 \, {\left (7 \, a^{2} \cosh \left (x\right )^{2} + a^{2}\right )} \sinh \left (x\right )^{6} + 6 \, a^{2} \cosh \left (x\right )^{4} + 8 \, {\left (7 \, a^{2} \cosh \left (x\right )^{3} + 3 \, a^{2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (35 \, a^{2} \cosh \left (x\right )^{4} + 30 \, a^{2} \cosh \left (x\right )^{2} + 3 \, a^{2}\right )} \sinh \left (x\right )^{4} + 4 \, a^{2} \cosh \left (x\right )^{2} + 8 \, {\left (7 \, a^{2} \cosh \left (x\right )^{5} + 10 \, a^{2} \cosh \left (x\right )^{3} + 3 \, a^{2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, a^{2} \cosh \left (x\right )^{6} + 15 \, a^{2} \cosh \left (x\right )^{4} + 9 \, a^{2} \cosh \left (x\right )^{2} + a^{2}\right )} \sinh \left (x\right )^{2} + a^{2} + 8 \, {\left (a^{2} \cosh \left (x\right )^{7} + 3 \, a^{2} \cosh \left (x\right )^{5} + 3 \, a^{2} \cosh \left (x\right )^{3} + a^{2} \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)^3)^(3/2),x, algorithm="fricas")

[Out]

2/21*(5*(sqrt(2)*cosh(x)^8 + 8*sqrt(2)*cosh(x)*sinh(x)^7 + sqrt(2)*sinh(x)^8 + 4*(7*sqrt(2)*cosh(x)^2 + sqrt(2
))*sinh(x)^6 + 4*sqrt(2)*cosh(x)^6 + 8*(7*sqrt(2)*cosh(x)^3 + 3*sqrt(2)*cosh(x))*sinh(x)^5 + 2*(35*sqrt(2)*cos
h(x)^4 + 30*sqrt(2)*cosh(x)^2 + 3*sqrt(2))*sinh(x)^4 + 6*sqrt(2)*cosh(x)^4 + 8*(7*sqrt(2)*cosh(x)^5 + 10*sqrt(
2)*cosh(x)^3 + 3*sqrt(2)*cosh(x))*sinh(x)^3 + 4*(7*sqrt(2)*cosh(x)^6 + 15*sqrt(2)*cosh(x)^4 + 9*sqrt(2)*cosh(x
)^2 + sqrt(2))*sinh(x)^2 + 4*sqrt(2)*cosh(x)^2 + 8*(sqrt(2)*cosh(x)^7 + 3*sqrt(2)*cosh(x)^5 + 3*sqrt(2)*cosh(x
)^3 + sqrt(2)*cosh(x))*sinh(x) + sqrt(2))*sqrt(a)*weierstrassPInverse(-4, 0, cosh(x) + sinh(x)) + 2*(5*cosh(x)
^7 + 35*cosh(x)*sinh(x)^6 + 5*sinh(x)^7 + (105*cosh(x)^2 + 17)*sinh(x)^5 + 17*cosh(x)^5 + 5*(35*cosh(x)^3 + 17
*cosh(x))*sinh(x)^4 + (175*cosh(x)^4 + 170*cosh(x)^2 - 17)*sinh(x)^3 - 17*cosh(x)^3 + (105*cosh(x)^5 + 170*cos
h(x)^3 - 51*cosh(x))*sinh(x)^2 + (35*cosh(x)^6 + 85*cosh(x)^4 - 51*cosh(x)^2 - 5)*sinh(x) - 5*cosh(x))*sqrt(a*
cosh(x)))/(a^2*cosh(x)^8 + 8*a^2*cosh(x)*sinh(x)^7 + a^2*sinh(x)^8 + 4*a^2*cosh(x)^6 + 4*(7*a^2*cosh(x)^2 + a^
2)*sinh(x)^6 + 6*a^2*cosh(x)^4 + 8*(7*a^2*cosh(x)^3 + 3*a^2*cosh(x))*sinh(x)^5 + 2*(35*a^2*cosh(x)^4 + 30*a^2*
cosh(x)^2 + 3*a^2)*sinh(x)^4 + 4*a^2*cosh(x)^2 + 8*(7*a^2*cosh(x)^5 + 10*a^2*cosh(x)^3 + 3*a^2*cosh(x))*sinh(x
)^3 + 4*(7*a^2*cosh(x)^6 + 15*a^2*cosh(x)^4 + 9*a^2*cosh(x)^2 + a^2)*sinh(x)^2 + a^2 + 8*(a^2*cosh(x)^7 + 3*a^
2*cosh(x)^5 + 3*a^2*cosh(x)^3 + a^2*cosh(x))*sinh(x))

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(x)**3)**(3/2),x)

[Out]

Timed out

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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)^3)^(3/2),x, algorithm="giac")

[Out]

integrate((a*cosh(x)^3)^(-3/2), 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 )}^3\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*cosh(x)^3)^(3/2),x)

[Out]

int(1/(a*cosh(x)^3)^(3/2), x)

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