3.2.50 \(\int \frac {1}{(a \sinh ^3(x))^{3/2}} \, dx\) [150]

Optimal. Leaf size=87 \[ \frac {10 \cosh (x)}{21 a \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}(x)}{7 a \sqrt {a \sinh ^3(x)}}+\frac {10 i F\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sqrt {i \sinh (x)} \sinh (x)}{21 a \sqrt {a \sinh ^3(x)}} \]

[Out]

10/21*cosh(x)/a/(a*sinh(x)^3)^(1/2)-2/7*coth(x)*csch(x)/a/(a*sinh(x)^3)^(1/2)+10/21*I*(sin(1/4*Pi+1/2*I*x)^2)^
(1/2)/sin(1/4*Pi+1/2*I*x)*EllipticF(cos(1/4*Pi+1/2*I*x),2^(1/2))*sinh(x)*(I*sinh(x))^(1/2)/a/(a*sinh(x)^3)^(1/
2)

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Rubi [A]
time = 0.03, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3286, 2716, 2721, 2720} \begin {gather*} \frac {10 \cosh (x)}{21 a \sqrt {a \sinh ^3(x)}}+\frac {10 i \sqrt {i \sinh (x)} \sinh (x) F\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{21 a \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}(x)}{7 a \sqrt {a \sinh ^3(x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 \sinh ^3(x)\right )^{3/2}} \, dx &=\frac {\sinh ^{\frac {3}{2}}(x) \int \frac {1}{\sinh ^{\frac {9}{2}}(x)} \, dx}{a \sqrt {a \sinh ^3(x)}}\\ &=-\frac {2 \coth (x) \text {csch}(x)}{7 a \sqrt {a \sinh ^3(x)}}-\frac {\left (5 \sinh ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sinh ^{\frac {5}{2}}(x)} \, dx}{7 a \sqrt {a \sinh ^3(x)}}\\ &=\frac {10 \cosh (x)}{21 a \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}(x)}{7 a \sqrt {a \sinh ^3(x)}}+\frac {\left (5 \sinh ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sqrt {\sinh (x)}} \, dx}{21 a \sqrt {a \sinh ^3(x)}}\\ &=\frac {10 \cosh (x)}{21 a \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}(x)}{7 a \sqrt {a \sinh ^3(x)}}+\frac {\left (5 \sqrt {i \sinh (x)} \sinh (x)\right ) \int \frac {1}{\sqrt {i \sinh (x)}} \, dx}{21 a \sqrt {a \sinh ^3(x)}}\\ &=\frac {10 \cosh (x)}{21 a \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}(x)}{7 a \sqrt {a \sinh ^3(x)}}+\frac {10 i F\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sqrt {i \sinh (x)} \sinh (x)}{21 a \sqrt {a \sinh ^3(x)}}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 53, normalized size = 0.61 \begin {gather*} \frac {2 \left (5 \cosh (x)-3 \coth (x) \text {csch}(x)+5 F\left (\left .\frac {1}{4} (\pi -2 i x)\right |2\right ) (i \sinh (x))^{3/2}\right )}{21 a \sqrt {a \sinh ^3(x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.78, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a \left (\sinh ^{3}\left (x \right )\right )\right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(a*sinh(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*sinh(x)^3)^(3/2),x, algorithm="maxima")

[Out]

integrate((a*sinh(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 = 0.11, size = 639, normalized size = 7.34 \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 \sinh \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*sinh(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*cosh
(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*s
inh(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*c
osh(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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a \sinh ^{3}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*sinh(x)**3)**(-3/2), x)

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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*sinh(x)^3)^(3/2),x, algorithm="giac")

[Out]

integrate((a*sinh(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 {sinh}\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*sinh(x)^3)^(3/2),x)

[Out]

int(1/(a*sinh(x)^3)^(3/2), x)

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