3.148 \(\int \sqrt{a \sinh ^3(x)} \, dx\)

Optimal. Leaf size=62 \[ \frac{2}{3} \coth (x) \sqrt{a \sinh ^3(x)}-\frac{2}{3} i \sqrt{i \sinh (x)} \text{csch}^2(x) \text{EllipticF}\left (\frac{\pi }{4}-\frac{i x}{2},2\right ) \sqrt{a \sinh ^3(x)} \]

[Out]

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

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Rubi [A]  time = 0.0326089, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3207, 2635, 2642, 2641} \[ \frac{2}{3} \coth (x) \sqrt{a \sinh ^3(x)}-\frac{2}{3} i \sqrt{i \sinh (x)} \text{csch}^2(x) F\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right ) \sqrt{a \sinh ^3(x)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*Sinh[x]^3],x]

[Out]

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

Rule 3207

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
]])

Rule 2635

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

Rule 2642

Int[1/Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[c + d*x]]/Sqrt[b*Sin[c + d*x]], Int[1/Sqr
t[Sin[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

Rule 2641

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

Rubi steps

\begin{align*} \int \sqrt{a \sinh ^3(x)} \, dx &=\frac{\sqrt{a \sinh ^3(x)} \int \sinh ^{\frac{3}{2}}(x) \, dx}{\sinh ^{\frac{3}{2}}(x)}\\ &=\frac{2}{3} \coth (x) \sqrt{a \sinh ^3(x)}-\frac{\sqrt{a \sinh ^3(x)} \int \frac{1}{\sqrt{\sinh (x)}} \, dx}{3 \sinh ^{\frac{3}{2}}(x)}\\ &=\frac{2}{3} \coth (x) \sqrt{a \sinh ^3(x)}-\frac{1}{3} \left (\text{csch}^2(x) \sqrt{i \sinh (x)} \sqrt{a \sinh ^3(x)}\right ) \int \frac{1}{\sqrt{i \sinh (x)}} \, dx\\ &=\frac{2}{3} \coth (x) \sqrt{a \sinh ^3(x)}-\frac{2}{3} i \text{csch}^2(x) F\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right ) \sqrt{i \sinh (x)} \sqrt{a \sinh ^3(x)}\\ \end{align*}

Mathematica [C]  time = 0.0834606, size = 60, normalized size = 0.97 \[ \frac{2}{3} \sqrt{a \sinh ^3(x)} \left (\coth (x)-\sqrt{2} \text{csch}^2(x) \sqrt{-\sinh (x) (\sinh (x)+\cosh (x))} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\cosh (2 x)+\sinh (2 x)\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*Sinh[x]^3],x]

[Out]

(2*Sqrt[a*Sinh[x]^3]*(Coth[x] - Sqrt[2]*Csch[x]^2*Hypergeometric2F1[1/4, 1/2, 5/4, Cosh[2*x] + Sinh[2*x]]*Sqrt
[-(Sinh[x]*(Cosh[x] + Sinh[x]))]))/3

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \left ( \sinh \left ( x \right ) \right ) ^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \sinh \left (x\right )^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*sinh(x)^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{a \sinh \left (x\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sinh(x)^3)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(a*sinh(x)^3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \sinh ^{3}{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*sinh(x)**3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \sinh \left (x\right )^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sinh(x)^3)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(a*sinh(x)^3), x)