3.130 \(\int \sqrt{a \cosh ^3(x)} \, dx\)

Optimal. Leaf size=48 \[ \frac{2}{3} \tanh (x) \sqrt{a \cosh ^3(x)}-\frac{2 i \text{EllipticF}\left (\frac{i x}{2},2\right ) \sqrt{a \cosh ^3(x)}}{3 \cosh ^{\frac{3}{2}}(x)} \]

[Out]

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

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Rubi [A]  time = 0.0240773, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3207, 2635, 2641} \[ \frac{2}{3} \tanh (x) \sqrt{a \cosh ^3(x)}-\frac{2 i F\left (\left .\frac{i x}{2}\right |2\right ) \sqrt{a \cosh ^3(x)}}{3 \cosh ^{\frac{3}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((-2*I)/3)*Sqrt[a*Cosh[x]^3]*EllipticF[(I/2)*x, 2])/Cosh[x]^(3/2) + (2*Sqrt[a*Cosh[x]^3]*Tanh[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 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 \cosh ^3(x)} \, dx &=\frac{\sqrt{a \cosh ^3(x)} \int \cosh ^{\frac{3}{2}}(x) \, dx}{\cosh ^{\frac{3}{2}}(x)}\\ &=\frac{2}{3} \sqrt{a \cosh ^3(x)} \tanh (x)+\frac{\sqrt{a \cosh ^3(x)} \int \frac{1}{\sqrt{\cosh (x)}} \, dx}{3 \cosh ^{\frac{3}{2}}(x)}\\ &=-\frac{2 i \sqrt{a \cosh ^3(x)} F\left (\left .\frac{i x}{2}\right |2\right )}{3 \cosh ^{\frac{3}{2}}(x)}+\frac{2}{3} \sqrt{a \cosh ^3(x)} \tanh (x)\\ \end{align*}

Mathematica [C]  time = 0.0513766, size = 59, normalized size = 1.23 \[ \frac{2}{3} \sqrt{a \cosh ^3(x)} \left (\text{sech}^2(x) \sqrt{\sinh (2 x)+\cosh (2 x)+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\cosh (2 x)-\sinh (2 x)\right )+\tanh (x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \left ( \cosh \left ( x \right ) \right ) ^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*cosh(x)^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(a*cosh(x)^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*cosh(x)^3), x)