3.332 \(\int (\frac{x^2}{\cosh ^{\frac{3}{2}}(x)}+x^2 \sqrt{\cosh (x)}) \, dx\)

Optimal. Leaf size=36 \[ \frac{2 x^2 \sinh (x)}{\sqrt{\cosh (x)}}-8 x \sqrt{\cosh (x)}-16 i E\left (\left .\frac{i x}{2}\right |2\right ) \]

[Out]

-8*x*Sqrt[Cosh[x]] - (16*I)*EllipticE[(I/2)*x, 2] + (2*x^2*Sinh[x])/Sqrt[Cosh[x]]

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Rubi [A]  time = 0.0941286, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3316, 2639} \[ \frac{2 x^2 \sinh (x)}{\sqrt{\cosh (x)}}-8 x \sqrt{\cosh (x)}-16 i E\left (\left .\frac{i x}{2}\right |2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2/Cosh[x]^(3/2) + x^2*Sqrt[Cosh[x]],x]

[Out]

-8*x*Sqrt[Cosh[x]] - (16*I)*EllipticE[(I/2)*x, 2] + (2*x^2*Sinh[x])/Sqrt[Cosh[x]]

Rule 3316

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((c + d*x)^m*Cos[e + f*
x]*(b*Sin[e + f*x])^(n + 1))/(b*f*(n + 1)), x] + (Dist[(n + 2)/(b^2*(n + 1)), Int[(c + d*x)^m*(b*Sin[e + f*x])
^(n + 2), x], x] + Dist[(d^2*m*(m - 1))/(b^2*f^2*(n + 1)*(n + 2)), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^(n +
2), x], x] - Simp[(d*m*(c + d*x)^(m - 1)*(b*Sin[e + f*x])^(n + 2))/(b^2*f^2*(n + 1)*(n + 2)), x]) /; FreeQ[{b
, c, d, e, f}, x] && LtQ[n, -1] && NeQ[n, -2] && GtQ[m, 1]

Rule 2639

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

Rubi steps

\begin{align*} \int \left (\frac{x^2}{\cosh ^{\frac{3}{2}}(x)}+x^2 \sqrt{\cosh (x)}\right ) \, dx &=\int \frac{x^2}{\cosh ^{\frac{3}{2}}(x)} \, dx+\int x^2 \sqrt{\cosh (x)} \, dx\\ &=-8 x \sqrt{\cosh (x)}+\frac{2 x^2 \sinh (x)}{\sqrt{\cosh (x)}}+8 \int \sqrt{\cosh (x)} \, dx\\ &=-8 x \sqrt{\cosh (x)}-16 i E\left (\left .\frac{i x}{2}\right |2\right )+\frac{2 x^2 \sinh (x)}{\sqrt{\cosh (x)}}\\ \end{align*}

Mathematica [C]  time = 0.179557, size = 76, normalized size = 2.11 \[ \frac{4 \sqrt{\cosh (x)} (\sinh (x)+\cosh (x)) \left (8 \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 x}\right ) (\sinh (x)-\cosh (x)) \sqrt{\sinh (2 x)+\cosh (2 x)+1}+x^2 \sinh (x)-4 (x-2) \cosh (x)\right )}{e^{2 x}+1} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/Cosh[x]^(3/2) + x^2*Sqrt[Cosh[x]],x]

[Out]

(4*Sqrt[Cosh[x]]*(Cosh[x] + Sinh[x])*(-4*(-2 + x)*Cosh[x] + x^2*Sinh[x] + 8*Hypergeometric2F1[-1/4, 1/2, 3/4,
-E^(2*x)]*(-Cosh[x] + Sinh[x])*Sqrt[1 + Cosh[2*x] + Sinh[2*x]]))/(1 + E^(2*x))

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Maple [F]  time = 0.063, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ( \cosh \left ( x \right ) \right ) ^{-{\frac{3}{2}}}}+{x}^{2}\sqrt{\cosh \left ( x \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sqrt{\cosh \left (x\right )} + \frac{x^{2}}{\cosh \left (x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*sqrt(cosh(x)) + x^2/cosh(x)^(3/2), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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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(x**2/cosh(x)**(3/2)+x**2*cosh(x)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sqrt{\cosh \left (x\right )} + \frac{x^{2}}{\cosh \left (x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*sqrt(cosh(x)) + x^2/cosh(x)^(3/2), x)