\(\int (\frac {x^2}{\cosh ^{\frac {3}{2}}(x)}+x^2 \sqrt {\cosh (x)}) \, dx\) [74]

Optimal result

Integrand size = 21, antiderivative size = 36 \[ \int \left (\frac {x^2}{\cosh ^{\frac {3}{2}}(x)}+x^2 \sqrt {\cosh (x)}\right ) \, dx=-8 x \sqrt {\cosh (x)}-16 i E\left (\left .\frac {i x}{2}\right |2\right )+\frac {2 x^2 \sinh (x)}{\sqrt {\cosh (x)}} \]

[Out]

-16*I*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticE(I*sinh(1/2*x),2^(1/2))+2*x^2*sinh(x)/cosh(x)^(1/2)-8*x*cosh(
x)^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3397, 2719} \[ \int \left (\frac {x^2}{\cosh ^{\frac {3}{2}}(x)}+x^2 \sqrt {\cosh (x)}\right ) \, dx=\frac {2 x^2 \sinh (x)}{\sqrt {\cosh (x)}}-8 x \sqrt {\cosh (x)}-16 i E\left (\left .\frac {i x}{2}\right |2\right ) \]

[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 2719

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

Rule 3397

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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.78 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.11 \[ \int \left (\frac {x^2}{\cosh ^{\frac {3}{2}}(x)}+x^2 \sqrt {\cosh (x)}\right ) \, dx=\frac {4 \sqrt {\cosh (x)} (\cosh (x)+\sinh (x)) \left (-4 (-2+x) \cosh (x)+x^2 \sinh (x)+8 \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-e^{2 x}\right ) (-\cosh (x)+\sinh (x)) \sqrt {1+\cosh (2 x)+\sinh (2 x)}\right )}{1+e^{2 x}} \]

[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))

Maple [F]

\[\int \left (\frac {x^{2}}{\cosh \left (x \right )^{\frac {3}{2}}}+x^{2} \sqrt {\cosh \left (x \right )}\right )d x\]

[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)

Fricas [F(-2)]

Exception generated. \[ \int \left (\frac {x^2}{\cosh ^{\frac {3}{2}}(x)}+x^2 \sqrt {\cosh (x)}\right ) \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [F]

\[ \int \left (\frac {x^2}{\cosh ^{\frac {3}{2}}(x)}+x^2 \sqrt {\cosh (x)}\right ) \, dx=\int \frac {x^{2} \left (\cosh ^{2}{\left (x \right )} + 1\right )}{\cosh ^{\frac {3}{2}}{\left (x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \left (\frac {x^2}{\cosh ^{\frac {3}{2}}(x)}+x^2 \sqrt {\cosh (x)}\right ) \, dx=\int { x^{2} \sqrt {\cosh \left (x\right )} + \frac {x^{2}}{\cosh \left (x\right )^{\frac {3}{2}}} \,d x } \]

[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)

Giac [F]

\[ \int \left (\frac {x^2}{\cosh ^{\frac {3}{2}}(x)}+x^2 \sqrt {\cosh (x)}\right ) \, dx=\int { x^{2} \sqrt {\cosh \left (x\right )} + \frac {x^{2}}{\cosh \left (x\right )^{\frac {3}{2}}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \left (\frac {x^2}{\cosh ^{\frac {3}{2}}(x)}+x^2 \sqrt {\cosh (x)}\right ) \, dx=\int x^2\,\sqrt {\mathrm {cosh}\left (x\right )}+\frac {x^2}{{\mathrm {cosh}\left (x\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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