3.93 \(\int (\frac{x^2}{\cos ^{\frac{3}{2}}(x)}+x^2 \sqrt{\cos (x)}) \, dx\)

Optimal. Leaf size=32 \[ \frac{2 x^2 \sin (x)}{\sqrt{\cos (x)}}+8 x \sqrt{\cos (x)}-16 E\left (\left .\frac{x}{2}\right |2\right ) \]

[Out]

8*x*Sqrt[Cos[x]] - 16*EllipticE[x/2, 2] + (2*x^2*Sin[x])/Sqrt[Cos[x]]

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Rubi [A]  time = 0.0859179, antiderivative size = 32, 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} \[ \frac{2 x^2 \sin (x)}{\sqrt{\cos (x)}}+8 x \sqrt{\cos (x)}-16 E\left (\left .\frac{x}{2}\right |2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

8*x*Sqrt[Cos[x]] - 16*EllipticE[x/2, 2] + (2*x^2*Sin[x])/Sqrt[Cos[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}{\cos ^{\frac{3}{2}}(x)}+x^2 \sqrt{\cos (x)}\right ) \, dx &=\int \frac{x^2}{\cos ^{\frac{3}{2}}(x)} \, dx+\int x^2 \sqrt{\cos (x)} \, dx\\ &=8 x \sqrt{\cos (x)}+\frac{2 x^2 \sin (x)}{\sqrt{\cos (x)}}-8 \int \sqrt{\cos (x)} \, dx\\ &=8 x \sqrt{\cos (x)}-16 E\left (\left .\frac{x}{2}\right |2\right )+\frac{2 x^2 \sin (x)}{\sqrt{\cos (x)}}\\ \end{align*}

Mathematica [A]  time = 0.145364, size = 29, normalized size = 0.91 \[ 2 \left (\frac{x (x \sin (x)+4 \cos (x))}{\sqrt{\cos (x)}}-8 E\left (\left .\frac{x}{2}\right |2\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*(-8*EllipticE[x/2, 2] + (x*(4*Cos[x] + x*Sin[x]))/Sqrt[Cos[x]])

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Maple [F]  time = 0.226, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ( \cos \left ( x \right ) \right ) ^{-{\frac{3}{2}}}}+{x}^{2}\sqrt{\cos \left ( x \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \left (\cos ^{2}{\left (x \right )} + 1\right )}{\cos ^{\frac{3}{2}}{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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