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]

-16*(cos(1/2*x)^2)^(1/2)/cos(1/2*x)*EllipticE(sin(1/2*x),2^(1/2))+2*x^2*sin(x)/cos(x)^(1/2)+8*x*cos(x)^(1/2)

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Rubi [A]  time = 0.09, antiderivative size = 32, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.15, 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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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \sqrt {\cos \relax (x)} + \frac {x^{2}}{\cos \relax (x)^{\frac {3}{2}}}\,{d x} \]

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)

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maple [F]  time = 0.26, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\cos \relax (x )^{\frac {3}{2}}}+x^{2} \left (\sqrt {\cos }\relax (x )\right )\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \sqrt {\cos \relax (x)} + \frac {x^{2}}{\cos \relax (x)^{\frac {3}{2}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int x^2\,\sqrt {\cos \relax (x)}+\frac {x^2}{{\cos \relax (x)}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \left (\cos ^{2}{\relax (x )} + 1\right )}{\cos ^{\frac {3}{2}}{\relax (x )}}\, dx \]

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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