3.68 \(\int (\frac{x^2}{\sin ^{\frac{3}{2}}(e+f x)}+x^2 \sqrt{\sin (e+f x)}) \, dx\)

Optimal. Leaf size=62 \[ \frac{8 x \sqrt{\sin (e+f x)}}{f^2}-\frac{16 E\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right )}{f^3}-\frac{2 x^2 \cos (e+f x)}{f \sqrt{\sin (e+f x)}} \]

[Out]

(-16*EllipticE[(e - Pi/2 + f*x)/2, 2])/f^3 - (2*x^2*Cos[e + f*x])/(f*Sqrt[Sin[e + f*x]]) + (8*x*Sqrt[Sin[e + f
*x]])/f^2

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Rubi [A]  time = 0.106662, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {3316, 2639} \[ \frac{8 x \sqrt{\sin (e+f x)}}{f^2}-\frac{16 E\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right )}{f^3}-\frac{2 x^2 \cos (e+f x)}{f \sqrt{\sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/Sin[e + f*x]^(3/2) + x^2*Sqrt[Sin[e + f*x]],x]

[Out]

(-16*EllipticE[(e - Pi/2 + f*x)/2, 2])/f^3 - (2*x^2*Cos[e + f*x])/(f*Sqrt[Sin[e + f*x]]) + (8*x*Sqrt[Sin[e + f
*x]])/f^2

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}{\sin ^{\frac{3}{2}}(e+f x)}+x^2 \sqrt{\sin (e+f x)}\right ) \, dx &=\int \frac{x^2}{\sin ^{\frac{3}{2}}(e+f x)} \, dx+\int x^2 \sqrt{\sin (e+f x)} \, dx\\ &=-\frac{2 x^2 \cos (e+f x)}{f \sqrt{\sin (e+f x)}}+\frac{8 x \sqrt{\sin (e+f x)}}{f^2}-\frac{8 \int \sqrt{\sin (e+f x)} \, dx}{f^2}\\ &=-\frac{16 E\left (\left .\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )\right |2\right )}{f^3}-\frac{2 x^2 \cos (e+f x)}{f \sqrt{\sin (e+f x)}}+\frac{8 x \sqrt{\sin (e+f x)}}{f^2}\\ \end{align*}

Mathematica [C]  time = 4.19399, size = 185, normalized size = 2.98 \[ -\frac{\sec (e) \left (\left (f^2 x^2-8\right ) \cos (2 e+f x)-8 f x \cos (e) \sin (e+f x)+\left (f^2 x^2+8\right ) \cos (f x)\right )}{f^3 \sqrt{\sin (e+f x)}}+\frac{8 \sec (e) e^{-i f x} \sqrt{2-2 e^{2 i (e+f x)}} \left (3 \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};e^{2 i (e+f x)}\right )+e^{2 i f x} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};e^{2 i (e+f x)}\right )\right )}{3 f^3 \sqrt{-i e^{-i (e+f x)} \left (-1+e^{2 i (e+f x)}\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sin[e + f*x]^(3/2) + x^2*Sqrt[Sin[e + f*x]],x]

[Out]

(8*Sqrt[2 - 2*E^((2*I)*(e + f*x))]*(3*Hypergeometric2F1[-1/4, 1/2, 3/4, E^((2*I)*(e + f*x))] + E^((2*I)*f*x)*H
ypergeometric2F1[1/2, 3/4, 7/4, E^((2*I)*(e + f*x))])*Sec[e])/(3*E^(I*f*x)*Sqrt[((-I)*(-1 + E^((2*I)*(e + f*x)
)))/E^(I*(e + f*x))]*f^3) - (Sec[e]*((8 + f^2*x^2)*Cos[f*x] + (-8 + f^2*x^2)*Cos[2*e + f*x] - 8*f*x*Cos[e]*Sin
[e + f*x]))/(f^3*Sqrt[Sin[e + f*x]])

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Maple [F]  time = 0.103, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}+{x}^{2}\sqrt{\sin \left ( fx+e \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/sin(f*x+e)^(3/2)+x^2*sin(f*x+e)^(1/2),x)

[Out]

int(x^2/sin(f*x+e)^(3/2)+x^2*sin(f*x+e)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*sqrt(sin(f*x + e)) + x^2/sin(f*x + e)^(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/sin(f*x+e)^(3/2)+x^2*sin(f*x+e)^(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 (\sin ^{2}{\left (e + f x \right )} + 1\right )}{\sin ^{\frac{3}{2}}{\left (e + f x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/sin(f*x+e)**(3/2)+x**2*sin(f*x+e)**(1/2),x)

[Out]

Integral(x**2*(sin(e + f*x)**2 + 1)/sin(e + f*x)**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*sqrt(sin(f*x + e)) + x^2/sin(f*x + e)^(3/2), x)