3.11 \(\int x^2 (a+b \csc (c+d x^2))^2 \, dx\)

Optimal. Leaf size=20 \[ \text{Unintegrable}\left (x^2 \left (a+b \csc \left (c+d x^2\right )\right )^2,x\right ) \]

[Out]

Unintegrable[x^2*(a + b*Csc[c + d*x^2])^2, x]

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Rubi [A]  time = 0.0225364, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int x^2 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^2*(a + b*Csc[c + d*x^2])^2,x]

[Out]

Defer[Int][x^2*(a + b*Csc[c + d*x^2])^2, x]

Rubi steps

\begin{align*} \int x^2 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx &=\int x^2 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx\\ \end{align*}

Mathematica [A]  time = 13.5025, size = 0, normalized size = 0. \[ \int x^2 \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x^2*(a + b*Csc[c + d*x^2])^2,x]

[Out]

Integrate[x^2*(a + b*Csc[c + d*x^2])^2, x]

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Maple [A]  time = 0.294, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b\csc \left ( d{x}^{2}+c \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*csc(d*x^2+c))^2,x)

[Out]

int(x^2*(a+b*csc(d*x^2+c))^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \, a^{2} x^{3} - \frac{b^{2} x \sin \left (2 \, d x^{2} + 2 \, c\right ) - \frac{1}{2} \,{\left (d \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + d \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} - 2 \, d \cos \left (2 \, d x^{2} + 2 \, c\right ) + d\right )} \int \frac{{\left (4 \, a b d x^{2} - b^{2}\right )} \sin \left (d x^{2} + c\right )}{d \cos \left (d x^{2} + c\right )^{2} + d \sin \left (d x^{2} + c\right )^{2} + 2 \, d \cos \left (d x^{2} + c\right ) + d}\,{d x} - \frac{1}{2} \,{\left (d \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + d \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} - 2 \, d \cos \left (2 \, d x^{2} + 2 \, c\right ) + d\right )} \int \frac{{\left (4 \, a b d x^{2} + b^{2}\right )} \sin \left (d x^{2} + c\right )}{d \cos \left (d x^{2} + c\right )^{2} + d \sin \left (d x^{2} + c\right )^{2} - 2 \, d \cos \left (d x^{2} + c\right ) + d}\,{d x}}{d \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + d \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} - 2 \, d \cos \left (2 \, d x^{2} + 2 \, c\right ) + d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*csc(d*x^2+c))^2,x, algorithm="maxima")

[Out]

1/3*a^2*x^3 - (b^2*x*sin(2*d*x^2 + 2*c) - (d*cos(2*d*x^2 + 2*c)^2 + d*sin(2*d*x^2 + 2*c)^2 - 2*d*cos(2*d*x^2 +
2*c) + d)*integrate(1/2*(4*a*b*d*x^2 - b^2)*sin(d*x^2 + c)/(d*cos(d*x^2 + c)^2 + d*sin(d*x^2 + c)^2 + 2*d*cos
(d*x^2 + c) + d), x) - (d*cos(2*d*x^2 + 2*c)^2 + d*sin(2*d*x^2 + 2*c)^2 - 2*d*cos(2*d*x^2 + 2*c) + d)*integrat
e(1/2*(4*a*b*d*x^2 + b^2)*sin(d*x^2 + c)/(d*cos(d*x^2 + c)^2 + d*sin(d*x^2 + c)^2 - 2*d*cos(d*x^2 + c) + d), x
))/(d*cos(2*d*x^2 + 2*c)^2 + d*sin(2*d*x^2 + 2*c)^2 - 2*d*cos(2*d*x^2 + 2*c) + d)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{2} x^{2} \csc \left (d x^{2} + c\right )^{2} + 2 \, a b x^{2} \csc \left (d x^{2} + c\right ) + a^{2} x^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*csc(d*x^2+c))^2,x, algorithm="fricas")

[Out]

integral(b^2*x^2*csc(d*x^2 + c)^2 + 2*a*b*x^2*csc(d*x^2 + c) + a^2*x^2, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \left (a + b \csc{\left (c + d x^{2} \right )}\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*csc(d*x**2+c))**2,x)

[Out]

Integral(x**2*(a + b*csc(c + d*x**2))**2, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \csc \left (d x^{2} + c\right ) + a\right )}^{2} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*csc(d*x^2+c))^2,x, algorithm="giac")

[Out]

integrate((b*csc(d*x^2 + c) + a)^2*x^2, x)