3.40 \(\int \frac{x^2}{a+b \sin (c+d x^2)} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0288569, 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 = {} \[ \int \frac{x^2}{a+b \sin \left (c+d x^2\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^2/(a + b*Sin[c + d*x^2]),x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.364938, size = 0, normalized size = 0. \[ \int \frac{x^2}{a+b \sin \left (c+d x^2\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.035, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{a+b\sin \left ( d{x}^{2}+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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