3.1.42 \(\int \frac {1}{x^2 (a+b \sin (c+d x^2))} \, dx\) [42]

Optimal. Leaf size=21 \[ \text {Int}\left (\frac {1}{x^2 \left (a+b \sin \left (c+d x^2\right )\right )},x\right ) \]

[Out]

Unintegrable(1/x^2/(a+b*sin(d*x^2+c)),x)

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Rubi [A]
time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{x^2 \left (a+b \sin \left (c+d x^2\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/(x^2*(a + b*Sin[c + d*x^2])),x]

[Out]

Defer[Int][1/(x^2*(a + b*Sin[c + d*x^2])), x]

Rubi steps

\begin {align*} \int \frac {1}{x^2 \left (a+b \sin \left (c+d x^2\right )\right )} \, dx &=\int \frac {1}{x^2 \left (a+b \sin \left (c+d x^2\right )\right )} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^2 \left (a+b \sin \left (c+d x^2\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[1/(x^2*(a + b*Sin[c + d*x^2])),x]

[Out]

Integrate[1/(x^2*(a + b*Sin[c + d*x^2])), x]

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Maple [A]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{2} \left (a +b \sin \left (d \,x^{2}+c \right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(a+b*sin(d*x^2+c)),x)

[Out]

int(1/x^2/(a+b*sin(d*x^2+c)),x)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*sin(d*x^2+c)),x, algorithm="maxima")

[Out]

integrate(1/((b*sin(d*x^2 + c) + a)*x^2), x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*sin(d*x^2+c)),x, algorithm="fricas")

[Out]

integral(1/(b*x^2*sin(d*x^2 + c) + a*x^2), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} \left (a + b \sin {\left (c + d x^{2} \right )}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(a+b*sin(d*x**2+c)),x)

[Out]

Integral(1/(x**2*(a + b*sin(c + d*x**2))), x)

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*sin(d*x^2+c)),x, algorithm="giac")

[Out]

integrate(1/((b*sin(d*x^2 + c) + a)*x^2), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {1}{x^2\,\left (a+b\,\sin \left (d\,x^2+c\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b*sin(c + d*x^2))),x)

[Out]

int(1/(x^2*(a + b*sin(c + d*x^2))), x)

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