3.18 \(\int \frac{1}{x^2 (a+b \tan (c+d x^2))} \, dx\)

Optimal. Leaf size=20 \[ \text{Unintegrable}\left (\frac{1}{x^2 \left (a+b \tan \left (c+d x^2\right )\right )},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0242795, 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 = {} \[ \int \frac{1}{x^2 \left (a+b \tan \left (c+d x^2\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x^2*(a + b*Tan[c + d*x^2])),x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 2.08864, size = 0, normalized size = 0. \[ \int \frac{1}{x^2 \left (a+b \tan \left (c+d x^2\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.189, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ( a+b\tan \left ( d{x}^{2}+c \right ) \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b x^{2} \tan \left (d x^{2} + c\right ) + a x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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