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

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

[Out]

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

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Rubi [A]  time = 0.0255398, 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 = {} \[ \int \frac{(d x)^m}{\left (a+b \tan ^{-1}\left (c x^2\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.32898, size = 0, normalized size = 0. \[ \int \frac{(d x)^m}{\left (a+b \tan ^{-1}\left (c x^2\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.171, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx \right ) ^{m}}{ \left ( a+b\arctan \left ( c{x}^{2} \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((c^2*d^m*x^4 + d^m)*x^m - 2*(b^2*c*x*arctan(c*x^2) + a*b*c*x)*integrate(1/2*((c^2*d^m*m + 3*c^2*d^m)*x^4
+ d^m*m - d^m)*x^m/(b^2*c*x^2*arctan(c*x^2) + a*b*c*x^2), x))/(b^2*c*x*arctan(c*x^2) + a*b*c*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [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((d*x)**m/(a+b*atan(c*x**2))**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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