3.131 \(\int \frac {(d x)^m}{(a+b \tan ^{-1}(c x^3))^2} \, dx\)

Optimal. Leaf size=21 \[ \text {Int}\left (\frac {(d x)^m}{\left (a+b \tan ^{-1}\left (c x^3\right )\right )^2},x\right ) \]

[Out]

Unintegrable((d*x)^m/(a+b*arctan(c*x^3))^2,x)

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Rubi [A]  time = 0.03, 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 = {} \[ \int \frac {(d x)^m}{\left (a+b \tan ^{-1}\left (c x^3\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(d*x)^m/(a + b*ArcTan[c*x^3])^2,x]

[Out]

Defer[Int][(d*x)^m/(a + b*ArcTan[c*x^3])^2, x]

Rubi steps

\begin {align*} \int \frac {(d x)^m}{\left (a+b \tan ^{-1}\left (c x^3\right )\right )^2} \, dx &=\int \frac {(d x)^m}{\left (a+b \tan ^{-1}\left (c x^3\right )\right )^2} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.36, size = 0, normalized size = 0.00 \[ \int \frac {(d x)^m}{\left (a+b \tan ^{-1}\left (c x^3\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(d*x)^m/(a + b*ArcTan[c*x^3])^2,x]

[Out]

Integrate[(d*x)^m/(a + b*ArcTan[c*x^3])^2, x]

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fricas [A]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (d x\right )^{m}}{b^{2} \arctan \left (c x^{3}\right )^{2} + 2 \, a b \arctan \left (c x^{3}\right ) + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m/(a+b*arctan(c*x^3))^2,x, algorithm="fricas")

[Out]

integral((d*x)^m/(b^2*arctan(c*x^3)^2 + 2*a*b*arctan(c*x^3) + a^2), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x\right )^{m}}{{\left (b \arctan \left (c x^{3}\right ) + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m/(a+b*arctan(c*x^3))^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x \right )^{m}}{\left (a +b \arctan \left (c \,x^{3}\right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m/(a+b*arctan(c*x^3))^2,x)

[Out]

int((d*x)^m/(a+b*arctan(c*x^3))^2,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (c^{2} d^{m} x^{6} + d^{m}\right )} x^{m} - {\left (b^{2} c x^{2} \arctan \left (c x^{3}\right ) + a b c x^{2}\right )} \int \frac {{\left ({\left (c^{2} d^{m} m + 4 \, c^{2} d^{m}\right )} x^{6} + d^{m} m - 2 \, d^{m}\right )} x^{m}}{b^{2} c x^{3} \arctan \left (c x^{3}\right ) + a b c x^{3}}\,{d x}}{3 \, {\left (b^{2} c x^{2} \arctan \left (c x^{3}\right ) + a b c x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m/(a+b*arctan(c*x^3))^2,x, algorithm="maxima")

[Out]

-1/3*((c^2*d^m*x^6 + d^m)*x^m - 3*(b^2*c*x^2*arctan(c*x^3) + a*b*c*x^2)*integrate(1/3*((c^2*d^m*m + 4*c^2*d^m)
*x^6 + d^m*m - 2*d^m)*x^m/(b^2*c*x^3*arctan(c*x^3) + a*b*c*x^3), x))/(b^2*c*x^2*arctan(c*x^3) + a*b*c*x^2)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {{\left (d\,x\right )}^m}{{\left (a+b\,\mathrm {atan}\left (c\,x^3\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m/(a + b*atan(c*x^3))^2,x)

[Out]

int((d*x)^m/(a + b*atan(c*x^3))^2, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m/(a+b*atan(c*x**3))**2,x)

[Out]

Timed out

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