3.1.95 \(\int \frac {(d x)^m}{(a+b \text {ArcTan}(c x^2))^2} \, dx\) [95]

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

[Out]

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

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Rubi [A]
time = 0.02, 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 {(d x)^m}{\left (a+b \text {ArcTan}\left (c x^2\right )\right )^2} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 0.23, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d x)^m}{\left (a+b \text {ArcTan}\left (c x^2\right )\right )^2} \, dx \end {gather*}

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 = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (d x \right )^{m}}{\left (a +b \arctan \left (c \,x^{2}\right )\right )^{2}}\, dx\]

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 = 0.00, 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((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 = 0.00, 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((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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, 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((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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m/(a + b*atan(c*x^2))^2,x)

[Out]

int((d*x)^m/(a + b*atan(c*x^2))^2, x)

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