3.5.65 \(\int \frac {x^m \text {ArcTan}(a x)^3}{(c+a^2 c x^2)^{3/2}} \, dx\) [465]

Optimal. Leaf size=27 \[ \text {Int}\left (\frac {x^m \text {ArcTan}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.08, 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 {x^m \text {ArcTan}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.46, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^m \text {ArcTan}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 2.17, size = 0, normalized size = 0.00 \[\int \frac {x^{m} \arctan \left (a x \right )^{3}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

integral(sqrt(a^2*c*x^2 + c)*x^m*arctan(a*x)^3/(a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{m} \operatorname {atan}^{3}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**m*atan(a*x)**3/(c*(a**2*x**2 + 1))**(3/2), 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(x^m*arctan(a*x)^3/(a^2*c*x^2+c)^(3/2),x, algorithm="giac")

[Out]

sage0*x

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x^m\,{\mathrm {atan}\left (a\,x\right )}^3}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^m*atan(a*x)^3)/(c + a^2*c*x^2)^(3/2),x)

[Out]

int((x^m*atan(a*x)^3)/(c + a^2*c*x^2)^(3/2), x)

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