3.5.81 \(\int \frac {x^3}{(c+a^2 c x^2)^2 \text {ArcTan}(a x)} \, dx\) [481]

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {x^3}{\left (c+a^2 c x^2\right )^2 \text {ArcTan}(a x)},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.05, 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^3}{\left (c+a^2 c x^2\right )^2 \text {ArcTan}(a x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 5.33, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \text {ArcTan}(a x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 3.61, size = 0, normalized size = 0.00 \[\int \frac {x^{3}}{\left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {x^{3}}{a^{4} x^{4} \operatorname {atan}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname {atan}{\left (a x \right )} + \operatorname {atan}{\left (a x \right )}}\, dx}{c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3/(a**4*x**4*atan(a*x) + 2*a**2*x**2*atan(a*x) + atan(a*x)), x)/c**2

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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^3/(a^2*c*x^2+c)^2/arctan(a*x),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^3}{\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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