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

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {x^m \left (a^2 c x^2+c\right )^3}{\tan ^{-1}(a x)^3},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.06, 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 = {} \[ \int \frac {x^m \left (c+a^2 c x^2\right )^3}{\tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.86, size = 0, normalized size = 0.00 \[ \int \frac {x^m \left (c+a^2 c x^2\right )^3}{\tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}\right )} x^{m}}{\arctan \left (a x\right )^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^6*c^3*x^6 + 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 + c^3)*x^m/arctan(a*x)^3, x)

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

[Out]

Timed out

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maple [A]  time = 1.82, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \left (a^{2} c \,x^{2}+c \right )^{3}}{\arctan \left (a x \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ c^{3} \left (\int \frac {x^{m}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {3 a^{2} x^{2} x^{m}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {3 a^{4} x^{4} x^{m}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {a^{6} x^{6} x^{m}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**3*(Integral(x**m/atan(a*x)**3, x) + Integral(3*a**2*x**2*x**m/atan(a*x)**3, x) + Integral(3*a**4*x**4*x**m/
atan(a*x)**3, x) + Integral(a**6*x**6*x**m/atan(a*x)**3, x))

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