3.6.98 \(\int \frac {x^m (c+a^2 c x^2)^3}{\text {ArcTan}(a x)^2} \, dx\) [598]

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

[Out]

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

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Rubi [A]
time = 0.04, 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 \left (c+a^2 c x^2\right )^3}{\text {ArcTan}(a x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.49, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^m \left (c+a^2 c x^2\right )^3}{\text {ArcTan}(a x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.80, size = 0, normalized size = 0.00 \[\int \frac {x^{m} \left (a^{2} c \,x^{2}+c \right )^{3}}{\arctan \left (a x \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-((a^8*c^3*x^8 + 4*a^6*c^3*x^6 + 6*a^4*c^3*x^4 + 4*a^2*c^3*x^2 + c^3)*x^m - arctan(a*x)*integrate(((a^8*c^3*m
+ 8*a^8*c^3)*x^8 + 4*(a^6*c^3*m + 6*a^6*c^3)*x^6 + 6*(a^4*c^3*m + 4*a^4*c^3)*x^4 + c^3*m + 4*(a^2*c^3*m + 2*a^
2*c^3)*x^2)*x^m/(x*arctan(a*x)), x))/(a*arctan(a*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*(a^2*c*x^2+c)^3/arctan(a*x)^2,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)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**3*(Integral(x**m/atan(a*x)**2, x) + Integral(3*a**2*x**2*x**m/atan(a*x)**2, x) + Integral(3*a**4*x**4*x**m/
atan(a*x)**2, x) + Integral(a**6*x**6*x**m/atan(a*x)**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*(a^2*c*x^2+c)^3/arctan(a*x)^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\,{\left (c\,a^2\,x^2+c\right )}^3}{{\mathrm {atan}\left (a\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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