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

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {x \left (a^2 c x^2+c\right )^3}{\tan ^{-1}(a x)^2},x\right ) \]

[Out]

Unintegrable(x*(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 = {} \[ \int \frac {x \left (c+a^2 c x^2\right )^3}{\tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (c+a^2 c x^2\right )^3}{\tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{6} c^{3} x^{7} + 3 \, a^{4} c^{3} x^{5} + 3 \, a^{2} c^{3} x^{3} + c^{3} x}{\arctan \left (a x\right )^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^6*c^3*x^7 + 3*a^4*c^3*x^5 + 3*a^2*c^3*x^3 + c^3*x)/arctan(a*x)^2, x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [A]  time = 2.09, size = 0, normalized size = 0.00 \[ \int \frac {x \left (a^{2} c \,x^{2}+c \right )^{3}}{\arctan \left (a x \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x*(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 \[ -\frac {a^{8} c^{3} x^{9} + 4 \, a^{6} c^{3} x^{7} + 6 \, a^{4} c^{3} x^{5} + 4 \, a^{2} c^{3} x^{3} + c^{3} x - \mathit {sage}_{0} x \arctan \left (a x\right )}{a \arctan \left (a x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a^8*c^3*x^9 + 4*a^6*c^3*x^7 + 6*a^4*c^3*x^5 + 4*a^2*c^3*x^3 + c^3*x - arctan(a*x)*integrate((9*a^8*c^3*x^8 +
28*a^6*c^3*x^6 + 30*a^4*c^3*x^4 + 12*a^2*c^3*x^2 + c^3)/arctan(a*x), x))/(a*arctan(a*x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ c^{3} \left (\int \frac {x}{\operatorname {atan}^{2}{\left (a x \right )}}\, dx + \int \frac {3 a^{2} x^{3}}{\operatorname {atan}^{2}{\left (a x \right )}}\, dx + \int \frac {3 a^{4} x^{5}}{\operatorname {atan}^{2}{\left (a x \right )}}\, dx + \int \frac {a^{6} x^{7}}{\operatorname {atan}^{2}{\left (a x \right )}}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**3*(Integral(x/atan(a*x)**2, x) + Integral(3*a**2*x**3/atan(a*x)**2, x) + Integral(3*a**4*x**5/atan(a*x)**2,
x) + Integral(a**6*x**7/atan(a*x)**2, x))

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