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

Optimal. Leaf size=22 \[ \text {Int}\left (\frac {\left (a^2 c x^2+c\right )^2}{\tan ^{-1}(a x)^3},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.02, 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 = {} \[ \int \frac {\left (c+a^2 c x^2\right )^2}{\tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.75, size = 0, normalized size = 0.00 \[ \int \frac {\left (c+a^2 c x^2\right )^2}{\tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}}{\arctan \left (a x\right )^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)/arctan(a*x)^3, 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((a^2*c*x^2+c)^2/arctan(a*x)^3,x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 1.69, size = 0, normalized size = 0.00 \[ \int \frac {\left (a^{2} c \,x^{2}+c \right )^{2}}{\arctan \left (a x \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {a^{6} c^{2} x^{6} + 3 \, a^{4} c^{2} x^{4} + 3 \, a^{2} c^{2} x^{2} - 6 \, a \arctan \left (a x\right )^{2} \int \frac {7 \, a^{6} c^{2} x^{6} + 15 \, a^{4} c^{2} x^{4} + 9 \, a^{2} c^{2} x^{2} + c^{2}}{\arctan \left (a x\right )}\,{d x} + c^{2} + 6 \, {\left (a^{7} c^{2} x^{7} + 3 \, a^{5} c^{2} x^{5} + 3 \, a^{3} c^{2} x^{3} + a c^{2} x\right )} \arctan \left (a x\right )}{2 \, a \arctan \left (a x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(a^6*c^2*x^6 + 3*a^4*c^2*x^4 + 3*a^2*c^2*x^2 - 2*a*arctan(a*x)^2*integrate(3*(7*a^6*c^2*x^6 + 15*a^4*c^2*
x^4 + 9*a^2*c^2*x^2 + c^2)/arctan(a*x), x) + c^2 + 6*(a^7*c^2*x^7 + 3*a^5*c^2*x^5 + 3*a^3*c^2*x^3 + a*c^2*x)*a
rctan(a*x))/(a*arctan(a*x)^2)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {{\left (c\,a^2\,x^2+c\right )}^2}{{\mathrm {atan}\left (a\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + a^2*c*x^2)^2/atan(a*x)^3,x)

[Out]

int((c + a^2*c*x^2)^2/atan(a*x)^3, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ c^{2} \left (\int \frac {2 a^{2} x^{2}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {a^{4} x^{4}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {1}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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