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

Optimal. Leaf size=20 \[ \text {Int}\left (\frac {a^2 c x^2+c}{\tan ^{-1}(a x)^3},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.01, 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 = {} \[ \int \frac {c+a^2 c x^2}{\tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(c + a^2*c*x^2)/ArcTan[a*x]^3,x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.26, size = 0, normalized size = 0.00 \[ \int \frac {c+a^2 c x^2}{\tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} c x^{2} + c}{\arctan \left (a x\right )^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

sage0*x

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maple [A]  time = 1.46, size = 0, normalized size = 0.00 \[ \int \frac {a^{2} c \,x^{2}+c}{\arctan \left (a x \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + a^2*c*x^2)/atan(a*x)^3,x)

[Out]

int((c + a^2*c*x^2)/atan(a*x)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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