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

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

[Out]

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

________________________________________________________________________________________

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

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 1.25, size = 0, normalized size = 0.00 \[ \int \frac {\left (c+a^2 c x^2\right )^3}{\tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}}{\arctan \left (a x\right )^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

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)^3/arctan(a*x)^3,x, algorithm="giac")

[Out]

sage0*x

________________________________________________________________________________________

maple [A]  time = 2.04, size = 0, normalized size = 0.00 \[ \int \frac {\left (a^{2} c \,x^{2}+c \right )^{3}}{\arctan \left (a x \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {a^{8} c^{3} x^{8} + 4 \, a^{6} c^{3} x^{6} + 6 \, a^{4} c^{3} x^{4} + 4 \, a^{2} c^{3} x^{2} - 8 \, a \arctan \left (a x\right )^{2} \int \frac {9 \, a^{8} c^{3} x^{8} + 28 \, a^{6} c^{3} x^{6} + 30 \, a^{4} c^{3} x^{4} + 12 \, a^{2} c^{3} x^{2} + c^{3}}{\arctan \left (a x\right )}\,{d x} + c^{3} + 8 \, {\left (a^{9} c^{3} x^{9} + 4 \, a^{7} c^{3} x^{7} + 6 \, a^{5} c^{3} x^{5} + 4 \, a^{3} c^{3} x^{3} + a c^{3} x\right )} \arctan \left (a x\right )}{2 \, a \arctan \left (a x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(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 - 2*a*arctan(a*x)^2*integrate(4*(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) + c^3 + 8*(a^9*c^3*x^9 + 4*a^7*c^
3*x^7 + 6*a^5*c^3*x^5 + 4*a^3*c^3*x^3 + a*c^3*x)*arctan(a*x))/(a*arctan(a*x)^2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + a^2*c*x^2)^3/atan(a*x)^3,x)

[Out]

int((c + a^2*c*x^2)^3/atan(a*x)^3, x)

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ c^{3} \left (\int \frac {3 a^{2} x^{2}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {3 a^{4} x^{4}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {a^{6} x^{6}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {1}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________