3.7.14 \(\int \frac {(c+a^2 c x^2)^2}{\text {ArcTan}(a x)^3} \, dx\) [614]

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

[Out]

Unintegrable((a^2*c*x^2+c)^2/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 = {} \begin {gather*} \int \frac {\left (c+a^2 c x^2\right )^2}{\text {ArcTan}(a x)^3} \, dx \end {gather*}

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*}

________________________________________________________________________________________

Mathematica [A]
time = 0.50, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c+a^2 c x^2\right )^2}{\text {ArcTan}(a x)^3} \, dx \end {gather*}

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]

________________________________________________________________________________________

Maple [A]
time = 2.27, 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)

________________________________________________________________________________________

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((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)

________________________________________________________________________________________

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((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)

________________________________________________________________________________________

Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 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 ) \end {gather*}

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))

________________________________________________________________________________________

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

[Out]

sage0*x

________________________________________________________________________________________

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

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)

________________________________________________________________________________________