3.6.41 \(\int \frac {(c+a^2 c x^2)^3}{\text {ArcTan}(a x)^2} \, dx\) [541]

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

[Out]

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

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 1.89, size = 0, normalized size = 0.00 \[\int \frac {\left (a^{2} c \,x^{2}+c \right )^{3}}{\arctan \left (a x \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((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 \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)^3/arctan(a*x)^2,x, algorithm="maxima")

[Out]

-(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 + c^3 - a*arctan(a*x)*integrate(8*(a^7*c^3*x^7 +
3*a^5*c^3*x^5 + 3*a^3*c^3*x^3 + a*c^3*x)/arctan(a*x), x))/(a*arctan(a*x))

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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)^3/arctan(a*x)^2,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)^2, x)

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

[Out]

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

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

[Out]

sage0*x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + a^2*c*x^2)^3/atan(a*x)^2,x)

[Out]

int((c + a^2*c*x^2)^3/atan(a*x)^2, x)

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