3.7.16 \(\int \frac {x (c+a^2 c x^2)^3}{\text {ArcTan}(a x)^3} \, dx\) [616]

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {x \left (c+a^2 c x^2\right )^3}{\text {ArcTan}(a x)^3},x\right ) \]

[Out]

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

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.70, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (c+a^2 c x^2\right )^3}{\text {ArcTan}(a x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 3.78, size = 0, normalized size = 0.00 \[\int \frac {x \left (a^{2} c \,x^{2}+c \right )^{3}}{\arctan \left (a x \right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-1/2*(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 - 2*a^2*arctan(a*x)^2*integrate((4
5*a^8*c^3*x^9 + 148*a^6*c^3*x^7 + 174*a^4*c^3*x^5 + 84*a^2*c^3*x^3 + 13*c^3*x)/arctan(a*x), x) + (9*a^10*c^3*x
^10 + 37*a^8*c^3*x^8 + 58*a^6*c^3*x^6 + 42*a^4*c^3*x^4 + 13*a^2*c^3*x^2 + c^3)*arctan(a*x))/(a^2*arctan(a*x)^2
)

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

[Out]

integral((a^6*c^3*x^7 + 3*a^4*c^3*x^5 + 3*a^2*c^3*x^3 + c^3*x)/arctan(a*x)^3, x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} c^{3} \left (\int \frac {x}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {3 a^{2} x^{3}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {3 a^{4} x^{5}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx + \int \frac {a^{6} x^{7}}{\operatorname {atan}^{3}{\left (a x \right )}}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**3*(Integral(x/atan(a*x)**3, x) + Integral(3*a**2*x**3/atan(a*x)**3, x) + Integral(3*a**4*x**5/atan(a*x)**3,
x) + Integral(a**6*x**7/atan(a*x)**3, 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(x*(a^2*c*x^2+c)^3/arctan(a*x)^3,x, algorithm="giac")

[Out]

sage0*x

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x\,{\left (c\,a^2\,x^2+c\right )}^3}{{\mathrm {atan}\left (a\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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