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

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{1}{x \left (a^2 c x^2+c\right )^3 \tan ^{-1}(a x)},x\right ) \]

[Out]

Unintegrable[1/(x*(c + a^2*c*x^2)^3*ArcTan[a*x]), x]

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Rubi [A]  time = 0.064984, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{x \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x*(c + a^2*c*x^2)^3*ArcTan[a*x]),x]

[Out]

Defer[Int][1/(x*(c + a^2*c*x^2)^3*ArcTan[a*x]), x]

Rubi steps

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

Mathematica [A]  time = 0.877712, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x*(c + a^2*c*x^2)^3*ArcTan[a*x]),x]

[Out]

Integrate[1/(x*(c + a^2*c*x^2)^3*ArcTan[a*x]), x]

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Maple [A]  time = 0.323, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ({a}^{2}c{x}^{2}+c \right ) ^{3}\arctan \left ( ax \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(a^2*c*x^2+c)^3/arctan(a*x),x)

[Out]

int(1/x/(a^2*c*x^2+c)^3/arctan(a*x),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} c x^{2} + c\right )}^{3} x \arctan \left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a^2*c*x^2+c)^3/arctan(a*x),x, algorithm="maxima")

[Out]

integrate(1/((a^2*c*x^2 + c)^3*x*arctan(a*x)), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{{\left (a^{6} c^{3} x^{7} + 3 \, a^{4} c^{3} x^{5} + 3 \, a^{2} c^{3} x^{3} + c^{3} x\right )} \arctan \left (a x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a^2*c*x^2+c)^3/arctan(a*x),x, algorithm="fricas")

[Out]

integral(1/((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)), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{a^{6} x^{7} \operatorname{atan}{\left (a x \right )} + 3 a^{4} x^{5} \operatorname{atan}{\left (a x \right )} + 3 a^{2} x^{3} \operatorname{atan}{\left (a x \right )} + x \operatorname{atan}{\left (a x \right )}}\, dx}{c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a**2*c*x**2+c)**3/atan(a*x),x)

[Out]

Integral(1/(a**6*x**7*atan(a*x) + 3*a**4*x**5*atan(a*x) + 3*a**2*x**3*atan(a*x) + x*atan(a*x)), x)/c**3

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} c x^{2} + c\right )}^{3} x \arctan \left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a^2*c*x^2+c)^3/arctan(a*x),x, algorithm="giac")

[Out]

integrate(1/((a^2*c*x^2 + c)^3*x*arctan(a*x)), x)