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

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {1}{x \left (a^2 c x^2+c\right ) \tan ^{-1}(a x)},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.07, 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 = {} \[ \int \frac {1}{x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (a^{2} c x^{3} + c x\right )} \arctan \left (a x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

sage0*x

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maple [A]  time = 0.25, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a^{2} c x^{2} + c\right )} x \arctan \left (a x\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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