3.5.78 \(\int \frac {1}{x (c+a^2 c x^2) \text {ArcTan}(a x)} \, dx\) [478]

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

[Out]

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

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Rubi [A]
time = 0.05, 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 {1}{x \left (c+a^2 c x^2\right ) \text {ArcTan}(a x)} \, dx \end {gather*}

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 = 0.14, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (c+a^2 c x^2\right ) \text {ArcTan}(a x)} \, dx \end {gather*}

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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Maple [A]
time = 3.19, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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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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(1/x/(a^2*c*x^2+c)/arctan(a*x),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 {1}{x\,\mathrm {atan}\left (a\,x\right )\,\left (c\,a^2\,x^2+c\right )} \,d x \end {gather*}

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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