3.6.48 \(\int \frac {1}{x^2 (c+a^2 c x^2) \text {ArcTan}(a x)^2} \, dx\) [548]

Optimal. Leaf size=39 \[ -\frac {1}{a c x^2 \text {ArcTan}(a x)}-\frac {2 \text {Int}\left (\frac {1}{x^3 \text {ArcTan}(a x)},x\right )}{a c} \]

[Out]

-1/a/c/x^2/arctan(a*x)-2*Unintegrable(1/x^3/arctan(a*x),x)/a/c

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Rubi [A]
time = 0.06, 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^2 \left (c+a^2 c x^2\right ) \text {ArcTan}(a x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.47, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^2 \left (c+a^2 c x^2\right ) \text {ArcTan}(a x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.31, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{2} \left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

integral(1/((a^2*c*x^4 + c*x^2)*arctan(a*x)^2), 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^{4} \operatorname {atan}^{2}{\left (a x \right )} + x^{2} \operatorname {atan}^{2}{\left (a x \right )}}\, dx}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(a**2*x**4*atan(a*x)**2 + x**2*atan(a*x)**2), 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^2/(a^2*c*x^2+c)/arctan(a*x)^2,x, algorithm="giac")

[Out]

sage0*x

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{x^2\,{\mathrm {atan}\left (a\,x\right )}^2\,\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^2*atan(a*x)^2*(c + a^2*c*x^2)),x)

[Out]

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

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