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

Optimal. Leaf size=32 \[ \frac {\text {Int}\left (\frac {1}{\tan ^{-1}(a x)},x\right )}{a c}-\frac {x}{a c \tan ^{-1}(a x)} \]

[Out]

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

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

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.30, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x}{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

sage0*x

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maple [A]  time = 0.24, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x/(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 \[ \frac {\mathit {sage}_{0} x \arctan \left (a x\right ) - x}{a c \arctan \left (a x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(arctan(a*x)*integrate(1/arctan(a*x), x) - x)/(a*c*arctan(a*x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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