3.7.21 \(\int \frac {x}{(c+a^2 c x^2) \text {ArcTan}(a x)^3} \, dx\) [621]

Optimal. Leaf size=37 \[ -\frac {x}{2 a c \text {ArcTan}(a x)^2}+\frac {\text {Int}\left (\frac {1}{\text {ArcTan}(a x)^2},x\right )}{2 a c} \]

[Out]

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

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Rubi [A]
time = 0.03, 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 {x}{\left (c+a^2 c x^2\right ) \text {ArcTan}(a x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.36, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\left (c+a^2 c x^2\right ) \text {ArcTan}(a x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {x}{\left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/(a**2*x**2*atan(a*x)**3 + atan(a*x)**3), 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(x/(a^2*c*x^2+c)/arctan(a*x)^3,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 {x}{{\mathrm {atan}\left (a\,x\right )}^3\,\left (c\,a^2\,x^2+c\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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