3.627 \(\int \frac{x^3}{(c+a^2 c x^2)^2 \tan ^{-1}(a x)^3} \, dx\)

Optimal. Leaf size=115 \[ \frac{\text{Unintegrable}\left (\frac{1}{\tan ^{-1}(a x)^2},x\right )}{2 a^3 c^2}+\frac{\text{Si}\left (2 \tan ^{-1}(a x)\right )}{a^4 c^2}+\frac{x}{2 a^3 c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}+\frac{1-a^2 x^2}{2 a^4 c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}-\frac{x}{2 a^3 c^2 \tan ^{-1}(a x)^2} \]

[Out]

-x/(2*a^3*c^2*ArcTan[a*x]^2) + x/(2*a^3*c^2*(1 + a^2*x^2)*ArcTan[a*x]^2) + (1 - a^2*x^2)/(2*a^4*c^2*(1 + a^2*x
^2)*ArcTan[a*x]) + SinIntegral[2*ArcTan[a*x]]/(a^4*c^2) + Unintegrable[ArcTan[a*x]^(-2), x]/(2*a^3*c^2)

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Rubi [A]  time = 0.243868, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{x^3}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-x/(2*a^3*c^2*ArcTan[a*x]^2) + x/(2*a^3*c^2*(1 + a^2*x^2)*ArcTan[a*x]^2) + (1 - a^2*x^2)/(2*a^4*c^2*(1 + a^2*x
^2)*ArcTan[a*x]) + SinIntegral[2*ArcTan[a*x]]/(a^4*c^2) + Defer[Int][ArcTan[a*x]^(-2), x]/(2*a^3*c^2)

Rubi steps

\begin{align*} \int \frac{x^3}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx &=-\frac{\int \frac{x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx}{a^2}+\frac{\int \frac{x}{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3} \, dx}{a^2 c}\\ &=-\frac{x}{2 a^3 c^2 \tan ^{-1}(a x)^2}+\frac{x}{2 a^3 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac{1-a^2 x^2}{2 a^4 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{2 \int \frac{x}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)} \, dx}{a^2}+\frac{\int \frac{1}{\tan ^{-1}(a x)^2} \, dx}{2 a^3 c^2}\\ &=-\frac{x}{2 a^3 c^2 \tan ^{-1}(a x)^2}+\frac{x}{2 a^3 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac{1-a^2 x^2}{2 a^4 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{2 \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^2}+\frac{\int \frac{1}{\tan ^{-1}(a x)^2} \, dx}{2 a^3 c^2}\\ &=-\frac{x}{2 a^3 c^2 \tan ^{-1}(a x)^2}+\frac{x}{2 a^3 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac{1-a^2 x^2}{2 a^4 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{2 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 x} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^2}+\frac{\int \frac{1}{\tan ^{-1}(a x)^2} \, dx}{2 a^3 c^2}\\ &=-\frac{x}{2 a^3 c^2 \tan ^{-1}(a x)^2}+\frac{x}{2 a^3 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac{1-a^2 x^2}{2 a^4 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\sin (2 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^2}+\frac{\int \frac{1}{\tan ^{-1}(a x)^2} \, dx}{2 a^3 c^2}\\ &=-\frac{x}{2 a^3 c^2 \tan ^{-1}(a x)^2}+\frac{x}{2 a^3 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}+\frac{1-a^2 x^2}{2 a^4 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}+\frac{\text{Si}\left (2 \tan ^{-1}(a x)\right )}{a^4 c^2}+\frac{\int \frac{1}{\tan ^{-1}(a x)^2} \, dx}{2 a^3 c^2}\\ \end{align*}

Mathematica [A]  time = 9.38248, size = 0, normalized size = 0. \[ \int \frac{x^3}{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.525, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{ \left ({a}^{2}c{x}^{2}+c \right ) ^{2} \left ( \arctan \left ( ax \right ) \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a x^{3} +{\left (a^{2} x^{4} + 3 \, x^{2}\right )} \arctan \left (a x\right ) - \frac{2 \,{\left (a^{4} c^{2} x^{2} + a^{2} c^{2}\right )}{\left (a^{4} \int \frac{x^{5}}{a^{4} x^{4} \arctan \left (a x\right ) + 2 \, a^{2} x^{2} \arctan \left (a x\right ) + \arctan \left (a x\right )}\,{d x} + 2 \, a^{2} \int \frac{x^{3}}{a^{4} x^{4} \arctan \left (a x\right ) + 2 \, a^{2} x^{2} \arctan \left (a x\right ) + \arctan \left (a x\right )}\,{d x} + 3 \, \int \frac{x}{a^{4} x^{4} \arctan \left (a x\right ) + 2 \, a^{2} x^{2} \arctan \left (a x\right ) + \arctan \left (a x\right )}\,{d x}\right )} \arctan \left (a x\right )^{2}}{a^{2} c^{2}}}{2 \,{\left (a^{4} c^{2} x^{2} + a^{2} c^{2}\right )} \arctan \left (a x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(a*x^3 - 2*(a^4*c^2*x^2 + a^2*c^2)*arctan(a*x)^2*integrate((a^4*x^5 + 2*a^2*x^3 + 3*x)/((a^6*c^2*x^4 + 2*
a^4*c^2*x^2 + a^2*c^2)*arctan(a*x)), x) + (a^2*x^4 + 3*x^2)*arctan(a*x))/((a^4*c^2*x^2 + a^2*c^2)*arctan(a*x)^
2)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{3}}{{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \arctan \left (a x\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^3/((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)*arctan(a*x)^3), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{x^{3}}{a^{4} x^{4} \operatorname{atan}^{3}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname{atan}^{3}{\left (a x \right )} + \operatorname{atan}^{3}{\left (a x \right )}}\, dx}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3/(a**4*x**4*atan(a*x)**3 + 2*a**2*x**2*atan(a*x)**3 + atan(a*x)**3), x)/c**2

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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