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

Optimal. Leaf size=135 \[ \frac{\text{Unintegrable}\left (\frac{x}{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3},x\right )}{a^2 c}+\frac{\sqrt{a^2 x^2+1} \text{Si}\left (\tan ^{-1}(a x)\right )}{2 a^4 c \sqrt{a^2 c x^2+c}}+\frac{x}{2 a^3 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}+\frac{1}{2 a^4 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)} \]

[Out]

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

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Rubi [A]  time = 0.50715, 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 = {} \[ \int \frac{x^3}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 3.37592, size = 0, normalized size = 0. \[ \int \frac{x^3}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.299, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{ \left ( \arctan \left ( ax \right ) \right ) ^{3}} \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{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)^(3/2)/arctan(a*x)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a^{2} c x^{2} + c} 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)^(3/2)/arctan(a*x)^3,x, algorithm="fricas")

[Out]

integral(sqrt(a^2*c*x^2 + c)*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*} \int \frac{x^{3}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}} \operatorname{atan}^{3}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3/((c*(a**2*x**2 + 1))**(3/2)*atan(a*x)**3), x)

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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 )}^{\frac{3}{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)^(3/2)/arctan(a*x)^3,x, algorithm="giac")

[Out]

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