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

Optimal. Leaf size=199 \[ \frac{a \text{Unintegrable}\left (\frac{1}{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right )}{2 c}+\frac{\text{Unintegrable}\left (\frac{1}{x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3},x\right )}{c}+\frac{a \sqrt{a^2 c x^2+c}}{2 c^2 x \tan ^{-1}(a x)^2}-\frac{a^2 \sqrt{a^2 x^2+1} \text{Si}\left (\tan ^{-1}(a x)\right )}{2 c \sqrt{a^2 c x^2+c}}-\frac{a^3 x}{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}-\frac{a^2}{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)} \]

[Out]

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

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Rubi [A]  time = 0.877665, 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 = {} \[ \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 3.13534, size = 0, normalized size = 0. \[ \int \frac{1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3),x]

[Out]

Integrate[1/(x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3), x]

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Maple [A]  time = 0.764, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{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(1/x^3/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^3,x)

[Out]

int(1/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{1}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} x^{3} \arctan \left (a x\right )^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((a^2*c*x^2 + c)^(3/2)*x^3*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}}{{\left (a^{4} c^{2} x^{7} + 2 \, a^{2} c^{2} x^{5} + c^{2} x^{3}\right )} \arctan \left (a x\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/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)/((a^4*c^2*x^7 + 2*a^2*c^2*x^5 + c^2*x^3)*arctan(a*x)^3), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{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(1/x**3/(a**2*c*x**2+c)**(3/2)/atan(a*x)**3,x)

[Out]

Integral(1/(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{1}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} x^{3} \arctan \left (a x\right )^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((a^2*c*x^2 + c)^(3/2)*x^3*arctan(a*x)^3), x)