Optimal. Leaf size=130 \[ \frac{\text{Unintegrable}\left (\frac{1}{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3},x\right )}{c}+\frac{a \sqrt{a^2 x^2+1} \text{CosIntegral}\left (\tan ^{-1}(a x)\right )}{2 c \sqrt{a^2 c x^2+c}}-\frac{a^2 x}{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}+\frac{a}{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.462229, 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 = {} \[ \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3} \, dx &=-\left (a^2 \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3} \, dx\right )+\frac{\int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx}{c}\\ &=\frac{a}{2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}+\frac{1}{2} a^3 \int \frac{x}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2} \, dx+\frac{\int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx}{c}\\ &=\frac{a}{2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac{a^2 x}{2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}+\frac{1}{2} a^2 \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx+\frac{\int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx}{c}\\ &=\frac{a}{2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac{a^2 x}{2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}+\frac{\int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx}{c}+\frac{\left (a^2 \sqrt{1+a^2 x^2}\right ) \int \frac{1}{\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}{2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac{a^2 x}{2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}+\frac{\int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx}{c}+\frac{\left (a \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 c \sqrt{c+a^2 c x^2}}\\ &=\frac{a}{2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac{a^2 x}{2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}+\frac{a \sqrt{1+a^2 x^2} \text{Ci}\left (\tan ^{-1}(a x)\right )}{2 c \sqrt{c+a^2 c x^2}}+\frac{\int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx}{c}\\ \end{align*}
Mathematica [A] time = 2.47986, size = 0, normalized size = 0. \[ \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.595, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \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]
[Out]
________________________________________________________________________________________
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^{2} \arctan \left (a x\right )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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^{6} + 2 \, a^{2} c^{2} x^{4} + c^{2} x^{2}\right )} \arctan \left (a x\right )^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \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]
[Out]
________________________________________________________________________________________
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^{2} \arctan \left (a x\right )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]