Optimal. Leaf size=138 \[ \frac{3}{2} i a^2 c^3 \text{PolyLog}(2,-i a x)-\frac{3}{2} i a^2 c^3 \text{PolyLog}(2,i a x)-\frac{1}{12} a^5 c^3 x^3+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)-\frac{5}{4} a^3 c^3 x+\frac{3}{4} a^2 c^3 \tan ^{-1}(a x)-\frac{c^3 \tan ^{-1}(a x)}{2 x^2}-\frac{a c^3}{2 x} \]
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Rubi [A] time = 0.152538, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4948, 4852, 325, 203, 4848, 2391, 321, 302} \[ \frac{3}{2} i a^2 c^3 \text{PolyLog}(2,-i a x)-\frac{3}{2} i a^2 c^3 \text{PolyLog}(2,i a x)-\frac{1}{12} a^5 c^3 x^3+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)-\frac{5}{4} a^3 c^3 x+\frac{3}{4} a^2 c^3 \tan ^{-1}(a x)-\frac{c^3 \tan ^{-1}(a x)}{2 x^2}-\frac{a c^3}{2 x} \]
Antiderivative was successfully verified.
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Rule 4948
Rule 4852
Rule 325
Rule 203
Rule 4848
Rule 2391
Rule 321
Rule 302
Rubi steps
\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)}{x^3} \, dx &=\int \left (\frac{c^3 \tan ^{-1}(a x)}{x^3}+\frac{3 a^2 c^3 \tan ^{-1}(a x)}{x}+3 a^4 c^3 x \tan ^{-1}(a x)+a^6 c^3 x^3 \tan ^{-1}(a x)\right ) \, dx\\ &=c^3 \int \frac{\tan ^{-1}(a x)}{x^3} \, dx+\left (3 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)}{x} \, dx+\left (3 a^4 c^3\right ) \int x \tan ^{-1}(a x) \, dx+\left (a^6 c^3\right ) \int x^3 \tan ^{-1}(a x) \, dx\\ &=-\frac{c^3 \tan ^{-1}(a x)}{2 x^2}+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)+\frac{1}{2} \left (a c^3\right ) \int \frac{1}{x^2 \left (1+a^2 x^2\right )} \, dx+\frac{1}{2} \left (3 i a^2 c^3\right ) \int \frac{\log (1-i a x)}{x} \, dx-\frac{1}{2} \left (3 i a^2 c^3\right ) \int \frac{\log (1+i a x)}{x} \, dx-\frac{1}{2} \left (3 a^5 c^3\right ) \int \frac{x^2}{1+a^2 x^2} \, dx-\frac{1}{4} \left (a^7 c^3\right ) \int \frac{x^4}{1+a^2 x^2} \, dx\\ &=-\frac{a c^3}{2 x}-\frac{3}{2} a^3 c^3 x-\frac{c^3 \tan ^{-1}(a x)}{2 x^2}+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)+\frac{3}{2} i a^2 c^3 \text{Li}_2(-i a x)-\frac{3}{2} i a^2 c^3 \text{Li}_2(i a x)-\frac{1}{2} \left (a^3 c^3\right ) \int \frac{1}{1+a^2 x^2} \, dx+\frac{1}{2} \left (3 a^3 c^3\right ) \int \frac{1}{1+a^2 x^2} \, dx-\frac{1}{4} \left (a^7 c^3\right ) \int \left (-\frac{1}{a^4}+\frac{x^2}{a^2}+\frac{1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx\\ &=-\frac{a c^3}{2 x}-\frac{5}{4} a^3 c^3 x-\frac{1}{12} a^5 c^3 x^3+a^2 c^3 \tan ^{-1}(a x)-\frac{c^3 \tan ^{-1}(a x)}{2 x^2}+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)+\frac{3}{2} i a^2 c^3 \text{Li}_2(-i a x)-\frac{3}{2} i a^2 c^3 \text{Li}_2(i a x)-\frac{1}{4} \left (a^3 c^3\right ) \int \frac{1}{1+a^2 x^2} \, dx\\ &=-\frac{a c^3}{2 x}-\frac{5}{4} a^3 c^3 x-\frac{1}{12} a^5 c^3 x^3+\frac{3}{4} a^2 c^3 \tan ^{-1}(a x)-\frac{c^3 \tan ^{-1}(a x)}{2 x^2}+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)+\frac{3}{2} i a^2 c^3 \text{Li}_2(-i a x)-\frac{3}{2} i a^2 c^3 \text{Li}_2(i a x)\\ \end{align*}
Mathematica [C] time = 0.0433007, size = 154, normalized size = 1.12 \[ -\frac{a c^3 \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-a^2 x^2\right )}{2 x}+\frac{3}{2} i a^2 c^3 \text{PolyLog}(2,-i a x)-\frac{3}{2} i a^2 c^3 \text{PolyLog}(2,i a x)-\frac{1}{12} a^5 c^3 x^3+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)-\frac{5}{4} a^3 c^3 x+\frac{5}{4} a^2 c^3 \tan ^{-1}(a x)-\frac{c^3 \tan ^{-1}(a x)}{2 x^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.044, size = 177, normalized size = 1.3 \begin{align*}{\frac{{a}^{6}{c}^{3}{x}^{4}\arctan \left ( ax \right ) }{4}}+{\frac{3\,{a}^{4}{c}^{3}{x}^{2}\arctan \left ( ax \right ) }{2}}-{\frac{{c}^{3}\arctan \left ( ax \right ) }{2\,{x}^{2}}}+3\,{a}^{2}{c}^{3}\arctan \left ( ax \right ) \ln \left ( ax \right ) -{\frac{{a}^{5}{c}^{3}{x}^{3}}{12}}-{\frac{5\,{a}^{3}{c}^{3}x}{4}}+{\frac{3\,{a}^{2}{c}^{3}\arctan \left ( ax \right ) }{4}}-{\frac{a{c}^{3}}{2\,x}}+{\frac{3\,i}{2}}{a}^{2}{c}^{3}\ln \left ( ax \right ) \ln \left ( 1+iax \right ) -{\frac{3\,i}{2}}{a}^{2}{c}^{3}\ln \left ( ax \right ) \ln \left ( 1-iax \right ) +{\frac{3\,i}{2}}{a}^{2}{c}^{3}{\it dilog} \left ( 1+iax \right ) -{\frac{3\,i}{2}}{a}^{2}{c}^{3}{\it dilog} \left ( 1-iax \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.69919, size = 220, normalized size = 1.59 \begin{align*} -\frac{a^{5} c^{3} x^{5} + 15 \, a^{3} c^{3} x^{3} + 9 \, \pi a^{2} c^{3} x^{2} \log \left (a^{2} x^{2} + 1\right ) - 36 \, a^{2} c^{3} x^{2} \arctan \left (a x\right ) \log \left (x{\left | a \right |}\right ) + 18 i \, a^{2} c^{3} x^{2}{\rm Li}_2\left (i \, a x + 1\right ) - 18 i \, a^{2} c^{3} x^{2}{\rm Li}_2\left (-i \, a x + 1\right ) + 6 \, a c^{3} x - 3 \,{\left (a^{6} c^{3} x^{6} + 6 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2}{\left (4 i \, \arctan \left (0, a\right ) + 1\right )} - 2 \, c^{3}\right )} \arctan \left (a x\right )}{12 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}\right )} \arctan \left (a x\right )}{x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} c^{3} \left (\int \frac{\operatorname{atan}{\left (a x \right )}}{x^{3}}\, dx + \int \frac{3 a^{2} \operatorname{atan}{\left (a x \right )}}{x}\, dx + \int 3 a^{4} x \operatorname{atan}{\left (a x \right )}\, dx + \int a^{6} x^{3} \operatorname{atan}{\left (a x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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