Optimal. Leaf size=207 \[ \frac {1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^2-a^3 c^2 x \tan ^{-1}(a x)-a^2 c^2 \text {Li}_3\left (1-\frac {2}{i a x+1}\right )+a^2 c^2 \text {Li}_3\left (\frac {2}{i a x+1}-1\right )-2 i a^2 c^2 \text {Li}_2\left (1-\frac {2}{i a x+1}\right ) \tan ^{-1}(a x)+2 i a^2 c^2 \text {Li}_2\left (\frac {2}{i a x+1}-1\right ) \tan ^{-1}(a x)+a^2 c^2 \log (x)+4 a^2 c^2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {c^2 \tan ^{-1}(a x)^2}{2 x^2}-\frac {a c^2 \tan ^{-1}(a x)}{x} \]
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Rubi [A] time = 0.45, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 15, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {4948, 4852, 4918, 266, 36, 29, 31, 4884, 4850, 4988, 4994, 6610, 4916, 4846, 260} \[ -a^2 c^2 \text {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+a^2 c^2 \text {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )-2 i a^2 c^2 \tan ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+2 i a^2 c^2 \tan ^{-1}(a x) \text {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )+\frac {1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^2+a^2 c^2 \log (x)-a^3 c^2 x \tan ^{-1}(a x)+4 a^2 c^2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {c^2 \tan ^{-1}(a x)^2}{2 x^2}-\frac {a c^2 \tan ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 260
Rule 266
Rule 4846
Rule 4850
Rule 4852
Rule 4884
Rule 4916
Rule 4918
Rule 4948
Rule 4988
Rule 4994
Rule 6610
Rubi steps
\begin {align*} \int \frac {\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2}{x^3} \, dx &=\int \left (\frac {c^2 \tan ^{-1}(a x)^2}{x^3}+\frac {2 a^2 c^2 \tan ^{-1}(a x)^2}{x}+a^4 c^2 x \tan ^{-1}(a x)^2\right ) \, dx\\ &=c^2 \int \frac {\tan ^{-1}(a x)^2}{x^3} \, dx+\left (2 a^2 c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{x} \, dx+\left (a^4 c^2\right ) \int x \tan ^{-1}(a x)^2 \, dx\\ &=-\frac {c^2 \tan ^{-1}(a x)^2}{2 x^2}+\frac {1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^2+4 a^2 c^2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+\left (a c^2\right ) \int \frac {\tan ^{-1}(a x)}{x^2 \left (1+a^2 x^2\right )} \, dx-\left (8 a^3 c^2\right ) \int \frac {\tan ^{-1}(a x) \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (a^5 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac {c^2 \tan ^{-1}(a x)^2}{2 x^2}+\frac {1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^2+4 a^2 c^2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+\left (a c^2\right ) \int \frac {\tan ^{-1}(a x)}{x^2} \, dx-\left (a^3 c^2\right ) \int \tan ^{-1}(a x) \, dx+\left (4 a^3 c^2\right ) \int \frac {\tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (4 a^3 c^2\right ) \int \frac {\tan ^{-1}(a x) \log \left (2-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac {a c^2 \tan ^{-1}(a x)}{x}-a^3 c^2 x \tan ^{-1}(a x)-\frac {c^2 \tan ^{-1}(a x)^2}{2 x^2}+\frac {1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^2+4 a^2 c^2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-2 i a^2 c^2 \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+2 i a^2 c^2 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )+\left (a^2 c^2\right ) \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx+\left (2 i a^3 c^2\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (2 i a^3 c^2\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\left (a^4 c^2\right ) \int \frac {x}{1+a^2 x^2} \, dx\\ &=-\frac {a c^2 \tan ^{-1}(a x)}{x}-a^3 c^2 x \tan ^{-1}(a x)-\frac {c^2 \tan ^{-1}(a x)^2}{2 x^2}+\frac {1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^2+4 a^2 c^2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+\frac {1}{2} a^2 c^2 \log \left (1+a^2 x^2\right )-2 i a^2 c^2 \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+2 i a^2 c^2 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-a^2 c^2 \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+a^2 c^2 \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )+\frac {1}{2} \left (a^2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {a c^2 \tan ^{-1}(a x)}{x}-a^3 c^2 x \tan ^{-1}(a x)-\frac {c^2 \tan ^{-1}(a x)^2}{2 x^2}+\frac {1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^2+4 a^2 c^2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+\frac {1}{2} a^2 c^2 \log \left (1+a^2 x^2\right )-2 i a^2 c^2 \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+2 i a^2 c^2 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-a^2 c^2 \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+a^2 c^2 \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )+\frac {1}{2} \left (a^2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{2} \left (a^4 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac {a c^2 \tan ^{-1}(a x)}{x}-a^3 c^2 x \tan ^{-1}(a x)-\frac {c^2 \tan ^{-1}(a x)^2}{2 x^2}+\frac {1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^2+4 a^2 c^2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+a^2 c^2 \log (x)-2 i a^2 c^2 \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+2 i a^2 c^2 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-a^2 c^2 \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+a^2 c^2 \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )\\ \end {align*}
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Mathematica [A] time = 0.33, size = 226, normalized size = 1.09 \[ a^2 c^2 \left (\log \left (\frac {a x}{\sqrt {a^2 x^2+1}}\right )+\frac {1}{2} \log \left (a^2 x^2+1\right )+\frac {1}{2} a^2 x^2 \tan ^{-1}(a x)^2-\frac {\tan ^{-1}(a x)^2}{2 a^2 x^2}+2 i \tan ^{-1}(a x) \text {Li}_2\left (e^{-2 i \tan ^{-1}(a x)}\right )+2 i \tan ^{-1}(a x) \text {Li}_2\left (-e^{2 i \tan ^{-1}(a x)}\right )+\text {Li}_3\left (e^{-2 i \tan ^{-1}(a x)}\right )-\text {Li}_3\left (-e^{2 i \tan ^{-1}(a x)}\right )+\frac {4}{3} i \tan ^{-1}(a x)^3-a x \tan ^{-1}(a x)-\frac {\tan ^{-1}(a x)}{a x}+2 \tan ^{-1}(a x)^2 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )-2 \tan ^{-1}(a x)^2 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-\frac {i \pi ^3}{12}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.03, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \arctan \left (a x\right )^{2}}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 5.53, size = 1255, normalized size = 6.06 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^2}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ c^{2} \left (\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{x^{3}}\, dx + \int \frac {2 a^{2} \operatorname {atan}^{2}{\left (a x \right )}}{x}\, dx + \int a^{4} x \operatorname {atan}^{2}{\left (a x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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