Optimal. Leaf size=100 \[ -\frac{1}{3} i b^2 c \text{PolyLog}\left (2,-1+\frac{2}{1-i c x^3}\right )-\frac{1}{3} i c \left (a+b \tan ^{-1}\left (c x^3\right )\right )^2-\frac{\left (a+b \tan ^{-1}\left (c x^3\right )\right )^2}{3 x^3}+\frac{2}{3} b c \log \left (2-\frac{2}{1-i c x^3}\right ) \left (a+b \tan ^{-1}\left (c x^3\right )\right ) \]
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Rubi [B] time = 0.660795, antiderivative size = 290, normalized size of antiderivative = 2.9, number of steps used = 24, number of rules used = 13, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.812, Rules used = {5035, 2454, 2397, 2392, 2391, 2395, 36, 29, 31, 2439, 2416, 2394, 2393} \[ \frac{1}{3} i b^2 c \text{PolyLog}\left (2,-i c x^3\right )-\frac{1}{3} i b^2 c \text{PolyLog}\left (2,i c x^3\right )-\frac{1}{6} i b^2 c \text{PolyLog}\left (2,\frac{1}{2} \left (1-i c x^3\right )\right )+\frac{1}{6} i b^2 c \text{PolyLog}\left (2,\frac{1}{2} \left (1+i c x^3\right )\right )+\frac{1}{6} i b c \log \left (\frac{1}{2} \left (1+i c x^3\right )\right ) \left (2 i a-b \log \left (1-i c x^3\right )\right )+\frac{b \log \left (1+i c x^3\right ) \left (2 i a-b \log \left (1-i c x^3\right )\right )}{6 x^3}-\frac{\left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{12 x^3}+2 a b c \log (x)+\frac{b^2 \left (1+i c x^3\right ) \log ^2\left (1+i c x^3\right )}{12 x^3}+\frac{1}{6} i b^2 c \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right ) \]
Warning: Unable to verify antiderivative.
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Rule 5035
Rule 2454
Rule 2397
Rule 2392
Rule 2391
Rule 2395
Rule 36
Rule 29
Rule 31
Rule 2439
Rule 2416
Rule 2394
Rule 2393
Rubi steps
\begin{align*} \int \frac{\left (a+b \tan ^{-1}\left (c x^3\right )\right )^2}{x^4} \, dx &=\int \left (\frac{\left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{4 x^4}+\frac{b \left (-2 i a+b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{2 x^4}-\frac{b^2 \log ^2\left (1+i c x^3\right )}{4 x^4}\right ) \, dx\\ &=\frac{1}{4} \int \frac{\left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{x^4} \, dx+\frac{1}{2} b \int \frac{\left (-2 i a+b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{x^4} \, dx-\frac{1}{4} b^2 \int \frac{\log ^2\left (1+i c x^3\right )}{x^4} \, dx\\ &=\frac{1}{12} \operatorname{Subst}\left (\int \frac{(2 a+i b \log (1-i c x))^2}{x^2} \, dx,x,x^3\right )+\frac{1}{6} b \operatorname{Subst}\left (\int \frac{(-2 i a+b \log (1-i c x)) \log (1+i c x)}{x^2} \, dx,x,x^3\right )-\frac{1}{12} b^2 \operatorname{Subst}\left (\int \frac{\log ^2(1+i c x)}{x^2} \, dx,x,x^3\right )\\ &=-\frac{\left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{12 x^3}+\frac{b \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{6 x^3}+\frac{b^2 \left (1+i c x^3\right ) \log ^2\left (1+i c x^3\right )}{12 x^3}+\frac{1}{6} (i b c) \operatorname{Subst}\left (\int \frac{-2 i a+b \log (1-i c x)}{x (1+i c x)} \, dx,x,x^3\right )+\frac{1}{6} (b c) \operatorname{Subst}\left (\int \frac{2 a+i b \log (1-i c x)}{x} \, dx,x,x^3\right )-\frac{1}{6} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{x} \, dx,x,x^3\right )-\frac{1}{6} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{x (1-i c x)} \, dx,x,x^3\right )\\ &=a b c \log (x)-\frac{\left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{12 x^3}+\frac{b \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{6 x^3}+\frac{b^2 \left (1+i c x^3\right ) \log ^2\left (1+i c x^3\right )}{12 x^3}+\frac{1}{6} i b^2 c \text{Li}_2\left (-i c x^3\right )+\frac{1}{6} (i b c) \operatorname{Subst}\left (\int \left (\frac{-2 i a+b \log (1-i c x)}{x}-\frac{c (-2 i a+b \log (1-i c x))}{-i+c x}\right ) \, dx,x,x^3\right )+\frac{1}{6} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1-i c x)}{x} \, dx,x,x^3\right )-\frac{1}{6} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{\log (1+i c x)}{x}-\frac{c \log (1+i c x)}{i+c x}\right ) \, dx,x,x^3\right )\\ &=a b c \log (x)-\frac{\left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{12 x^3}+\frac{b \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{6 x^3}+\frac{b^2 \left (1+i c x^3\right ) \log ^2\left (1+i c x^3\right )}{12 x^3}+\frac{1}{6} i b^2 c \text{Li}_2\left (-i c x^3\right )-\frac{1}{6} i b^2 c \text{Li}_2\left (i c x^3\right )+\frac{1}{6} (i b c) \operatorname{Subst}\left (\int \frac{-2 i a+b \log (1-i c x)}{x} \, dx,x,x^3\right )-\frac{1}{6} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{x} \, dx,x,x^3\right )-\frac{1}{6} \left (i b c^2\right ) \operatorname{Subst}\left (\int \frac{-2 i a+b \log (1-i c x)}{-i+c x} \, dx,x,x^3\right )+\frac{1}{6} \left (i b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{i+c x} \, dx,x,x^3\right )\\ &=2 a b c \log (x)-\frac{\left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{12 x^3}+\frac{1}{6} i b c \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )+\frac{1}{6} i b^2 c \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )+\frac{b \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{6 x^3}+\frac{b^2 \left (1+i c x^3\right ) \log ^2\left (1+i c x^3\right )}{12 x^3}+\frac{1}{3} i b^2 c \text{Li}_2\left (-i c x^3\right )-\frac{1}{6} i b^2 c \text{Li}_2\left (i c x^3\right )+\frac{1}{6} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1-i c x)}{x} \, dx,x,x^3\right )+\frac{1}{6} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{2} i (-i+c x)\right )}{1-i c x} \, dx,x,x^3\right )+\frac{1}{6} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{1}{2} i (i+c x)\right )}{1+i c x} \, dx,x,x^3\right )\\ &=2 a b c \log (x)-\frac{\left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{12 x^3}+\frac{1}{6} i b c \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )+\frac{1}{6} i b^2 c \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )+\frac{b \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{6 x^3}+\frac{b^2 \left (1+i c x^3\right ) \log ^2\left (1+i c x^3\right )}{12 x^3}+\frac{1}{3} i b^2 c \text{Li}_2\left (-i c x^3\right )-\frac{1}{3} i b^2 c \text{Li}_2\left (i c x^3\right )+\frac{1}{6} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1-i c x^3\right )-\frac{1}{6} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1+i c x^3\right )\\ &=2 a b c \log (x)-\frac{\left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{12 x^3}+\frac{1}{6} i b c \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )+\frac{1}{6} i b^2 c \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )+\frac{b \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{6 x^3}+\frac{b^2 \left (1+i c x^3\right ) \log ^2\left (1+i c x^3\right )}{12 x^3}+\frac{1}{3} i b^2 c \text{Li}_2\left (-i c x^3\right )-\frac{1}{3} i b^2 c \text{Li}_2\left (i c x^3\right )-\frac{1}{6} i b^2 c \text{Li}_2\left (\frac{1}{2} \left (1-i c x^3\right )\right )+\frac{1}{6} i b^2 c \text{Li}_2\left (\frac{1}{2} \left (1+i c x^3\right )\right )\\ \end{align*}
Mathematica [A] time = 0.154743, size = 125, normalized size = 1.25 \[ \frac{-i b^2 c x^3 \text{PolyLog}\left (2,e^{2 i \tan ^{-1}\left (c x^3\right )}\right )-a \left (a+b c x^3 \log \left (c^2 x^6+1\right )-2 b c x^3 \log \left (c x^3\right )\right )+2 b \tan ^{-1}\left (c x^3\right ) \left (-a+b c x^3 \log \left (1-e^{2 i \tan ^{-1}\left (c x^3\right )}\right )\right )+b^2 \left (-1-i c x^3\right ) \tan ^{-1}\left (c x^3\right )^2}{3 x^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.408, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\arctan \left ( c{x}^{3} \right ) \right ) ^{2}}{{x}^{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{3} \,{\left (c{\left (\log \left (c^{2} x^{6} + 1\right ) - \log \left (x^{6}\right )\right )} + \frac{2 \, \arctan \left (c x^{3}\right )}{x^{3}}\right )} a b + \frac{\frac{1}{4} \,{\left (12 \, x^{3} \int -\frac{12 \, c^{2} x^{6} \log \left (c^{2} x^{6} + 1\right ) - 56 \, c x^{3} \arctan \left (c x^{3}\right ) - 36 \,{\left (c^{2} x^{6} + 1\right )} \arctan \left (c x^{3}\right )^{2} - 3 \,{\left (c^{2} x^{6} + 1\right )} \log \left (c^{2} x^{6} + 1\right )^{2}}{4 \,{\left (c^{2} x^{10} + x^{4}\right )}}\,{d x} - 28 \, \arctan \left (c x^{3}\right )^{2} + 3 \, \log \left (c^{2} x^{6} + 1\right )^{2}\right )} b^{2}}{48 \, x^{3}} - \frac{a^{2}}{3 \, x^{3}} \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{b^{2} \arctan \left (c x^{3}\right )^{2} + 2 \, a b \arctan \left (c x^{3}\right ) + a^{2}}{x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (b \arctan \left (c x^{3}\right ) + a\right )}^{2}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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