Optimal. Leaf size=154 \[ \frac {2}{3} \left (a+b \text {ArcTan}\left (c x^3\right )\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x^3}\right )-\frac {1}{3} i b \left (a+b \text {ArcTan}\left (c x^3\right )\right ) \text {PolyLog}\left (2,1-\frac {2}{1+i c x^3}\right )+\frac {1}{3} i b \left (a+b \text {ArcTan}\left (c x^3\right )\right ) \text {PolyLog}\left (2,-1+\frac {2}{1+i c x^3}\right )-\frac {1}{6} b^2 \text {PolyLog}\left (3,1-\frac {2}{1+i c x^3}\right )+\frac {1}{6} b^2 \text {PolyLog}\left (3,-1+\frac {2}{1+i c x^3}\right ) \]
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Rubi [A]
time = 0.21, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4944, 4942,
5108, 5004, 5114, 6745} \begin {gather*} -\frac {1}{3} i b \text {Li}_2\left (1-\frac {2}{i c x^3+1}\right ) \left (a+b \text {ArcTan}\left (c x^3\right )\right )+\frac {1}{3} i b \text {Li}_2\left (\frac {2}{i c x^3+1}-1\right ) \left (a+b \text {ArcTan}\left (c x^3\right )\right )+\frac {2}{3} \tanh ^{-1}\left (1-\frac {2}{1+i c x^3}\right ) \left (a+b \text {ArcTan}\left (c x^3\right )\right )^2-\frac {1}{6} b^2 \text {Li}_3\left (1-\frac {2}{i c x^3+1}\right )+\frac {1}{6} b^2 \text {Li}_3\left (\frac {2}{i c x^3+1}-1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 4942
Rule 4944
Rule 5004
Rule 5108
Rule 5114
Rule 6745
Rubi steps
\begin {align*} \int \frac {\left (a+b \tan ^{-1}\left (c x^3\right )\right )^2}{x} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x} \, dx,x,x^3\right )\\ &=\frac {2}{3} \left (a+b \tan ^{-1}\left (c x^3\right )\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x^3}\right )-\frac {1}{3} (4 b c) \text {Subst}\left (\int \frac {\left (a+b \tan ^{-1}(c x)\right ) \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx,x,x^3\right )\\ &=\frac {2}{3} \left (a+b \tan ^{-1}\left (c x^3\right )\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x^3}\right )+\frac {1}{3} (2 b c) \text {Subst}\left (\int \frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx,x,x^3\right )-\frac {1}{3} (2 b c) \text {Subst}\left (\int \frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx,x,x^3\right )\\ &=\frac {2}{3} \left (a+b \tan ^{-1}\left (c x^3\right )\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x^3}\right )-\frac {1}{3} i b \left (a+b \tan ^{-1}\left (c x^3\right )\right ) \text {Li}_2\left (1-\frac {2}{1+i c x^3}\right )+\frac {1}{3} i b \left (a+b \tan ^{-1}\left (c x^3\right )\right ) \text {Li}_2\left (-1+\frac {2}{1+i c x^3}\right )+\frac {1}{3} \left (i b^2 c\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx,x,x^3\right )-\frac {1}{3} \left (i b^2 c\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-1+\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx,x,x^3\right )\\ &=\frac {2}{3} \left (a+b \tan ^{-1}\left (c x^3\right )\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x^3}\right )-\frac {1}{3} i b \left (a+b \tan ^{-1}\left (c x^3\right )\right ) \text {Li}_2\left (1-\frac {2}{1+i c x^3}\right )+\frac {1}{3} i b \left (a+b \tan ^{-1}\left (c x^3\right )\right ) \text {Li}_2\left (-1+\frac {2}{1+i c x^3}\right )-\frac {1}{6} b^2 \text {Li}_3\left (1-\frac {2}{1+i c x^3}\right )+\frac {1}{6} b^2 \text {Li}_3\left (-1+\frac {2}{1+i c x^3}\right )\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 167, normalized size = 1.08 \begin {gather*} \frac {1}{6} \left (4 \left (a+b \text {ArcTan}\left (c x^3\right )\right )^2 \tanh ^{-1}\left (1+\frac {2 i}{-i+c x^3}\right )+b \left (2 i \left (a+b \text {ArcTan}\left (c x^3\right )\right ) \text {PolyLog}\left (2,\frac {i+c x^3}{i-c x^3}\right )-2 i \left (a+b \text {ArcTan}\left (c x^3\right )\right ) \text {PolyLog}\left (2,\frac {i+c x^3}{-i+c x^3}\right )+b \left (\text {PolyLog}\left (3,\frac {i+c x^3}{i-c x^3}\right )-\text {PolyLog}\left (3,\frac {i+c x^3}{-i+c x^3}\right )\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arctan \left (c \,x^{3}\right )\right )^{2}}{x}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (a + b \operatorname {atan}{\left (c x^{3} \right )}\right )^{2}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x^3\right )\right )}^2}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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