Optimal. Leaf size=82 \[ -i c \left (a+b \tan ^{-1}(c x)\right )^2-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x}+2 b c \log \left (2-\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )-i b^2 c \text {Li}_2\left (\frac {2}{1-i c x}-1\right ) \]
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Rubi [A] time = 0.15, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4852, 4924, 4868, 2447} \[ -i b^2 c \text {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )-i c \left (a+b \tan ^{-1}(c x)\right )^2-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x}+2 b c \log \left (2-\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right ) \]
Antiderivative was successfully verified.
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Rule 2447
Rule 4852
Rule 4868
Rule 4924
Rubi steps
\begin {align*} \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x^2} \, dx &=-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x}+(2 b c) \int \frac {a+b \tan ^{-1}(c x)}{x \left (1+c^2 x^2\right )} \, dx\\ &=-i c \left (a+b \tan ^{-1}(c x)\right )^2-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x}+(2 i b c) \int \frac {a+b \tan ^{-1}(c x)}{x (i+c x)} \, dx\\ &=-i c \left (a+b \tan ^{-1}(c x)\right )^2-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x}+2 b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-i c x}\right )-\left (2 b^2 c^2\right ) \int \frac {\log \left (2-\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx\\ &=-i c \left (a+b \tan ^{-1}(c x)\right )^2-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x}+2 b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-i c x}\right )-i b^2 c \text {Li}_2\left (-1+\frac {2}{1-i c x}\right )\\ \end {align*}
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Mathematica [A] time = 0.15, size = 102, normalized size = 1.24 \[ \frac {-a \left (a+b c x \log \left (c^2 x^2+1\right )-2 b c x \log (c x)\right )+2 b \tan ^{-1}(c x) \left (-a+b c x \log \left (1-e^{2 i \tan ^{-1}(c x)}\right )\right )-i b^2 c x \text {Li}_2\left (e^{2 i \tan ^{-1}(c x)}\right )+b^2 (-1-i c x) \tan ^{-1}(c x)^2}{x} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \arctan \left (c x\right )^{2} + 2 \, a b \arctan \left (c x\right ) + a^{2}}{x^{2}}, x\right ) \]
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 [B] time = 0.02, size = 323, normalized size = 3.94 \[ -\frac {a^{2}}{x}-\frac {b^{2} \arctan \left (c x \right )^{2}}{x}+2 c \,b^{2} \ln \left (c x \right ) \arctan \left (c x \right )-c \,b^{2} \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )-i c \,b^{2} \ln \left (c x \right ) \ln \left (-i c x +1\right )-i c \,b^{2} \dilog \left (-i c x +1\right )+\frac {i c \,b^{2} \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{2}+i c \,b^{2} \ln \left (c x \right ) \ln \left (i c x +1\right )-\frac {i c \,b^{2} \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{2}+\frac {i c \,b^{2} \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{2}+\frac {i c \,b^{2} \dilog \left (-\frac {i \left (c x +i\right )}{2}\right )}{2}-\frac {i c \,b^{2} \ln \left (c x +i\right )^{2}}{4}-\frac {i c \,b^{2} \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{2}+\frac {i c \,b^{2} \ln \left (c x -i\right )^{2}}{4}-\frac {i c \,b^{2} \dilog \left (\frac {i \left (c x -i\right )}{2}\right )}{2}+i c \,b^{2} \dilog \left (i c x +1\right )-\frac {2 a b \arctan \left (c x \right )}{x}+2 c a b \ln \left (c x \right )-c a b \ln \left (c^{2} x^{2}+1\right ) \]
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 = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{x^2} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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