Optimal. Leaf size=159 \[ -\frac {b c d (a+b \text {ArcTan}(c x))}{x}+\frac {1}{2} c^2 d (a+b \text {ArcTan}(c x))^2-\frac {d (a+b \text {ArcTan}(c x))^2}{2 x^2}-\frac {i c d (a+b \text {ArcTan}(c x))^2}{x}+b^2 c^2 d \log (x)-\frac {1}{2} b^2 c^2 d \log \left (1+c^2 x^2\right )+2 i b c^2 d (a+b \text {ArcTan}(c x)) \log \left (2-\frac {2}{1-i c x}\right )+b^2 c^2 d \text {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right ) \]
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Rubi [A]
time = 0.25, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {4996, 4946,
5038, 272, 36, 29, 31, 5004, 5044, 4988, 2497} \begin {gather*} \frac {1}{2} c^2 d (a+b \text {ArcTan}(c x))^2+2 i b c^2 d \log \left (2-\frac {2}{1-i c x}\right ) (a+b \text {ArcTan}(c x))-\frac {d (a+b \text {ArcTan}(c x))^2}{2 x^2}-\frac {i c d (a+b \text {ArcTan}(c x))^2}{x}-\frac {b c d (a+b \text {ArcTan}(c x))}{x}+b^2 c^2 d \text {Li}_2\left (\frac {2}{1-i c x}-1\right )-\frac {1}{2} b^2 c^2 d \log \left (c^2 x^2+1\right )+b^2 c^2 d \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 272
Rule 2497
Rule 4946
Rule 4988
Rule 4996
Rule 5004
Rule 5038
Rule 5044
Rubi steps
\begin {align*} \int \frac {(d+i c d x) \left (a+b \tan ^{-1}(c x)\right )^2}{x^3} \, dx &=\int \left (\frac {d \left (a+b \tan ^{-1}(c x)\right )^2}{x^3}+\frac {i c d \left (a+b \tan ^{-1}(c x)\right )^2}{x^2}\right ) \, dx\\ &=d \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x^3} \, dx+(i c d) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x^2} \, dx\\ &=-\frac {d \left (a+b \tan ^{-1}(c x)\right )^2}{2 x^2}-\frac {i c d \left (a+b \tan ^{-1}(c x)\right )^2}{x}+(b c d) \int \frac {a+b \tan ^{-1}(c x)}{x^2 \left (1+c^2 x^2\right )} \, dx+\left (2 i b c^2 d\right ) \int \frac {a+b \tan ^{-1}(c x)}{x \left (1+c^2 x^2\right )} \, dx\\ &=c^2 d \left (a+b \tan ^{-1}(c x)\right )^2-\frac {d \left (a+b \tan ^{-1}(c x)\right )^2}{2 x^2}-\frac {i c d \left (a+b \tan ^{-1}(c x)\right )^2}{x}+(b c d) \int \frac {a+b \tan ^{-1}(c x)}{x^2} \, dx-\left (2 b c^2 d\right ) \int \frac {a+b \tan ^{-1}(c x)}{x (i+c x)} \, dx-\left (b c^3 d\right ) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx\\ &=-\frac {b c d \left (a+b \tan ^{-1}(c x)\right )}{x}+\frac {1}{2} c^2 d \left (a+b \tan ^{-1}(c x)\right )^2-\frac {d \left (a+b \tan ^{-1}(c x)\right )^2}{2 x^2}-\frac {i c d \left (a+b \tan ^{-1}(c x)\right )^2}{x}+2 i b c^2 d \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-i c x}\right )+\left (b^2 c^2 d\right ) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx-\left (2 i b^2 c^3 d\right ) \int \frac {\log \left (2-\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx\\ &=-\frac {b c d \left (a+b \tan ^{-1}(c x)\right )}{x}+\frac {1}{2} c^2 d \left (a+b \tan ^{-1}(c x)\right )^2-\frac {d \left (a+b \tan ^{-1}(c x)\right )^2}{2 x^2}-\frac {i c d \left (a+b \tan ^{-1}(c x)\right )^2}{x}+2 i b c^2 d \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-i c x}\right )+b^2 c^2 d \text {Li}_2\left (-1+\frac {2}{1-i c x}\right )+\frac {1}{2} \left (b^2 c^2 d\right ) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {b c d \left (a+b \tan ^{-1}(c x)\right )}{x}+\frac {1}{2} c^2 d \left (a+b \tan ^{-1}(c x)\right )^2-\frac {d \left (a+b \tan ^{-1}(c x)\right )^2}{2 x^2}-\frac {i c d \left (a+b \tan ^{-1}(c x)\right )^2}{x}+2 i b c^2 d \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-i c x}\right )+b^2 c^2 d \text {Li}_2\left (-1+\frac {2}{1-i c x}\right )+\frac {1}{2} \left (b^2 c^2 d\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{2} \left (b^2 c^4 d\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )\\ &=-\frac {b c d \left (a+b \tan ^{-1}(c x)\right )}{x}+\frac {1}{2} c^2 d \left (a+b \tan ^{-1}(c x)\right )^2-\frac {d \left (a+b \tan ^{-1}(c x)\right )^2}{2 x^2}-\frac {i c d \left (a+b \tan ^{-1}(c x)\right )^2}{x}+b^2 c^2 d \log (x)-\frac {1}{2} b^2 c^2 d \log \left (1+c^2 x^2\right )+2 i b c^2 d \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-i c x}\right )+b^2 c^2 d \text {Li}_2\left (-1+\frac {2}{1-i c x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 190, normalized size = 1.19 \begin {gather*} -\frac {d \left (a^2+2 i a^2 c x+2 a b c x-b^2 (-i+c x)^2 \text {ArcTan}(c x)^2+2 b \text {ArcTan}(c x) \left (a+2 i a c x+b c x+a c^2 x^2-2 i b c^2 x^2 \log \left (1-e^{2 i \text {ArcTan}(c x)}\right )\right )-4 i a b c^2 x^2 \log (c x)-2 b^2 c^2 x^2 \log \left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )+2 i a b c^2 x^2 \log \left (1+c^2 x^2\right )-2 b^2 c^2 x^2 \text {PolyLog}\left (2,e^{2 i \text {ArcTan}(c x)}\right )\right )}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 447 vs. \(2 (149 ) = 298\).
time = 0.55, size = 448, normalized size = 2.82
method | result | size |
derivativedivides | \(c^{2} \left (-i d \,b^{2} \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )-i d a b \ln \left (c^{2} x^{2}+1\right )-\frac {i d \,b^{2} \arctan \left (c x \right )^{2}}{c x}+\frac {d \,b^{2} \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{2}-\frac {d \,b^{2} \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{2}-\frac {d \,b^{2} \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{2}+\frac {d \,b^{2} \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{2}-\frac {d a b \arctan \left (c x \right )}{c^{2} x^{2}}-d a b \arctan \left (c x \right )-\frac {d a b}{c x}-\frac {d \,b^{2} \arctan \left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {d \,b^{2} \arctan \left (c x \right )}{c x}-\frac {d \,b^{2} \arctan \left (c x \right )^{2}}{2}-\frac {d \,b^{2} \ln \left (c^{2} x^{2}+1\right )}{2}+d \,b^{2} \ln \left (c x \right )+2 i d \,b^{2} \arctan \left (c x \right ) \ln \left (c x \right )+2 i d a b \ln \left (c x \right )-d \,b^{2} \ln \left (c x \right ) \ln \left (i c x +1\right )+d \,b^{2} \ln \left (c x \right ) \ln \left (-i c x +1\right )-\frac {d \,b^{2} \ln \left (c x -i\right )^{2}}{4}+\frac {d \,b^{2} \ln \left (c x +i\right )^{2}}{4}-\frac {d \,b^{2} \dilog \left (-\frac {i \left (c x +i\right )}{2}\right )}{2}+\frac {d \,b^{2} \dilog \left (\frac {i \left (c x -i\right )}{2}\right )}{2}-\frac {2 i d a b \arctan \left (c x \right )}{c x}+d \,a^{2} \left (-\frac {1}{2 c^{2} x^{2}}-\frac {i}{c x}\right )-d \,b^{2} \dilog \left (i c x +1\right )+d \,b^{2} \dilog \left (-i c x +1\right )\right )\) | \(448\) |
default | \(c^{2} \left (-i d \,b^{2} \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )-i d a b \ln \left (c^{2} x^{2}+1\right )-\frac {i d \,b^{2} \arctan \left (c x \right )^{2}}{c x}+\frac {d \,b^{2} \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{2}-\frac {d \,b^{2} \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{2}-\frac {d \,b^{2} \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{2}+\frac {d \,b^{2} \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{2}-\frac {d a b \arctan \left (c x \right )}{c^{2} x^{2}}-d a b \arctan \left (c x \right )-\frac {d a b}{c x}-\frac {d \,b^{2} \arctan \left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {d \,b^{2} \arctan \left (c x \right )}{c x}-\frac {d \,b^{2} \arctan \left (c x \right )^{2}}{2}-\frac {d \,b^{2} \ln \left (c^{2} x^{2}+1\right )}{2}+d \,b^{2} \ln \left (c x \right )+2 i d \,b^{2} \arctan \left (c x \right ) \ln \left (c x \right )+2 i d a b \ln \left (c x \right )-d \,b^{2} \ln \left (c x \right ) \ln \left (i c x +1\right )+d \,b^{2} \ln \left (c x \right ) \ln \left (-i c x +1\right )-\frac {d \,b^{2} \ln \left (c x -i\right )^{2}}{4}+\frac {d \,b^{2} \ln \left (c x +i\right )^{2}}{4}-\frac {d \,b^{2} \dilog \left (-\frac {i \left (c x +i\right )}{2}\right )}{2}+\frac {d \,b^{2} \dilog \left (\frac {i \left (c x -i\right )}{2}\right )}{2}-\frac {2 i d a b \arctan \left (c x \right )}{c x}+d \,a^{2} \left (-\frac {1}{2 c^{2} x^{2}}-\frac {i}{c x}\right )-d \,b^{2} \dilog \left (i c x +1\right )+d \,b^{2} \dilog \left (-i c x +1\right )\right )\) | \(448\) |
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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\right )\right )}^2\,\left (d+c\,d\,x\,1{}\mathrm {i}\right )}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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