Optimal. Leaf size=416 \[ -\frac {b c d^3 (a+b \text {ArcTan}(c x))}{x}+\frac {7}{2} c^2 d^3 (a+b \text {ArcTan}(c x))^2-\frac {d^3 (a+b \text {ArcTan}(c x))^2}{2 x^2}-\frac {3 i c d^3 (a+b \text {ArcTan}(c x))^2}{x}-i c^3 d^3 x (a+b \text {ArcTan}(c x))^2-6 c^2 d^3 (a+b \text {ArcTan}(c x))^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )+b^2 c^2 d^3 \log (x)-2 i b c^2 d^3 (a+b \text {ArcTan}(c x)) \log \left (\frac {2}{1+i c x}\right )-\frac {1}{2} b^2 c^2 d^3 \log \left (1+c^2 x^2\right )+6 i b c^2 d^3 (a+b \text {ArcTan}(c x)) \log \left (2-\frac {2}{1-i c x}\right )+3 b^2 c^2 d^3 \text {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )+b^2 c^2 d^3 \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )+3 i b c^2 d^3 (a+b \text {ArcTan}(c x)) \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )-3 i b c^2 d^3 (a+b \text {ArcTan}(c x)) \text {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )+\frac {3}{2} b^2 c^2 d^3 \text {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )-\frac {3}{2} b^2 c^2 d^3 \text {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right ) \]
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Rubi [A]
time = 0.54, antiderivative size = 416, normalized size of antiderivative = 1.00, number of steps
used = 25, number of rules used = 20, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {4996, 4930,
5040, 4964, 2449, 2352, 4946, 5038, 272, 36, 29, 31, 5004, 5044, 4988, 2497, 4942, 5108, 5114,
6745} \begin {gather*} -i c^3 d^3 x (a+b \text {ArcTan}(c x))^2+3 i b c^2 d^3 \text {Li}_2\left (1-\frac {2}{i c x+1}\right ) (a+b \text {ArcTan}(c x))-3 i b c^2 d^3 \text {Li}_2\left (\frac {2}{i c x+1}-1\right ) (a+b \text {ArcTan}(c x))+\frac {7}{2} c^2 d^3 (a+b \text {ArcTan}(c x))^2-2 i b c^2 d^3 \log \left (\frac {2}{1+i c x}\right ) (a+b \text {ArcTan}(c x))+6 i b c^2 d^3 \log \left (2-\frac {2}{1-i c x}\right ) (a+b \text {ArcTan}(c x))-6 c^2 d^3 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right ) (a+b \text {ArcTan}(c x))^2-\frac {d^3 (a+b \text {ArcTan}(c x))^2}{2 x^2}-\frac {3 i c d^3 (a+b \text {ArcTan}(c x))^2}{x}-\frac {b c d^3 (a+b \text {ArcTan}(c x))}{x}+3 b^2 c^2 d^3 \text {Li}_2\left (\frac {2}{1-i c x}-1\right )+b^2 c^2 d^3 \text {Li}_2\left (1-\frac {2}{i c x+1}\right )+\frac {3}{2} b^2 c^2 d^3 \text {Li}_3\left (1-\frac {2}{i c x+1}\right )-\frac {3}{2} b^2 c^2 d^3 \text {Li}_3\left (\frac {2}{i c x+1}-1\right )-\frac {1}{2} b^2 c^2 d^3 \log \left (c^2 x^2+1\right )+b^2 c^2 d^3 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 272
Rule 2352
Rule 2449
Rule 2497
Rule 4930
Rule 4942
Rule 4946
Rule 4964
Rule 4988
Rule 4996
Rule 5004
Rule 5038
Rule 5040
Rule 5044
Rule 5108
Rule 5114
Rule 6745
Rubi steps
\begin {align*} \int \frac {(d+i c d x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x^3} \, dx &=\int \left (-i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x^3}+\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x^2}-\frac {3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x}\right ) \, dx\\ &=d^3 \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x^3} \, dx+\left (3 i c d^3\right ) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x^2} \, dx-\left (3 c^2 d^3\right ) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x} \, dx-\left (i c^3 d^3\right ) \int \left (a+b \tan ^{-1}(c x)\right )^2 \, dx\\ &=-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x}-i c^3 d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-6 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )+\left (b c d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{x^2 \left (1+c^2 x^2\right )} \, dx+\left (6 i b c^2 d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{x \left (1+c^2 x^2\right )} \, dx+\left (12 b c^3 d^3\right ) \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+\left (2 i b c^4 d^3\right ) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=4 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x}-i c^3 d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-6 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )+\left (b c d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{x^2} \, dx-\left (6 b c^2 d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{x (i+c x)} \, dx-\left (2 i b c^3 d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx-\left (b c^3 d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx-\left (6 b c^3 d^3\right ) \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+\left (6 b c^3 d^3\right ) \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\\ &=-\frac {b c d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}+\frac {7}{2} c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x}-i c^3 d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-6 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )-2 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )+6 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-i c x}\right )+3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )-3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )+\left (b^2 c^2 d^3\right ) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx+\left (2 i b^2 c^3 d^3\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\left (3 i b^2 c^3 d^3\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx+\left (3 i b^2 c^3 d^3\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\left (6 i b^2 c^3 d^3\right ) \int \frac {\log \left (2-\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx\\ &=-\frac {b c d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}+\frac {7}{2} c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x}-i c^3 d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-6 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )-2 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )+6 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-i c x}\right )+3 b^2 c^2 d^3 \text {Li}_2\left (-1+\frac {2}{1-i c x}\right )+3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )-3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )+\frac {3}{2} b^2 c^2 d^3 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )-\frac {3}{2} b^2 c^2 d^3 \text {Li}_3\left (-1+\frac {2}{1+i c x}\right )+\frac {1}{2} \left (b^2 c^2 d^3\right ) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )+\left (2 b^2 c^2 d^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )\\ &=-\frac {b c d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}+\frac {7}{2} c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x}-i c^3 d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-6 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )-2 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )+6 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-i c x}\right )+3 b^2 c^2 d^3 \text {Li}_2\left (-1+\frac {2}{1-i c x}\right )+b^2 c^2 d^3 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )+3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )-3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )+\frac {3}{2} b^2 c^2 d^3 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )-\frac {3}{2} b^2 c^2 d^3 \text {Li}_3\left (-1+\frac {2}{1+i c x}\right )+\frac {1}{2} \left (b^2 c^2 d^3\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{2} \left (b^2 c^4 d^3\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )\\ &=-\frac {b c d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}+\frac {7}{2} c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x}-i c^3 d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-6 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )+b^2 c^2 d^3 \log (x)-2 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )-\frac {1}{2} b^2 c^2 d^3 \log \left (1+c^2 x^2\right )+6 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-i c x}\right )+3 b^2 c^2 d^3 \text {Li}_2\left (-1+\frac {2}{1-i c x}\right )+b^2 c^2 d^3 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )+3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )-3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )+\frac {3}{2} b^2 c^2 d^3 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )-\frac {3}{2} b^2 c^2 d^3 \text {Li}_3\left (-1+\frac {2}{1+i c x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.69, size = 500, normalized size = 1.20 \begin {gather*} \frac {1}{2} d^3 \left (-\frac {a^2}{x^2}-\frac {6 i a^2 c}{x}-2 i a^2 c^3 x-\frac {2 a b (\text {ArcTan}(c x)+c x (1+c x \text {ArcTan}(c x)))}{x^2}-6 a^2 c^2 \log (x)-\frac {b^2 \left (2 c x \text {ArcTan}(c x)+\left (1+c^2 x^2\right ) \text {ArcTan}(c x)^2-2 c^2 x^2 \log \left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )\right )}{x^2}-2 i a b c^2 \left (2 c x \text {ArcTan}(c x)-\log \left (1+c^2 x^2\right )\right )-\frac {6 i a b c \left (2 \text {ArcTan}(c x)+c x \left (-2 \log (c x)+\log \left (1+c^2 x^2\right )\right )\right )}{x}-2 i b^2 c^2 \left (\text {ArcTan}(c x) \left ((-i+c x) \text {ArcTan}(c x)+2 \log \left (1+e^{2 i \text {ArcTan}(c x)}\right )\right )-i \text {PolyLog}\left (2,-e^{2 i \text {ArcTan}(c x)}\right )\right )+\frac {6 b^2 c \left (\text {ArcTan}(c x) \left ((-i+c x) \text {ArcTan}(c x)+2 i c x \log \left (1-e^{2 i \text {ArcTan}(c x)}\right )\right )+c x \text {PolyLog}\left (2,e^{2 i \text {ArcTan}(c x)}\right )\right )}{x}-6 i a b c^2 (\text {PolyLog}(2,-i c x)-\text {PolyLog}(2,i c x))+6 b^2 c^2 \left (\frac {i \pi ^3}{24}-\frac {2}{3} i \text {ArcTan}(c x)^3-\text {ArcTan}(c x)^2 \log \left (1-e^{-2 i \text {ArcTan}(c x)}\right )+\text {ArcTan}(c x)^2 \log \left (1+e^{2 i \text {ArcTan}(c x)}\right )-i \text {ArcTan}(c x) \text {PolyLog}\left (2,e^{-2 i \text {ArcTan}(c x)}\right )-i \text {ArcTan}(c x) \text {PolyLog}\left (2,-e^{2 i \text {ArcTan}(c x)}\right )-\frac {1}{2} \text {PolyLog}\left (3,e^{-2 i \text {ArcTan}(c x)}\right )+\frac {1}{2} \text {PolyLog}\left (3,-e^{2 i \text {ArcTan}(c x)}\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 7.02, size = 1746, normalized size = 4.20
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1746\) |
default | \(\text {Expression too large to display}\) | \(1746\) |
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 = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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