Optimal. Leaf size=637 \[ \frac {2 (a+b \text {ArcTan}(c x))^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )}{d}+\frac {(a+b \text {ArcTan}(c x))^2 \log \left (\frac {2}{1-i c x}\right )}{d}-\frac {(a+b \text {ArcTan}(c x))^2 \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 d}-\frac {(a+b \text {ArcTan}(c x))^2 \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 d}-\frac {i b (a+b \text {ArcTan}(c x)) \text {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{d}-\frac {i b (a+b \text {ArcTan}(c x)) \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{d}+\frac {i b (a+b \text {ArcTan}(c x)) \text {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d}+\frac {i b (a+b \text {ArcTan}(c x)) \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 d}+\frac {i b (a+b \text {ArcTan}(c x)) \text {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 d}+\frac {b^2 \text {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 d}-\frac {b^2 \text {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 d}+\frac {b^2 \text {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d}-\frac {b^2 \text {PolyLog}\left (3,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 d}-\frac {b^2 \text {PolyLog}\left (3,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 d} \]
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Rubi [A]
time = 0.50, antiderivative size = 637, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {5100, 4942,
5108, 5004, 5114, 6745, 4968} \begin {gather*} \frac {i b (a+b \text {ArcTan}(c x)) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 d}+\frac {i b (a+b \text {ArcTan}(c x)) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right )}{2 d}-\frac {(a+b \text {ArcTan}(c x))^2 \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}-i \sqrt {e}\right )}\right )}{2 d}-\frac {(a+b \text {ArcTan}(c x))^2 \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}+i \sqrt {e}\right )}\right )}{2 d}-\frac {i b \text {Li}_2\left (1-\frac {2}{1-i c x}\right ) (a+b \text {ArcTan}(c x))}{d}-\frac {i b \text {Li}_2\left (1-\frac {2}{i c x+1}\right ) (a+b \text {ArcTan}(c x))}{d}+\frac {i b \text {Li}_2\left (\frac {2}{i c x+1}-1\right ) (a+b \text {ArcTan}(c x))}{d}+\frac {\log \left (\frac {2}{1-i c x}\right ) (a+b \text {ArcTan}(c x))^2}{d}+\frac {2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right ) (a+b \text {ArcTan}(c x))^2}{d}-\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 d}-\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right )}{4 d}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 d}-\frac {b^2 \text {Li}_3\left (1-\frac {2}{i c x+1}\right )}{2 d}+\frac {b^2 \text {Li}_3\left (\frac {2}{i c x+1}-1\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 4942
Rule 4968
Rule 5004
Rule 5100
Rule 5108
Rule 5114
Rule 6745
Rubi steps
\begin {align*} \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x \left (d+e x^2\right )} \, dx &=\int \left (\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{d x}-\frac {e x \left (a+b \tan ^{-1}(c x)\right )^2}{d \left (d+e x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x} \, dx}{d}-\frac {e \int \frac {x \left (a+b \tan ^{-1}(c x)\right )^2}{d+e x^2} \, dx}{d}\\ &=\frac {2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )}{d}-\frac {(4 b c) \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}{d}-\frac {e \int \left (-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{d}\\ &=\frac {2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )}{d}+\frac {(2 b c) \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}{d}-\frac {(2 b c) \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}{d}+\frac {\sqrt {e} \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 d}-\frac {\sqrt {e} \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 d}\\ &=\frac {2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )}{d}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{d}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 d}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 d}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{d}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{d}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )}{d}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 d}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 d}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 d}-\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 d}-\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 d}+\frac {\left (i b^2 c\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d}-\frac {\left (i b^2 c\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d}\\ &=\frac {2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )}{d}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-i c x}\right )}{d}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 d}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 d}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{d}-\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{d}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )}{d}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 d}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 d}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1-i c x}\right )}{2 d}-\frac {b^2 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 d}+\frac {b^2 \text {Li}_3\left (-1+\frac {2}{1+i c x}\right )}{2 d}-\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 d}-\frac {b^2 \text {Li}_3\left (1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 d}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(1412\) vs. \(2(637)=1274\).
time = 4.87, size = 1412, normalized size = 2.22 \begin {gather*} \frac {24 a^2 \log (x)-12 a^2 \log \left (d+e x^2\right )-24 a b \left (-i \text {ArcTan}(c x)^2+2 i \text {ArcSin}\left (\sqrt {\frac {c^2 d}{c^2 d-e}}\right ) \text {ArcTan}\left (\frac {c e x}{\sqrt {c^2 d e}}\right )-2 \text {ArcTan}(c x) \log \left (1-e^{2 i \text {ArcTan}(c x)}\right )+\left (-\text {ArcSin}\left (\sqrt {\frac {c^2 d}{c^2 d-e}}\right )+\text {ArcTan}(c x)\right ) \log \left (1+\frac {\left (c^2 d+e+2 \sqrt {c^2 d e}\right ) e^{2 i \text {ArcTan}(c x)}}{c^2 d-e}\right )+\left (\text {ArcSin}\left (\sqrt {\frac {c^2 d}{c^2 d-e}}\right )+\text {ArcTan}(c x)\right ) \log \left (\frac {-2 \sqrt {c^2 d e} e^{2 i \text {ArcTan}(c x)}+e \left (-1+e^{2 i \text {ArcTan}(c x)}\right )+c^2 d \left (1+e^{2 i \text {ArcTan}(c x)}\right )}{c^2 d-e}\right )+i \left (\text {ArcTan}(c x)^2+\text {PolyLog}\left (2,e^{2 i \text {ArcTan}(c x)}\right )\right )-\frac {1}{2} i \left (\text {PolyLog}\left (2,\frac {\left (-c^2 d-e+2 \sqrt {c^2 d e}\right ) e^{2 i \text {ArcTan}(c x)}}{c^2 d-e}\right )+\text {PolyLog}\left (2,-\frac {\left (c^2 d+e+2 \sqrt {c^2 d e}\right ) e^{2 i \text {ArcTan}(c x)}}{c^2 d-e}\right )\right )\right )+b^2 \left (-i \pi ^3+16 i \text {ArcTan}(c x)^3+24 \text {ArcTan}(c x)^2 \log \left (1-e^{-2 i \text {ArcTan}(c x)}\right )-12 \text {ArcTan}(c x)^2 \log \left (1+\frac {\left (c \sqrt {d}-\sqrt {e}\right ) e^{2 i \text {ArcTan}(c x)}}{c \sqrt {d}+\sqrt {e}}\right )-12 \text {ArcTan}(c x)^2 \log \left (1+\frac {\left (c \sqrt {d}+\sqrt {e}\right ) e^{2 i \text {ArcTan}(c x)}}{c \sqrt {d}-\sqrt {e}}\right )+12 \text {ArcTan}(c x)^2 \log \left (1+\frac {\left (c^2 d+e-2 \sqrt {c^2 d e}\right ) e^{2 i \text {ArcTan}(c x)}}{c^2 d-e}\right )+24 \text {ArcSin}\left (\sqrt {\frac {c^2 d}{c^2 d-e}}\right ) \text {ArcTan}(c x) \log \left (1+\frac {\left (c^2 d+e+2 \sqrt {c^2 d e}\right ) e^{2 i \text {ArcTan}(c x)}}{c^2 d-e}\right )-12 \text {ArcTan}(c x)^2 \log \left (1+\frac {\left (c^2 d+e+2 \sqrt {c^2 d e}\right ) e^{2 i \text {ArcTan}(c x)}}{c^2 d-e}\right )-24 \text {ArcSin}\left (\sqrt {\frac {c^2 d}{c^2 d-e}}\right ) \text {ArcTan}(c x) \log \left (\frac {-2 \sqrt {c^2 d e} e^{2 i \text {ArcTan}(c x)}+e \left (-1+e^{2 i \text {ArcTan}(c x)}\right )+c^2 d \left (1+e^{2 i \text {ArcTan}(c x)}\right )}{c^2 d-e}\right )-24 \text {ArcTan}(c x)^2 \log \left (\frac {-2 \sqrt {c^2 d e} e^{2 i \text {ArcTan}(c x)}+e \left (-1+e^{2 i \text {ArcTan}(c x)}\right )+c^2 d \left (1+e^{2 i \text {ArcTan}(c x)}\right )}{c^2 d-e}\right )+24 \text {ArcSin}\left (\sqrt {\frac {c^2 d}{c^2 d-e}}\right ) \text {ArcTan}(c x) \log \left (\frac {2 i c^2 d-2 i \sqrt {c^2 d e}+2 c \left (-e+\sqrt {c^2 d e}\right ) x}{\left (c^2 d-e\right ) (i+c x)}\right )+12 \text {ArcTan}(c x)^2 \log \left (\frac {2 i c^2 d-2 i \sqrt {c^2 d e}+2 c \left (-e+\sqrt {c^2 d e}\right ) x}{\left (c^2 d-e\right ) (i+c x)}\right )-24 \text {ArcSin}\left (\sqrt {\frac {c^2 d}{c^2 d-e}}\right ) \text {ArcTan}(c x) \log \left (1+\frac {\left (c^2 d+e+2 \sqrt {c^2 d e}\right ) (\cos (2 \text {ArcTan}(c x))+i \sin (2 \text {ArcTan}(c x)))}{c^2 d-e}\right )+12 \text {ArcTan}(c x)^2 \log \left (1+\frac {\left (c^2 d+e+2 \sqrt {c^2 d e}\right ) (\cos (2 \text {ArcTan}(c x))+i \sin (2 \text {ArcTan}(c x)))}{c^2 d-e}\right )+24 i \text {ArcTan}(c x) \text {PolyLog}\left (2,e^{-2 i \text {ArcTan}(c x)}\right )+12 i \text {ArcTan}(c x) \text {PolyLog}\left (2,\frac {\left (-c \sqrt {d}+\sqrt {e}\right ) e^{2 i \text {ArcTan}(c x)}}{c \sqrt {d}+\sqrt {e}}\right )+12 i \text {ArcTan}(c x) \text {PolyLog}\left (2,-\frac {\left (c \sqrt {d}+\sqrt {e}\right ) e^{2 i \text {ArcTan}(c x)}}{c \sqrt {d}-\sqrt {e}}\right )+12 \text {PolyLog}\left (3,e^{-2 i \text {ArcTan}(c x)}\right )-6 \text {PolyLog}\left (3,\frac {\left (-c \sqrt {d}+\sqrt {e}\right ) e^{2 i \text {ArcTan}(c x)}}{c \sqrt {d}+\sqrt {e}}\right )-6 \text {PolyLog}\left (3,-\frac {\left (c \sqrt {d}+\sqrt {e}\right ) e^{2 i \text {ArcTan}(c x)}}{c \sqrt {d}-\sqrt {e}}\right )\right )}{24 d} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 1.28, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arctan \left (c x \right )\right )^{2}}{x \left (e \,x^{2}+d \right )}\, dx\]
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 \right )}\right )^{2}}{x \left (d + e x^{2}\right )}\, 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 = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{x\,\left (e\,x^2+d\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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