Optimal. Leaf size=496 \[ \frac {c^2 (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^2}{2 e \left (c^2 d^2+e^2\right )^2}-\frac {b c \left (a+b \tan ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) (d+e x)}+\frac {i c^3 d \left (a+b \tan ^{-1}(c x)\right )^2}{\left (c^2 d^2+e^2\right )^2}-\frac {2 b c^3 d \log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c^3 d \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c^3 d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 e (d+e x)^2}-\frac {b^2 c^2 e \log \left (c^2 x^2+1\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {b^2 c^2 e \log (d+e x)}{\left (c^2 d^2+e^2\right )^2}+\frac {i b^2 c^3 d \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {i b^2 c^3 d \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {i b^2 c^3 d \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {b^2 c^3 d \tan ^{-1}(c x)}{\left (c^2 d^2+e^2\right )^2} \]
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Rubi [A] time = 0.54, antiderivative size = 496, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 15, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {4864, 4862, 706, 31, 635, 203, 260, 4856, 2402, 2315, 2447, 4984, 4884, 4920, 4854} \[ \frac {i b^2 c^3 d \text {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {i b^2 c^3 d \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {i b^2 c^3 d \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {i c^3 d \left (a+b \tan ^{-1}(c x)\right )^2}{\left (c^2 d^2+e^2\right )^2}+\frac {c^2 (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^2}{2 e \left (c^2 d^2+e^2\right )^2}-\frac {b c \left (a+b \tan ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) (d+e x)}-\frac {2 b c^3 d \log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c^3 d \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c^3 d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 e (d+e x)^2}-\frac {b^2 c^2 e \log \left (c^2 x^2+1\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {b^2 c^2 e \log (d+e x)}{\left (c^2 d^2+e^2\right )^2}+\frac {b^2 c^3 d \tan ^{-1}(c x)}{\left (c^2 d^2+e^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 203
Rule 260
Rule 635
Rule 706
Rule 2315
Rule 2402
Rule 2447
Rule 4854
Rule 4856
Rule 4862
Rule 4864
Rule 4884
Rule 4920
Rule 4984
Rubi steps
\begin {align*} \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{(d+e x)^3} \, dx &=-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {(b c) \int \left (\frac {e^2 \left (a+b \tan ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) (d+e x)^2}+\frac {2 c^2 d e^2 \left (a+b \tan ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right )^2 (d+e x)}+\frac {\left (c^4 d^2-c^2 e^2-2 c^4 d e x\right ) \left (a+b \tan ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right )^2 \left (1+c^2 x^2\right )}\right ) \, dx}{e}\\ &=-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {(b c) \int \frac {\left (c^4 d^2-c^2 e^2-2 c^4 d e x\right ) \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{e \left (c^2 d^2+e^2\right )^2}+\frac {\left (2 b c^3 d e\right ) \int \frac {a+b \tan ^{-1}(c x)}{d+e x} \, dx}{\left (c^2 d^2+e^2\right )^2}+\frac {(b c e) \int \frac {a+b \tan ^{-1}(c x)}{(d+e x)^2} \, dx}{c^2 d^2+e^2}\\ &=-\frac {b c \left (a+b \tan ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) (d+e x)}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 e (d+e x)^2}-\frac {2 b c^3 d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c^3 d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {\left (2 b^2 c^4 d\right ) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{\left (c^2 d^2+e^2\right )^2}-\frac {\left (2 b^2 c^4 d\right ) \int \frac {\log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{\left (c^2 d^2+e^2\right )^2}+\frac {(b c) \int \left (\frac {c^4 d^2 \left (1-\frac {e^2}{c^2 d^2}\right ) \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}-\frac {2 c^4 d e x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}\right ) \, dx}{e \left (c^2 d^2+e^2\right )^2}+\frac {\left (b^2 c^2\right ) \int \frac {1}{(d+e x) \left (1+c^2 x^2\right )} \, dx}{c^2 d^2+e^2}\\ &=-\frac {b c \left (a+b \tan ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) (d+e x)}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 e (d+e x)^2}-\frac {2 b c^3 d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c^3 d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {i b^2 c^3 d \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {\left (b^2 c^2\right ) \int \frac {c^2 d-c^2 e x}{1+c^2 x^2} \, dx}{\left (c^2 d^2+e^2\right )^2}+\frac {\left (2 i b^2 c^3 d\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {\left (2 b c^5 d\right ) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{\left (c^2 d^2+e^2\right )^2}+\frac {\left (b^2 c^2 e^2\right ) \int \frac {1}{d+e x} \, dx}{\left (c^2 d^2+e^2\right )^2}+\frac {\left (b c^3 (c d-e) (c d+e)\right ) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{e \left (c^2 d^2+e^2\right )^2}\\ &=-\frac {b c \left (a+b \tan ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) (d+e x)}+\frac {i c^3 d \left (a+b \tan ^{-1}(c x)\right )^2}{\left (c^2 d^2+e^2\right )^2}+\frac {c^2 (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^2}{2 e \left (c^2 d^2+e^2\right )^2}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 e (d+e x)^2}-\frac {2 b c^3 d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {b^2 c^2 e \log (d+e x)}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c^3 d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {i b^2 c^3 d \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {i b^2 c^3 d \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {\left (2 b c^4 d\right ) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx}{\left (c^2 d^2+e^2\right )^2}+\frac {\left (b^2 c^4 d\right ) \int \frac {1}{1+c^2 x^2} \, dx}{\left (c^2 d^2+e^2\right )^2}-\frac {\left (b^2 c^4 e\right ) \int \frac {x}{1+c^2 x^2} \, dx}{\left (c^2 d^2+e^2\right )^2}\\ &=\frac {b^2 c^3 d \tan ^{-1}(c x)}{\left (c^2 d^2+e^2\right )^2}-\frac {b c \left (a+b \tan ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) (d+e x)}+\frac {i c^3 d \left (a+b \tan ^{-1}(c x)\right )^2}{\left (c^2 d^2+e^2\right )^2}+\frac {c^2 (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^2}{2 e \left (c^2 d^2+e^2\right )^2}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 e (d+e x)^2}-\frac {2 b c^3 d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c^3 d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {b^2 c^2 e \log (d+e x)}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c^3 d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {b^2 c^2 e \log \left (1+c^2 x^2\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {i b^2 c^3 d \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {i b^2 c^3 d \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {\left (2 b^2 c^4 d\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{\left (c^2 d^2+e^2\right )^2}\\ &=\frac {b^2 c^3 d \tan ^{-1}(c x)}{\left (c^2 d^2+e^2\right )^2}-\frac {b c \left (a+b \tan ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) (d+e x)}+\frac {i c^3 d \left (a+b \tan ^{-1}(c x)\right )^2}{\left (c^2 d^2+e^2\right )^2}+\frac {c^2 (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^2}{2 e \left (c^2 d^2+e^2\right )^2}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 e (d+e x)^2}-\frac {2 b c^3 d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c^3 d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {b^2 c^2 e \log (d+e x)}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c^3 d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {b^2 c^2 e \log \left (1+c^2 x^2\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {i b^2 c^3 d \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {i b^2 c^3 d \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {\left (2 i b^2 c^3 d\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}\\ &=\frac {b^2 c^3 d \tan ^{-1}(c x)}{\left (c^2 d^2+e^2\right )^2}-\frac {b c \left (a+b \tan ^{-1}(c x)\right )}{\left (c^2 d^2+e^2\right ) (d+e x)}+\frac {i c^3 d \left (a+b \tan ^{-1}(c x)\right )^2}{\left (c^2 d^2+e^2\right )^2}+\frac {c^2 (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^2}{2 e \left (c^2 d^2+e^2\right )^2}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 e (d+e x)^2}-\frac {2 b c^3 d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c^3 d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {b^2 c^2 e \log (d+e x)}{\left (c^2 d^2+e^2\right )^2}+\frac {2 b c^3 d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {b^2 c^2 e \log \left (1+c^2 x^2\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {i b^2 c^3 d \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{\left (c^2 d^2+e^2\right )^2}+\frac {i b^2 c^3 d \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{\left (c^2 d^2+e^2\right )^2}-\frac {i b^2 c^3 d \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{\left (c^2 d^2+e^2\right )^2}\\ \end {align*}
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Mathematica [A] time = 7.08, size = 479, normalized size = 0.97 \[ -\frac {a^2}{2 e (d+e x)^2}+\frac {a b \left (c (d+e x) \left (c^2 \left (-d^2\right )-c^2 d \log \left (c^2 x^2+1\right ) (d+e x)+2 c^2 d (d+e x) \log (c (d+e x))-e^2\right )+\tan ^{-1}(c x) \left (c^4 d^2 x (2 d+e x)-c^2 e \left (3 d^2+2 d e x+e^2 x^2\right )-e^3\right )\right )}{\left (c^2 d^2+e^2\right )^2 (d+e x)^2}+\frac {b^2 c^2 \left (-\frac {2 c d \left (-\frac {1}{2} \pi \log \left (c^2 x^2+1\right )+i \text {Li}_2\left (e^{2 i \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )-i \tan ^{-1}(c x) \left (\pi -2 \tan ^{-1}\left (\frac {c d}{e}\right )\right )-2 \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right ) \log \left (1-e^{2 i \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )}\right )+2 \tan ^{-1}\left (\frac {c d}{e}\right ) \log \left (\sin \left (\tan ^{-1}\left (\frac {c d}{e}\right )+\tan ^{-1}(c x)\right )\right )-\pi \log \left (1+e^{-2 i \tan ^{-1}(c x)}\right )\right )}{c^2 d^2+e^2}-\frac {2 \tan ^{-1}(c x)^2 e^{i \tan ^{-1}\left (\frac {c d}{e}\right )}}{e \sqrt {\frac {c^2 d^2}{e^2}+1}}-\frac {e \left (c^2 x^2+1\right ) \tan ^{-1}(c x)^2}{c^2 (d+e x)^2}+\frac {2 c d e \log \left (\frac {c (d+e x)}{\sqrt {c^2 x^2+1}}\right )-2 e^2 \tan ^{-1}(c x)}{c^3 d^3+c d e^2}+\frac {2 x \tan ^{-1}(c x) \left (c d \tan ^{-1}(c x)+e\right )}{c d (d+e x)}\right )}{2 \left (c^2 d^2+e^2\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.29, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \arctan \left (c x\right )^{2} + 2 \, a b \arctan \left (c x\right ) + a^{2}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]
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 = 0.11, size = 961, normalized size = 1.94 \[ -\frac {b^{2} c^{2} e \ln \left (c^{2} x^{2}+1\right )}{2 \left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {c^{2} b^{2} e \ln \left (c e x +d c \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {c^{2} b^{2} \arctan \left (c x \right )^{2}}{2 \left (c e x +d c \right )^{2} e}-\frac {c^{2} a b}{\left (c^{2} d^{2}+e^{2}\right ) \left (c e x +d c \right )}-\frac {c^{2} b^{2} e \arctan \left (c x \right )^{2}}{2 \left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {c^{2} b^{2} \arctan \left (c x \right )}{\left (c^{2} d^{2}+e^{2}\right ) \left (c e x +d c \right )}+\frac {i c^{3} b^{2} d \ln \left (c e x +d c \right ) \ln \left (\frac {-c e x +i e}{d c +i e}\right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {i c^{3} b^{2} d \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{2 \left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {i c^{3} b^{2} d \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{2 \left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {i c^{3} b^{2} d \ln \left (c e x +d c \right ) \ln \left (\frac {c e x +i e}{-d c +i e}\right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {i c^{3} b^{2} d \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{2 \left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {i c^{3} b^{2} d \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{2 \left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {c^{4} a b \arctan \left (c x \right ) d^{2}}{e \left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {b^{2} c^{3} d \arctan \left (c x \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {c^{2} a^{2}}{2 \left (c e x +d c \right )^{2} e}-\frac {i c^{3} b^{2} d \dilog \left (\frac {i \left (c x -i\right )}{2}\right )}{2 \left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {i c^{3} b^{2} d \ln \left (c x -i\right )^{2}}{4 \left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {i c^{3} b^{2} d \ln \left (c x +i\right )^{2}}{4 \left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {i c^{3} b^{2} d \dilog \left (\frac {-c e x +i e}{d c +i e}\right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {i c^{3} b^{2} d \dilog \left (\frac {c e x +i e}{-d c +i e}\right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {i c^{3} b^{2} d \dilog \left (-\frac {i \left (c x +i\right )}{2}\right )}{2 \left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {2 c^{3} a b d \ln \left (c e x +d c \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {c^{3} a b d \ln \left (c^{2} x^{2}+1\right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {c^{2} a b e \arctan \left (c x \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {c^{4} b^{2} \arctan \left (c x \right )^{2} d^{2}}{2 e \left (c^{2} d^{2}+e^{2}\right )^{2}}+\frac {2 c^{3} b^{2} \arctan \left (c x \right ) d \ln \left (c e x +d c \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {c^{2} a b \arctan \left (c x \right )}{\left (c e x +d c \right )^{2} e}-\frac {c^{3} b^{2} \arctan \left (c x \right ) d \ln \left (c^{2} x^{2}+1\right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}} \]
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 = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{{\left (d+e\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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