Optimal. Leaf size=293 \[ -\frac {b c}{2 d x}-\frac {b c^2 \text {ArcTan}(c x)}{2 d}-\frac {a+b \text {ArcTan}(c x)}{2 d x^2}+\frac {e (a+b \text {ArcTan}(c x))}{d^2 x}-\frac {b c e \log (x)}{d^2}+\frac {a e^2 \log (x)}{d^3}+\frac {e^2 (a+b \text {ArcTan}(c x)) \log \left (\frac {2}{1-i c x}\right )}{d^3}-\frac {e^2 (a+b \text {ArcTan}(c x)) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{d^3}+\frac {b c e \log \left (1+c^2 x^2\right )}{2 d^2}+\frac {i b e^2 \text {PolyLog}(2,-i c x)}{2 d^3}-\frac {i b e^2 \text {PolyLog}(2,i c x)}{2 d^3}-\frac {i b e^2 \text {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 d^3}+\frac {i b e^2 \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 d^3} \]
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Rubi [A]
time = 0.20, antiderivative size = 293, normalized size of antiderivative = 1.00, number
of steps used = 17, number of rules used = 14, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules
used = {4996, 4946, 331, 209, 272, 36, 29, 31, 4940, 2438, 4966, 2449, 2352, 2497}
\begin {gather*} \frac {e^2 \log \left (\frac {2}{1-i c x}\right ) (a+b \text {ArcTan}(c x))}{d^3}-\frac {e^2 (a+b \text {ArcTan}(c x)) \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{d^3}+\frac {e (a+b \text {ArcTan}(c x))}{d^2 x}-\frac {a+b \text {ArcTan}(c x)}{2 d x^2}+\frac {a e^2 \log (x)}{d^3}-\frac {b c^2 \text {ArcTan}(c x)}{2 d}+\frac {b c e \log \left (c^2 x^2+1\right )}{2 d^2}+\frac {i b e^2 \text {Li}_2(-i c x)}{2 d^3}-\frac {i b e^2 \text {Li}_2(i c x)}{2 d^3}-\frac {i b e^2 \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 d^3}+\frac {i b e^2 \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 d^3}-\frac {b c e \log (x)}{d^2}-\frac {b c}{2 d x} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 209
Rule 272
Rule 331
Rule 2352
Rule 2438
Rule 2449
Rule 2497
Rule 4940
Rule 4946
Rule 4966
Rule 4996
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}(c x)}{x^3 (d+e x)} \, dx &=\int \left (\frac {a+b \tan ^{-1}(c x)}{d x^3}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{d^2 x^2}+\frac {e^2 \left (a+b \tan ^{-1}(c x)\right )}{d^3 x}-\frac {e^3 \left (a+b \tan ^{-1}(c x)\right )}{d^3 (d+e x)}\right ) \, dx\\ &=\frac {\int \frac {a+b \tan ^{-1}(c x)}{x^3} \, dx}{d}-\frac {e \int \frac {a+b \tan ^{-1}(c x)}{x^2} \, dx}{d^2}+\frac {e^2 \int \frac {a+b \tan ^{-1}(c x)}{x} \, dx}{d^3}-\frac {e^3 \int \frac {a+b \tan ^{-1}(c x)}{d+e x} \, dx}{d^3}\\ &=-\frac {a+b \tan ^{-1}(c x)}{2 d x^2}+\frac {e \left (a+b \tan ^{-1}(c x)\right )}{d^2 x}+\frac {a e^2 \log (x)}{d^3}+\frac {e^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{d^3}-\frac {e^2 \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 )}{d^3}+\frac {(b c) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx}{2 d}-\frac {(b c e) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx}{d^2}+\frac {\left (i b e^2\right ) \int \frac {\log (1-i c x)}{x} \, dx}{2 d^3}-\frac {\left (i b e^2\right ) \int \frac {\log (1+i c x)}{x} \, dx}{2 d^3}-\frac {\left (b c e^2\right ) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{d^3}+\frac {\left (b c e^2\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}{d^3}\\ &=-\frac {b c}{2 d x}-\frac {a+b \tan ^{-1}(c x)}{2 d x^2}+\frac {e \left (a+b \tan ^{-1}(c x)\right )}{d^2 x}+\frac {a e^2 \log (x)}{d^3}+\frac {e^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{d^3}-\frac {e^2 \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 )}{d^3}+\frac {i b e^2 \text {Li}_2(-i c x)}{2 d^3}-\frac {i b e^2 \text {Li}_2(i c x)}{2 d^3}+\frac {i b e^2 \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 d^3}-\frac {\left (b c^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{2 d}-\frac {(b c e) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 d^2}-\frac {\left (i b e^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i c x}\right )}{d^3}\\ &=-\frac {b c}{2 d x}-\frac {b c^2 \tan ^{-1}(c x)}{2 d}-\frac {a+b \tan ^{-1}(c x)}{2 d x^2}+\frac {e \left (a+b \tan ^{-1}(c x)\right )}{d^2 x}+\frac {a e^2 \log (x)}{d^3}+\frac {e^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{d^3}-\frac {e^2 \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 )}{d^3}+\frac {i b e^2 \text {Li}_2(-i c x)}{2 d^3}-\frac {i b e^2 \text {Li}_2(i c x)}{2 d^3}-\frac {i b e^2 \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 d^3}+\frac {i b e^2 \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 d^3}-\frac {(b c e) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 d^2}+\frac {\left (b c^3 e\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )}{2 d^2}\\ &=-\frac {b c}{2 d x}-\frac {b c^2 \tan ^{-1}(c x)}{2 d}-\frac {a+b \tan ^{-1}(c x)}{2 d x^2}+\frac {e \left (a+b \tan ^{-1}(c x)\right )}{d^2 x}-\frac {b c e \log (x)}{d^2}+\frac {a e^2 \log (x)}{d^3}+\frac {e^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{d^3}-\frac {e^2 \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 )}{d^3}+\frac {b c e \log \left (1+c^2 x^2\right )}{2 d^2}+\frac {i b e^2 \text {Li}_2(-i c x)}{2 d^3}-\frac {i b e^2 \text {Li}_2(i c x)}{2 d^3}-\frac {i b e^2 \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 d^3}+\frac {i b e^2 \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 d^3}\\ \end {align*}
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Mathematica [A]
time = 2.17, size = 441, normalized size = 1.51 \begin {gather*} -\frac {\frac {a d^3}{x^2}-\frac {2 a d^2 e}{x}-2 a d e^2 \log (x)+2 a d e^2 \log (d+e x)+\frac {b \left (\frac {c^2 d^3}{x}+c^3 d^3 \text {ArcTan}(c x)+i c d e^2 \pi \text {ArcTan}(c x)+\frac {c d^3 \text {ArcTan}(c x)}{x^2}-\frac {2 c d^2 e \text {ArcTan}(c x)}{x}-2 i c d e^2 \text {ArcTan}\left (\frac {c d}{e}\right ) \text {ArcTan}(c x)+i c d e^2 \text {ArcTan}(c x)^2+e^3 \text {ArcTan}(c x)^2-\sqrt {1+\frac {c^2 d^2}{e^2}} e^3 e^{i \text {ArcTan}\left (\frac {c d}{e}\right )} \text {ArcTan}(c x)^2+c d e^2 \pi \log \left (1+e^{-2 i \text {ArcTan}(c x)}\right )-2 c d e^2 \text {ArcTan}(c x) \log \left (1-e^{2 i \text {ArcTan}(c x)}\right )+2 c d e^2 \text {ArcTan}\left (\frac {c d}{e}\right ) \log \left (1-e^{2 i \left (\text {ArcTan}\left (\frac {c d}{e}\right )+\text {ArcTan}(c x)\right )}\right )+2 c d e^2 \text {ArcTan}(c x) \log \left (1-e^{2 i \left (\text {ArcTan}\left (\frac {c d}{e}\right )+\text {ArcTan}(c x)\right )}\right )+2 c^2 d^2 e \log \left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )+\frac {1}{2} c d e^2 \pi \log \left (1+c^2 x^2\right )-2 c d e^2 \text {ArcTan}\left (\frac {c d}{e}\right ) \log \left (\sin \left (\text {ArcTan}\left (\frac {c d}{e}\right )+\text {ArcTan}(c x)\right )\right )+i c d e^2 \text {PolyLog}\left (2,e^{2 i \text {ArcTan}(c x)}\right )-i c d e^2 \text {PolyLog}\left (2,e^{2 i \left (\text {ArcTan}\left (\frac {c d}{e}\right )+\text {ArcTan}(c x)\right )}\right )\right )}{c}}{2 d^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 448, normalized size = 1.53
method | result | size |
derivativedivides | \(c^{2} \left (-\frac {a \,e^{2} \ln \left (c e x +c d \right )}{c^{2} d^{3}}-\frac {a}{2 d \,c^{2} x^{2}}+\frac {a \,e^{2} \ln \left (c x \right )}{c^{2} d^{3}}+\frac {a e}{c^{2} d^{2} x}-\frac {b \arctan \left (c x \right ) e^{2} \ln \left (c e x +c d \right )}{c^{2} d^{3}}-\frac {b \arctan \left (c x \right )}{2 d \,c^{2} x^{2}}+\frac {b \arctan \left (c x \right ) e^{2} \ln \left (c x \right )}{c^{2} d^{3}}+\frac {b \arctan \left (c x \right ) e}{c^{2} d^{2} x}+\frac {i b \,e^{2} \dilog \left (\frac {c e x +i e}{-c d +i e}\right )}{2 c^{2} d^{3}}+\frac {i b \,e^{2} \ln \left (c e x +c d \right ) \ln \left (\frac {c e x +i e}{-c d +i e}\right )}{2 c^{2} d^{3}}-\frac {i b \,e^{2} \dilog \left (\frac {-c e x +i e}{c d +i e}\right )}{2 c^{2} d^{3}}+\frac {i b \,e^{2} \ln \left (c x \right ) \ln \left (i c x +1\right )}{2 c^{2} d^{3}}-\frac {i b \,e^{2} \dilog \left (-i c x +1\right )}{2 c^{2} d^{3}}-\frac {i b \,e^{2} \ln \left (c e x +c d \right ) \ln \left (\frac {-c e x +i e}{c d +i e}\right )}{2 c^{2} d^{3}}-\frac {i b \,e^{2} \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2 c^{2} d^{3}}+\frac {i b \,e^{2} \dilog \left (i c x +1\right )}{2 c^{2} d^{3}}+\frac {b e \ln \left (c^{2} x^{2}+1\right )}{2 c \,d^{2}}-\frac {b \arctan \left (c x \right )}{2 d}-\frac {b}{2 d c x}-\frac {b e \ln \left (c x \right )}{c \,d^{2}}\right )\) | \(448\) |
default | \(c^{2} \left (-\frac {a \,e^{2} \ln \left (c e x +c d \right )}{c^{2} d^{3}}-\frac {a}{2 d \,c^{2} x^{2}}+\frac {a \,e^{2} \ln \left (c x \right )}{c^{2} d^{3}}+\frac {a e}{c^{2} d^{2} x}-\frac {b \arctan \left (c x \right ) e^{2} \ln \left (c e x +c d \right )}{c^{2} d^{3}}-\frac {b \arctan \left (c x \right )}{2 d \,c^{2} x^{2}}+\frac {b \arctan \left (c x \right ) e^{2} \ln \left (c x \right )}{c^{2} d^{3}}+\frac {b \arctan \left (c x \right ) e}{c^{2} d^{2} x}+\frac {i b \,e^{2} \dilog \left (\frac {c e x +i e}{-c d +i e}\right )}{2 c^{2} d^{3}}+\frac {i b \,e^{2} \ln \left (c e x +c d \right ) \ln \left (\frac {c e x +i e}{-c d +i e}\right )}{2 c^{2} d^{3}}-\frac {i b \,e^{2} \dilog \left (\frac {-c e x +i e}{c d +i e}\right )}{2 c^{2} d^{3}}+\frac {i b \,e^{2} \ln \left (c x \right ) \ln \left (i c x +1\right )}{2 c^{2} d^{3}}-\frac {i b \,e^{2} \dilog \left (-i c x +1\right )}{2 c^{2} d^{3}}-\frac {i b \,e^{2} \ln \left (c e x +c d \right ) \ln \left (\frac {-c e x +i e}{c d +i e}\right )}{2 c^{2} d^{3}}-\frac {i b \,e^{2} \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2 c^{2} d^{3}}+\frac {i b \,e^{2} \dilog \left (i c x +1\right )}{2 c^{2} d^{3}}+\frac {b e \ln \left (c^{2} x^{2}+1\right )}{2 c \,d^{2}}-\frac {b \arctan \left (c x \right )}{2 d}-\frac {b}{2 d c x}-\frac {b e \ln \left (c x \right )}{c \,d^{2}}\right )\) | \(448\) |
risch | \(-\frac {i b \,e^{2} \ln \left (-i c x +1\right ) \ln \left (\frac {-i c d +\left (-i c x +1\right ) e -e}{-i c d -e}\right )}{2 d^{3}}-\frac {i b \,e^{2} \dilog \left (\frac {-i c d +\left (-i c x +1\right ) e -e}{-i c d -e}\right )}{2 d^{3}}-\frac {i b e \ln \left (i c x +1\right )}{2 d^{2} x}-\frac {i b \dilog \left (-i c x +1\right ) e^{2}}{2 d^{3}}-\frac {b c}{2 d x}-\frac {b \,c^{2} \arctan \left (c x \right )}{2 d}+\frac {i b \,e^{2} \ln \left (i c x +1\right ) \ln \left (\frac {i c d +\left (i c x +1\right ) e -e}{i c d -e}\right )}{2 d^{3}}+\frac {i b \ln \left (i c x +1\right )}{4 d \,x^{2}}+\frac {b c e \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}-\frac {c b e \ln \left (-i c x \right )}{2 d^{2}}+\frac {a \,e^{2} \ln \left (-i c x \right )}{d^{3}}-\frac {b c e \ln \left (i c x \right )}{2 d^{2}}-\frac {a \,e^{2} \ln \left (i c d -\left (-i c x +1\right ) e +e \right )}{d^{3}}+\frac {i b e \ln \left (-i c x +1\right )}{2 d^{2} x}-\frac {i b \ln \left (-i c x +1\right )}{4 d \,x^{2}}-\frac {a}{2 d \,x^{2}}+\frac {i b \dilog \left (i c x +1\right ) e^{2}}{2 d^{3}}+\frac {i c^{2} b \ln \left (-i c x \right )}{4 d}+\frac {i b \,e^{2} \dilog \left (\frac {i c d +\left (i c x +1\right ) e -e}{i c d -e}\right )}{2 d^{3}}+\frac {a e}{x \,d^{2}}-\frac {i b \,c^{2} \ln \left (i c x \right )}{4 d}\) | \(452\) |
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 {a + b \operatorname {atan}{\left (c x \right )}}{x^{3} \left (d + e x\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 {a+b\,\mathrm {atan}\left (c\,x\right )}{x^3\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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