Optimal. Leaf size=228 \[ -\frac {3 b d^2 e x}{2 c}+\frac {3 b d e^2 x}{4 c^3}-\frac {b e^3 x}{6 c^5}-\frac {b d e^2 x^3}{4 c}+\frac {b e^3 x^3}{18 c^3}-\frac {b e^3 x^5}{30 c}+\frac {3 b d^2 e \text {ArcTan}(c x)}{2 c^2}-\frac {3 b d e^2 \text {ArcTan}(c x)}{4 c^4}+\frac {b e^3 \text {ArcTan}(c x)}{6 c^6}+\frac {3}{2} d^2 e x^2 (a+b \text {ArcTan}(c x))+\frac {3}{4} d e^2 x^4 (a+b \text {ArcTan}(c x))+\frac {1}{6} e^3 x^6 (a+b \text {ArcTan}(c x))+a d^3 \log (x)+\frac {1}{2} i b d^3 \text {PolyLog}(2,-i c x)-\frac {1}{2} i b d^3 \text {PolyLog}(2,i c x) \]
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Rubi [A]
time = 0.15, antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5100, 4940,
2438, 4946, 327, 209, 308} \begin {gather*} \frac {3}{2} d^2 e x^2 (a+b \text {ArcTan}(c x))+\frac {3}{4} d e^2 x^4 (a+b \text {ArcTan}(c x))+\frac {1}{6} e^3 x^6 (a+b \text {ArcTan}(c x))+a d^3 \log (x)+\frac {b e^3 \text {ArcTan}(c x)}{6 c^6}-\frac {3 b d e^2 \text {ArcTan}(c x)}{4 c^4}+\frac {3 b d^2 e \text {ArcTan}(c x)}{2 c^2}-\frac {b e^3 x}{6 c^5}+\frac {3 b d e^2 x}{4 c^3}+\frac {b e^3 x^3}{18 c^3}+\frac {1}{2} i b d^3 \text {Li}_2(-i c x)-\frac {1}{2} i b d^3 \text {Li}_2(i c x)-\frac {3 b d^2 e x}{2 c}-\frac {b d e^2 x^3}{4 c}-\frac {b e^3 x^5}{30 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 308
Rule 327
Rule 2438
Rule 4940
Rule 4946
Rule 5100
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^3 \left (a+b \tan ^{-1}(c x)\right )}{x} \, dx &=\int \left (\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}+3 d^2 e x \left (a+b \tan ^{-1}(c x)\right )+3 d e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )+e^3 x^5 \left (a+b \tan ^{-1}(c x)\right )\right ) \, dx\\ &=d^3 \int \frac {a+b \tan ^{-1}(c x)}{x} \, dx+\left (3 d^2 e\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx+\left (3 d e^2\right ) \int x^3 \left (a+b \tan ^{-1}(c x)\right ) \, dx+e^3 \int x^5 \left (a+b \tan ^{-1}(c x)\right ) \, dx\\ &=\frac {3}{2} d^2 e x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac {3}{4} d e^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{6} e^3 x^6 \left (a+b \tan ^{-1}(c x)\right )+a d^3 \log (x)+\frac {1}{2} \left (i b d^3\right ) \int \frac {\log (1-i c x)}{x} \, dx-\frac {1}{2} \left (i b d^3\right ) \int \frac {\log (1+i c x)}{x} \, dx-\frac {1}{2} \left (3 b c d^2 e\right ) \int \frac {x^2}{1+c^2 x^2} \, dx-\frac {1}{4} \left (3 b c d e^2\right ) \int \frac {x^4}{1+c^2 x^2} \, dx-\frac {1}{6} \left (b c e^3\right ) \int \frac {x^6}{1+c^2 x^2} \, dx\\ &=-\frac {3 b d^2 e x}{2 c}+\frac {3}{2} d^2 e x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac {3}{4} d e^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{6} e^3 x^6 \left (a+b \tan ^{-1}(c x)\right )+a d^3 \log (x)+\frac {1}{2} i b d^3 \text {Li}_2(-i c x)-\frac {1}{2} i b d^3 \text {Li}_2(i c x)+\frac {\left (3 b d^2 e\right ) \int \frac {1}{1+c^2 x^2} \, dx}{2 c}-\frac {1}{4} \left (3 b c d e^2\right ) \int \left (-\frac {1}{c^4}+\frac {x^2}{c^2}+\frac {1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx-\frac {1}{6} \left (b c e^3\right ) \int \left (\frac {1}{c^6}-\frac {x^2}{c^4}+\frac {x^4}{c^2}-\frac {1}{c^6 \left (1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac {3 b d^2 e x}{2 c}+\frac {3 b d e^2 x}{4 c^3}-\frac {b e^3 x}{6 c^5}-\frac {b d e^2 x^3}{4 c}+\frac {b e^3 x^3}{18 c^3}-\frac {b e^3 x^5}{30 c}+\frac {3 b d^2 e \tan ^{-1}(c x)}{2 c^2}+\frac {3}{2} d^2 e x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac {3}{4} d e^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{6} e^3 x^6 \left (a+b \tan ^{-1}(c x)\right )+a d^3 \log (x)+\frac {1}{2} i b d^3 \text {Li}_2(-i c x)-\frac {1}{2} i b d^3 \text {Li}_2(i c x)-\frac {\left (3 b d e^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{4 c^3}+\frac {\left (b e^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{6 c^5}\\ &=-\frac {3 b d^2 e x}{2 c}+\frac {3 b d e^2 x}{4 c^3}-\frac {b e^3 x}{6 c^5}-\frac {b d e^2 x^3}{4 c}+\frac {b e^3 x^3}{18 c^3}-\frac {b e^3 x^5}{30 c}+\frac {3 b d^2 e \tan ^{-1}(c x)}{2 c^2}-\frac {3 b d e^2 \tan ^{-1}(c x)}{4 c^4}+\frac {b e^3 \tan ^{-1}(c x)}{6 c^6}+\frac {3}{2} d^2 e x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac {3}{4} d e^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{6} e^3 x^6 \left (a+b \tan ^{-1}(c x)\right )+a d^3 \log (x)+\frac {1}{2} i b d^3 \text {Li}_2(-i c x)-\frac {1}{2} i b d^3 \text {Li}_2(i c x)\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 190, normalized size = 0.83 \begin {gather*} \frac {3}{2} a d^2 e x^2+\frac {3}{4} a d e^2 x^4+\frac {1}{6} a e^3 x^6+\frac {3 b d^2 e \left (-c x+\left (1+c^2 x^2\right ) \text {ArcTan}(c x)\right )}{2 c^2}+\frac {b d e^2 \left (3 c x-c^3 x^3+3 \left (-1+c^4 x^4\right ) \text {ArcTan}(c x)\right )}{4 c^4}+\frac {b e^3 \left (-15 c x+5 c^3 x^3-3 c^5 x^5+15 \left (1+c^6 x^6\right ) \text {ArcTan}(c x)\right )}{90 c^6}+a d^3 \log (x)+\frac {1}{2} i b d^3 (\text {PolyLog}(2,-i c x)-\text {PolyLog}(2,i c x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 272, normalized size = 1.19
method | result | size |
derivativedivides | \(\frac {3 a \,d^{2} e \,x^{2}}{2}+\frac {3 a d \,e^{2} x^{4}}{4}+\frac {a \,e^{3} x^{6}}{6}+a \,d^{3} \ln \left (c x \right )+\frac {3 b \arctan \left (c x \right ) d^{2} e \,x^{2}}{2}+\frac {3 b \arctan \left (c x \right ) d \,e^{2} x^{4}}{4}+\frac {b \arctan \left (c x \right ) e^{3} x^{6}}{6}+b \arctan \left (c x \right ) d^{3} \ln \left (c x \right )-\frac {3 b \,d^{2} e x}{2 c}-\frac {b d \,e^{2} x^{3}}{4 c}-\frac {b \,e^{3} x^{5}}{30 c}+\frac {3 b d \,e^{2} x}{4 c^{3}}+\frac {b \,e^{3} x^{3}}{18 c^{3}}-\frac {b \,e^{3} x}{6 c^{5}}+\frac {3 b \,d^{2} e \arctan \left (c x \right )}{2 c^{2}}-\frac {3 b d \,e^{2} \arctan \left (c x \right )}{4 c^{4}}+\frac {b \,e^{3} \arctan \left (c x \right )}{6 c^{6}}-\frac {i b \,d^{3} \dilog \left (-i c x +1\right )}{2}+\frac {i b \,d^{3} \dilog \left (i c x +1\right )}{2}-\frac {i b \,d^{3} \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}+\frac {i b \,d^{3} \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}\) | \(272\) |
default | \(\frac {3 a \,d^{2} e \,x^{2}}{2}+\frac {3 a d \,e^{2} x^{4}}{4}+\frac {a \,e^{3} x^{6}}{6}+a \,d^{3} \ln \left (c x \right )+\frac {3 b \arctan \left (c x \right ) d^{2} e \,x^{2}}{2}+\frac {3 b \arctan \left (c x \right ) d \,e^{2} x^{4}}{4}+\frac {b \arctan \left (c x \right ) e^{3} x^{6}}{6}+b \arctan \left (c x \right ) d^{3} \ln \left (c x \right )-\frac {3 b \,d^{2} e x}{2 c}-\frac {b d \,e^{2} x^{3}}{4 c}-\frac {b \,e^{3} x^{5}}{30 c}+\frac {3 b d \,e^{2} x}{4 c^{3}}+\frac {b \,e^{3} x^{3}}{18 c^{3}}-\frac {b \,e^{3} x}{6 c^{5}}+\frac {3 b \,d^{2} e \arctan \left (c x \right )}{2 c^{2}}-\frac {3 b d \,e^{2} \arctan \left (c x \right )}{4 c^{4}}+\frac {b \,e^{3} \arctan \left (c x \right )}{6 c^{6}}-\frac {i b \,d^{3} \dilog \left (-i c x +1\right )}{2}+\frac {i b \,d^{3} \dilog \left (i c x +1\right )}{2}-\frac {i b \,d^{3} \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}+\frac {i b \,d^{3} \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}\) | \(272\) |
risch | \(-\frac {i b \,e^{3} \ln \left (c^{2} x^{2}+1\right )}{24 c^{6}}-\frac {i b \,e^{3} \ln \left (i c x +1\right ) x^{6}}{12}+a \,d^{3} \ln \left (-i c x \right )-\frac {3 i b \,d^{2} e \left (-i c x +1\right )^{2} \ln \left (-i c x +1\right )}{4 c^{2}}-\frac {3 i b d \,e^{2} \left (-i c x +1\right )^{3} \ln \left (-i c x +1\right )}{2 c^{4}}+\frac {9 i b d \,e^{2} \left (-i c x +1\right )^{2} \ln \left (-i c x +1\right )}{4 c^{4}}-\frac {3 i b \ln \left (-i c x +1\right ) \left (-i c x +1\right ) d \,e^{2}}{2 c^{4}}+\frac {3 i b \ln \left (-i c x +1\right ) \left (-i c x +1\right ) d^{2} e}{2 c^{2}}+\frac {a \,e^{3}}{6 c^{6}}+\frac {i b \ln \left (-i c x +1\right ) \left (-i c x +1\right ) e^{3}}{2 c^{6}}-\frac {i b \ln \left (-i c x +1\right ) \left (-i c x +1\right )^{6} e^{3}}{12 c^{6}}-\frac {5 i b \ln \left (-i c x +1\right ) \left (-i c x +1\right )^{4} e^{3}}{4 c^{6}}-\frac {3 i b \,e^{2} d \ln \left (i c x +1\right ) x^{4}}{8}+\frac {3 i b d \,e^{2} \ln \left (c^{2} x^{2}+1\right )}{16 c^{4}}-\frac {3 i b \,d^{2} e \ln \left (c^{2} x^{2}+1\right )}{8 c^{2}}-\frac {3 i b e \,d^{2} \ln \left (i c x +1\right ) x^{2}}{4}-\frac {5 i b \ln \left (-i c x +1\right ) \left (-i c x +1\right )^{2} e^{3}}{4 c^{6}}+\frac {i b \ln \left (-i c x +1\right ) \left (-i c x +1\right )^{5} e^{3}}{2 c^{6}}+\frac {5 i b \ln \left (-i c x +1\right ) \left (-i c x +1\right )^{3} e^{3}}{3 c^{6}}+\frac {3 b \,d^{2} e \arctan \left (c x \right )}{4 c^{2}}-\frac {3 b d \,e^{2} \arctan \left (c x \right )}{8 c^{4}}-\frac {3 b \,d^{2} e x}{2 c}+\frac {3 b d \,e^{2} x}{4 c^{3}}-\frac {b d \,e^{2} x^{3}}{4 c}-\frac {b \,e^{3} x}{6 c^{5}}+\frac {b \,e^{3} x^{3}}{18 c^{3}}-\frac {b \,e^{3} x^{5}}{30 c}+\frac {b \,e^{3} \arctan \left (c x \right )}{12 c^{6}}-\frac {i b \,d^{3} \dilog \left (-i c x +1\right )}{2}+\frac {i b \,d^{3} \dilog \left (i c x +1\right )}{2}-\frac {3 a d \,e^{2}}{4 c^{4}}+\frac {3 a \,d^{2} e}{2 c^{2}}+\frac {a \,e^{3} x^{6}}{6}+\frac {3 a d \,e^{2} x^{4}}{4}+\frac {3 a \,d^{2} e \,x^{2}}{2}+\frac {3 i b d \,e^{2} \left (-i c x +1\right )^{4} \ln \left (-i c x +1\right )}{8 c^{4}}\) | \(653\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.61, size = 244, normalized size = 1.07 \begin {gather*} \frac {1}{6} \, a x^{6} e^{3} + \frac {3}{4} \, a d x^{4} e^{2} + \frac {3}{2} \, a d^{2} x^{2} e + a d^{3} \log \left (x\right ) - \frac {6 \, b c^{5} x^{5} e^{3} + 45 \, \pi b c^{6} d^{3} \log \left (c^{2} x^{2} + 1\right ) - 180 \, b c^{6} d^{3} \arctan \left (c x\right ) \log \left (c x\right ) + 90 i \, b c^{6} d^{3} {\rm Li}_2\left (i \, c x + 1\right ) - 90 i \, b c^{6} d^{3} {\rm Li}_2\left (-i \, c x + 1\right ) + 5 \, {\left (9 \, b c^{5} d e^{2} - 2 \, b c^{3} e^{3}\right )} x^{3} + 15 \, {\left (18 \, b c^{5} d^{2} e - 9 \, b c^{3} d e^{2} + 2 \, b c e^{3}\right )} x - 15 \, {\left (2 \, b c^{6} x^{6} e^{3} + 9 \, b c^{6} d x^{4} e^{2} + 18 \, b c^{6} d^{2} x^{2} e + 18 \, b c^{4} d^{2} e - 9 \, b c^{2} d e^{2} + 2 \, b e^{3}\right )} \arctan \left (c x\right )}{180 \, c^{6}} \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 ) \left (d + e x^{2}\right )^{3}}{x}\, 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 [B]
time = 0.78, size = 232, normalized size = 1.02 \begin {gather*} \left \{\begin {array}{cl} \frac {a\,e^3\,x^6}{6}+a\,d^3\,\ln \left (x\right )+\frac {3\,a\,d^2\,e\,x^2}{2}+\frac {3\,a\,d\,e^2\,x^4}{4} & \text {\ if\ \ }c=0\\ \frac {a\,e^3\,x^6}{6}+a\,d^3\,\ln \left (x\right )-\frac {b\,e^3\,\left (\frac {x}{c^4}-\frac {\mathrm {atan}\left (c\,x\right )}{c^5}+\frac {x^5}{5}-\frac {x^3}{3\,c^2}\right )}{6\,c}-3\,b\,d^2\,e\,\left (\frac {x}{2\,c}-\mathrm {atan}\left (c\,x\right )\,\left (\frac {1}{2\,c^2}+\frac {x^2}{2}\right )\right )+\frac {3\,a\,d^2\,e\,x^2}{2}+\frac {3\,a\,d\,e^2\,x^4}{4}-3\,b\,d\,e^2\,\left (\frac {3\,\mathrm {atan}\left (c\,x\right )-3\,c\,x+c^3\,x^3}{12\,c^4}-\frac {x^4\,\mathrm {atan}\left (c\,x\right )}{4}\right )+\frac {b\,e^3\,x^6\,\mathrm {atan}\left (c\,x\right )}{6}-\frac {b\,d^3\,{\mathrm {Li}}_{\mathrm {2}}\left (1-c\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\frac {b\,d^3\,{\mathrm {Li}}_{\mathrm {2}}\left (1+c\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} & \text {\ if\ \ }c\neq 0 \end {array}\right . \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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