Optimal. Leaf size=77 \[ \frac {1}{2} e x^2 \left (a+b \tan ^{-1}(c x)\right )+a d \log (x)+\frac {b e \tan ^{-1}(c x)}{2 c^2}+\frac {1}{2} i b d \text {Li}_2(-i c x)-\frac {1}{2} i b d \text {Li}_2(i c x)-\frac {b e x}{2 c} \]
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Rubi [A] time = 0.09, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {4980, 4848, 2391, 4852, 321, 203} \[ \frac {1}{2} i b d \text {PolyLog}(2,-i c x)-\frac {1}{2} i b d \text {PolyLog}(2,i c x)+\frac {1}{2} e x^2 \left (a+b \tan ^{-1}(c x)\right )+a d \log (x)+\frac {b e \tan ^{-1}(c x)}{2 c^2}-\frac {b e x}{2 c} \]
Antiderivative was successfully verified.
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Rule 203
Rule 321
Rule 2391
Rule 4848
Rule 4852
Rule 4980
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right ) \left (a+b \tan ^{-1}(c x)\right )}{x} \, dx &=\int \left (\frac {d \left (a+b \tan ^{-1}(c x)\right )}{x}+e x \left (a+b \tan ^{-1}(c x)\right )\right ) \, dx\\ &=d \int \frac {a+b \tan ^{-1}(c x)}{x} \, dx+e \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx\\ &=\frac {1}{2} e x^2 \left (a+b \tan ^{-1}(c x)\right )+a d \log (x)+\frac {1}{2} (i b d) \int \frac {\log (1-i c x)}{x} \, dx-\frac {1}{2} (i b d) \int \frac {\log (1+i c x)}{x} \, dx-\frac {1}{2} (b c e) \int \frac {x^2}{1+c^2 x^2} \, dx\\ &=-\frac {b e x}{2 c}+\frac {1}{2} e x^2 \left (a+b \tan ^{-1}(c x)\right )+a d \log (x)+\frac {1}{2} i b d \text {Li}_2(-i c x)-\frac {1}{2} i b d \text {Li}_2(i c x)+\frac {(b e) \int \frac {1}{1+c^2 x^2} \, dx}{2 c}\\ &=-\frac {b e x}{2 c}+\frac {b e \tan ^{-1}(c x)}{2 c^2}+\frac {1}{2} e x^2 \left (a+b \tan ^{-1}(c x)\right )+a d \log (x)+\frac {1}{2} i b d \text {Li}_2(-i c x)-\frac {1}{2} i b d \text {Li}_2(i c x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 83, normalized size = 1.08 \[ a d \log (x)+\frac {1}{2} a e x^2+\frac {b e \tan ^{-1}(c x)}{2 c^2}+\frac {1}{2} i b d \text {Li}_2(-i c x)-\frac {1}{2} i b d \text {Li}_2(i c x)+\frac {1}{2} b e x^2 \tan ^{-1}(c x)-\frac {b e x}{2 c} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a e x^{2} + a d + {\left (b e x^{2} + b d\right )} \arctan \left (c x\right )}{x}, x\right ) \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 117, normalized size = 1.52 \[ \frac {a \,x^{2} e}{2}+d a \ln \left (c x \right )+\frac {\arctan \left (c x \right ) b e \,x^{2}}{2}+b \arctan \left (c x \right ) d \ln \left (c x \right )+\frac {b e \arctan \left (c x \right )}{2 c^{2}}-\frac {b e x}{2 c}+\frac {i b d \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}-\frac {i b d \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}+\frac {i b d \dilog \left (i c x +1\right )}{2}-\frac {i b d \dilog \left (-i c x +1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 104, normalized size = 1.35 \[ \frac {1}{2} \, a e x^{2} + a d \log \relax (x) - \frac {\pi b c^{2} d \log \left (c^{2} x^{2} + 1\right ) - 4 \, b c^{2} d \arctan \left (c x\right ) \log \left (c x\right ) + 2 i \, b c^{2} d {\rm Li}_2\left (i \, c x + 1\right ) - 2 i \, b c^{2} d {\rm Li}_2\left (-i \, c x + 1\right ) + 2 \, b c e x - 2 \, {\left (b c^{2} e x^{2} + b e\right )} \arctan \left (c x\right )}{4 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.68, size = 88, normalized size = 1.14 \[ \left \{\begin {array}{cl} \frac {a\,\left (e\,x^2+2\,d\,\ln \relax (x)\right )}{2} & \text {\ if\ \ }c=0\\ \frac {a\,\left (e\,x^2+2\,d\,\ln \relax (x)\right )}{2}-b\,e\,\left (\frac {x}{2\,c}-\mathrm {atan}\left (c\,x\right )\,\left (\frac {1}{2\,c^2}+\frac {x^2}{2}\right )\right )-\frac {b\,d\,\left ({\mathrm {Li}}_{\mathrm {2}}\left (1-c\,x\,1{}\mathrm {i}\right )-{\mathrm {Li}}_{\mathrm {2}}\left (1+c\,x\,1{}\mathrm {i}\right )\right )\,1{}\mathrm {i}}{2} & \text {\ if\ \ }c\neq 0 \end {array}\right . \]
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 \[ \int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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