Optimal. Leaf size=282 \[ -\frac {1}{2} a e \text {Li}_2\left (-c^2 x^2\right )+a d \log (x)-\frac {1}{2} i b e \text {Li}_2(-i c x) \left (-\log \left (c^2 x^2+1\right )+\log (1-i c x)+\log (1+i c x)\right )+\frac {1}{2} i b e \text {Li}_2(i c x) \left (-\log \left (c^2 x^2+1\right )+\log (1-i c x)+\log (1+i c x)\right )+\frac {1}{2} i b d \text {Li}_2(-i c x)-\frac {1}{2} i b d \text {Li}_2(i c x)-i b e \text {Li}_3(1-i c x)+i b e \text {Li}_3(i c x+1)+i b e \text {Li}_2(1-i c x) \log (1-i c x)-i b e \text {Li}_2(i c x+1) \log (1+i c x)+\frac {1}{2} i b e \log (i c x) \log ^2(1-i c x)-\frac {1}{2} i b e \log (-i c x) \log ^2(1+i c x) \]
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Rubi [A] time = 0.34, antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {5015, 4848, 2391, 5013, 5011, 2396, 2433, 2374, 6589} \[ -\frac {1}{2} a e \text {PolyLog}\left (2,-c^2 x^2\right )-\frac {1}{2} i b e \left (-\log \left (c^2 x^2+1\right )+\log (1-i c x)+\log (1+i c x)\right ) \text {PolyLog}(2,-i c x)+\frac {1}{2} i b e \left (-\log \left (c^2 x^2+1\right )+\log (1-i c x)+\log (1+i c x)\right ) \text {PolyLog}(2,i c x)+\frac {1}{2} i b d \text {PolyLog}(2,-i c x)-\frac {1}{2} i b d \text {PolyLog}(2,i c x)-i b e \text {PolyLog}(3,1-i c x)+i b e \text {PolyLog}(3,1+i c x)+i b e \log (1-i c x) \text {PolyLog}(2,1-i c x)-i b e \log (1+i c x) \text {PolyLog}(2,1+i c x)+a d \log (x)+\frac {1}{2} i b e \log (i c x) \log ^2(1-i c x)-\frac {1}{2} i b e \log (-i c x) \log ^2(1+i c x) \]
Antiderivative was successfully verified.
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Rule 2374
Rule 2391
Rule 2396
Rule 2433
Rule 4848
Rule 5011
Rule 5013
Rule 5015
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x} \, dx &=d \int \frac {a+b \tan ^{-1}(c x)}{x} \, dx+e \int \frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (1+c^2 x^2\right )}{x} \, dx\\ &=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+(a e) \int \frac {\log \left (1+c^2 x^2\right )}{x} \, dx+(b e) \int \frac {\tan ^{-1}(c x) \log \left (1+c^2 x^2\right )}{x} \, dx\\ &=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 {1}{2} a e \text {Li}_2\left (-c^2 x^2\right )+\frac {1}{2} (i b e) \int \frac {\log ^2(1-i c x)}{x} \, dx-\frac {1}{2} (i b e) \int \frac {\log ^2(1+i c x)}{x} \, dx+\left (b e \left (-\log (1-i c x)-\log (1+i c x)+\log \left (1+c^2 x^2\right )\right )\right ) \int \frac {\tan ^{-1}(c x)}{x} \, dx\\ &=a d \log (x)+\frac {1}{2} i b e \log (i c x) \log ^2(1-i c x)-\frac {1}{2} i b e \log (-i c x) \log ^2(1+i c 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 {1}{2} a e \text {Li}_2\left (-c^2 x^2\right )-(b c e) \int \frac {\log (i c x) \log (1-i c x)}{1-i c x} \, dx-(b c e) \int \frac {\log (-i c x) \log (1+i c x)}{1+i c x} \, dx+\frac {1}{2} \left (i b e \left (-\log (1-i c x)-\log (1+i c x)+\log \left (1+c^2 x^2\right )\right )\right ) \int \frac {\log (1-i c x)}{x} \, dx-\frac {1}{2} \left (i b e \left (-\log (1-i c x)-\log (1+i c x)+\log \left (1+c^2 x^2\right )\right )\right ) \int \frac {\log (1+i c x)}{x} \, dx\\ &=a d \log (x)+\frac {1}{2} i b e \log (i c x) \log ^2(1-i c x)-\frac {1}{2} i b e \log (-i c x) \log ^2(1+i c x)+\frac {1}{2} i b d \text {Li}_2(-i c x)-\frac {1}{2} i b e \left (\log (1-i c x)+\log (1+i c x)-\log \left (1+c^2 x^2\right )\right ) \text {Li}_2(-i c x)-\frac {1}{2} i b d \text {Li}_2(i c x)+\frac {1}{2} i b e \left (\log (1-i c x)+\log (1+i c x)-\log \left (1+c^2 x^2\right )\right ) \text {Li}_2(i c x)-\frac {1}{2} a e \text {Li}_2\left (-c^2 x^2\right )+(i b e) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (-i c \left (\frac {i}{c}-\frac {i x}{c}\right )\right )}{x} \, dx,x,1+i c x\right )-(i b e) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (i c \left (-\frac {i}{c}+\frac {i x}{c}\right )\right )}{x} \, dx,x,1-i c x\right )\\ &=a d \log (x)+\frac {1}{2} i b e \log (i c x) \log ^2(1-i c x)-\frac {1}{2} i b e \log (-i c x) \log ^2(1+i c x)+\frac {1}{2} i b d \text {Li}_2(-i c x)-\frac {1}{2} i b e \left (\log (1-i c x)+\log (1+i c x)-\log \left (1+c^2 x^2\right )\right ) \text {Li}_2(-i c x)-\frac {1}{2} i b d \text {Li}_2(i c x)+\frac {1}{2} i b e \left (\log (1-i c x)+\log (1+i c x)-\log \left (1+c^2 x^2\right )\right ) \text {Li}_2(i c x)-\frac {1}{2} a e \text {Li}_2\left (-c^2 x^2\right )+i b e \log (1-i c x) \text {Li}_2(1-i c x)-i b e \log (1+i c x) \text {Li}_2(1+i c x)-(i b e) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1-i c x\right )+(i b e) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1+i c x\right )\\ &=a d \log (x)+\frac {1}{2} i b e \log (i c x) \log ^2(1-i c x)-\frac {1}{2} i b e \log (-i c x) \log ^2(1+i c x)+\frac {1}{2} i b d \text {Li}_2(-i c x)-\frac {1}{2} i b e \left (\log (1-i c x)+\log (1+i c x)-\log \left (1+c^2 x^2\right )\right ) \text {Li}_2(-i c x)-\frac {1}{2} i b d \text {Li}_2(i c x)+\frac {1}{2} i b e \left (\log (1-i c x)+\log (1+i c x)-\log \left (1+c^2 x^2\right )\right ) \text {Li}_2(i c x)-\frac {1}{2} a e \text {Li}_2\left (-c^2 x^2\right )+i b e \log (1-i c x) \text {Li}_2(1-i c x)-i b e \log (1+i c x) \text {Li}_2(1+i c x)-i b e \text {Li}_3(1-i c x)+i b e \text {Li}_3(1+i c x)\\ \end {align*}
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Mathematica [F] time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b d \arctan \left (c x\right ) + a d + {\left (b e \arctan \left (c x\right ) + a e\right )} \log \left (c^{2} x^{2} + 1\right )}{x}, x\right ) \]
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 [C] time = 6.09, size = 6931, normalized size = 24.58 \[ \text {output too large to display} \]
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 \[ a d \log \relax (x) + \frac {1}{2} \, \int \frac {2 \, {\left (b d \arctan \left (c x\right ) + {\left (b e \arctan \left (c x\right ) + a e\right )} \log \left (c^{2} x^{2} + 1\right )\right )}}{x}\,{d x} \]
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 \[ \int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (c^2\,x^2+1\right )\right )}{x} \,d x \]
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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