Optimal. Leaf size=156 \[ \frac {d \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{e^2}-\frac {d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e^2}+\frac {a x}{e}+\frac {b \log \left (1-c^2 x^2\right )}{2 c e}-\frac {b d \text {Li}_2\left (1-\frac {2}{c x+1}\right )}{2 e^2}+\frac {b d \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{2 e^2}+\frac {b x \tanh ^{-1}(c x)}{e} \]
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Rubi [A] time = 0.15, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {5940, 5910, 260, 5920, 2402, 2315, 2447} \[ -\frac {b d \text {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{2 e^2}+\frac {b d \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{2 e^2}+\frac {d \log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{e^2}-\frac {d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e^2}+\frac {a x}{e}+\frac {b \log \left (1-c^2 x^2\right )}{2 c e}+\frac {b x \tanh ^{-1}(c x)}{e} \]
Antiderivative was successfully verified.
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Rule 260
Rule 2315
Rule 2402
Rule 2447
Rule 5910
Rule 5920
Rule 5940
Rubi steps
\begin {align*} \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{d+e x} \, dx &=\int \left (\frac {a+b \tanh ^{-1}(c x)}{e}-\frac {d \left (a+b \tanh ^{-1}(c x)\right )}{e (d+e x)}\right ) \, dx\\ &=\frac {\int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{e}-\frac {d \int \frac {a+b \tanh ^{-1}(c x)}{d+e x} \, dx}{e}\\ &=\frac {a x}{e}+\frac {d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{e^2}-\frac {d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^2}-\frac {(b c d) \int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{e^2}+\frac {(b c d) \int \frac {\log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{1-c^2 x^2} \, dx}{e^2}+\frac {b \int \tanh ^{-1}(c x) \, dx}{e}\\ &=\frac {a x}{e}+\frac {b x \tanh ^{-1}(c x)}{e}+\frac {d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{e^2}-\frac {d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^2}+\frac {b d \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e^2}-\frac {(b d) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c x}\right )}{e^2}-\frac {(b c) \int \frac {x}{1-c^2 x^2} \, dx}{e}\\ &=\frac {a x}{e}+\frac {b x \tanh ^{-1}(c x)}{e}+\frac {d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{e^2}-\frac {d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^2}+\frac {b \log \left (1-c^2 x^2\right )}{2 c e}-\frac {b d \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{2 e^2}+\frac {b d \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e^2}\\ \end {align*}
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Mathematica [C] time = 2.47, size = 315, normalized size = 2.02 \[ \frac {-2 a d \log (d+e x)+2 a e x+\frac {b \left (e \sqrt {1-\frac {c^2 d^2}{e^2}} \tanh ^{-1}(c x)^2 e^{-\tanh ^{-1}\left (\frac {c d}{e}\right )}+\frac {1}{2} i \pi c d \log \left (1-c^2 x^2\right )+e \log \left (1-c^2 x^2\right )+c d \text {Li}_2\left (e^{-2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )-2 c d \tanh ^{-1}(c x) \tanh ^{-1}\left (\frac {c d}{e}\right )-2 c d \tanh ^{-1}(c x) \log \left (1-e^{-2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )-2 c d \tanh ^{-1}\left (\frac {c d}{e}\right ) \log \left (1-e^{-2 \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )}\right )+2 c d \tanh ^{-1}\left (\frac {c d}{e}\right ) \log \left (i \sinh \left (\tanh ^{-1}\left (\frac {c d}{e}\right )+\tanh ^{-1}(c x)\right )\right )-c d \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )+c d \tanh ^{-1}(c x)^2-i \pi c d \tanh ^{-1}(c x)+2 c d \tanh ^{-1}(c x) \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )+i \pi c d \log \left (e^{2 \tanh ^{-1}(c x)}+1\right )-e \tanh ^{-1}(c x)^2+2 c e x \tanh ^{-1}(c x)\right )}{c}}{2 e^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x \operatorname {artanh}\left (c x\right ) + a x}{e x + d}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} x}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 217, normalized size = 1.39 \[ \frac {a x}{e}-\frac {a d \ln \left (c x e +c d \right )}{e^{2}}+\frac {b x \arctanh \left (c x \right )}{e}-\frac {b \arctanh \left (c x \right ) d \ln \left (c x e +c d \right )}{e^{2}}+\frac {b d \ln \left (c x e +c d \right ) \ln \left (\frac {c x e +e}{-c d +e}\right )}{2 e^{2}}+\frac {b d \dilog \left (\frac {c x e +e}{-c d +e}\right )}{2 e^{2}}-\frac {b d \ln \left (c x e +c d \right ) \ln \left (\frac {c x e -e}{-c d -e}\right )}{2 e^{2}}-\frac {b d \dilog \left (\frac {c x e -e}{-c d -e}\right )}{2 e^{2}}+\frac {b \ln \left (c^{2} d^{2}-2 c d \left (c x e +c d \right )+\left (c x e +c d \right )^{2}-e^{2}\right )}{2 c e} \]
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 \[ a {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} + \frac {1}{2} \, b \int \frac {x {\left (\log \left (c x + 1\right ) - \log \left (-c x + 1\right )\right )}}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}{d+e\,x} \,d x \]
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 \[ \int \frac {x \left (a + b \operatorname {atanh}{\left (c x \right )}\right )}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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