Optimal. Leaf size=83 \[ -\frac {e \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 b d}+\frac {e \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{d}+\frac {b e \text {Li}_2\left (-\frac {c+d x+1}{-c-d x+1}\right )}{2 d} \]
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Rubi [A] time = 0.26, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {12, 5984, 5918, 2402, 2315} \[ \frac {b e \text {PolyLog}\left (2,-\frac {c+d x+1}{-c-d x+1}\right )}{2 d}-\frac {e \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 b d}+\frac {e \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2315
Rule 2402
Rule 5918
Rule 5984
Rubi steps
\begin {align*} \int \frac {(c e+d e x) \left (a+b \tanh ^{-1}(c+d x)\right )}{1-(c+d x)^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {e x \left (a+b \tanh ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e \operatorname {Subst}\left (\int \frac {x \left (a+b \tanh ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{d}\\ &=-\frac {e \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 b d}+\frac {e \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(x)}{1-x} \, dx,x,c+d x\right )}{d}\\ &=-\frac {e \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 b d}+\frac {e \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d}-\frac {(b e) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{d}\\ &=-\frac {e \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 b d}+\frac {e \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d}+\frac {(b e) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c-d x}\right )}{d}\\ &=-\frac {e \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 b d}+\frac {e \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d}+\frac {b e \text {Li}_2\left (1-\frac {2}{1-c-d x}\right )}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 114, normalized size = 1.37 \[ -\frac {e \left (4 a \log (-c-d x+1)+4 a \log (c+d x+1)-2 b \text {Li}_2\left (\frac {1}{2} (-c-d x+1)\right )+2 b \text {Li}_2\left (\frac {1}{2} (c+d x+1)\right )-b \log ^2(-c-d x+1)+b \log ^2(c+d x+1)+b \log (4) \log (c+d x-1)-b \log (4) \log (c+d x+1)\right )}{8 d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {a d e x + a c e + {\left (b d e x + b c e\right )} \operatorname {artanh}\left (d x + c\right )}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}, x\right ) \]
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 [B] time = 0.06, size = 194, normalized size = 2.34 \[ -\frac {a e \ln \left (d x +c -1\right )}{2 d}-\frac {a e \ln \left (d x +c +1\right )}{2 d}-\frac {b e \arctanh \left (d x +c \right ) \ln \left (d x +c -1\right )}{2 d}-\frac {b e \arctanh \left (d x +c \right ) \ln \left (d x +c +1\right )}{2 d}-\frac {b e \ln \left (d x +c -1\right )^{2}}{8 d}+\frac {b e \dilog \left (\frac {1}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{2 d}+\frac {b e \ln \left (d x +c -1\right ) \ln \left (\frac {1}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {b e \ln \left (d x +c +1\right )^{2}}{8 d}-\frac {b e \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right ) \ln \left (d x +c +1\right )}{4 d}+\frac {b e \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{4 d} \]
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 \[ \frac {1}{2} \, b c e {\left (\frac {\log \left (d x + c + 1\right )}{d} - \frac {\log \left (d x + c - 1\right )}{d}\right )} \operatorname {artanh}\left (d x + c\right ) - \frac {1}{2} \, a d e {\left (\frac {{\left (c + 1\right )} \log \left (d x + c + 1\right )}{d^{2}} - \frac {{\left (c - 1\right )} \log \left (d x + c - 1\right )}{d^{2}}\right )} + \frac {1}{2} \, a c e {\left (\frac {\log \left (d x + c + 1\right )}{d} - \frac {\log \left (d x + c - 1\right )}{d}\right )} + \frac {1}{8} \, b d e {\left (\frac {2 \, {\left (c + 1\right )} \log \left (d x + c + 1\right ) \log \left (-d x - c + 1\right ) - {\left (c - 1\right )} \log \left (-d x - c + 1\right )^{2}}{d^{2}} - 4 \, \int \frac {{\left (c^{2} + {\left (c d + 3 \, d\right )} x + 2 \, c + 1\right )} \log \left (d x + c + 1\right )}{2 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d - d\right )}}\,{d x}\right )} - \frac {{\left (\log \left (d x + c + 1\right )^{2} - 2 \, \log \left (d x + c + 1\right ) \log \left (d x + c - 1\right ) + \log \left (d x + c - 1\right )^{2}\right )} b c e}{8 \, d} \]
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 \[ \int -\frac {\left (c\,e+d\,e\,x\right )\,\left (a+b\,\mathrm {atanh}\left (c+d\,x\right )\right )}{{\left (c+d\,x\right )}^2-1} \,d x \]
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 \[ - e \left (\int \frac {a c}{c^{2} + 2 c d x + d^{2} x^{2} - 1}\, dx + \int \frac {a d x}{c^{2} + 2 c d x + d^{2} x^{2} - 1}\, dx + \int \frac {b c \operatorname {atanh}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2} - 1}\, dx + \int \frac {b d x \operatorname {atanh}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2} - 1}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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