Optimal. Leaf size=61 \[ \frac {d^2 \left (c^2 x^2-1\right ) \left (a+b \tanh ^{-1}(c x)\right )}{x}+c d^2 (2 a+b) \log (x)-b c d^2 \text {Li}_2(-c x)+b c d^2 \text {Li}_2(c x) \]
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Rubi [A] time = 0.13, antiderivative size = 80, normalized size of antiderivative = 1.31, number of steps used = 11, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {5940, 5910, 260, 5916, 266, 36, 29, 31, 5912} \[ -b c d^2 \text {PolyLog}(2,-c x)+b c d^2 \text {PolyLog}(2,c x)-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+a c^2 d^2 x+2 a c d^2 \log (x)+b c^2 d^2 x \tanh ^{-1}(c x)+b c d^2 \log (x) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 260
Rule 266
Rule 5910
Rule 5912
Rule 5916
Rule 5940
Rubi steps
\begin {align*} \int \frac {(d+c d x)^2 \left (a+b \tanh ^{-1}(c x)\right )}{x^2} \, dx &=\int \left (c^2 d^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}+\frac {2 c d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}\right ) \, dx\\ &=d^2 \int \frac {a+b \tanh ^{-1}(c x)}{x^2} \, dx+\left (2 c d^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx+\left (c^2 d^2\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx\\ &=a c^2 d^2 x-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+2 a c d^2 \log (x)-b c d^2 \text {Li}_2(-c x)+b c d^2 \text {Li}_2(c x)+\left (b c d^2\right ) \int \frac {1}{x \left (1-c^2 x^2\right )} \, dx+\left (b c^2 d^2\right ) \int \tanh ^{-1}(c x) \, dx\\ &=a c^2 d^2 x+b c^2 d^2 x \tanh ^{-1}(c x)-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+2 a c d^2 \log (x)-b c d^2 \text {Li}_2(-c x)+b c d^2 \text {Li}_2(c x)+\frac {1}{2} \left (b c d^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )-\left (b c^3 d^2\right ) \int \frac {x}{1-c^2 x^2} \, dx\\ &=a c^2 d^2 x+b c^2 d^2 x \tanh ^{-1}(c x)-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+2 a c d^2 \log (x)+\frac {1}{2} b c d^2 \log \left (1-c^2 x^2\right )-b c d^2 \text {Li}_2(-c x)+b c d^2 \text {Li}_2(c x)+\frac {1}{2} \left (b c d^2\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{2} \left (b c^3 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x} \, dx,x,x^2\right )\\ &=a c^2 d^2 x+b c^2 d^2 x \tanh ^{-1}(c x)-\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+2 a c d^2 \log (x)+b c d^2 \log (x)-b c d^2 \text {Li}_2(-c x)+b c d^2 \text {Li}_2(c x)\\ \end {align*}
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Mathematica [A] time = 0.11, size = 73, normalized size = 1.20 \[ \frac {d^2 \left (a c^2 x^2+2 a c x \log (x)-a+b c^2 x^2 \tanh ^{-1}(c x)-b c x \text {Li}_2(-c x)+b c x \text {Li}_2(c x)+b c x \log (c x)-b \tanh ^{-1}(c x)\right )}{x} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a c^{2} d^{2} x^{2} + 2 \, a c d^{2} x + a d^{2} + {\left (b c^{2} d^{2} x^{2} + 2 \, b c d^{2} x + b d^{2}\right )} \operatorname {artanh}\left (c x\right )}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.80, size = 410, normalized size = 6.72 \[ \frac {1}{6} \, {\left (\frac {6 \, a d^{2}}{\frac {{\left (c x + 1\right )} c^{2}}{c x - 1} + c^{2}} + \frac {5 \, b d^{2} \log \left (-\frac {c x + 1}{c x - 1} + 1\right )}{c^{2}} + \frac {3 \, b d^{2} \log \left (-\frac {c x + 1}{c x - 1} - 1\right )}{c^{2}} + {\left (\frac {3 \, b d^{2}}{\frac {{\left (c x + 1\right )} c^{2}}{c x - 1} + c^{2}} - \frac {\frac {3 \, {\left (c x + 1\right )}^{2} b d^{2}}{{\left (c x - 1\right )}^{2}} - \frac {12 \, {\left (c x + 1\right )} b d^{2}}{c x - 1} + 5 \, b d^{2}}{\frac {{\left (c x + 1\right )}^{3} c^{2}}{{\left (c x - 1\right )}^{3}} - \frac {3 \, {\left (c x + 1\right )}^{2} c^{2}}{{\left (c x - 1\right )}^{2}} + \frac {3 \, {\left (c x + 1\right )} c^{2}}{c x - 1} - c^{2}}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right ) - \frac {8 \, b d^{2} \log \left (-\frac {c x + 1}{c x - 1}\right )}{c^{2}} - \frac {2 \, {\left (\frac {3 \, {\left (c x + 1\right )}^{2} a d^{2}}{{\left (c x - 1\right )}^{2}} - \frac {12 \, {\left (c x + 1\right )} a d^{2}}{c x - 1} + 5 \, a d^{2} - \frac {{\left (c x + 1\right )}^{2} b d^{2}}{{\left (c x - 1\right )}^{2}} + \frac {{\left (c x + 1\right )} b d^{2}}{c x - 1}\right )}}{\frac {{\left (c x + 1\right )}^{3} c^{2}}{{\left (c x - 1\right )}^{3}} - \frac {3 \, {\left (c x + 1\right )}^{2} c^{2}}{{\left (c x - 1\right )}^{2}} + \frac {3 \, {\left (c x + 1\right )} c^{2}}{c x - 1} - c^{2}}\right )} c^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 123, normalized size = 2.02 \[ d^{2} a \,c^{2} x +2 c \,d^{2} a \ln \left (c x \right )-\frac {d^{2} a}{x}+b \,c^{2} d^{2} x \arctanh \left (c x \right )+2 c \,d^{2} b \arctanh \left (c x \right ) \ln \left (c x \right )-\frac {d^{2} b \arctanh \left (c x \right )}{x}+c \,d^{2} b \ln \left (c x \right )-c \,d^{2} b \dilog \left (c x \right )-c \,d^{2} b \dilog \left (c x +1\right )-c \,d^{2} b \ln \left (c x \right ) \ln \left (c x +1\right ) \]
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 \[ a c^{2} d^{2} x + \frac {1}{2} \, {\left (2 \, c x \operatorname {artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} b c d^{2} + b c d^{2} \int \frac {\log \left (c x + 1\right ) - \log \left (-c x + 1\right )}{x}\,{d x} + 2 \, a c d^{2} \log \relax (x) - \frac {1}{2} \, {\left (c {\left (\log \left (c^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x}\right )} b d^{2} - \frac {a d^{2}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )\,{\left (d+c\,d\,x\right )}^2}{x^2} \,d x \]
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 \[ d^{2} \left (\int a c^{2}\, dx + \int \frac {a}{x^{2}}\, dx + \int \frac {2 a c}{x}\, dx + \int b c^{2} \operatorname {atanh}{\left (c x \right )}\, dx + \int \frac {b \operatorname {atanh}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {2 b c \operatorname {atanh}{\left (c x \right )}}{x}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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