Optimal. Leaf size=106 \[ \frac{b \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right )}{2 c^2 d^2}+\frac{a+b \tanh ^{-1}(c x)}{c^2 d^2 (c x+1)}-\frac{\log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d^2}+\frac{b}{2 c^2 d^2 (c x+1)}-\frac{b \tanh ^{-1}(c x)}{2 c^2 d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.143107, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {5940, 5926, 627, 44, 207, 5918, 2402, 2315} \[ \frac{b \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right )}{2 c^2 d^2}+\frac{a+b \tanh ^{-1}(c x)}{c^2 d^2 (c x+1)}-\frac{\log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d^2}+\frac{b}{2 c^2 d^2 (c x+1)}-\frac{b \tanh ^{-1}(c x)}{2 c^2 d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5940
Rule 5926
Rule 627
Rule 44
Rule 207
Rule 5918
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{x \left (a+b \tanh ^{-1}(c x)\right )}{(d+c d x)^2} \, dx &=\int \left (-\frac{a+b \tanh ^{-1}(c x)}{c d^2 (1+c x)^2}+\frac{a+b \tanh ^{-1}(c x)}{c d^2 (1+c x)}\right ) \, dx\\ &=-\frac{\int \frac{a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx}{c d^2}+\frac{\int \frac{a+b \tanh ^{-1}(c x)}{1+c x} \, dx}{c d^2}\\ &=\frac{a+b \tanh ^{-1}(c x)}{c^2 d^2 (1+c x)}-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{c^2 d^2}-\frac{b \int \frac{1}{(1+c x) \left (1-c^2 x^2\right )} \, dx}{c d^2}+\frac{b \int \frac{\log \left (\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx}{c d^2}\\ &=\frac{a+b \tanh ^{-1}(c x)}{c^2 d^2 (1+c x)}-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{c^2 d^2}+\frac{b \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+c x}\right )}{c^2 d^2}-\frac{b \int \frac{1}{(1-c x) (1+c x)^2} \, dx}{c d^2}\\ &=\frac{a+b \tanh ^{-1}(c x)}{c^2 d^2 (1+c x)}-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{c^2 d^2}+\frac{b \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{2 c^2 d^2}-\frac{b \int \left (\frac{1}{2 (1+c x)^2}-\frac{1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx}{c d^2}\\ &=\frac{b}{2 c^2 d^2 (1+c x)}+\frac{a+b \tanh ^{-1}(c x)}{c^2 d^2 (1+c x)}-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{c^2 d^2}+\frac{b \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{2 c^2 d^2}+\frac{b \int \frac{1}{-1+c^2 x^2} \, dx}{2 c d^2}\\ &=\frac{b}{2 c^2 d^2 (1+c x)}-\frac{b \tanh ^{-1}(c x)}{2 c^2 d^2}+\frac{a+b \tanh ^{-1}(c x)}{c^2 d^2 (1+c x)}-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{c^2 d^2}+\frac{b \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{2 c^2 d^2}\\ \end{align*}
Mathematica [A] time = 0.378704, size = 99, normalized size = 0.93 \[ \frac{b \left (2 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )-\sinh \left (2 \tanh ^{-1}(c x)\right )+\cosh \left (2 \tanh ^{-1}(c x)\right )+2 \tanh ^{-1}(c x) \left (-2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )-\sinh \left (2 \tanh ^{-1}(c x)\right )+\cosh \left (2 \tanh ^{-1}(c x)\right )\right )\right )+\frac{4 a}{c x+1}+4 a \log (c x+1)}{4 c^2 d^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.046, size = 192, normalized size = 1.8 \begin{align*}{\frac{a}{{c}^{2}{d}^{2} \left ( cx+1 \right ) }}+{\frac{a\ln \left ( cx+1 \right ) }{{c}^{2}{d}^{2}}}+{\frac{b{\it Artanh} \left ( cx \right ) }{{c}^{2}{d}^{2} \left ( cx+1 \right ) }}+{\frac{b{\it Artanh} \left ( cx \right ) \ln \left ( cx+1 \right ) }{{c}^{2}{d}^{2}}}+{\frac{b\ln \left ( cx-1 \right ) }{4\,{c}^{2}{d}^{2}}}+{\frac{b}{2\,{c}^{2}{d}^{2} \left ( cx+1 \right ) }}-{\frac{b\ln \left ( cx+1 \right ) }{4\,{c}^{2}{d}^{2}}}-{\frac{b}{2\,{c}^{2}{d}^{2}}\ln \left ( -{\frac{cx}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{cx}{2}} \right ) }+{\frac{b\ln \left ( cx+1 \right ) }{2\,{c}^{2}{d}^{2}}\ln \left ( -{\frac{cx}{2}}+{\frac{1}{2}} \right ) }-{\frac{b}{2\,{c}^{2}{d}^{2}}{\it dilog} \left ({\frac{1}{2}}+{\frac{cx}{2}} \right ) }-{\frac{b \left ( \ln \left ( cx+1 \right ) \right ) ^{2}}{4\,{c}^{2}{d}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{8} \,{\left (8 \, c^{2} \int \frac{x^{2} \log \left (c x + 1\right )}{c^{4} d^{2} x^{3} + c^{3} d^{2} x^{2} - c^{2} d^{2} x - c d^{2}}\,{d x} - c{\left (\frac{2}{c^{4} d^{2} x + c^{3} d^{2}} + \frac{\log \left (c x + 1\right )}{c^{3} d^{2}} - \frac{\log \left (c x - 1\right )}{c^{3} d^{2}}\right )} + 4 \, c \int \frac{x \log \left (c x + 1\right )}{c^{4} d^{2} x^{3} + c^{3} d^{2} x^{2} - c^{2} d^{2} x - c d^{2}}\,{d x} - \frac{4 \,{\left ({\left (c x + 1\right )} \log \left (c x + 1\right ) + 1\right )} \log \left (-c x + 1\right )}{c^{3} d^{2} x + c^{2} d^{2}} + \frac{2}{c^{3} d^{2} x + c^{2} d^{2}} - \frac{\log \left (c x + 1\right )}{c^{2} d^{2}} + \frac{\log \left (c x - 1\right )}{c^{2} d^{2}} + 4 \, \int \frac{\log \left (c x + 1\right )}{c^{4} d^{2} x^{3} + c^{3} d^{2} x^{2} - c^{2} d^{2} x - c d^{2}}\,{d x}\right )} b + a{\left (\frac{1}{c^{3} d^{2} x + c^{2} d^{2}} + \frac{\log \left (c x + 1\right )}{c^{2} d^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b x \operatorname{artanh}\left (c x\right ) + a x}{c^{2} d^{2} x^{2} + 2 \, c d^{2} x + d^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a x}{c^{2} x^{2} + 2 c x + 1}\, dx + \int \frac{b x \operatorname{atanh}{\left (c x \right )}}{c^{2} x^{2} + 2 c x + 1}\, dx}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{artanh}\left (c x\right ) + a\right )} x}{{\left (c d x + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]