Optimal. Leaf size=152 \[ -\frac{1}{2} b d^3 \text{PolyLog}(2,-c x)+\frac{1}{2} b d^3 \text{PolyLog}(2,c x)+\frac{1}{3} c^3 d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{2} c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+3 a c d^3 x+a d^3 \log (x)+\frac{1}{6} b c^2 d^3 x^2+\frac{5}{3} b d^3 \log \left (1-c^2 x^2\right )+\frac{3}{2} b c d^3 x-\frac{3}{2} b d^3 \tanh ^{-1}(c x)+3 b c d^3 x \tanh ^{-1}(c x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.165891, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {5940, 5910, 260, 5912, 5916, 321, 206, 266, 43} \[ -\frac{1}{2} b d^3 \text{PolyLog}(2,-c x)+\frac{1}{2} b d^3 \text{PolyLog}(2,c x)+\frac{1}{3} c^3 d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3}{2} c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+3 a c d^3 x+a d^3 \log (x)+\frac{1}{6} b c^2 d^3 x^2+\frac{5}{3} b d^3 \log \left (1-c^2 x^2\right )+\frac{3}{2} b c d^3 x-\frac{3}{2} b d^3 \tanh ^{-1}(c x)+3 b c d^3 x \tanh ^{-1}(c x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5940
Rule 5910
Rule 260
Rule 5912
Rule 5916
Rule 321
Rule 206
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+c d x)^3 \left (a+b \tanh ^{-1}(c x)\right )}{x} \, dx &=\int \left (3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+3 c^2 d^3 x \left (a+b \tanh ^{-1}(c x)\right )+c^3 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )\right ) \, dx\\ &=d^3 \int \frac{a+b \tanh ^{-1}(c x)}{x} \, dx+\left (3 c d^3\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx+\left (3 c^2 d^3\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx+\left (c^3 d^3\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx\\ &=3 a c d^3 x+\frac{3}{2} c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{3} c^3 d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+a d^3 \log (x)-\frac{1}{2} b d^3 \text{Li}_2(-c x)+\frac{1}{2} b d^3 \text{Li}_2(c x)+\left (3 b c d^3\right ) \int \tanh ^{-1}(c x) \, dx-\frac{1}{2} \left (3 b c^3 d^3\right ) \int \frac{x^2}{1-c^2 x^2} \, dx-\frac{1}{3} \left (b c^4 d^3\right ) \int \frac{x^3}{1-c^2 x^2} \, dx\\ &=3 a c d^3 x+\frac{3}{2} b c d^3 x+3 b c d^3 x \tanh ^{-1}(c x)+\frac{3}{2} c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{3} c^3 d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+a d^3 \log (x)-\frac{1}{2} b d^3 \text{Li}_2(-c x)+\frac{1}{2} b d^3 \text{Li}_2(c x)-\frac{1}{2} \left (3 b c d^3\right ) \int \frac{1}{1-c^2 x^2} \, dx-\left (3 b c^2 d^3\right ) \int \frac{x}{1-c^2 x^2} \, dx-\frac{1}{6} \left (b c^4 d^3\right ) \operatorname{Subst}\left (\int \frac{x}{1-c^2 x} \, dx,x,x^2\right )\\ &=3 a c d^3 x+\frac{3}{2} b c d^3 x-\frac{3}{2} b d^3 \tanh ^{-1}(c x)+3 b c d^3 x \tanh ^{-1}(c x)+\frac{3}{2} c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{3} c^3 d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+a d^3 \log (x)+\frac{3}{2} b d^3 \log \left (1-c^2 x^2\right )-\frac{1}{2} b d^3 \text{Li}_2(-c x)+\frac{1}{2} b d^3 \text{Li}_2(c x)-\frac{1}{6} \left (b c^4 d^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^2}-\frac{1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=3 a c d^3 x+\frac{3}{2} b c d^3 x+\frac{1}{6} b c^2 d^3 x^2-\frac{3}{2} b d^3 \tanh ^{-1}(c x)+3 b c d^3 x \tanh ^{-1}(c x)+\frac{3}{2} c^2 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{3} c^3 d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+a d^3 \log (x)+\frac{5}{3} b d^3 \log \left (1-c^2 x^2\right )-\frac{1}{2} b d^3 \text{Li}_2(-c x)+\frac{1}{2} b d^3 \text{Li}_2(c x)\\ \end{align*}
Mathematica [A] time = 0.13589, size = 148, normalized size = 0.97 \[ \frac{1}{12} d^3 \left (-6 b \text{PolyLog}(2,-c x)+6 b \text{PolyLog}(2,c x)+4 a c^3 x^3+18 a c^2 x^2+36 a c x+12 a \log (x)+2 b c^2 x^2+18 b \log \left (1-c^2 x^2\right )+2 b \log \left (c^2 x^2-1\right )+4 b c^3 x^3 \tanh ^{-1}(c x)+18 b c^2 x^2 \tanh ^{-1}(c x)+18 b c x+9 b \log (1-c x)-9 b \log (c x+1)+36 b c x \tanh ^{-1}(c x)\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.042, size = 182, normalized size = 1.2 \begin{align*}{\frac{{d}^{3}a{c}^{3}{x}^{3}}{3}}+{\frac{3\,{d}^{3}a{c}^{2}{x}^{2}}{2}}+3\,ac{d}^{3}x+{d}^{3}a\ln \left ( cx \right ) +{\frac{{d}^{3}b{\it Artanh} \left ( cx \right ){c}^{3}{x}^{3}}{3}}+{\frac{3\,{d}^{3}b{\it Artanh} \left ( cx \right ){c}^{2}{x}^{2}}{2}}+3\,bc{d}^{3}x{\it Artanh} \left ( cx \right ) +{d}^{3}b{\it Artanh} \left ( cx \right ) \ln \left ( cx \right ) -{\frac{{d}^{3}b{\it dilog} \left ( cx \right ) }{2}}-{\frac{{d}^{3}b{\it dilog} \left ( cx+1 \right ) }{2}}-{\frac{{d}^{3}b\ln \left ( cx \right ) \ln \left ( cx+1 \right ) }{2}}+{\frac{b{c}^{2}{d}^{3}{x}^{2}}{6}}+{\frac{3\,bc{d}^{3}x}{2}}+{\frac{29\,{d}^{3}b\ln \left ( cx-1 \right ) }{12}}+{\frac{11\,{d}^{3}b\ln \left ( cx+1 \right ) }{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.47818, size = 308, normalized size = 2.03 \begin{align*} \frac{1}{3} \, a c^{3} d^{3} x^{3} + \frac{3}{2} \, a c^{2} d^{3} x^{2} + \frac{1}{6} \, b c^{2} d^{3} x^{2} + 3 \, a c d^{3} x + \frac{3}{2} \, b c d^{3} x + \frac{3}{2} \,{\left (2 \, c x \operatorname{artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} b d^{3} - \frac{1}{2} \,{\left (\log \left (c x\right ) \log \left (-c x + 1\right ) +{\rm Li}_2\left (-c x + 1\right )\right )} b d^{3} + \frac{1}{2} \,{\left (\log \left (c x + 1\right ) \log \left (-c x\right ) +{\rm Li}_2\left (c x + 1\right )\right )} b d^{3} - \frac{7}{12} \, b d^{3} \log \left (c x + 1\right ) + \frac{11}{12} \, b d^{3} \log \left (c x - 1\right ) + a d^{3} \log \left (x\right ) + \frac{1}{12} \,{\left (2 \, b c^{3} d^{3} x^{3} + 9 \, b c^{2} d^{3} x^{2}\right )} \log \left (c x + 1\right ) - \frac{1}{12} \,{\left (2 \, b c^{3} d^{3} x^{3} + 9 \, b c^{2} d^{3} x^{2}\right )} \log \left (-c x + 1\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{a c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{3} x^{2} + 3 \, a c d^{3} x + a d^{3} +{\left (b c^{3} d^{3} x^{3} + 3 \, b c^{2} d^{3} x^{2} + 3 \, b c d^{3} x + b d^{3}\right )} \operatorname{artanh}\left (c x\right )}{x}, 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*} d^{3} \left (\int 3 a c\, dx + \int \frac{a}{x}\, dx + \int 3 a c^{2} x\, dx + \int a c^{3} x^{2}\, dx + \int 3 b c \operatorname{atanh}{\left (c x \right )}\, dx + \int \frac{b \operatorname{atanh}{\left (c x \right )}}{x}\, dx + \int 3 b c^{2} x \operatorname{atanh}{\left (c x \right )}\, dx + \int b c^{3} x^{2} \operatorname{atanh}{\left (c x \right )}\, dx\right ) \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 (c d x + d\right )}^{3}{\left (b \operatorname{artanh}\left (c x\right ) + a\right )}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]