Optimal. Leaf size=178 \[ -2 b c d^4 \text{PolyLog}(2,-c x)+2 b c d^4 \text{PolyLog}(2,c x)+\frac{1}{3} c^4 d^4 x^3 \left (a+b \tanh ^{-1}(c x)\right )+2 c^3 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac{d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+6 a c^2 d^4 x+4 a c d^4 \log (x)+\frac{1}{6} b c^3 d^4 x^2+\frac{8}{3} b c d^4 \log \left (1-c^2 x^2\right )+2 b c^2 d^4 x+6 b c^2 d^4 x \tanh ^{-1}(c x)+b c d^4 \log (x)-2 b c d^4 \tanh ^{-1}(c x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.199811, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 12, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {5940, 5910, 260, 5916, 266, 36, 29, 31, 5912, 321, 206, 43} \[ -2 b c d^4 \text{PolyLog}(2,-c x)+2 b c d^4 \text{PolyLog}(2,c x)+\frac{1}{3} c^4 d^4 x^3 \left (a+b \tanh ^{-1}(c x)\right )+2 c^3 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac{d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+6 a c^2 d^4 x+4 a c d^4 \log (x)+\frac{1}{6} b c^3 d^4 x^2+\frac{8}{3} b c d^4 \log \left (1-c^2 x^2\right )+2 b c^2 d^4 x+6 b c^2 d^4 x \tanh ^{-1}(c x)+b c d^4 \log (x)-2 b c d^4 \tanh ^{-1}(c x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5940
Rule 5910
Rule 260
Rule 5916
Rule 266
Rule 36
Rule 29
Rule 31
Rule 5912
Rule 321
Rule 206
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+c d x)^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^2} \, dx &=\int \left (6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}+\frac{4 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+4 c^3 d^4 x \left (a+b \tanh ^{-1}(c x)\right )+c^4 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )\right ) \, dx\\ &=d^4 \int \frac{a+b \tanh ^{-1}(c x)}{x^2} \, dx+\left (4 c d^4\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x} \, dx+\left (6 c^2 d^4\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx+\left (4 c^3 d^4\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx+\left (c^4 d^4\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx\\ &=6 a c^2 d^4 x-\frac{d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+2 c^3 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{3} c^4 d^4 x^3 \left (a+b \tanh ^{-1}(c x)\right )+4 a c d^4 \log (x)-2 b c d^4 \text{Li}_2(-c x)+2 b c d^4 \text{Li}_2(c x)+\left (b c d^4\right ) \int \frac{1}{x \left (1-c^2 x^2\right )} \, dx+\left (6 b c^2 d^4\right ) \int \tanh ^{-1}(c x) \, dx-\left (2 b c^4 d^4\right ) \int \frac{x^2}{1-c^2 x^2} \, dx-\frac{1}{3} \left (b c^5 d^4\right ) \int \frac{x^3}{1-c^2 x^2} \, dx\\ &=6 a c^2 d^4 x+2 b c^2 d^4 x+6 b c^2 d^4 x \tanh ^{-1}(c x)-\frac{d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+2 c^3 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{3} c^4 d^4 x^3 \left (a+b \tanh ^{-1}(c x)\right )+4 a c d^4 \log (x)-2 b c d^4 \text{Li}_2(-c x)+2 b c d^4 \text{Li}_2(c x)+\frac{1}{2} \left (b c d^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )-\left (2 b c^2 d^4\right ) \int \frac{1}{1-c^2 x^2} \, dx-\left (6 b c^3 d^4\right ) \int \frac{x}{1-c^2 x^2} \, dx-\frac{1}{6} \left (b c^5 d^4\right ) \operatorname{Subst}\left (\int \frac{x}{1-c^2 x} \, dx,x,x^2\right )\\ &=6 a c^2 d^4 x+2 b c^2 d^4 x-2 b c d^4 \tanh ^{-1}(c x)+6 b c^2 d^4 x \tanh ^{-1}(c x)-\frac{d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+2 c^3 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{3} c^4 d^4 x^3 \left (a+b \tanh ^{-1}(c x)\right )+4 a c d^4 \log (x)+3 b c d^4 \log \left (1-c^2 x^2\right )-2 b c d^4 \text{Li}_2(-c x)+2 b c d^4 \text{Li}_2(c x)+\frac{1}{2} \left (b c d^4\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )+\frac{1}{2} \left (b c^3 d^4\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x} \, dx,x,x^2\right )-\frac{1}{6} \left (b c^5 d^4\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 )\\ &=6 a c^2 d^4 x+2 b c^2 d^4 x+\frac{1}{6} b c^3 d^4 x^2-2 b c d^4 \tanh ^{-1}(c x)+6 b c^2 d^4 x \tanh ^{-1}(c x)-\frac{d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x}+2 c^3 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{3} c^4 d^4 x^3 \left (a+b \tanh ^{-1}(c x)\right )+4 a c d^4 \log (x)+b c d^4 \log (x)+\frac{8}{3} b c d^4 \log \left (1-c^2 x^2\right )-2 b c d^4 \text{Li}_2(-c x)+2 b c d^4 \text{Li}_2(c x)\\ \end{align*}
Mathematica [A] time = 0.175252, size = 194, normalized size = 1.09 \[ \frac{d^4 \left (-12 b c x \text{PolyLog}(2,-c x)+12 b c x \text{PolyLog}(2,c x)+2 a c^4 x^4+12 a c^3 x^3+36 a c^2 x^2+24 a c x \log (x)-6 a+b c^3 x^3+12 b c^2 x^2+15 b c x \log \left (1-c^2 x^2\right )+b c x \log \left (c^2 x^2-1\right )+2 b c^4 x^4 \tanh ^{-1}(c x)+12 b c^3 x^3 \tanh ^{-1}(c x)+36 b c^2 x^2 \tanh ^{-1}(c x)+6 b c x \log (c x)+6 b c x \log (1-c x)-6 b c x \log (c x+1)-6 b \tanh ^{-1}(c x)\right )}{6 x} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.049, size = 229, normalized size = 1.3 \begin{align*}{\frac{{d}^{4}a{c}^{4}{x}^{3}}{3}}+2\,{d}^{4}a{c}^{3}{x}^{2}+6\,a{c}^{2}{d}^{4}x-{\frac{{d}^{4}a}{x}}+4\,c{d}^{4}a\ln \left ( cx \right ) +{\frac{{d}^{4}b{\it Artanh} \left ( cx \right ){c}^{4}{x}^{3}}{3}}+2\,{d}^{4}b{\it Artanh} \left ( cx \right ){c}^{3}{x}^{2}+6\,b{c}^{2}{d}^{4}x{\it Artanh} \left ( cx \right ) -{\frac{{d}^{4}b{\it Artanh} \left ( cx \right ) }{x}}+4\,c{d}^{4}b{\it Artanh} \left ( cx \right ) \ln \left ( cx \right ) -2\,c{d}^{4}b{\it dilog} \left ( cx \right ) -2\,c{d}^{4}b{\it dilog} \left ( cx+1 \right ) -2\,c{d}^{4}b\ln \left ( cx \right ) \ln \left ( cx+1 \right ) +{\frac{b{c}^{3}{d}^{4}{x}^{2}}{6}}+2\,b{c}^{2}{d}^{4}x+{\frac{11\,c{d}^{4}b\ln \left ( cx-1 \right ) }{3}}+c{d}^{4}b\ln \left ( cx \right ) +{\frac{5\,c{d}^{4}b\ln \left ( cx+1 \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.46349, size = 379, normalized size = 2.13 \begin{align*} \frac{1}{3} \, a c^{4} d^{4} x^{3} + 2 \, a c^{3} d^{4} x^{2} + \frac{1}{6} \, b c^{3} d^{4} x^{2} + 6 \, a c^{2} d^{4} x + 2 \, b c^{2} d^{4} x + 3 \,{\left (2 \, c x \operatorname{artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} b c d^{4} - 2 \,{\left (\log \left (c x\right ) \log \left (-c x + 1\right ) +{\rm Li}_2\left (-c x + 1\right )\right )} b c d^{4} + 2 \,{\left (\log \left (c x + 1\right ) \log \left (-c x\right ) +{\rm Li}_2\left (c x + 1\right )\right )} b c d^{4} - \frac{5}{6} \, b c d^{4} \log \left (c x + 1\right ) + \frac{7}{6} \, b c d^{4} \log \left (c x - 1\right ) + 4 \, a c d^{4} \log \left (x\right ) - \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^{4} - \frac{a d^{4}}{x} + \frac{1}{6} \,{\left (b c^{4} d^{4} x^{3} + 6 \, b c^{3} d^{4} x^{2}\right )} \log \left (c x + 1\right ) - \frac{1}{6} \,{\left (b c^{4} d^{4} x^{3} + 6 \, b c^{3} d^{4} 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^{4} d^{4} x^{4} + 4 \, a c^{3} d^{4} x^{3} + 6 \, a c^{2} d^{4} x^{2} + 4 \, a c d^{4} x + a d^{4} +{\left (b c^{4} d^{4} x^{4} + 4 \, b c^{3} d^{4} x^{3} + 6 \, b c^{2} d^{4} x^{2} + 4 \, b c d^{4} x + b d^{4}\right )} \operatorname{artanh}\left (c x\right )}{x^{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*} d^{4} \left (\int 6 a c^{2}\, dx + \int \frac{a}{x^{2}}\, dx + \int \frac{4 a c}{x}\, dx + \int 4 a c^{3} x\, dx + \int a c^{4} x^{2}\, dx + \int 6 b c^{2} \operatorname{atanh}{\left (c x \right )}\, dx + \int \frac{b \operatorname{atanh}{\left (c x \right )}}{x^{2}}\, dx + \int \frac{4 b c \operatorname{atanh}{\left (c x \right )}}{x}\, dx + \int 4 b c^{3} x \operatorname{atanh}{\left (c x \right )}\, dx + \int b c^{4} 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 )}^{4}{\left (b \operatorname{artanh}\left (c x\right ) + a\right )}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]