Optimal. Leaf size=361 \[ -3 b c d^3 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+3 b c d^3 \text{PolyLog}\left (2,\frac{2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )-3 b^2 c d^3 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )-b^2 c d^3 \text{PolyLog}\left (2,\frac{2}{c x+1}-1\right )+\frac{3}{2} b^2 c d^3 \text{PolyLog}\left (3,1-\frac{2}{1-c x}\right )-\frac{3}{2} b^2 c d^3 \text{PolyLog}\left (3,\frac{2}{1-c x}-1\right )+\frac{1}{2} c^3 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+a b c^2 d^3 x+3 c^2 d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{7}{2} c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+6 c d^3 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2-6 b c d^3 \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+2 b c d^3 \log \left (2-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{2} b^2 c d^3 \log \left (1-c^2 x^2\right )+b^2 c^2 d^3 x \tanh ^{-1}(c x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.776101, antiderivative size = 361, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 17, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.773, Rules used = {5940, 5910, 5984, 5918, 2402, 2315, 5916, 5988, 5932, 2447, 5914, 6052, 5948, 6058, 6610, 5980, 260} \[ -3 b c d^3 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+3 b c d^3 \text{PolyLog}\left (2,\frac{2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )-3 b^2 c d^3 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )-b^2 c d^3 \text{PolyLog}\left (2,\frac{2}{c x+1}-1\right )+\frac{3}{2} b^2 c d^3 \text{PolyLog}\left (3,1-\frac{2}{1-c x}\right )-\frac{3}{2} b^2 c d^3 \text{PolyLog}\left (3,\frac{2}{1-c x}-1\right )+\frac{1}{2} c^3 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+a b c^2 d^3 x+3 c^2 d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{7}{2} c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+6 c d^3 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2-6 b c d^3 \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+2 b c d^3 \log \left (2-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{2} b^2 c d^3 \log \left (1-c^2 x^2\right )+b^2 c^2 d^3 x \tanh ^{-1}(c x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5940
Rule 5910
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rule 5916
Rule 5988
Rule 5932
Rule 2447
Rule 5914
Rule 6052
Rule 5948
Rule 6058
Rule 6610
Rule 5980
Rule 260
Rubi steps
\begin{align*} \int \frac{(d+c d x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x^2} \, dx &=\int \left (3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x^2}+\frac{3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+c^3 d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^3 \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{x^2} \, dx+\left (3 c d^3\right ) \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{x} \, dx+\left (3 c^2 d^3\right ) \int \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+\left (c^3 d^3\right ) \int x \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx\\ &=-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+3 c^2 d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{2} c^3 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+6 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )+\left (2 b c d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx-\left (12 b c^2 d^3\right ) \int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (6 b c^3 d^3\right ) \int \frac{x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\left (b c^4 d^3\right ) \int \frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=4 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+3 c^2 d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{2} c^3 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+6 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )+\left (2 b c d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x (1+c x)} \, dx+\left (b c^2 d^3\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\left (b c^2 d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx-\left (6 b c^2 d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{1-c x} \, dx+\left (6 b c^2 d^3\right ) \int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (6 b c^2 d^3\right ) \int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=a b c^2 d^3 x+\frac{7}{2} c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+3 c^2 d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{2} c^3 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+6 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )-6 b c d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )+2 b c d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )-3 b c d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )+3 b c d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1-c x}\right )+\left (b^2 c^2 d^3\right ) \int \tanh ^{-1}(c x) \, dx-\left (2 b^2 c^2 d^3\right ) \int \frac{\log \left (2-\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx+\left (3 b^2 c^2 d^3\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (3 b^2 c^2 d^3\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx+\left (6 b^2 c^2 d^3\right ) \int \frac{\log \left (\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=a b c^2 d^3 x+b^2 c^2 d^3 x \tanh ^{-1}(c x)+\frac{7}{2} c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+3 c^2 d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{2} c^3 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+6 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )-6 b c d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )+2 b c d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )-3 b c d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )+3 b c d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1-c x}\right )-b^2 c d^3 \text{Li}_2\left (-1+\frac{2}{1+c x}\right )+\frac{3}{2} b^2 c d^3 \text{Li}_3\left (1-\frac{2}{1-c x}\right )-\frac{3}{2} b^2 c d^3 \text{Li}_3\left (-1+\frac{2}{1-c x}\right )-\left (6 b^2 c d^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-c x}\right )-\left (b^2 c^3 d^3\right ) \int \frac{x}{1-c^2 x^2} \, dx\\ &=a b c^2 d^3 x+b^2 c^2 d^3 x \tanh ^{-1}(c x)+\frac{7}{2} c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+3 c^2 d^3 x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{2} c^3 d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+6 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-c x}\right )-6 b c d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )+\frac{1}{2} b^2 c d^3 \log \left (1-c^2 x^2\right )+2 b c d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )-3 b^2 c d^3 \text{Li}_2\left (1-\frac{2}{1-c x}\right )-3 b c d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-c x}\right )+3 b c d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1-c x}\right )-b^2 c d^3 \text{Li}_2\left (-1+\frac{2}{1+c x}\right )+\frac{3}{2} b^2 c d^3 \text{Li}_3\left (1-\frac{2}{1-c x}\right )-\frac{3}{2} b^2 c d^3 \text{Li}_3\left (-1+\frac{2}{1-c x}\right )\\ \end{align*}
Mathematica [C] time = 0.635039, size = 479, normalized size = 1.33 \[ \frac{d^3 \left (-24 a b c x \text{PolyLog}(2,-c x)+24 a b c x \text{PolyLog}(2,c x)+24 b^2 c x \left (\tanh ^{-1}(c x)+1\right ) \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )-8 b^2 c x \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )+24 b^2 c x \tanh ^{-1}(c x) \text{PolyLog}\left (2,e^{2 \tanh ^{-1}(c x)}\right )+12 b^2 c x \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )-12 b^2 c x \text{PolyLog}\left (3,e^{2 \tanh ^{-1}(c x)}\right )+4 a^2 c^3 x^3+24 a^2 c^2 x^2+24 a^2 c x \log (x)-8 a^2+8 a b c^2 x^2+16 a b c x \log \left (1-c^2 x^2\right )+8 a b c^3 x^3 \tanh ^{-1}(c x)+48 a b c^2 x^2 \tanh ^{-1}(c x)+16 a b c x \log (c x)+4 a b c x \log (1-c x)-4 a b c x \log (c x+1)-16 a b \tanh ^{-1}(c x)+4 b^2 c x \log \left (1-c^2 x^2\right )+4 b^2 c^3 x^3 \tanh ^{-1}(c x)^2+24 b^2 c^2 x^2 \tanh ^{-1}(c x)^2+8 b^2 c^2 x^2 \tanh ^{-1}(c x)+i \pi ^3 b^2 c x-16 b^2 c x \tanh ^{-1}(c x)^3-20 b^2 c x \tanh ^{-1}(c x)^2-8 b^2 \tanh ^{-1}(c x)^2+16 b^2 c x \tanh ^{-1}(c x) \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )-24 b^2 c x \tanh ^{-1}(c x)^2 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )-48 b^2 c x \tanh ^{-1}(c x) \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )+24 b^2 c x \tanh ^{-1}(c x)^2 \log \left (1-e^{2 \tanh ^{-1}(c x)}\right )\right )}{8 x} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 1.339, size = 1270, normalized size = 3.5 \begin{align*} \text{result too large to display} \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}{2} \, a^{2} c^{3} d^{3} x^{2} + 3 \, a^{2} c^{2} d^{3} x + 3 \,{\left (2 \, c x \operatorname{artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} a b c d^{3} + 3 \, a^{2} c d^{3} \log \left (x\right ) -{\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 )} a b d^{3} - \frac{a^{2} d^{3}}{x} + \frac{{\left (b^{2} c^{3} d^{3} x^{3} + 6 \, b^{2} c^{2} d^{3} x^{2} - 2 \, b^{2} d^{3}\right )} \log \left (-c x + 1\right )^{2}}{8 \, x} - \int -\frac{{\left (b^{2} c^{4} d^{3} x^{4} + 2 \, b^{2} c^{3} d^{3} x^{3} - 2 \, b^{2} c d^{3} x - b^{2} d^{3}\right )} \log \left (c x + 1\right )^{2} + 4 \,{\left (a b c^{4} d^{3} x^{4} - a b c^{3} d^{3} x^{3} + 3 \, a b c^{2} d^{3} x^{2} - 3 \, a b c d^{3} x\right )} \log \left (c x + 1\right ) -{\left (12 \, a b c^{2} d^{3} x^{2} +{\left (4 \, a b c^{4} d^{3} + b^{2} c^{4} d^{3}\right )} x^{4} - 2 \,{\left (2 \, a b c^{3} d^{3} - 3 \, b^{2} c^{3} d^{3}\right )} x^{3} - 2 \,{\left (6 \, a b c d^{3} + b^{2} c d^{3}\right )} x + 2 \,{\left (b^{2} c^{4} d^{3} x^{4} + 2 \, b^{2} c^{3} d^{3} x^{3} - 2 \, b^{2} c d^{3} x - b^{2} d^{3}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{4 \,{\left (c x^{3} - x^{2}\right )}}\,{d x} \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^{2} c^{3} d^{3} x^{3} + 3 \, a^{2} c^{2} d^{3} x^{2} + 3 \, a^{2} c d^{3} x + a^{2} d^{3} +{\left (b^{2} c^{3} d^{3} x^{3} + 3 \, b^{2} c^{2} d^{3} x^{2} + 3 \, b^{2} c d^{3} x + b^{2} d^{3}\right )} \operatorname{artanh}\left (c x\right )^{2} + 2 \,{\left (a b c^{3} d^{3} x^{3} + 3 \, a b c^{2} d^{3} x^{2} + 3 \, a b c d^{3} x + a b d^{3}\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^{3} \left (\int 3 a^{2} c^{2}\, dx + \int \frac{a^{2}}{x^{2}}\, dx + \int \frac{3 a^{2} c}{x}\, dx + \int a^{2} c^{3} x\, dx + \int 3 b^{2} c^{2} \operatorname{atanh}^{2}{\left (c x \right )}\, dx + \int \frac{b^{2} \operatorname{atanh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int 6 a b c^{2} \operatorname{atanh}{\left (c x \right )}\, dx + \int \frac{2 a b \operatorname{atanh}{\left (c x \right )}}{x^{2}}\, dx + \int \frac{3 b^{2} c \operatorname{atanh}^{2}{\left (c x \right )}}{x}\, dx + \int b^{2} c^{3} x \operatorname{atanh}^{2}{\left (c x \right )}\, dx + \int \frac{6 a b c \operatorname{atanh}{\left (c x \right )}}{x}\, dx + \int 2 a b c^{3} x \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 )}^{2}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]