Optimal. Leaf size=181 \[ \frac{3 b \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right )}{2 c^4 d^2}+\frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 c^2 d^2}+\frac{a+b \tanh ^{-1}(c x)}{c^4 d^2 (c x+1)}-\frac{3 \log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^4 d^2}-\frac{2 a x}{c^3 d^2}-\frac{b \log \left (1-c^2 x^2\right )}{c^4 d^2}+\frac{b x}{2 c^3 d^2}+\frac{b}{2 c^4 d^2 (c x+1)}-\frac{2 b x \tanh ^{-1}(c x)}{c^3 d^2}-\frac{b \tanh ^{-1}(c x)}{c^4 d^2} \]
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Rubi [A] time = 0.222001, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.65, Rules used = {5940, 5910, 260, 5916, 321, 206, 5926, 627, 44, 207, 5918, 2402, 2315} \[ \frac{3 b \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right )}{2 c^4 d^2}+\frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 c^2 d^2}+\frac{a+b \tanh ^{-1}(c x)}{c^4 d^2 (c x+1)}-\frac{3 \log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^4 d^2}-\frac{2 a x}{c^3 d^2}-\frac{b \log \left (1-c^2 x^2\right )}{c^4 d^2}+\frac{b x}{2 c^3 d^2}+\frac{b}{2 c^4 d^2 (c x+1)}-\frac{2 b x \tanh ^{-1}(c x)}{c^3 d^2}-\frac{b \tanh ^{-1}(c x)}{c^4 d^2} \]
Antiderivative was successfully verified.
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Rule 5940
Rule 5910
Rule 260
Rule 5916
Rule 321
Rule 206
Rule 5926
Rule 627
Rule 44
Rule 207
Rule 5918
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b \tanh ^{-1}(c x)\right )}{(d+c d x)^2} \, dx &=\int \left (-\frac{2 \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d^2}+\frac{x \left (a+b \tanh ^{-1}(c x)\right )}{c^2 d^2}-\frac{a+b \tanh ^{-1}(c x)}{c^3 d^2 (1+c x)^2}+\frac{3 \left (a+b \tanh ^{-1}(c x)\right )}{c^3 d^2 (1+c x)}\right ) \, dx\\ &=-\frac{\int \frac{a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx}{c^3 d^2}-\frac{2 \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c^3 d^2}+\frac{3 \int \frac{a+b \tanh ^{-1}(c x)}{1+c x} \, dx}{c^3 d^2}+\frac{\int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c^2 d^2}\\ &=-\frac{2 a x}{c^3 d^2}+\frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 c^2 d^2}+\frac{a+b \tanh ^{-1}(c x)}{c^4 d^2 (1+c x)}-\frac{3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{c^4 d^2}-\frac{b \int \frac{1}{(1+c x) \left (1-c^2 x^2\right )} \, dx}{c^3 d^2}-\frac{(2 b) \int \tanh ^{-1}(c x) \, dx}{c^3 d^2}+\frac{(3 b) \int \frac{\log \left (\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx}{c^3 d^2}-\frac{b \int \frac{x^2}{1-c^2 x^2} \, dx}{2 c d^2}\\ &=-\frac{2 a x}{c^3 d^2}+\frac{b x}{2 c^3 d^2}-\frac{2 b x \tanh ^{-1}(c x)}{c^3 d^2}+\frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 c^2 d^2}+\frac{a+b \tanh ^{-1}(c x)}{c^4 d^2 (1+c x)}-\frac{3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{c^4 d^2}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+c x}\right )}{c^4 d^2}-\frac{b \int \frac{1}{1-c^2 x^2} \, dx}{2 c^3 d^2}-\frac{b \int \frac{1}{(1-c x) (1+c x)^2} \, dx}{c^3 d^2}+\frac{(2 b) \int \frac{x}{1-c^2 x^2} \, dx}{c^2 d^2}\\ &=-\frac{2 a x}{c^3 d^2}+\frac{b x}{2 c^3 d^2}-\frac{b \tanh ^{-1}(c x)}{2 c^4 d^2}-\frac{2 b x \tanh ^{-1}(c x)}{c^3 d^2}+\frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 c^2 d^2}+\frac{a+b \tanh ^{-1}(c x)}{c^4 d^2 (1+c x)}-\frac{3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{c^4 d^2}-\frac{b \log \left (1-c^2 x^2\right )}{c^4 d^2}+\frac{3 b \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{2 c^4 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^3 d^2}\\ &=-\frac{2 a x}{c^3 d^2}+\frac{b x}{2 c^3 d^2}+\frac{b}{2 c^4 d^2 (1+c x)}-\frac{b \tanh ^{-1}(c x)}{2 c^4 d^2}-\frac{2 b x \tanh ^{-1}(c x)}{c^3 d^2}+\frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 c^2 d^2}+\frac{a+b \tanh ^{-1}(c x)}{c^4 d^2 (1+c x)}-\frac{3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{c^4 d^2}-\frac{b \log \left (1-c^2 x^2\right )}{c^4 d^2}+\frac{3 b \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{2 c^4 d^2}+\frac{b \int \frac{1}{-1+c^2 x^2} \, dx}{2 c^3 d^2}\\ &=-\frac{2 a x}{c^3 d^2}+\frac{b x}{2 c^3 d^2}+\frac{b}{2 c^4 d^2 (1+c x)}-\frac{b \tanh ^{-1}(c x)}{c^4 d^2}-\frac{2 b x \tanh ^{-1}(c x)}{c^3 d^2}+\frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )}{2 c^2 d^2}+\frac{a+b \tanh ^{-1}(c x)}{c^4 d^2 (1+c x)}-\frac{3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )}{c^4 d^2}-\frac{b \log \left (1-c^2 x^2\right )}{c^4 d^2}+\frac{3 b \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{2 c^4 d^2}\\ \end{align*}
Mathematica [A] time = 0.701472, size = 142, normalized size = 0.78 \[ \frac{b \left (6 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )-4 \log \left (1-c^2 x^2\right )+2 \tanh ^{-1}(c x) \left (c^2 x^2-4 c x-6 \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )-\sinh \left (2 \tanh ^{-1}(c x)\right )+\cosh \left (2 \tanh ^{-1}(c x)\right )-1\right )+2 c x-\sinh \left (2 \tanh ^{-1}(c x)\right )+\cosh \left (2 \tanh ^{-1}(c x)\right )\right )+2 a c^2 x^2-8 a c x+\frac{4 a}{c x+1}+12 a \log (c x+1)}{4 c^4 d^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.064, size = 265, normalized size = 1.5 \begin{align*}{\frac{a{x}^{2}}{2\,{c}^{2}{d}^{2}}}-2\,{\frac{ax}{{c}^{3}{d}^{2}}}+{\frac{a}{{d}^{2}{c}^{4} \left ( cx+1 \right ) }}+3\,{\frac{a\ln \left ( cx+1 \right ) }{{d}^{2}{c}^{4}}}+{\frac{b{\it Artanh} \left ( cx \right ){x}^{2}}{2\,{c}^{2}{d}^{2}}}-2\,{\frac{bx{\it Artanh} \left ( cx \right ) }{{c}^{3}{d}^{2}}}+{\frac{b{\it Artanh} \left ( cx \right ) }{{d}^{2}{c}^{4} \left ( cx+1 \right ) }}+3\,{\frac{b{\it Artanh} \left ( cx \right ) \ln \left ( cx+1 \right ) }{{d}^{2}{c}^{4}}}-{\frac{3\,b}{2\,{d}^{2}{c}^{4}}\ln \left ( -{\frac{cx}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{cx}{2}} \right ) }+{\frac{3\,b\ln \left ( cx+1 \right ) }{2\,{d}^{2}{c}^{4}}\ln \left ( -{\frac{cx}{2}}+{\frac{1}{2}} \right ) }-{\frac{3\,b}{2\,{d}^{2}{c}^{4}}{\it dilog} \left ({\frac{1}{2}}+{\frac{cx}{2}} \right ) }-{\frac{3\,b \left ( \ln \left ( cx+1 \right ) \right ) ^{2}}{4\,{d}^{2}{c}^{4}}}+{\frac{bx}{2\,{c}^{3}{d}^{2}}}+{\frac{b}{2\,{d}^{2}{c}^{4}}}-{\frac{b\ln \left ( cx-1 \right ) }{2\,{d}^{2}{c}^{4}}}+{\frac{b}{2\,{d}^{2}{c}^{4} \left ( cx+1 \right ) }}-{\frac{3\,b\ln \left ( cx+1 \right ) }{2\,{d}^{2}{c}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b x^{3} \operatorname{artanh}\left (c x\right ) + a x^{3}}{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.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a x^{3}}{c^{2} x^{2} + 2 c x + 1}\, dx + \int \frac{b x^{3} \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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{artanh}\left (c x\right ) + a\right )} x^{3}}{{\left (c d x + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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