Optimal. Leaf size=250 \[ -\frac{b c^2 \text{PolyLog}\left (2,\frac{2}{c x+1}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d}+\frac{b^2 c^2 \text{PolyLog}\left (2,\frac{2}{c x+1}-1\right )}{d}-\frac{b^2 c^2 \text{PolyLog}\left (3,\frac{2}{c x+1}-1\right )}{2 d}-\frac{c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d}+\frac{c^2 \log \left (2-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{d}-\frac{2 b c^2 \log \left (2-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d x^2}+\frac{c \left (a+b \tanh ^{-1}(c x)\right )^2}{d x}-\frac{b c \left (a+b \tanh ^{-1}(c x)\right )}{d x}-\frac{b^2 c^2 \log \left (1-c^2 x^2\right )}{2 d}+\frac{b^2 c^2 \log (x)}{d} \]
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Rubi [A] time = 0.634345, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 13, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.591, Rules used = {5934, 5916, 5982, 266, 36, 29, 31, 5948, 5988, 5932, 2447, 6056, 6610} \[ -\frac{b c^2 \text{PolyLog}\left (2,\frac{2}{c x+1}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d}+\frac{b^2 c^2 \text{PolyLog}\left (2,\frac{2}{c x+1}-1\right )}{d}-\frac{b^2 c^2 \text{PolyLog}\left (3,\frac{2}{c x+1}-1\right )}{2 d}-\frac{c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d}+\frac{c^2 \log \left (2-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{d}-\frac{2 b c^2 \log \left (2-\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d x^2}+\frac{c \left (a+b \tanh ^{-1}(c x)\right )^2}{d x}-\frac{b c \left (a+b \tanh ^{-1}(c x)\right )}{d x}-\frac{b^2 c^2 \log \left (1-c^2 x^2\right )}{2 d}+\frac{b^2 c^2 \log (x)}{d} \]
Antiderivative was successfully verified.
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Rule 5934
Rule 5916
Rule 5982
Rule 266
Rule 36
Rule 29
Rule 31
Rule 5948
Rule 5988
Rule 5932
Rule 2447
Rule 6056
Rule 6610
Rubi steps
\begin{align*} \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{x^3 (d+c d x)} \, dx &=-\left (c \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{x^2 (d+c d x)} \, dx\right )+\frac{\int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{x^3} \, dx}{d}\\ &=-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d x^2}+c^2 \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{x (d+c d x)} \, dx-\frac{c \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{x^2} \, dx}{d}+\frac{(b c) \int \frac{a+b \tanh ^{-1}(c x)}{x^2 \left (1-c^2 x^2\right )} \, dx}{d}\\ &=-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d x^2}+\frac{c \left (a+b \tanh ^{-1}(c x)\right )^2}{d x}+\frac{c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (2-\frac{2}{1+c x}\right )}{d}+\frac{(b c) \int \frac{a+b \tanh ^{-1}(c x)}{x^2} \, dx}{d}-\frac{\left (2 b c^2\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx}{d}+\frac{\left (b c^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{d}-\frac{\left (2 b c^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}{d}\\ &=-\frac{b c \left (a+b \tanh ^{-1}(c x)\right )}{d x}-\frac{c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d x^2}+\frac{c \left (a+b \tanh ^{-1}(c x)\right )^2}{d x}+\frac{c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (2-\frac{2}{1+c x}\right )}{d}-\frac{b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1+c x}\right )}{d}-\frac{\left (2 b c^2\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x (1+c x)} \, dx}{d}+\frac{\left (b^2 c^2\right ) \int \frac{1}{x \left (1-c^2 x^2\right )} \, dx}{d}+\frac{\left (b^2 c^3\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx}{d}\\ &=-\frac{b c \left (a+b \tanh ^{-1}(c x)\right )}{d x}-\frac{c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d x^2}+\frac{c \left (a+b \tanh ^{-1}(c x)\right )^2}{d x}-\frac{2 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )}{d}+\frac{c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (2-\frac{2}{1+c x}\right )}{d}-\frac{b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1+c x}\right )}{d}-\frac{b^2 c^2 \text{Li}_3\left (-1+\frac{2}{1+c x}\right )}{2 d}+\frac{\left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )}{2 d}+\frac{\left (2 b^2 c^3\right ) \int \frac{\log \left (2-\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx}{d}\\ &=-\frac{b c \left (a+b \tanh ^{-1}(c x)\right )}{d x}-\frac{c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d x^2}+\frac{c \left (a+b \tanh ^{-1}(c x)\right )^2}{d x}-\frac{2 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )}{d}+\frac{c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (2-\frac{2}{1+c x}\right )}{d}+\frac{b^2 c^2 \text{Li}_2\left (-1+\frac{2}{1+c x}\right )}{d}-\frac{b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1+c x}\right )}{d}-\frac{b^2 c^2 \text{Li}_3\left (-1+\frac{2}{1+c x}\right )}{2 d}+\frac{\left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )}{2 d}+\frac{\left (b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x} \, dx,x,x^2\right )}{2 d}\\ &=-\frac{b c \left (a+b \tanh ^{-1}(c x)\right )}{d x}-\frac{c^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 d}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{2 d x^2}+\frac{c \left (a+b \tanh ^{-1}(c x)\right )^2}{d x}+\frac{b^2 c^2 \log (x)}{d}-\frac{b^2 c^2 \log \left (1-c^2 x^2\right )}{2 d}-\frac{2 b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+c x}\right )}{d}+\frac{c^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (2-\frac{2}{1+c x}\right )}{d}+\frac{b^2 c^2 \text{Li}_2\left (-1+\frac{2}{1+c x}\right )}{d}-\frac{b c^2 \left (a+b \tanh ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1+c x}\right )}{d}-\frac{b^2 c^2 \text{Li}_3\left (-1+\frac{2}{1+c x}\right )}{2 d}\\ \end{align*}
Mathematica [C] time = 1.0157, size = 317, normalized size = 1.27 \[ \frac{\frac{2 a b \left (-c^2 x^2 \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )-c x \left (2 c x \log \left (\frac{c x}{\sqrt{1-c^2 x^2}}\right )+1\right )+\tanh ^{-1}(c x) \left (c^2 x^2+2 c^2 x^2 \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )+2 c x-1\right )\right )}{x^2}+2 b^2 c^2 \left (\tanh ^{-1}(c x) \text{PolyLog}\left (2,e^{2 \tanh ^{-1}(c x)}\right )+\text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )-\frac{1}{2} \text{PolyLog}\left (3,e^{2 \tanh ^{-1}(c x)}\right )+\log \left (\frac{c x}{\sqrt{1-c^2 x^2}}\right )-\frac{\tanh ^{-1}(c x)^2}{2 c^2 x^2}-\frac{2}{3} \tanh ^{-1}(c x)^3+\frac{\tanh ^{-1}(c x)^2}{c x}-\frac{1}{2} \tanh ^{-1}(c x)^2-\frac{\tanh ^{-1}(c x)}{c x}+\tanh ^{-1}(c x)^2 \log \left (1-e^{2 \tanh ^{-1}(c x)}\right )-2 \tanh ^{-1}(c x) \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )+\frac{i \pi ^3}{24}\right )+2 a^2 c^2 \log (x)-2 a^2 c^2 \log (c x+1)+\frac{2 a^2 c}{x}-\frac{a^2}{x^2}}{2 d} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 1.205, size = 1841, normalized size = 7.4 \begin{align*} \text{result too large to display} \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*} -\frac{1}{2} \,{\left (\frac{2 \, c^{2} \log \left (c x + 1\right )}{d} - \frac{2 \, c^{2} \log \left (x\right )}{d} - \frac{2 \, c x - 1}{d x^{2}}\right )} a^{2} - \frac{{\left (2 \, b^{2} c^{2} x^{2} \log \left (c x + 1\right ) - 2 \, b^{2} c x + b^{2}\right )} \log \left (-c x + 1\right )^{2}}{8 \, d x^{2}} + \int \frac{{\left (b^{2} c x - b^{2}\right )} \log \left (c x + 1\right )^{2} + 4 \,{\left (a b c x - a b\right )} \log \left (c x + 1\right ) -{\left (2 \, b^{2} c^{3} x^{3} + b^{2} c^{2} x^{2} - 4 \, a b +{\left (4 \, a b c - b^{2} c\right )} x - 2 \,{\left (b^{2} c^{4} x^{4} + b^{2} c^{3} x^{3} - b^{2} c x + b^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{4 \,{\left (c^{2} d x^{5} - d x^{3}\right )}}\,{d x} \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^{2} \operatorname{artanh}\left (c x\right )^{2} + 2 \, a b \operatorname{artanh}\left (c x\right ) + a^{2}}{c d x^{4} + d x^{3}}, 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^{2}}{c x^{4} + x^{3}}\, dx + \int \frac{b^{2} \operatorname{atanh}^{2}{\left (c x \right )}}{c x^{4} + x^{3}}\, dx + \int \frac{2 a b \operatorname{atanh}{\left (c x \right )}}{c x^{4} + x^{3}}\, dx}{d} \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 )}^{2}}{{\left (c d x + d\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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