Optimal. Leaf size=479 \[ -\frac{14}{15} b^2 c^6 d^3 \text{PolyLog}(2,-c x)+\frac{14}{15} b^2 c^6 d^3 \text{PolyLog}(2,c x)+\frac{37}{40} b^2 c^6 d^3 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )-\frac{1}{120} b^2 c^6 d^3 \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right )-\frac{14 b c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^2}-\frac{c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac{11 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 x^3}-\frac{3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac{3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{10 x^4}+\frac{28}{15} a b c^6 d^3 \log (x)-\frac{11 b c^5 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x}+\frac{37}{20} b c^6 d^3 \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{60} b c^6 d^3 \log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac{3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}-\frac{b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^5}-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 x^6}-\frac{61 b^2 c^4 d^3}{180 x^2}-\frac{b^2 c^3 d^3}{10 x^3}-\frac{b^2 c^2 d^3}{60 x^4}-\frac{113}{90} b^2 c^6 d^3 \log \left (1-c^2 x^2\right )-\frac{37 b^2 c^5 d^3}{30 x}+\frac{113}{45} b^2 c^6 d^3 \log (x)+\frac{37}{30} b^2 c^6 d^3 \tanh ^{-1}(c x) \]
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Rubi [A] time = 0.509132, antiderivative size = 479, normalized size of antiderivative = 1., number of steps used = 29, number of rules used = 14, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636, Rules used = {43, 5938, 5916, 266, 44, 325, 206, 36, 29, 31, 5912, 5918, 2402, 2315} \[ -\frac{14}{15} b^2 c^6 d^3 \text{PolyLog}(2,-c x)+\frac{14}{15} b^2 c^6 d^3 \text{PolyLog}(2,c x)+\frac{37}{40} b^2 c^6 d^3 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )-\frac{1}{120} b^2 c^6 d^3 \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right )-\frac{14 b c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^2}-\frac{c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac{11 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 x^3}-\frac{3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac{3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{10 x^4}+\frac{28}{15} a b c^6 d^3 \log (x)-\frac{11 b c^5 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x}+\frac{37}{20} b c^6 d^3 \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{60} b c^6 d^3 \log \left (\frac{2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac{3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}-\frac{b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^5}-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 x^6}-\frac{61 b^2 c^4 d^3}{180 x^2}-\frac{b^2 c^3 d^3}{10 x^3}-\frac{b^2 c^2 d^3}{60 x^4}-\frac{113}{90} b^2 c^6 d^3 \log \left (1-c^2 x^2\right )-\frac{37 b^2 c^5 d^3}{30 x}+\frac{113}{45} b^2 c^6 d^3 \log (x)+\frac{37}{30} b^2 c^6 d^3 \tanh ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 43
Rule 5938
Rule 5916
Rule 266
Rule 44
Rule 325
Rule 206
Rule 36
Rule 29
Rule 31
Rule 5912
Rule 5918
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{(d+c d x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x^7} \, dx &=-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 x^6}-\frac{3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}-\frac{3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac{c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-(2 b c) \int \left (-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}-\frac{3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac{11 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{12 x^4}-\frac{14 c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^3}-\frac{11 c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{12 x^2}-\frac{14 c^5 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x}+\frac{37 c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{40 (-1+c x)}+\frac{c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{120 (1+c x)}\right ) \, dx\\ &=-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 x^6}-\frac{3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}-\frac{3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac{c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{3} \left (b c d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x^6} \, dx+\frac{1}{5} \left (6 b c^2 d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x^5} \, dx+\frac{1}{6} \left (11 b c^3 d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x^4} \, dx+\frac{1}{15} \left (28 b c^4 d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x^3} \, dx+\frac{1}{6} \left (11 b c^5 d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x^2} \, dx+\frac{1}{15} \left (28 b c^6 d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x} \, dx-\frac{1}{60} \left (b c^7 d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{1+c x} \, dx-\frac{1}{20} \left (37 b c^7 d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{-1+c x} \, dx\\ &=-\frac{b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^5}-\frac{3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{10 x^4}-\frac{11 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 x^3}-\frac{14 b c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^2}-\frac{11 b c^5 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x}-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 x^6}-\frac{3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}-\frac{3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac{c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}+\frac{28}{15} a b c^6 d^3 \log (x)+\frac{37}{20} b c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )+\frac{1}{60} b c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )-\frac{14}{15} b^2 c^6 d^3 \text{Li}_2(-c x)+\frac{14}{15} b^2 c^6 d^3 \text{Li}_2(c x)+\frac{1}{15} \left (b^2 c^2 d^3\right ) \int \frac{1}{x^5 \left (1-c^2 x^2\right )} \, dx+\frac{1}{10} \left (3 b^2 c^3 d^3\right ) \int \frac{1}{x^4 \left (1-c^2 x^2\right )} \, dx+\frac{1}{18} \left (11 b^2 c^4 d^3\right ) \int \frac{1}{x^3 \left (1-c^2 x^2\right )} \, dx+\frac{1}{15} \left (14 b^2 c^5 d^3\right ) \int \frac{1}{x^2 \left (1-c^2 x^2\right )} \, dx+\frac{1}{6} \left (11 b^2 c^6 d^3\right ) \int \frac{1}{x \left (1-c^2 x^2\right )} \, dx-\frac{1}{60} \left (b^2 c^7 d^3\right ) \int \frac{\log \left (\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx-\frac{1}{20} \left (37 b^2 c^7 d^3\right ) \int \frac{\log \left (\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac{b^2 c^3 d^3}{10 x^3}-\frac{14 b^2 c^5 d^3}{15 x}-\frac{b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^5}-\frac{3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{10 x^4}-\frac{11 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 x^3}-\frac{14 b c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^2}-\frac{11 b c^5 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x}-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 x^6}-\frac{3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}-\frac{3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac{c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}+\frac{28}{15} a b c^6 d^3 \log (x)+\frac{37}{20} b c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )+\frac{1}{60} b c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )-\frac{14}{15} b^2 c^6 d^3 \text{Li}_2(-c x)+\frac{14}{15} b^2 c^6 d^3 \text{Li}_2(c x)+\frac{1}{30} \left (b^2 c^2 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^3 \left (1-c^2 x\right )} \, dx,x,x^2\right )+\frac{1}{36} \left (11 b^2 c^4 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-c^2 x\right )} \, dx,x,x^2\right )+\frac{1}{10} \left (3 b^2 c^5 d^3\right ) \int \frac{1}{x^2 \left (1-c^2 x^2\right )} \, dx-\frac{1}{60} \left (b^2 c^6 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+c x}\right )+\frac{1}{12} \left (11 b^2 c^6 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )+\frac{1}{20} \left (37 b^2 c^6 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-c x}\right )+\frac{1}{15} \left (14 b^2 c^7 d^3\right ) \int \frac{1}{1-c^2 x^2} \, dx\\ &=-\frac{b^2 c^3 d^3}{10 x^3}-\frac{37 b^2 c^5 d^3}{30 x}+\frac{14}{15} b^2 c^6 d^3 \tanh ^{-1}(c x)-\frac{b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^5}-\frac{3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{10 x^4}-\frac{11 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 x^3}-\frac{14 b c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^2}-\frac{11 b c^5 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x}-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 x^6}-\frac{3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}-\frac{3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac{c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}+\frac{28}{15} a b c^6 d^3 \log (x)+\frac{37}{20} b c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )+\frac{1}{60} b c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )-\frac{14}{15} b^2 c^6 d^3 \text{Li}_2(-c x)+\frac{14}{15} b^2 c^6 d^3 \text{Li}_2(c x)+\frac{37}{40} b^2 c^6 d^3 \text{Li}_2\left (1-\frac{2}{1-c x}\right )-\frac{1}{120} b^2 c^6 d^3 \text{Li}_2\left (1-\frac{2}{1+c x}\right )+\frac{1}{30} \left (b^2 c^2 d^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^3}+\frac{c^2}{x^2}+\frac{c^4}{x}-\frac{c^6}{-1+c^2 x}\right ) \, dx,x,x^2\right )+\frac{1}{36} \left (11 b^2 c^4 d^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^2}+\frac{c^2}{x}-\frac{c^4}{-1+c^2 x}\right ) \, dx,x,x^2\right )+\frac{1}{12} \left (11 b^2 c^6 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )+\frac{1}{10} \left (3 b^2 c^7 d^3\right ) \int \frac{1}{1-c^2 x^2} \, dx+\frac{1}{12} \left (11 b^2 c^8 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-c^2 x} \, dx,x,x^2\right )\\ &=-\frac{b^2 c^2 d^3}{60 x^4}-\frac{b^2 c^3 d^3}{10 x^3}-\frac{61 b^2 c^4 d^3}{180 x^2}-\frac{37 b^2 c^5 d^3}{30 x}+\frac{37}{30} b^2 c^6 d^3 \tanh ^{-1}(c x)-\frac{b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^5}-\frac{3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{10 x^4}-\frac{11 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 x^3}-\frac{14 b c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{15 x^2}-\frac{11 b c^5 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x}-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 x^6}-\frac{3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 x^5}-\frac{3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac{c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}+\frac{28}{15} a b c^6 d^3 \log (x)+\frac{113}{45} b^2 c^6 d^3 \log (x)+\frac{37}{20} b c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )+\frac{1}{60} b c^6 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1+c x}\right )-\frac{113}{90} b^2 c^6 d^3 \log \left (1-c^2 x^2\right )-\frac{14}{15} b^2 c^6 d^3 \text{Li}_2(-c x)+\frac{14}{15} b^2 c^6 d^3 \text{Li}_2(c x)+\frac{37}{40} b^2 c^6 d^3 \text{Li}_2\left (1-\frac{2}{1-c x}\right )-\frac{1}{120} b^2 c^6 d^3 \text{Li}_2\left (1-\frac{2}{1+c x}\right )\\ \end{align*}
Mathematica [A] time = 1.37705, size = 402, normalized size = 0.84 \[ -\frac{d^3 \left (168 b^2 c^6 x^6 \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )+60 a^2 c^3 x^3+135 a^2 c^2 x^2+108 a^2 c x+30 a^2+330 a b c^5 x^5+168 a b c^4 x^4+110 a b c^3 x^3+54 a b c^2 x^2-336 a b c^6 x^6 \log (c x)+165 a b c^6 x^6 \log (1-c x)-165 a b c^6 x^6 \log (c x+1)+168 a b c^6 x^6 \log \left (1-c^2 x^2\right )+2 b \tanh ^{-1}(c x) \left (3 a \left (20 c^3 x^3+45 c^2 x^2+36 c x+10\right )+b c x \left (-111 c^5 x^5+165 c^4 x^4+84 c^3 x^3+55 c^2 x^2+27 c x+6\right )-168 b c^6 x^6 \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )\right )+12 a b c x-64 b^2 c^6 x^6+222 b^2 c^5 x^5+61 b^2 c^4 x^4+18 b^2 c^3 x^3+3 b^2 c^2 x^2-452 b^2 c^6 x^6 \log \left (\frac{c x}{\sqrt{1-c^2 x^2}}\right )+3 b^2 \left (-111 c^6 x^6+20 c^3 x^3+45 c^2 x^2+36 c x+10\right ) \tanh ^{-1}(c x)^2\right )}{180 x^6} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.073, size = 736, normalized size = 1.5 \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 [B] time = 3.17225, size = 1297, normalized size = 2.71 \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{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^{7}}, 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*} d^{3} \left (\int \frac{a^{2}}{x^{7}}\, dx + \int \frac{3 a^{2} c}{x^{6}}\, dx + \int \frac{3 a^{2} c^{2}}{x^{5}}\, dx + \int \frac{a^{2} c^{3}}{x^{4}}\, dx + \int \frac{b^{2} \operatorname{atanh}^{2}{\left (c x \right )}}{x^{7}}\, dx + \int \frac{2 a b \operatorname{atanh}{\left (c x \right )}}{x^{7}}\, dx + \int \frac{3 b^{2} c \operatorname{atanh}^{2}{\left (c x \right )}}{x^{6}}\, dx + \int \frac{3 b^{2} c^{2} \operatorname{atanh}^{2}{\left (c x \right )}}{x^{5}}\, dx + \int \frac{b^{2} c^{3} \operatorname{atanh}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int \frac{6 a b c \operatorname{atanh}{\left (c x \right )}}{x^{6}}\, dx + \int \frac{6 a b c^{2} \operatorname{atanh}{\left (c x \right )}}{x^{5}}\, dx + \int \frac{2 a b c^{3} \operatorname{atanh}{\left (c x \right )}}{x^{4}}\, dx\right ) \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 (c d x + d\right )}^{3}{\left (b \operatorname{artanh}\left (c x\right ) + a\right )}^{2}}{x^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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