Optimal. Leaf size=271 \[ -2 b^2 c^4 d^3 \text{PolyLog}(2,-c x)+2 b^2 c^4 d^3 \text{PolyLog}(2,c x)+2 b^2 c^4 d^3 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )-\frac{b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}+4 a b c^4 d^3 \log (x)-\frac{7 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x}+4 b c^4 d^3 \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac{b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^3}-\frac{d^3 (c x+1)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac{b^2 c^2 d^3}{12 x^2}-\frac{11}{6} b^2 c^4 d^3 \log \left (1-c^2 x^2\right )-\frac{b^2 c^3 d^3}{x}+\frac{11}{3} b^2 c^4 d^3 \log (x)+b^2 c^4 d^3 \tanh ^{-1}(c x) \]
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Rubi [A] time = 0.307835, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 14, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636, Rules used = {37, 5938, 5916, 266, 44, 325, 206, 36, 29, 31, 5912, 5918, 2402, 2315} \[ -2 b^2 c^4 d^3 \text{PolyLog}(2,-c x)+2 b^2 c^4 d^3 \text{PolyLog}(2,c x)+2 b^2 c^4 d^3 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )-\frac{b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}+4 a b c^4 d^3 \log (x)-\frac{7 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x}+4 b c^4 d^3 \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac{b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^3}-\frac{d^3 (c x+1)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-\frac{b^2 c^2 d^3}{12 x^2}-\frac{11}{6} b^2 c^4 d^3 \log \left (1-c^2 x^2\right )-\frac{b^2 c^3 d^3}{x}+\frac{11}{3} b^2 c^4 d^3 \log (x)+b^2 c^4 d^3 \tanh ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 37
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^5} \, dx &=-\frac{d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}-(2 b c) \int \left (-\frac{d^3 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac{c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x^3}-\frac{7 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^2}-\frac{2 c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac{2 c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{-1+c x}\right ) \, dx\\ &=-\frac{d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}+\frac{1}{2} \left (b c d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x^4} \, dx+\left (2 b c^2 d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x^3} \, dx+\frac{1}{2} \left (7 b c^3 d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x^2} \, dx+\left (4 b c^4 d^3\right ) \int \frac{a+b \tanh ^{-1}(c x)}{x} \, dx-\left (4 b c^5 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 )}{6 x^3}-\frac{b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}-\frac{7 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x}-\frac{d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}+4 a b c^4 d^3 \log (x)+4 b c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )-2 b^2 c^4 d^3 \text{Li}_2(-c x)+2 b^2 c^4 d^3 \text{Li}_2(c x)+\frac{1}{6} \left (b^2 c^2 d^3\right ) \int \frac{1}{x^3 \left (1-c^2 x^2\right )} \, dx+\left (b^2 c^3 d^3\right ) \int \frac{1}{x^2 \left (1-c^2 x^2\right )} \, dx+\frac{1}{2} \left (7 b^2 c^4 d^3\right ) \int \frac{1}{x \left (1-c^2 x^2\right )} \, dx-\left (4 b^2 c^5 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}{x}-\frac{b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^3}-\frac{b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}-\frac{7 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x}-\frac{d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}+4 a b c^4 d^3 \log (x)+4 b c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )-2 b^2 c^4 d^3 \text{Li}_2(-c x)+2 b^2 c^4 d^3 \text{Li}_2(c x)+\frac{1}{12} \left (b^2 c^2 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-c^2 x\right )} \, dx,x,x^2\right )+\frac{1}{4} \left (7 b^2 c^4 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )+\left (4 b^2 c^4 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^5 d^3\right ) \int \frac{1}{1-c^2 x^2} \, dx\\ &=-\frac{b^2 c^3 d^3}{x}+b^2 c^4 d^3 \tanh ^{-1}(c x)-\frac{b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^3}-\frac{b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}-\frac{7 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x}-\frac{d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}+4 a b c^4 d^3 \log (x)+4 b c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )-2 b^2 c^4 d^3 \text{Li}_2(-c x)+2 b^2 c^4 d^3 \text{Li}_2(c x)+2 b^2 c^4 d^3 \text{Li}_2\left (1-\frac{2}{1-c x}\right )+\frac{1}{12} \left (b^2 c^2 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}{4} \left (7 b^2 c^4 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )+\frac{1}{4} \left (7 b^2 c^6 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}{12 x^2}-\frac{b^2 c^3 d^3}{x}+b^2 c^4 d^3 \tanh ^{-1}(c x)-\frac{b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^3}-\frac{b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x^2}-\frac{7 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{2 x}-\frac{d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 x^4}+4 a b c^4 d^3 \log (x)+\frac{11}{3} b^2 c^4 d^3 \log (x)+4 b c^4 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )-\frac{11}{6} b^2 c^4 d^3 \log \left (1-c^2 x^2\right )-2 b^2 c^4 d^3 \text{Li}_2(-c x)+2 b^2 c^4 d^3 \text{Li}_2(c x)+2 b^2 c^4 d^3 \text{Li}_2\left (1-\frac{2}{1-c x}\right )\\ \end{align*}
Mathematica [A] time = 0.771332, size = 343, normalized size = 1.27 \[ -\frac{d^3 \left (24 b^2 c^4 x^4 \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )+12 a^2 c^3 x^3+18 a^2 c^2 x^2+12 a^2 c x+3 a^2+42 a b c^3 x^3+12 a b c^2 x^2-48 a b c^4 x^4 \log (c x)+21 a b c^4 x^4 \log (1-c x)-21 a b c^4 x^4 \log (c x+1)+24 a b c^4 x^4 \log \left (1-c^2 x^2\right )+2 b \tanh ^{-1}(c x) \left (3 a \left (4 c^3 x^3+6 c^2 x^2+4 c x+1\right )+b c x \left (-6 c^3 x^3+21 c^2 x^2+6 c x+1\right )-24 b c^4 x^4 \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )\right )+2 a b c x-b^2 c^4 x^4+12 b^2 c^3 x^3+b^2 c^2 x^2-44 b^2 c^4 x^4 \log \left (\frac{c x}{\sqrt{1-c^2 x^2}}\right )+3 b^2 \left (-15 c^4 x^4+4 c^3 x^3+6 c^2 x^2+4 c x+1\right ) \tanh ^{-1}(c x)^2\right )}{12 x^4} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.075, size = 646, normalized size = 2.4 \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 [B] time = 3.10516, size = 1098, normalized size = 4.05 \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^{5}}, 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^{5}}\, dx + \int \frac{3 a^{2} c}{x^{4}}\, dx + \int \frac{3 a^{2} c^{2}}{x^{3}}\, dx + \int \frac{a^{2} c^{3}}{x^{2}}\, dx + \int \frac{b^{2} \operatorname{atanh}^{2}{\left (c x \right )}}{x^{5}}\, dx + \int \frac{2 a b \operatorname{atanh}{\left (c x \right )}}{x^{5}}\, dx + \int \frac{3 b^{2} c \operatorname{atanh}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int \frac{3 b^{2} c^{2} \operatorname{atanh}^{2}{\left (c x \right )}}{x^{3}}\, dx + \int \frac{b^{2} c^{3} \operatorname{atanh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int \frac{6 a b c \operatorname{atanh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac{6 a b c^{2} \operatorname{atanh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac{2 a b c^{3} \operatorname{atanh}{\left (c x \right )}}{x^{2}}\, 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^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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