Optimal. Leaf size=88 \[ \frac{1}{6} c^2 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2-\frac{b c \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{3 x^3}-\frac{\left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2}{6 x^6}-\frac{1}{6} b^2 c^2 \log \left (1-c^2 x^6\right )+b^2 c^2 \log (x) \]
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Rubi [C] time = 1.06043, antiderivative size = 360, normalized size of antiderivative = 4.09, number of steps used = 46, number of rules used = 23, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.438, Rules used = {6099, 2454, 2398, 2411, 2347, 2344, 2301, 2316, 2315, 2314, 31, 2395, 44, 2439, 2416, 36, 29, 2392, 2391, 2394, 2393, 2410, 2390} \[ -\frac{1}{12} b^2 c^2 \text{PolyLog}\left (2,\frac{1}{2} \left (1-c x^3\right )\right )-\frac{1}{12} b^2 c^2 \text{PolyLog}\left (2,\frac{1}{2} \left (c x^3+1\right )\right )+\frac{1}{12} b c^2 \log \left (\frac{1}{2} \left (c x^3+1\right )\right ) \left (2 a-b \log \left (1-c x^3\right )\right )+\frac{1}{24} c^2 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac{b c \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{12 x^3}-\frac{b c \left (2 a-b \log \left (1-c x^3\right )\right )}{12 x^3}-\frac{b \log \left (c x^3+1\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{12 x^6}-\frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{24 x^6}+\frac{1}{24} b^2 c^2 \log ^2\left (c x^3+1\right )-\frac{1}{12} b^2 c^2 \log \left (1-c x^3\right )-\frac{1}{6} b^2 c^2 \log \left (c x^3+1\right )-\frac{1}{12} b^2 c^2 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (c x^3+1\right )+b^2 c^2 \log (x)-\frac{b^2 \log ^2\left (c x^3+1\right )}{24 x^6}-\frac{b^2 c \log \left (c x^3+1\right )}{6 x^3} \]
Warning: Unable to verify antiderivative.
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Rule 6099
Rule 2454
Rule 2398
Rule 2411
Rule 2347
Rule 2344
Rule 2301
Rule 2316
Rule 2315
Rule 2314
Rule 31
Rule 2395
Rule 44
Rule 2439
Rule 2416
Rule 36
Rule 29
Rule 2392
Rule 2391
Rule 2394
Rule 2393
Rule 2410
Rule 2390
Rubi steps
\begin{align*} \int \frac{\left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2}{x^7} \, dx &=\int \left (\frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{4 x^7}-\frac{b \left (-2 a+b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{2 x^7}+\frac{b^2 \log ^2\left (1+c x^3\right )}{4 x^7}\right ) \, dx\\ &=\frac{1}{4} \int \frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{x^7} \, dx-\frac{1}{2} b \int \frac{\left (-2 a+b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{x^7} \, dx+\frac{1}{4} b^2 \int \frac{\log ^2\left (1+c x^3\right )}{x^7} \, dx\\ &=\frac{1}{12} \operatorname{Subst}\left (\int \frac{(2 a-b \log (1-c x))^2}{x^3} \, dx,x,x^3\right )-\frac{1}{6} b \operatorname{Subst}\left (\int \frac{(-2 a+b \log (1-c x)) \log (1+c x)}{x^3} \, dx,x,x^3\right )+\frac{1}{12} b^2 \operatorname{Subst}\left (\int \frac{\log ^2(1+c x)}{x^3} \, dx,x,x^3\right )\\ &=-\frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{24 x^6}-\frac{b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{12 x^6}-\frac{b^2 \log ^2\left (1+c x^3\right )}{24 x^6}+\frac{1}{12} (b c) \operatorname{Subst}\left (\int \frac{2 a-b \log (1-c x)}{x^2 (1-c x)} \, dx,x,x^3\right )-\frac{1}{12} (b c) \operatorname{Subst}\left (\int \frac{-2 a+b \log (1-c x)}{x^2 (1+c x)} \, dx,x,x^3\right )+\frac{1}{12} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x^2 (1-c x)} \, dx,x,x^3\right )+\frac{1}{12} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x^2 (1+c x)} \, dx,x,x^3\right )\\ &=-\frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{24 x^6}-\frac{b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{12 x^6}-\frac{b^2 \log ^2\left (1+c x^3\right )}{24 x^6}-\frac{1}{12} b \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{x \left (\frac{1}{c}-\frac{x}{c}\right )^2} \, dx,x,1-c x^3\right )-\frac{1}{12} (b c) \operatorname{Subst}\left (\int \left (\frac{-2 a+b \log (1-c x)}{x^2}-\frac{c (-2 a+b \log (1-c x))}{x}+\frac{c^2 (-2 a+b \log (1-c x))}{1+c x}\right ) \, dx,x,x^3\right )+\frac{1}{12} \left (b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{\log (1+c x)}{x^2}+\frac{c \log (1+c x)}{x}-\frac{c^2 \log (1+c x)}{-1+c x}\right ) \, dx,x,x^3\right )+\frac{1}{12} \left (b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{\log (1+c x)}{x^2}-\frac{c \log (1+c x)}{x}+\frac{c^2 \log (1+c x)}{1+c x}\right ) \, dx,x,x^3\right )\\ &=-\frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{24 x^6}-\frac{b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{12 x^6}-\frac{b^2 \log ^2\left (1+c x^3\right )}{24 x^6}-\frac{1}{12} b \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{\left (\frac{1}{c}-\frac{x}{c}\right )^2} \, dx,x,1-c x^3\right )-\frac{1}{12} (b c) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{x \left (\frac{1}{c}-\frac{x}{c}\right )} \, dx,x,1-c x^3\right )-\frac{1}{12} (b c) \operatorname{Subst}\left (\int \frac{-2 a+b \log (1-c x)}{x^2} \, dx,x,x^3\right )+2 \left (\frac{1}{12} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x^2} \, dx,x,x^3\right )\right )+\frac{1}{12} \left (b c^2\right ) \operatorname{Subst}\left (\int \frac{-2 a+b \log (1-c x)}{x} \, dx,x,x^3\right )-\frac{1}{12} \left (b c^3\right ) \operatorname{Subst}\left (\int \frac{-2 a+b \log (1-c x)}{1+c x} \, dx,x,x^3\right )-\frac{1}{12} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{-1+c x} \, dx,x,x^3\right )+\frac{1}{12} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{1+c x} \, dx,x,x^3\right )\\ &=-\frac{1}{2} a b c^2 \log (x)-\frac{b c \left (2 a-b \log \left (1-c x^3\right )\right )}{12 x^3}-\frac{b c \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{12 x^3}-\frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{24 x^6}+\frac{1}{12} b c^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+c x^3\right )\right )-\frac{1}{12} b^2 c^2 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )-\frac{b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{12 x^6}-\frac{b^2 \log ^2\left (1+c x^3\right )}{24 x^6}-\frac{1}{12} (b c) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{\frac{1}{c}-\frac{x}{c}} \, dx,x,1-c x^3\right )-\frac{1}{12} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c}-\frac{x}{c}} \, dx,x,1-c x^3\right )-\frac{1}{12} \left (b c^2\right ) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{x} \, dx,x,1-c x^3\right )+\frac{1}{12} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x (1-c x)} \, dx,x,x^3\right )+2 \left (-\frac{b^2 c \log \left (1+c x^3\right )}{12 x^3}+\frac{1}{12} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x (1+c x)} \, dx,x,x^3\right )\right )+\frac{1}{12} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+c x^3\right )+\frac{1}{12} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-c x)}{x} \, dx,x,x^3\right )+\frac{1}{12} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{2} (1-c x)\right )}{1+c x} \, dx,x,x^3\right )-\frac{1}{12} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{2} (1+c x)\right )}{1-c x} \, dx,x,x^3\right )\\ &=\frac{1}{4} b^2 c^2 \log (x)-\frac{b c \left (2 a-b \log \left (1-c x^3\right )\right )}{12 x^3}-\frac{b c \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{12 x^3}+\frac{1}{24} c^2 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{24 x^6}+\frac{1}{12} b c^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+c x^3\right )\right )-\frac{1}{12} b^2 c^2 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )-\frac{b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{12 x^6}+\frac{1}{24} b^2 c^2 \log ^2\left (1+c x^3\right )-\frac{b^2 \log ^2\left (1+c x^3\right )}{24 x^6}-\frac{1}{12} b^2 c^2 \text{Li}_2\left (c x^3\right )+\frac{1}{12} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\frac{1}{c}-\frac{x}{c}} \, dx,x,1-c x^3\right )+\frac{1}{12} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^3\right )+\frac{1}{12} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1-c x^3\right )+\frac{1}{12} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1+c x^3\right )+\frac{1}{12} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x} \, dx,x,x^3\right )+2 \left (-\frac{b^2 c \log \left (1+c x^3\right )}{12 x^3}+\frac{1}{12} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^3\right )-\frac{1}{12} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+c x} \, dx,x,x^3\right )\right )\\ &=\frac{1}{2} b^2 c^2 \log (x)-\frac{1}{12} b^2 c^2 \log \left (1-c x^3\right )-\frac{b c \left (2 a-b \log \left (1-c x^3\right )\right )}{12 x^3}-\frac{b c \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{12 x^3}+\frac{1}{24} c^2 \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac{\left (2 a-b \log \left (1-c x^3\right )\right )^2}{24 x^6}+\frac{1}{12} b c^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+c x^3\right )\right )-\frac{1}{12} b^2 c^2 \log \left (\frac{1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )-\frac{b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{12 x^6}+\frac{1}{24} b^2 c^2 \log ^2\left (1+c x^3\right )-\frac{b^2 \log ^2\left (1+c x^3\right )}{24 x^6}+2 \left (\frac{1}{4} b^2 c^2 \log (x)-\frac{1}{12} b^2 c^2 \log \left (1+c x^3\right )-\frac{b^2 c \log \left (1+c x^3\right )}{12 x^3}\right )-\frac{1}{12} b^2 c^2 \text{Li}_2\left (\frac{1}{2} \left (1-c x^3\right )\right )-\frac{1}{12} b^2 c^2 \text{Li}_2\left (\frac{1}{2} \left (1+c x^3\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0831599, size = 111, normalized size = 1.26 \[ \frac{1}{6} \left (-\frac{a^2}{x^6}-b c^2 (a+b) \log \left (1-c x^3\right )+b c^2 (a-b) \log \left (c x^3+1\right )-\frac{2 a b c}{x^3}-\frac{2 b \tanh ^{-1}\left (c x^3\right ) \left (a+b c x^3\right )}{x^6}+\frac{b^2 \left (c^2 x^6-1\right ) \tanh ^{-1}\left (c x^3\right )^2}{x^6}+6 b^2 c^2 \log (x)\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.201, size = 257, normalized size = 2.9 \begin{align*}{\frac{{b}^{2} \left ({c}^{2}{x}^{6}-1 \right ) \left ( \ln \left ( c{x}^{3}+1 \right ) \right ) ^{2}}{24\,{x}^{6}}}-{\frac{b \left ({x}^{6}b\ln \left ( -c{x}^{3}+1 \right ){c}^{2}+2\,bc{x}^{3}-b\ln \left ( -c{x}^{3}+1 \right ) +2\,a \right ) \ln \left ( c{x}^{3}+1 \right ) }{12\,{x}^{6}}}-{\frac{-{b}^{2}{c}^{2}{x}^{6} \left ( \ln \left ( -c{x}^{3}+1 \right ) \right ) ^{2}+4\,b{c}^{2}\ln \left ( c{x}^{3}-1 \right ){x}^{6}a+4\,{b}^{2}{c}^{2}\ln \left ( c{x}^{3}-1 \right ){x}^{6}-4\,b{c}^{2}\ln \left ( c{x}^{3}+1 \right ){x}^{6}a+4\,{b}^{2}{c}^{2}\ln \left ( c{x}^{3}+1 \right ){x}^{6}-24\,{b}^{2}{c}^{2}\ln \left ( x \right ){x}^{6}-4\,{b}^{2}c{x}^{3}\ln \left ( -c{x}^{3}+1 \right ) +8\,abc{x}^{3}+{b}^{2} \left ( \ln \left ( -c{x}^{3}+1 \right ) \right ) ^{2}-4\,b\ln \left ( -c{x}^{3}+1 \right ) a+4\,{a}^{2}}{24\,{x}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.99658, size = 236, normalized size = 2.68 \begin{align*} \frac{1}{6} \,{\left ({\left (c \log \left (c x^{3} + 1\right ) - c \log \left (c x^{3} - 1\right ) - \frac{2}{x^{3}}\right )} c - \frac{2 \, \operatorname{artanh}\left (c x^{3}\right )}{x^{6}}\right )} a b + \frac{1}{24} \,{\left ({\left (2 \,{\left (\log \left (c x^{3} - 1\right ) - 2\right )} \log \left (c x^{3} + 1\right ) - \log \left (c x^{3} + 1\right )^{2} - \log \left (c x^{3} - 1\right )^{2} - 4 \, \log \left (c x^{3} - 1\right ) + 24 \, \log \left (x\right )\right )} c^{2} + 4 \,{\left (c \log \left (c x^{3} + 1\right ) - c \log \left (c x^{3} - 1\right ) - \frac{2}{x^{3}}\right )} c \operatorname{artanh}\left (c x^{3}\right )\right )} b^{2} - \frac{b^{2} \operatorname{artanh}\left (c x^{3}\right )^{2}}{6 \, x^{6}} - \frac{a^{2}}{6 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77071, size = 324, normalized size = 3.68 \begin{align*} \frac{24 \, b^{2} c^{2} x^{6} \log \left (x\right ) + 4 \,{\left (a b - b^{2}\right )} c^{2} x^{6} \log \left (c x^{3} + 1\right ) - 4 \,{\left (a b + b^{2}\right )} c^{2} x^{6} \log \left (c x^{3} - 1\right ) - 8 \, a b c x^{3} +{\left (b^{2} c^{2} x^{6} - b^{2}\right )} \log \left (-\frac{c x^{3} + 1}{c x^{3} - 1}\right )^{2} - 4 \, a^{2} - 4 \,{\left (b^{2} c x^{3} + a b\right )} \log \left (-\frac{c x^{3} + 1}{c x^{3} - 1}\right )}{24 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{3}\right ) + a\right )}^{2}}{x^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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