Optimal. Leaf size=145 \[ \frac{b x^3 \left (a+b \tanh ^{-1}(c x)\right )}{9 c^3}+\frac{a b x}{3 c^5}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^6}+\frac{1}{6} x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{b x^5 \left (a+b \tanh ^{-1}(c x)\right )}{15 c}+\frac{b^2 x^4}{60 c^2}+\frac{4 b^2 x^2}{45 c^4}+\frac{23 b^2 \log \left (1-c^2 x^2\right )}{90 c^6}+\frac{b^2 x \tanh ^{-1}(c x)}{3 c^5} \]
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Rubi [A] time = 0.326833, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5916, 5980, 266, 43, 5910, 260, 5948} \[ \frac{b x^3 \left (a+b \tanh ^{-1}(c x)\right )}{9 c^3}+\frac{a b x}{3 c^5}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^6}+\frac{1}{6} x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{b x^5 \left (a+b \tanh ^{-1}(c x)\right )}{15 c}+\frac{b^2 x^4}{60 c^2}+\frac{4 b^2 x^2}{45 c^4}+\frac{23 b^2 \log \left (1-c^2 x^2\right )}{90 c^6}+\frac{b^2 x \tanh ^{-1}(c x)}{3 c^5} \]
Antiderivative was successfully verified.
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Rule 5916
Rule 5980
Rule 266
Rule 43
Rule 5910
Rule 260
Rule 5948
Rubi steps
\begin{align*} \int x^5 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx &=\frac{1}{6} x^6 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{1}{3} (b c) \int \frac{x^6 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac{1}{6} x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{b \int x^4 \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{3 c}-\frac{b \int \frac{x^4 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 c}\\ &=\frac{b x^5 \left (a+b \tanh ^{-1}(c x)\right )}{15 c}+\frac{1}{6} x^6 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{1}{15} b^2 \int \frac{x^5}{1-c^2 x^2} \, dx+\frac{b \int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{3 c^3}-\frac{b \int \frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 c^3}\\ &=\frac{b x^3 \left (a+b \tanh ^{-1}(c x)\right )}{9 c^3}+\frac{b x^5 \left (a+b \tanh ^{-1}(c x)\right )}{15 c}+\frac{1}{6} x^6 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{1}{30} b^2 \operatorname{Subst}\left (\int \frac{x^2}{1-c^2 x} \, dx,x,x^2\right )+\frac{b \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{3 c^5}-\frac{b \int \frac{a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{3 c^5}-\frac{b^2 \int \frac{x^3}{1-c^2 x^2} \, dx}{9 c^2}\\ &=\frac{a b x}{3 c^5}+\frac{b x^3 \left (a+b \tanh ^{-1}(c x)\right )}{9 c^3}+\frac{b x^5 \left (a+b \tanh ^{-1}(c x)\right )}{15 c}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^6}+\frac{1}{6} x^6 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{1}{30} b^2 \operatorname{Subst}\left (\int \left (-\frac{1}{c^4}-\frac{x}{c^2}-\frac{1}{c^4 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )+\frac{b^2 \int \tanh ^{-1}(c x) \, dx}{3 c^5}-\frac{b^2 \operatorname{Subst}\left (\int \frac{x}{1-c^2 x} \, dx,x,x^2\right )}{18 c^2}\\ &=\frac{a b x}{3 c^5}+\frac{b^2 x^2}{30 c^4}+\frac{b^2 x^4}{60 c^2}+\frac{b^2 x \tanh ^{-1}(c x)}{3 c^5}+\frac{b x^3 \left (a+b \tanh ^{-1}(c x)\right )}{9 c^3}+\frac{b x^5 \left (a+b \tanh ^{-1}(c x)\right )}{15 c}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^6}+\frac{1}{6} x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{b^2 \log \left (1-c^2 x^2\right )}{30 c^6}-\frac{b^2 \int \frac{x}{1-c^2 x^2} \, dx}{3 c^4}-\frac{b^2 \operatorname{Subst}\left (\int \left (-\frac{1}{c^2}-\frac{1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{18 c^2}\\ &=\frac{a b x}{3 c^5}+\frac{4 b^2 x^2}{45 c^4}+\frac{b^2 x^4}{60 c^2}+\frac{b^2 x \tanh ^{-1}(c x)}{3 c^5}+\frac{b x^3 \left (a+b \tanh ^{-1}(c x)\right )}{9 c^3}+\frac{b x^5 \left (a+b \tanh ^{-1}(c x)\right )}{15 c}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^6}+\frac{1}{6} x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{23 b^2 \log \left (1-c^2 x^2\right )}{90 c^6}\\ \end{align*}
Mathematica [A] time = 0.0678579, size = 164, normalized size = 1.13 \[ \frac{30 a^2 c^6 x^6+12 a b c^5 x^5+20 a b c^3 x^3+4 b c x \tanh ^{-1}(c x) \left (15 a c^5 x^5+b \left (3 c^4 x^4+5 c^2 x^2+15\right )\right )+60 a b c x+2 b (15 a+23 b) \log (1-c x)-30 a b \log (c x+1)+3 b^2 c^4 x^4+16 b^2 c^2 x^2+30 b^2 \left (c^6 x^6-1\right ) \tanh ^{-1}(c x)^2+46 b^2 \log (c x+1)}{180 c^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.023, size = 314, normalized size = 2.2 \begin{align*}{\frac{{x}^{6}{a}^{2}}{6}}+{\frac{{b}^{2}{x}^{6} \left ({\it Artanh} \left ( cx \right ) \right ) ^{2}}{6}}+{\frac{{b}^{2}{\it Artanh} \left ( cx \right ){x}^{5}}{15\,c}}+{\frac{{b}^{2}{\it Artanh} \left ( cx \right ){x}^{3}}{9\,{c}^{3}}}+{\frac{{b}^{2}x{\it Artanh} \left ( cx \right ) }{3\,{c}^{5}}}+{\frac{{b}^{2}{\it Artanh} \left ( cx \right ) \ln \left ( cx-1 \right ) }{6\,{c}^{6}}}-{\frac{{b}^{2}{\it Artanh} \left ( cx \right ) \ln \left ( cx+1 \right ) }{6\,{c}^{6}}}+{\frac{{b}^{2} \left ( \ln \left ( cx-1 \right ) \right ) ^{2}}{24\,{c}^{6}}}-{\frac{{b}^{2}\ln \left ( cx-1 \right ) }{12\,{c}^{6}}\ln \left ({\frac{1}{2}}+{\frac{cx}{2}} \right ) }-{\frac{{b}^{2}\ln \left ( cx+1 \right ) }{12\,{c}^{6}}\ln \left ( -{\frac{cx}{2}}+{\frac{1}{2}} \right ) }+{\frac{{b}^{2}}{12\,{c}^{6}}\ln \left ( -{\frac{cx}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{cx}{2}} \right ) }+{\frac{{b}^{2} \left ( \ln \left ( cx+1 \right ) \right ) ^{2}}{24\,{c}^{6}}}+{\frac{{b}^{2}{x}^{4}}{60\,{c}^{2}}}+{\frac{4\,{b}^{2}{x}^{2}}{45\,{c}^{4}}}+{\frac{23\,{b}^{2}\ln \left ( cx-1 \right ) }{90\,{c}^{6}}}+{\frac{23\,{b}^{2}\ln \left ( cx+1 \right ) }{90\,{c}^{6}}}+{\frac{ab{x}^{6}{\it Artanh} \left ( cx \right ) }{3}}+{\frac{ab{x}^{5}}{15\,c}}+{\frac{ab{x}^{3}}{9\,{c}^{3}}}+{\frac{xab}{3\,{c}^{5}}}+{\frac{ab\ln \left ( cx-1 \right ) }{6\,{c}^{6}}}-{\frac{ab\ln \left ( cx+1 \right ) }{6\,{c}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01312, size = 290, normalized size = 2. \begin{align*} \frac{1}{6} \, b^{2} x^{6} \operatorname{artanh}\left (c x\right )^{2} + \frac{1}{6} \, a^{2} x^{6} + \frac{1}{90} \,{\left (30 \, x^{6} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{2 \,{\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac{15 \, \log \left (c x + 1\right )}{c^{7}} + \frac{15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} a b + \frac{1}{360} \,{\left (4 \, c{\left (\frac{2 \,{\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac{15 \, \log \left (c x + 1\right )}{c^{7}} + \frac{15 \, \log \left (c x - 1\right )}{c^{7}}\right )} \operatorname{artanh}\left (c x\right ) + \frac{6 \, c^{4} x^{4} + 32 \, c^{2} x^{2} - 2 \,{\left (15 \, \log \left (c x - 1\right ) - 46\right )} \log \left (c x + 1\right ) + 15 \, \log \left (c x + 1\right )^{2} + 15 \, \log \left (c x - 1\right )^{2} + 92 \, \log \left (c x - 1\right )}{c^{6}}\right )} b^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.18829, size = 437, normalized size = 3.01 \begin{align*} \frac{60 \, a^{2} c^{6} x^{6} + 24 \, a b c^{5} x^{5} + 6 \, b^{2} c^{4} x^{4} + 40 \, a b c^{3} x^{3} + 32 \, b^{2} c^{2} x^{2} + 120 \, a b c x + 15 \,{\left (b^{2} c^{6} x^{6} - b^{2}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )^{2} - 4 \,{\left (15 \, a b - 23 \, b^{2}\right )} \log \left (c x + 1\right ) + 4 \,{\left (15 \, a b + 23 \, b^{2}\right )} \log \left (c x - 1\right ) + 4 \,{\left (15 \, a b c^{6} x^{6} + 3 \, b^{2} c^{5} x^{5} + 5 \, b^{2} c^{3} x^{3} + 15 \, b^{2} c x\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{360 \, c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.11455, size = 211, normalized size = 1.46 \begin{align*} \begin{cases} \frac{a^{2} x^{6}}{6} + \frac{a b x^{6} \operatorname{atanh}{\left (c x \right )}}{3} + \frac{a b x^{5}}{15 c} + \frac{a b x^{3}}{9 c^{3}} + \frac{a b x}{3 c^{5}} - \frac{a b \operatorname{atanh}{\left (c x \right )}}{3 c^{6}} + \frac{b^{2} x^{6} \operatorname{atanh}^{2}{\left (c x \right )}}{6} + \frac{b^{2} x^{5} \operatorname{atanh}{\left (c x \right )}}{15 c} + \frac{b^{2} x^{4}}{60 c^{2}} + \frac{b^{2} x^{3} \operatorname{atanh}{\left (c x \right )}}{9 c^{3}} + \frac{4 b^{2} x^{2}}{45 c^{4}} + \frac{b^{2} x \operatorname{atanh}{\left (c x \right )}}{3 c^{5}} + \frac{23 b^{2} \log{\left (x - \frac{1}{c} \right )}}{45 c^{6}} - \frac{b^{2} \operatorname{atanh}^{2}{\left (c x \right )}}{6 c^{6}} + \frac{23 b^{2} \operatorname{atanh}{\left (c x \right )}}{45 c^{6}} & \text{for}\: c \neq 0 \\\frac{a^{2} x^{6}}{6} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30206, size = 259, normalized size = 1.79 \begin{align*} \frac{1}{6} \, a^{2} x^{6} + \frac{a b x^{5}}{15 \, c} + \frac{b^{2} x^{4}}{60 \, c^{2}} + \frac{1}{24} \,{\left (b^{2} x^{6} - \frac{b^{2}}{c^{6}}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )^{2} + \frac{a b x^{3}}{9 \, c^{3}} + \frac{1}{90} \,{\left (15 \, a b x^{6} + \frac{3 \, b^{2} x^{5}}{c} + \frac{5 \, b^{2} x^{3}}{c^{3}} + \frac{15 \, b^{2} x}{c^{5}}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right ) + \frac{4 \, b^{2} x^{2}}{45 \, c^{4}} + \frac{a b x}{3 \, c^{5}} - \frac{{\left (15 \, a b - 23 \, b^{2}\right )} \log \left (c x + 1\right )}{90 \, c^{6}} + \frac{{\left (15 \, a b + 23 \, b^{2}\right )} \log \left (c x - 1\right )}{90 \, c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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