Optimal. Leaf size=176 \[ -\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{12 c (c x+1)}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{12 c (c x+1)^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{9 c (c x+1)^3}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{24 c}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{3 c (c x+1)^3}-\frac{11 b^2}{144 c (c x+1)}-\frac{5 b^2}{144 c (c x+1)^2}-\frac{b^2}{54 c (c x+1)^3}+\frac{11 b^2 \tanh ^{-1}(c x)}{144 c} \]
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Rubi [A] time = 0.217757, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5928, 5926, 627, 44, 207, 5948} \[ -\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{12 c (c x+1)}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{12 c (c x+1)^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{9 c (c x+1)^3}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{24 c}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{3 c (c x+1)^3}-\frac{11 b^2}{144 c (c x+1)}-\frac{5 b^2}{144 c (c x+1)^2}-\frac{b^2}{54 c (c x+1)^3}+\frac{11 b^2 \tanh ^{-1}(c x)}{144 c} \]
Antiderivative was successfully verified.
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Rule 5928
Rule 5926
Rule 627
Rule 44
Rule 207
Rule 5948
Rubi steps
\begin{align*} \int \frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{(1+c x)^4} \, dx &=-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{3 c (1+c x)^3}+\frac{1}{3} (2 b) \int \left (\frac{a+b \tanh ^{-1}(c x)}{2 (1+c x)^4}+\frac{a+b \tanh ^{-1}(c x)}{4 (1+c x)^3}+\frac{a+b \tanh ^{-1}(c x)}{8 (1+c x)^2}-\frac{a+b \tanh ^{-1}(c x)}{8 \left (-1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{3 c (1+c x)^3}+\frac{1}{12} b \int \frac{a+b \tanh ^{-1}(c x)}{(1+c x)^2} \, dx-\frac{1}{12} b \int \frac{a+b \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx+\frac{1}{6} b \int \frac{a+b \tanh ^{-1}(c x)}{(1+c x)^3} \, dx+\frac{1}{3} b \int \frac{a+b \tanh ^{-1}(c x)}{(1+c x)^4} \, dx\\ &=-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{9 c (1+c x)^3}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{12 c (1+c x)^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{12 c (1+c x)}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{24 c}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{3 c (1+c x)^3}+\frac{1}{12} b^2 \int \frac{1}{(1+c x)^2 \left (1-c^2 x^2\right )} \, dx+\frac{1}{12} b^2 \int \frac{1}{(1+c x) \left (1-c^2 x^2\right )} \, dx+\frac{1}{9} b^2 \int \frac{1}{(1+c x)^3 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{9 c (1+c x)^3}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{12 c (1+c x)^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{12 c (1+c x)}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{24 c}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{3 c (1+c x)^3}+\frac{1}{12} b^2 \int \frac{1}{(1-c x) (1+c x)^3} \, dx+\frac{1}{12} b^2 \int \frac{1}{(1-c x) (1+c x)^2} \, dx+\frac{1}{9} b^2 \int \frac{1}{(1-c x) (1+c x)^4} \, dx\\ &=-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{9 c (1+c x)^3}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{12 c (1+c x)^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{12 c (1+c x)}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{24 c}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{3 c (1+c x)^3}+\frac{1}{12} b^2 \int \left (\frac{1}{2 (1+c x)^2}-\frac{1}{2 \left (-1+c^2 x^2\right )}\right ) \, dx+\frac{1}{12} b^2 \int \left (\frac{1}{2 (1+c x)^3}+\frac{1}{4 (1+c x)^2}-\frac{1}{4 \left (-1+c^2 x^2\right )}\right ) \, dx+\frac{1}{9} b^2 \int \left (\frac{1}{2 (1+c x)^4}+\frac{1}{4 (1+c x)^3}+\frac{1}{8 (1+c x)^2}-\frac{1}{8 \left (-1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac{b^2}{54 c (1+c x)^3}-\frac{5 b^2}{144 c (1+c x)^2}-\frac{11 b^2}{144 c (1+c x)}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{9 c (1+c x)^3}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{12 c (1+c x)^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{12 c (1+c x)}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{24 c}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{3 c (1+c x)^3}-\frac{1}{72} b^2 \int \frac{1}{-1+c^2 x^2} \, dx-\frac{1}{48} b^2 \int \frac{1}{-1+c^2 x^2} \, dx-\frac{1}{24} b^2 \int \frac{1}{-1+c^2 x^2} \, dx\\ &=-\frac{b^2}{54 c (1+c x)^3}-\frac{5 b^2}{144 c (1+c x)^2}-\frac{11 b^2}{144 c (1+c x)}+\frac{11 b^2 \tanh ^{-1}(c x)}{144 c}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{9 c (1+c x)^3}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{12 c (1+c x)^2}-\frac{b \left (a+b \tanh ^{-1}(c x)\right )}{12 c (1+c x)}+\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{24 c}-\frac{\left (a+b \tanh ^{-1}(c x)\right )^2}{3 c (1+c x)^3}\\ \end{align*}
Mathematica [A] time = 0.173419, size = 168, normalized size = 0.95 \[ -\frac{16 \left (18 a^2+6 a b+b^2\right )+24 b \tanh ^{-1}(c x) \left (24 a+b \left (3 c^2 x^2+9 c x+10\right )\right )+6 b (12 a+11 b) (c x+1)^2+6 b (12 a+5 b) (c x+1)+3 b (12 a+11 b) (c x+1)^3 \log (1-c x)-3 b (12 a+11 b) (c x+1)^3 \log (c x+1)-36 b^2 \left (c^3 x^3+3 c^2 x^2+3 c x-7\right ) \tanh ^{-1}(c x)^2}{864 c (c x+1)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.062, size = 386, normalized size = 2.2 \begin{align*} -{\frac{{a}^{2}}{3\,c \left ( cx+1 \right ) ^{3}}}-{\frac{{b}^{2} \left ({\it Artanh} \left ( cx \right ) \right ) ^{2}}{3\,c \left ( cx+1 \right ) ^{3}}}-{\frac{{b}^{2}{\it Artanh} \left ( cx \right ) \ln \left ( cx-1 \right ) }{24\,c}}-{\frac{{b}^{2}{\it Artanh} \left ( cx \right ) }{9\,c \left ( cx+1 \right ) ^{3}}}-{\frac{{b}^{2}{\it Artanh} \left ( cx \right ) }{12\,c \left ( cx+1 \right ) ^{2}}}-{\frac{{b}^{2}{\it Artanh} \left ( cx \right ) }{12\,c \left ( cx+1 \right ) }}+{\frac{{b}^{2}{\it Artanh} \left ( cx \right ) \ln \left ( cx+1 \right ) }{24\,c}}-{\frac{{b}^{2} \left ( \ln \left ( cx-1 \right ) \right ) ^{2}}{96\,c}}+{\frac{{b}^{2}\ln \left ( cx-1 \right ) }{48\,c}\ln \left ({\frac{1}{2}}+{\frac{cx}{2}} \right ) }-{\frac{{b}^{2}}{48\,c}\ln \left ( -{\frac{cx}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{cx}{2}} \right ) }+{\frac{{b}^{2}\ln \left ( cx+1 \right ) }{48\,c}\ln \left ( -{\frac{cx}{2}}+{\frac{1}{2}} \right ) }-{\frac{{b}^{2} \left ( \ln \left ( cx+1 \right ) \right ) ^{2}}{96\,c}}-{\frac{11\,{b}^{2}\ln \left ( cx-1 \right ) }{288\,c}}-{\frac{{b}^{2}}{54\,c \left ( cx+1 \right ) ^{3}}}-{\frac{5\,{b}^{2}}{144\,c \left ( cx+1 \right ) ^{2}}}-{\frac{11\,{b}^{2}}{144\,c \left ( cx+1 \right ) }}+{\frac{11\,{b}^{2}\ln \left ( cx+1 \right ) }{288\,c}}-{\frac{2\,ab{\it Artanh} \left ( cx \right ) }{3\,c \left ( cx+1 \right ) ^{3}}}-{\frac{ab\ln \left ( cx-1 \right ) }{24\,c}}-{\frac{ab}{9\,c \left ( cx+1 \right ) ^{3}}}-{\frac{ab}{12\,c \left ( cx+1 \right ) ^{2}}}-{\frac{ab}{12\,c \left ( cx+1 \right ) }}+{\frac{ab\ln \left ( cx+1 \right ) }{24\,c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04532, size = 601, normalized size = 3.41 \begin{align*} -\frac{1}{72} \,{\left (c{\left (\frac{2 \,{\left (3 \, c^{2} x^{2} + 9 \, c x + 10\right )}}{c^{5} x^{3} + 3 \, c^{4} x^{2} + 3 \, c^{3} x + c^{2}} - \frac{3 \, \log \left (c x + 1\right )}{c^{2}} + \frac{3 \, \log \left (c x - 1\right )}{c^{2}}\right )} + \frac{48 \, \operatorname{artanh}\left (c x\right )}{c^{4} x^{3} + 3 \, c^{3} x^{2} + 3 \, c^{2} x + c}\right )} a b - \frac{1}{864} \,{\left (12 \, c{\left (\frac{2 \,{\left (3 \, c^{2} x^{2} + 9 \, c x + 10\right )}}{c^{5} x^{3} + 3 \, c^{4} x^{2} + 3 \, c^{3} x + c^{2}} - \frac{3 \, \log \left (c x + 1\right )}{c^{2}} + \frac{3 \, \log \left (c x - 1\right )}{c^{2}}\right )} \operatorname{artanh}\left (c x\right ) + \frac{{\left (66 \, c^{2} x^{2} + 9 \,{\left (c^{3} x^{3} + 3 \, c^{2} x^{2} + 3 \, c x + 1\right )} \log \left (c x + 1\right )^{2} + 9 \,{\left (c^{3} x^{3} + 3 \, c^{2} x^{2} + 3 \, c x + 1\right )} \log \left (c x - 1\right )^{2} + 162 \, c x - 3 \,{\left (11 \, c^{3} x^{3} + 33 \, c^{2} x^{2} + 33 \, c x + 6 \,{\left (c^{3} x^{3} + 3 \, c^{2} x^{2} + 3 \, c x + 1\right )} \log \left (c x - 1\right ) + 11\right )} \log \left (c x + 1\right ) + 33 \,{\left (c^{3} x^{3} + 3 \, c^{2} x^{2} + 3 \, c x + 1\right )} \log \left (c x - 1\right ) + 112\right )} c^{2}}{c^{6} x^{3} + 3 \, c^{5} x^{2} + 3 \, c^{4} x + c^{3}}\right )} b^{2} - \frac{b^{2} \operatorname{artanh}\left (c x\right )^{2}}{3 \,{\left (c^{4} x^{3} + 3 \, c^{3} x^{2} + 3 \, c^{2} x + c\right )}} - \frac{a^{2}}{3 \,{\left (c^{4} x^{3} + 3 \, c^{3} x^{2} + 3 \, c^{2} x + c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96812, size = 458, normalized size = 2.6 \begin{align*} -\frac{6 \,{\left (12 \, a b + 11 \, b^{2}\right )} c^{2} x^{2} + 54 \,{\left (4 \, a b + 3 \, b^{2}\right )} c x - 9 \,{\left (b^{2} c^{3} x^{3} + 3 \, b^{2} c^{2} x^{2} + 3 \, b^{2} c x - 7 \, b^{2}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )^{2} + 288 \, a^{2} + 240 \, a b + 112 \, b^{2} - 3 \,{\left ({\left (12 \, a b + 11 \, b^{2}\right )} c^{3} x^{3} + 3 \,{\left (12 \, a b + 7 \, b^{2}\right )} c^{2} x^{2} + 3 \,{\left (12 \, a b - b^{2}\right )} c x - 84 \, a b - 29 \, b^{2}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{864 \,{\left (c^{4} x^{3} + 3 \, c^{3} x^{2} + 3 \, c^{2} x + c\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*} \int \frac{\left (a + b \operatorname{atanh}{\left (c x \right )}\right )^{2}}{\left (c x + 1\right )^{4}}\, dx \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 x + 1\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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