Optimal. Leaf size=107 \[ \frac{d^4 (c x+1)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 c}+\frac{b d^4 (c x+1)^4}{20 c}+\frac{2 b d^4 (c x+1)^3}{15 c}+\frac{2 b d^4 (c x+1)^2}{5 c}+\frac{16 b d^4 \log (1-c x)}{5 c}+\frac{8}{5} b d^4 x \]
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Rubi [A] time = 0.0548806, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {5926, 627, 43} \[ \frac{d^4 (c x+1)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 c}+\frac{b d^4 (c x+1)^4}{20 c}+\frac{2 b d^4 (c x+1)^3}{15 c}+\frac{2 b d^4 (c x+1)^2}{5 c}+\frac{16 b d^4 \log (1-c x)}{5 c}+\frac{8}{5} b d^4 x \]
Antiderivative was successfully verified.
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Rule 5926
Rule 627
Rule 43
Rubi steps
\begin{align*} \int (d+c d x)^4 \left (a+b \tanh ^{-1}(c x)\right ) \, dx &=\frac{d^4 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 c}-\frac{b \int \frac{(d+c d x)^5}{1-c^2 x^2} \, dx}{5 d}\\ &=\frac{d^4 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 c}-\frac{b \int \frac{(d+c d x)^4}{\frac{1}{d}-\frac{c x}{d}} \, dx}{5 d}\\ &=\frac{d^4 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 c}-\frac{b \int \left (-8 d^5+\frac{16 d^4}{\frac{1}{d}-\frac{c x}{d}}-4 d^4 (d+c d x)-2 d^3 (d+c d x)^2-d^2 (d+c d x)^3\right ) \, dx}{5 d}\\ &=\frac{8}{5} b d^4 x+\frac{2 b d^4 (1+c x)^2}{5 c}+\frac{2 b d^4 (1+c x)^3}{15 c}+\frac{b d^4 (1+c x)^4}{20 c}+\frac{d^4 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )}{5 c}+\frac{16 b d^4 \log (1-c x)}{5 c}\\ \end{align*}
Mathematica [A] time = 0.178214, size = 146, normalized size = 1.36 \[ \frac{d^4 \left (12 a c^5 x^5+60 a c^4 x^4+120 a c^3 x^3+120 a c^2 x^2+60 a c x+3 b c^4 x^4+20 b c^3 x^3+66 b c^2 x^2+6 b \log \left (1-c^2 x^2\right )+12 b c x \left (c^4 x^4+5 c^3 x^3+10 c^2 x^2+10 c x+5\right ) \tanh ^{-1}(c x)+180 b c x+180 b \log (1-c x)\right )}{60 c} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.026, size = 202, normalized size = 1.9 \begin{align*}{\frac{{c}^{4}{x}^{5}a{d}^{4}}{5}}+{c}^{3}{x}^{4}a{d}^{4}+2\,{c}^{2}{x}^{3}a{d}^{4}+2\,c{x}^{2}a{d}^{4}+xa{d}^{4}+{\frac{{d}^{4}a}{5\,c}}+{\frac{{c}^{4}{d}^{4}b{\it Artanh} \left ( cx \right ){x}^{5}}{5}}+{c}^{3}{d}^{4}b{\it Artanh} \left ( cx \right ){x}^{4}+2\,{c}^{2}{d}^{4}b{\it Artanh} \left ( cx \right ){x}^{3}+2\,c{d}^{4}b{\it Artanh} \left ( cx \right ){x}^{2}+{d}^{4}b{\it Artanh} \left ( cx \right ) x+{\frac{{d}^{4}b{\it Artanh} \left ( cx \right ) }{5\,c}}+{\frac{{c}^{3}{d}^{4}b{x}^{4}}{20}}+{\frac{{c}^{2}{d}^{4}b{x}^{3}}{3}}+{\frac{11\,c{d}^{4}b{x}^{2}}{10}}+3\,b{d}^{4}x+{\frac{16\,{d}^{4}b\ln \left ( cx-1 \right ) }{5\,c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.949735, size = 382, normalized size = 3.57 \begin{align*} \frac{1}{5} \, a c^{4} d^{4} x^{5} + a c^{3} d^{4} x^{4} + \frac{1}{20} \,{\left (4 \, x^{5} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac{2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} b c^{4} d^{4} + 2 \, a c^{2} d^{4} x^{3} + \frac{1}{6} \,{\left (6 \, x^{4} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{2 \,{\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac{3 \, \log \left (c x + 1\right )}{c^{5}} + \frac{3 \, \log \left (c x - 1\right )}{c^{5}}\right )}\right )} b c^{3} d^{4} +{\left (2 \, x^{3} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{x^{2}}{c^{2}} + \frac{\log \left (c^{2} x^{2} - 1\right )}{c^{4}}\right )}\right )} b c^{2} d^{4} + 2 \, a c d^{4} x^{2} +{\left (2 \, x^{2} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{2 \, x}{c^{2}} - \frac{\log \left (c x + 1\right )}{c^{3}} + \frac{\log \left (c x - 1\right )}{c^{3}}\right )}\right )} b c d^{4} + a d^{4} x + \frac{{\left (2 \, c x \operatorname{artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} b d^{4}}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.04621, size = 402, normalized size = 3.76 \begin{align*} \frac{12 \, a c^{5} d^{4} x^{5} + 3 \,{\left (20 \, a + b\right )} c^{4} d^{4} x^{4} + 20 \,{\left (6 \, a + b\right )} c^{3} d^{4} x^{3} + 6 \,{\left (20 \, a + 11 \, b\right )} c^{2} d^{4} x^{2} + 60 \,{\left (a + 3 \, b\right )} c d^{4} x + 6 \, b d^{4} \log \left (c x + 1\right ) + 186 \, b d^{4} \log \left (c x - 1\right ) + 6 \,{\left (b c^{5} d^{4} x^{5} + 5 \, b c^{4} d^{4} x^{4} + 10 \, b c^{3} d^{4} x^{3} + 10 \, b c^{2} d^{4} x^{2} + 5 \, b c d^{4} x\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{60 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.00716, size = 226, normalized size = 2.11 \begin{align*} \begin{cases} \frac{a c^{4} d^{4} x^{5}}{5} + a c^{3} d^{4} x^{4} + 2 a c^{2} d^{4} x^{3} + 2 a c d^{4} x^{2} + a d^{4} x + \frac{b c^{4} d^{4} x^{5} \operatorname{atanh}{\left (c x \right )}}{5} + b c^{3} d^{4} x^{4} \operatorname{atanh}{\left (c x \right )} + \frac{b c^{3} d^{4} x^{4}}{20} + 2 b c^{2} d^{4} x^{3} \operatorname{atanh}{\left (c x \right )} + \frac{b c^{2} d^{4} x^{3}}{3} + 2 b c d^{4} x^{2} \operatorname{atanh}{\left (c x \right )} + \frac{11 b c d^{4} x^{2}}{10} + b d^{4} x \operatorname{atanh}{\left (c x \right )} + 3 b d^{4} x + \frac{16 b d^{4} \log{\left (x - \frac{1}{c} \right )}}{5 c} + \frac{b d^{4} \operatorname{atanh}{\left (c x \right )}}{5 c} & \text{for}\: c \neq 0 \\a d^{4} x & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.28042, size = 261, normalized size = 2.44 \begin{align*} \frac{1}{5} \, a c^{4} d^{4} x^{5} + \frac{1}{20} \,{\left (20 \, a c^{3} d^{4} + b c^{3} d^{4}\right )} x^{4} + \frac{b d^{4} \log \left (c x + 1\right )}{10 \, c} + \frac{31 \, b d^{4} \log \left (c x - 1\right )}{10 \, c} + \frac{1}{3} \,{\left (6 \, a c^{2} d^{4} + b c^{2} d^{4}\right )} x^{3} + \frac{1}{10} \,{\left (20 \, a c d^{4} + 11 \, b c d^{4}\right )} x^{2} +{\left (a d^{4} + 3 \, b d^{4}\right )} x + \frac{1}{10} \,{\left (b c^{4} d^{4} x^{5} + 5 \, b c^{3} d^{4} x^{4} + 10 \, b c^{2} d^{4} x^{3} + 10 \, b c d^{4} x^{2} + 5 \, b d^{4} x\right )} \log \left (-\frac{c x + 1}{c x - 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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