Optimal. Leaf size=224 \[ \frac{1}{8} c^4 d^4 x^8 \left (a+b \tanh ^{-1}(c x)\right )+\frac{4}{7} c^3 d^4 x^7 \left (a+b \tanh ^{-1}(c x)\right )+c^2 d^4 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac{4}{5} c d^4 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{4} d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{56} b c^3 d^4 x^7+\frac{2}{21} b c^2 d^4 x^6+\frac{24 b d^4 x^2}{35 c^2}+\frac{11 b d^4 x}{8 c^3}+\frac{769 b d^4 \log (1-c x)}{560 c^4}-\frac{b d^4 \log (c x+1)}{560 c^4}+\frac{9}{40} b c d^4 x^5+\frac{11 b d^4 x^3}{24 c}+\frac{12}{35} b d^4 x^4 \]
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Rubi [A] time = 0.213476, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {43, 5936, 12, 1802, 633, 31} \[ \frac{1}{8} c^4 d^4 x^8 \left (a+b \tanh ^{-1}(c x)\right )+\frac{4}{7} c^3 d^4 x^7 \left (a+b \tanh ^{-1}(c x)\right )+c^2 d^4 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac{4}{5} c d^4 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{4} d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{56} b c^3 d^4 x^7+\frac{2}{21} b c^2 d^4 x^6+\frac{24 b d^4 x^2}{35 c^2}+\frac{11 b d^4 x}{8 c^3}+\frac{769 b d^4 \log (1-c x)}{560 c^4}-\frac{b d^4 \log (c x+1)}{560 c^4}+\frac{9}{40} b c d^4 x^5+\frac{11 b d^4 x^3}{24 c}+\frac{12}{35} b d^4 x^4 \]
Antiderivative was successfully verified.
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Rule 43
Rule 5936
Rule 12
Rule 1802
Rule 633
Rule 31
Rubi steps
\begin{align*} \int x^3 (d+c d x)^4 \left (a+b \tanh ^{-1}(c x)\right ) \, dx &=\frac{1}{4} d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{4}{5} c d^4 x^5 \left (a+b \tanh ^{-1}(c x)\right )+c^2 d^4 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac{4}{7} c^3 d^4 x^7 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{8} c^4 d^4 x^8 \left (a+b \tanh ^{-1}(c x)\right )-(b c) \int \frac{d^4 x^4 \left (70+224 c x+280 c^2 x^2+160 c^3 x^3+35 c^4 x^4\right )}{280 \left (1-c^2 x^2\right )} \, dx\\ &=\frac{1}{4} d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{4}{5} c d^4 x^5 \left (a+b \tanh ^{-1}(c x)\right )+c^2 d^4 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac{4}{7} c^3 d^4 x^7 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{8} c^4 d^4 x^8 \left (a+b \tanh ^{-1}(c x)\right )-\frac{1}{280} \left (b c d^4\right ) \int \frac{x^4 \left (70+224 c x+280 c^2 x^2+160 c^3 x^3+35 c^4 x^4\right )}{1-c^2 x^2} \, dx\\ &=\frac{1}{4} d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{4}{5} c d^4 x^5 \left (a+b \tanh ^{-1}(c x)\right )+c^2 d^4 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac{4}{7} c^3 d^4 x^7 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{8} c^4 d^4 x^8 \left (a+b \tanh ^{-1}(c x)\right )-\frac{1}{280} \left (b c d^4\right ) \int \left (-\frac{385}{c^4}-\frac{384 x}{c^3}-\frac{385 x^2}{c^2}-\frac{384 x^3}{c}-315 x^4-160 c x^5-35 c^2 x^6+\frac{385+384 c x}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx\\ &=\frac{11 b d^4 x}{8 c^3}+\frac{24 b d^4 x^2}{35 c^2}+\frac{11 b d^4 x^3}{24 c}+\frac{12}{35} b d^4 x^4+\frac{9}{40} b c d^4 x^5+\frac{2}{21} b c^2 d^4 x^6+\frac{1}{56} b c^3 d^4 x^7+\frac{1}{4} d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{4}{5} c d^4 x^5 \left (a+b \tanh ^{-1}(c x)\right )+c^2 d^4 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac{4}{7} c^3 d^4 x^7 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{8} c^4 d^4 x^8 \left (a+b \tanh ^{-1}(c x)\right )-\frac{\left (b d^4\right ) \int \frac{385+384 c x}{1-c^2 x^2} \, dx}{280 c^3}\\ &=\frac{11 b d^4 x}{8 c^3}+\frac{24 b d^4 x^2}{35 c^2}+\frac{11 b d^4 x^3}{24 c}+\frac{12}{35} b d^4 x^4+\frac{9}{40} b c d^4 x^5+\frac{2}{21} b c^2 d^4 x^6+\frac{1}{56} b c^3 d^4 x^7+\frac{1}{4} d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{4}{5} c d^4 x^5 \left (a+b \tanh ^{-1}(c x)\right )+c^2 d^4 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac{4}{7} c^3 d^4 x^7 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{8} c^4 d^4 x^8 \left (a+b \tanh ^{-1}(c x)\right )+\frac{\left (b d^4\right ) \int \frac{1}{-c-c^2 x} \, dx}{560 c^2}-\frac{\left (769 b d^4\right ) \int \frac{1}{c-c^2 x} \, dx}{560 c^2}\\ &=\frac{11 b d^4 x}{8 c^3}+\frac{24 b d^4 x^2}{35 c^2}+\frac{11 b d^4 x^3}{24 c}+\frac{12}{35} b d^4 x^4+\frac{9}{40} b c d^4 x^5+\frac{2}{21} b c^2 d^4 x^6+\frac{1}{56} b c^3 d^4 x^7+\frac{1}{4} d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{4}{5} c d^4 x^5 \left (a+b \tanh ^{-1}(c x)\right )+c^2 d^4 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac{4}{7} c^3 d^4 x^7 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{8} c^4 d^4 x^8 \left (a+b \tanh ^{-1}(c x)\right )+\frac{769 b d^4 \log (1-c x)}{560 c^4}-\frac{b d^4 \log (1+c x)}{560 c^4}\\ \end{align*}
Mathematica [A] time = 0.157504, size = 177, normalized size = 0.79 \[ \frac{d^4 \left (210 a c^8 x^8+960 a c^7 x^7+1680 a c^6 x^6+1344 a c^5 x^5+420 a c^4 x^4+30 b c^7 x^7+160 b c^6 x^6+378 b c^5 x^5+576 b c^4 x^4+770 b c^3 x^3+1152 b c^2 x^2+6 b c^4 x^4 \left (35 c^4 x^4+160 c^3 x^3+280 c^2 x^2+224 c x+70\right ) \tanh ^{-1}(c x)+2310 b c x+2307 b \log (1-c x)-3 b \log (c x+1)\right )}{1680 c^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 237, normalized size = 1.1 \begin{align*}{\frac{{c}^{4}{d}^{4}a{x}^{8}}{8}}+{\frac{4\,{c}^{3}{d}^{4}a{x}^{7}}{7}}+{c}^{2}{d}^{4}a{x}^{6}+{\frac{4\,c{d}^{4}a{x}^{5}}{5}}+{\frac{{d}^{4}a{x}^{4}}{4}}+{\frac{{c}^{4}{d}^{4}b{\it Artanh} \left ( cx \right ){x}^{8}}{8}}+{\frac{4\,{c}^{3}{d}^{4}b{\it Artanh} \left ( cx \right ){x}^{7}}{7}}+{c}^{2}{d}^{4}b{\it Artanh} \left ( cx \right ){x}^{6}+{\frac{4\,c{d}^{4}b{\it Artanh} \left ( cx \right ){x}^{5}}{5}}+{\frac{{d}^{4}b{\it Artanh} \left ( cx \right ){x}^{4}}{4}}+{\frac{b{c}^{3}{d}^{4}{x}^{7}}{56}}+{\frac{2\,b{c}^{2}{d}^{4}{x}^{6}}{21}}+{\frac{9\,bc{d}^{4}{x}^{5}}{40}}+{\frac{12\,b{d}^{4}{x}^{4}}{35}}+{\frac{11\,b{d}^{4}{x}^{3}}{24\,c}}+{\frac{24\,{d}^{4}b{x}^{2}}{35\,{c}^{2}}}+{\frac{11\,b{d}^{4}x}{8\,{c}^{3}}}+{\frac{769\,{d}^{4}b\ln \left ( cx-1 \right ) }{560\,{c}^{4}}}-{\frac{{d}^{4}b\ln \left ( cx+1 \right ) }{560\,{c}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.971307, size = 504, normalized size = 2.25 \begin{align*} \frac{1}{8} \, a c^{4} d^{4} x^{8} + \frac{4}{7} \, a c^{3} d^{4} x^{7} + a c^{2} d^{4} x^{6} + \frac{4}{5} \, a c d^{4} x^{5} + \frac{1}{1680} \,{\left (210 \, x^{8} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{2 \,{\left (15 \, c^{6} x^{7} + 21 \, c^{4} x^{5} + 35 \, c^{2} x^{3} + 105 \, x\right )}}{c^{8}} - \frac{105 \, \log \left (c x + 1\right )}{c^{9}} + \frac{105 \, \log \left (c x - 1\right )}{c^{9}}\right )}\right )} b c^{4} d^{4} + \frac{1}{21} \,{\left (12 \, x^{7} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{2 \, c^{4} x^{6} + 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} + \frac{6 \, \log \left (c^{2} x^{2} - 1\right )}{c^{8}}\right )}\right )} b c^{3} d^{4} + \frac{1}{4} \, a d^{4} x^{4} + \frac{1}{30} \,{\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 )} b c^{2} d^{4} + \frac{1}{5} \,{\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 d^{4} + \frac{1}{24} \,{\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 d^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.24534, size = 521, normalized size = 2.33 \begin{align*} \frac{210 \, a c^{8} d^{4} x^{8} + 30 \,{\left (32 \, a + b\right )} c^{7} d^{4} x^{7} + 80 \,{\left (21 \, a + 2 \, b\right )} c^{6} d^{4} x^{6} + 42 \,{\left (32 \, a + 9 \, b\right )} c^{5} d^{4} x^{5} + 12 \,{\left (35 \, a + 48 \, b\right )} c^{4} d^{4} x^{4} + 770 \, b c^{3} d^{4} x^{3} + 1152 \, b c^{2} d^{4} x^{2} + 2310 \, b c d^{4} x - 3 \, b d^{4} \log \left (c x + 1\right ) + 2307 \, b d^{4} \log \left (c x - 1\right ) + 3 \,{\left (35 \, b c^{8} d^{4} x^{8} + 160 \, b c^{7} d^{4} x^{7} + 280 \, b c^{6} d^{4} x^{6} + 224 \, b c^{5} d^{4} x^{5} + 70 \, b c^{4} d^{4} x^{4}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{1680 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.01342, size = 294, normalized size = 1.31 \begin{align*} \begin{cases} \frac{a c^{4} d^{4} x^{8}}{8} + \frac{4 a c^{3} d^{4} x^{7}}{7} + a c^{2} d^{4} x^{6} + \frac{4 a c d^{4} x^{5}}{5} + \frac{a d^{4} x^{4}}{4} + \frac{b c^{4} d^{4} x^{8} \operatorname{atanh}{\left (c x \right )}}{8} + \frac{4 b c^{3} d^{4} x^{7} \operatorname{atanh}{\left (c x \right )}}{7} + \frac{b c^{3} d^{4} x^{7}}{56} + b c^{2} d^{4} x^{6} \operatorname{atanh}{\left (c x \right )} + \frac{2 b c^{2} d^{4} x^{6}}{21} + \frac{4 b c d^{4} x^{5} \operatorname{atanh}{\left (c x \right )}}{5} + \frac{9 b c d^{4} x^{5}}{40} + \frac{b d^{4} x^{4} \operatorname{atanh}{\left (c x \right )}}{4} + \frac{12 b d^{4} x^{4}}{35} + \frac{11 b d^{4} x^{3}}{24 c} + \frac{24 b d^{4} x^{2}}{35 c^{2}} + \frac{11 b d^{4} x}{8 c^{3}} + \frac{48 b d^{4} \log{\left (x - \frac{1}{c} \right )}}{35 c^{4}} - \frac{b d^{4} \operatorname{atanh}{\left (c x \right )}}{280 c^{4}} & \text{for}\: c \neq 0 \\\frac{a d^{4} x^{4}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22285, size = 317, normalized size = 1.42 \begin{align*} \frac{1}{8} \, a c^{4} d^{4} x^{8} + \frac{1}{56} \,{\left (32 \, a c^{3} d^{4} + b c^{3} d^{4}\right )} x^{7} + \frac{11 \, b d^{4} x^{3}}{24 \, c} + \frac{1}{21} \,{\left (21 \, a c^{2} d^{4} + 2 \, b c^{2} d^{4}\right )} x^{6} + \frac{1}{40} \,{\left (32 \, a c d^{4} + 9 \, b c d^{4}\right )} x^{5} + \frac{24 \, b d^{4} x^{2}}{35 \, c^{2}} + \frac{1}{140} \,{\left (35 \, a d^{4} + 48 \, b d^{4}\right )} x^{4} + \frac{11 \, b d^{4} x}{8 \, c^{3}} - \frac{b d^{4} \log \left (c x + 1\right )}{560 \, c^{4}} + \frac{769 \, b d^{4} \log \left (c x - 1\right )}{560 \, c^{4}} + \frac{1}{560} \,{\left (35 \, b c^{4} d^{4} x^{8} + 160 \, b c^{3} d^{4} x^{7} + 280 \, b c^{2} d^{4} x^{6} + 224 \, b c d^{4} x^{5} + 70 \, b d^{4} x^{4}\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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