Optimal. Leaf size=224 \[ \frac {1}{8} c^4 d^4 x^8 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4}{7} c^3 d^4 x^7 \left (a+b \tanh ^{-1}(c x)\right )+c^2 d^4 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4}{5} c d^4 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{4} d^4 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {769 b d^4 \log (1-c x)}{560 c^4}-\frac {b d^4 \log (c x+1)}{560 c^4}+\frac {1}{56} b c^3 d^4 x^7+\frac {11 b d^4 x}{8 c^3}+\frac {2}{21} b c^2 d^4 x^6+\frac {24 b d^4 x^2}{35 c^2}+\frac {9}{40} b c d^4 x^5+\frac {11 b d^4 x^3}{24 c}+\frac {12}{35} b d^4 x^4 \]
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Rubi [A] time = 0.21, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 12
Rule 31
Rule 43
Rule 633
Rule 1802
Rule 5936
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*}
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Mathematica [A] time = 0.16, 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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fricas [A] time = 0.55, size = 222, normalized size = 0.99 \[ \frac {210 \, a c^{8} d^{4} x^{8} + 30 \, {\left (32 \, a + b\right )} c^{7} d^{4} x^{7} + 80 \, {\left (21 \, a + 2 \, b\right )} c^{6} d^{4} x^{6} + 42 \, {\left (32 \, a + 9 \, b\right )} c^{5} d^{4} x^{5} + 12 \, {\left (35 \, a + 48 \, b\right )} c^{4} d^{4} x^{4} + 770 \, b c^{3} d^{4} x^{3} + 1152 \, b c^{2} d^{4} x^{2} + 2310 \, b c d^{4} x - 3 \, b d^{4} \log \left (c x + 1\right ) + 2307 \, b d^{4} \log \left (c x - 1\right ) + 3 \, {\left (35 \, b c^{8} d^{4} x^{8} + 160 \, b c^{7} d^{4} x^{7} + 280 \, b c^{6} d^{4} x^{6} + 224 \, b c^{5} d^{4} x^{5} + 70 \, b c^{4} d^{4} x^{4}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{1680 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.34, size = 817, normalized size = 3.65 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 237, normalized size = 1.06 \[ \frac {c^{4} d^{4} a \,x^{8}}{8}+\frac {4 c^{3} d^{4} a \,x^{7}}{7}+c^{2} d^{4} a \,x^{6}+\frac {4 c \,d^{4} a \,x^{5}}{5}+\frac {d^{4} a \,x^{4}}{4}+\frac {c^{4} d^{4} b \arctanh \left (c x \right ) x^{8}}{8}+\frac {4 c^{3} d^{4} b \arctanh \left (c x \right ) x^{7}}{7}+c^{2} d^{4} b \arctanh \left (c x \right ) x^{6}+\frac {4 c \,d^{4} b \arctanh \left (c x \right ) x^{5}}{5}+\frac {d^{4} b \arctanh \left (c x \right ) x^{4}}{4}+\frac {b \,c^{3} d^{4} x^{7}}{56}+\frac {2 b \,c^{2} d^{4} x^{6}}{21}+\frac {9 b c \,d^{4} x^{5}}{40}+\frac {12 b \,d^{4} x^{4}}{35}+\frac {11 b \,d^{4} x^{3}}{24 c}+\frac {24 b \,d^{4} x^{2}}{35 c^{2}}+\frac {11 b \,d^{4} x}{8 c^{3}}+\frac {769 d^{4} b \ln \left (c x -1\right )}{560 c^{4}}-\frac {b \,d^{4} \ln \left (c x +1\right )}{560 c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 373, normalized size = 1.67 \[ \frac {1}{8} \, a c^{4} d^{4} x^{8} + \frac {4}{7} \, a c^{3} d^{4} x^{7} + a c^{2} d^{4} x^{6} + \frac {4}{5} \, a c d^{4} x^{5} + \frac {1}{1680} \, {\left (210 \, x^{8} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (15 \, c^{6} x^{7} + 21 \, c^{4} x^{5} + 35 \, c^{2} x^{3} + 105 \, x\right )}}{c^{8}} - \frac {105 \, \log \left (c x + 1\right )}{c^{9}} + \frac {105 \, \log \left (c x - 1\right )}{c^{9}}\right )}\right )} b c^{4} d^{4} + \frac {1}{21} \, {\left (12 \, x^{7} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, c^{4} x^{6} + 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} + \frac {6 \, \log \left (c^{2} x^{2} - 1\right )}{c^{8}}\right )}\right )} b c^{3} d^{4} + \frac {1}{4} \, a d^{4} x^{4} + \frac {1}{30} \, {\left (30 \, x^{6} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac {15 \, \log \left (c x + 1\right )}{c^{7}} + \frac {15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} b c^{2} d^{4} + \frac {1}{5} \, {\left (4 \, x^{5} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} b c d^{4} + \frac {1}{24} \, {\left (6 \, x^{4} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac {3 \, \log \left (c x + 1\right )}{c^{5}} + \frac {3 \, \log \left (c x - 1\right )}{c^{5}}\right )}\right )} b d^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.84, size = 337, normalized size = 1.50 \[ \frac {a\,d^4\,x^4}{4}+\frac {12\,b\,d^4\,x^4}{35}+a\,c^2\,d^4\,x^6+\frac {4\,a\,c^3\,d^4\,x^7}{7}+\frac {a\,c^4\,d^4\,x^8}{8}+\frac {11\,b\,d^4\,x^3}{24\,c}+\frac {24\,b\,d^4\,x^2}{35\,c^2}+\frac {2\,b\,c^2\,d^4\,x^6}{21}+\frac {b\,c^3\,d^4\,x^7}{56}+\frac {769\,b\,d^4\,\ln \left (c\,x-1\right )}{560\,c^4}-\frac {b\,d^4\,\ln \left (c\,x+1\right )}{560\,c^4}+\frac {b\,d^4\,x^4\,\ln \left (c\,x+1\right )}{8}-\frac {b\,d^4\,x^4\,\ln \left (1-c\,x\right )}{8}+\frac {4\,a\,c\,d^4\,x^5}{5}+\frac {11\,b\,d^4\,x}{8\,c^3}+\frac {9\,b\,c\,d^4\,x^5}{40}+\frac {b\,c^2\,d^4\,x^6\,\ln \left (c\,x+1\right )}{2}-\frac {b\,c^2\,d^4\,x^6\,\ln \left (1-c\,x\right )}{2}+\frac {2\,b\,c^3\,d^4\,x^7\,\ln \left (c\,x+1\right )}{7}-\frac {2\,b\,c^3\,d^4\,x^7\,\ln \left (1-c\,x\right )}{7}+\frac {b\,c^4\,d^4\,x^8\,\ln \left (c\,x+1\right )}{16}-\frac {b\,c^4\,d^4\,x^8\,\ln \left (1-c\,x\right )}{16}+\frac {2\,b\,c\,d^4\,x^5\,\ln \left (c\,x+1\right )}{5}-\frac {2\,b\,c\,d^4\,x^5\,\ln \left (1-c\,x\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.76, size = 294, normalized size = 1.31 \[ \begin {cases} \frac {a c^{4} d^{4} x^{8}}{8} + \frac {4 a c^{3} d^{4} x^{7}}{7} + a c^{2} d^{4} x^{6} + \frac {4 a c d^{4} x^{5}}{5} + \frac {a d^{4} x^{4}}{4} + \frac {b c^{4} d^{4} x^{8} \operatorname {atanh}{\left (c x \right )}}{8} + \frac {4 b c^{3} d^{4} x^{7} \operatorname {atanh}{\left (c x \right )}}{7} + \frac {b c^{3} d^{4} x^{7}}{56} + b c^{2} d^{4} x^{6} \operatorname {atanh}{\left (c x \right )} + \frac {2 b c^{2} d^{4} x^{6}}{21} + \frac {4 b c d^{4} x^{5} \operatorname {atanh}{\left (c x \right )}}{5} + \frac {9 b c d^{4} x^{5}}{40} + \frac {b d^{4} x^{4} \operatorname {atanh}{\left (c x \right )}}{4} + \frac {12 b d^{4} x^{4}}{35} + \frac {11 b d^{4} x^{3}}{24 c} + \frac {24 b d^{4} x^{2}}{35 c^{2}} + \frac {11 b d^{4} x}{8 c^{3}} + \frac {48 b d^{4} \log {\left (x - \frac {1}{c} \right )}}{35 c^{4}} - \frac {b d^{4} \operatorname {atanh}{\left (c x \right )}}{280 c^{4}} & \text {for}\: c \neq 0 \\\frac {a d^{4} x^{4}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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