Optimal. Leaf size=157 \[ \frac {1}{6} c^2 d^2 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac {2}{5} c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{4} d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {49 b d^2 \log (1-c x)}{120 c^4}-\frac {b d^2 \log (c x+1)}{120 c^4}+\frac {5 b d^2 x}{12 c^3}+\frac {b d^2 x^2}{5 c^2}+\frac {1}{30} b c d^2 x^5+\frac {5 b d^2 x^3}{36 c}+\frac {1}{10} b d^2 x^4 \]
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Rubi [A] time = 0.17, antiderivative size = 157, 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} \[ \frac {1}{6} c^2 d^2 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac {2}{5} c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{4} d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {b d^2 x^2}{5 c^2}+\frac {5 b d^2 x}{12 c^3}+\frac {49 b d^2 \log (1-c x)}{120 c^4}-\frac {b d^2 \log (c x+1)}{120 c^4}+\frac {1}{30} b c d^2 x^5+\frac {5 b d^2 x^3}{36 c}+\frac {1}{10} b d^2 x^4 \]
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)^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx &=\frac {1}{4} d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {2}{5} c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{6} c^2 d^2 x^6 \left (a+b \tanh ^{-1}(c x)\right )-(b c) \int \frac {d^2 x^4 \left (15+24 c x+10 c^2 x^2\right )}{60 \left (1-c^2 x^2\right )} \, dx\\ &=\frac {1}{4} d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {2}{5} c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{6} c^2 d^2 x^6 \left (a+b \tanh ^{-1}(c x)\right )-\frac {1}{60} \left (b c d^2\right ) \int \frac {x^4 \left (15+24 c x+10 c^2 x^2\right )}{1-c^2 x^2} \, dx\\ &=\frac {1}{4} d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {2}{5} c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{6} c^2 d^2 x^6 \left (a+b \tanh ^{-1}(c x)\right )-\frac {1}{60} \left (b c d^2\right ) \int \left (-\frac {25}{c^4}-\frac {24 x}{c^3}-\frac {25 x^2}{c^2}-\frac {24 x^3}{c}-10 x^4+\frac {25+24 c x}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx\\ &=\frac {5 b d^2 x}{12 c^3}+\frac {b d^2 x^2}{5 c^2}+\frac {5 b d^2 x^3}{36 c}+\frac {1}{10} b d^2 x^4+\frac {1}{30} b c d^2 x^5+\frac {1}{4} d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {2}{5} c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{6} c^2 d^2 x^6 \left (a+b \tanh ^{-1}(c x)\right )-\frac {\left (b d^2\right ) \int \frac {25+24 c x}{1-c^2 x^2} \, dx}{60 c^3}\\ &=\frac {5 b d^2 x}{12 c^3}+\frac {b d^2 x^2}{5 c^2}+\frac {5 b d^2 x^3}{36 c}+\frac {1}{10} b d^2 x^4+\frac {1}{30} b c d^2 x^5+\frac {1}{4} d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {2}{5} c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{6} c^2 d^2 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac {\left (b d^2\right ) \int \frac {1}{-c-c^2 x} \, dx}{120 c^2}-\frac {\left (49 b d^2\right ) \int \frac {1}{c-c^2 x} \, dx}{120 c^2}\\ &=\frac {5 b d^2 x}{12 c^3}+\frac {b d^2 x^2}{5 c^2}+\frac {5 b d^2 x^3}{36 c}+\frac {1}{10} b d^2 x^4+\frac {1}{30} b c d^2 x^5+\frac {1}{4} d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {2}{5} c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{6} c^2 d^2 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac {49 b d^2 \log (1-c x)}{120 c^4}-\frac {b d^2 \log (1+c x)}{120 c^4}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 125, normalized size = 0.80 \[ \frac {d^2 \left (60 a c^6 x^6+144 a c^5 x^5+90 a c^4 x^4+12 b c^5 x^5+36 b c^4 x^4+50 b c^3 x^3+72 b c^2 x^2+6 b c^4 x^4 \left (10 c^2 x^2+24 c x+15\right ) \tanh ^{-1}(c x)+150 b c x+147 b \log (1-c x)-3 b \log (c x+1)\right )}{360 c^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 162, normalized size = 1.03 \[ \frac {60 \, a c^{6} d^{2} x^{6} + 12 \, {\left (12 \, a + b\right )} c^{5} d^{2} x^{5} + 18 \, {\left (5 \, a + 2 \, b\right )} c^{4} d^{2} x^{4} + 50 \, b c^{3} d^{2} x^{3} + 72 \, b c^{2} d^{2} x^{2} + 150 \, b c d^{2} x - 3 \, b d^{2} \log \left (c x + 1\right ) + 147 \, b d^{2} \log \left (c x - 1\right ) + 3 \, {\left (10 \, b c^{6} d^{2} x^{6} + 24 \, b c^{5} d^{2} x^{5} + 15 \, b c^{4} d^{2} x^{4}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{360 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 620, normalized size = 3.95 \[ \frac {1}{45} \, c {\left (\frac {6 \, {\left (\frac {30 \, {\left (c x + 1\right )}^{5} b d^{2}}{{\left (c x - 1\right )}^{5}} - \frac {30 \, {\left (c x + 1\right )}^{4} b d^{2}}{{\left (c x - 1\right )}^{4}} + \frac {70 \, {\left (c x + 1\right )}^{3} b d^{2}}{{\left (c x - 1\right )}^{3}} - \frac {45 \, {\left (c x + 1\right )}^{2} b d^{2}}{{\left (c x - 1\right )}^{2}} + \frac {18 \, {\left (c x + 1\right )} b d^{2}}{c x - 1} - 3 \, b d^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{\frac {{\left (c x + 1\right )}^{6} c^{5}}{{\left (c x - 1\right )}^{6}} - \frac {6 \, {\left (c x + 1\right )}^{5} c^{5}}{{\left (c x - 1\right )}^{5}} + \frac {15 \, {\left (c x + 1\right )}^{4} c^{5}}{{\left (c x - 1\right )}^{4}} - \frac {20 \, {\left (c x + 1\right )}^{3} c^{5}}{{\left (c x - 1\right )}^{3}} + \frac {15 \, {\left (c x + 1\right )}^{2} c^{5}}{{\left (c x - 1\right )}^{2}} - \frac {6 \, {\left (c x + 1\right )} c^{5}}{c x - 1} + c^{5}} + \frac {\frac {360 \, {\left (c x + 1\right )}^{5} a d^{2}}{{\left (c x - 1\right )}^{5}} - \frac {360 \, {\left (c x + 1\right )}^{4} a d^{2}}{{\left (c x - 1\right )}^{4}} + \frac {840 \, {\left (c x + 1\right )}^{3} a d^{2}}{{\left (c x - 1\right )}^{3}} - \frac {540 \, {\left (c x + 1\right )}^{2} a d^{2}}{{\left (c x - 1\right )}^{2}} + \frac {216 \, {\left (c x + 1\right )} a d^{2}}{c x - 1} - 36 \, a d^{2} + \frac {162 \, {\left (c x + 1\right )}^{5} b d^{2}}{{\left (c x - 1\right )}^{5}} - \frac {531 \, {\left (c x + 1\right )}^{4} b d^{2}}{{\left (c x - 1\right )}^{4}} + \frac {818 \, {\left (c x + 1\right )}^{3} b d^{2}}{{\left (c x - 1\right )}^{3}} - \frac {696 \, {\left (c x + 1\right )}^{2} b d^{2}}{{\left (c x - 1\right )}^{2}} + \frac {300 \, {\left (c x + 1\right )} b d^{2}}{c x - 1} - 53 \, b d^{2}}{\frac {{\left (c x + 1\right )}^{6} c^{5}}{{\left (c x - 1\right )}^{6}} - \frac {6 \, {\left (c x + 1\right )}^{5} c^{5}}{{\left (c x - 1\right )}^{5}} + \frac {15 \, {\left (c x + 1\right )}^{4} c^{5}}{{\left (c x - 1\right )}^{4}} - \frac {20 \, {\left (c x + 1\right )}^{3} c^{5}}{{\left (c x - 1\right )}^{3}} + \frac {15 \, {\left (c x + 1\right )}^{2} c^{5}}{{\left (c x - 1\right )}^{2}} - \frac {6 \, {\left (c x + 1\right )} c^{5}}{c x - 1} + c^{5}} - \frac {18 \, b d^{2} \log \left (-\frac {c x + 1}{c x - 1} + 1\right )}{c^{5}} + \frac {18 \, b d^{2} \log \left (-\frac {c x + 1}{c x - 1}\right )}{c^{5}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 159, normalized size = 1.01 \[ \frac {c^{2} d^{2} a \,x^{6}}{6}+\frac {2 c \,d^{2} a \,x^{5}}{5}+\frac {a \,x^{4} d^{2}}{4}+\frac {c^{2} d^{2} b \arctanh \left (c x \right ) x^{6}}{6}+\frac {2 c \,d^{2} b \arctanh \left (c x \right ) x^{5}}{5}+\frac {d^{2} b \arctanh \left (c x \right ) x^{4}}{4}+\frac {b c \,d^{2} x^{5}}{30}+\frac {b \,d^{2} x^{4}}{10}+\frac {5 b \,d^{2} x^{3}}{36 c}+\frac {b \,d^{2} x^{2}}{5 c^{2}}+\frac {5 b \,d^{2} x}{12 c^{3}}+\frac {49 d^{2} b \ln \left (c x -1\right )}{120 c^{4}}-\frac {b \,d^{2} \ln \left (c x +1\right )}{120 c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 210, normalized size = 1.34 \[ \frac {1}{6} \, a c^{2} d^{2} x^{6} + \frac {2}{5} \, a c d^{2} x^{5} + \frac {1}{4} \, a d^{2} x^{4} + \frac {1}{180} \, {\left (30 \, x^{6} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac {15 \, \log \left (c x + 1\right )}{c^{7}} + \frac {15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} b c^{2} d^{2} + \frac {1}{10} \, {\left (4 \, x^{5} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} b c d^{2} + \frac {1}{24} \, {\left (6 \, x^{4} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac {3 \, \log \left (c x + 1\right )}{c^{5}} + \frac {3 \, \log \left (c x - 1\right )}{c^{5}}\right )}\right )} b d^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.04, size = 146, normalized size = 0.93 \[ \frac {\frac {b\,c^2\,d^2\,x^2}{5}-\frac {d^2\,\left (75\,b\,\mathrm {atanh}\left (c\,x\right )-36\,b\,\ln \left (c^2\,x^2-1\right )\right )}{180}+\frac {5\,b\,c^3\,d^2\,x^3}{36}+\frac {5\,b\,c\,d^2\,x}{12}}{c^4}+\frac {d^2\,\left (45\,a\,x^4+18\,b\,x^4+45\,b\,x^4\,\mathrm {atanh}\left (c\,x\right )\right )}{180}+\frac {c^2\,d^2\,\left (30\,a\,x^6+30\,b\,x^6\,\mathrm {atanh}\left (c\,x\right )\right )}{180}+\frac {c\,d^2\,\left (72\,a\,x^5+6\,b\,x^5+72\,b\,x^5\,\mathrm {atanh}\left (c\,x\right )\right )}{180} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.16, size = 196, normalized size = 1.25 \[ \begin {cases} \frac {a c^{2} d^{2} x^{6}}{6} + \frac {2 a c d^{2} x^{5}}{5} + \frac {a d^{2} x^{4}}{4} + \frac {b c^{2} d^{2} x^{6} \operatorname {atanh}{\left (c x \right )}}{6} + \frac {2 b c d^{2} x^{5} \operatorname {atanh}{\left (c x \right )}}{5} + \frac {b c d^{2} x^{5}}{30} + \frac {b d^{2} x^{4} \operatorname {atanh}{\left (c x \right )}}{4} + \frac {b d^{2} x^{4}}{10} + \frac {5 b d^{2} x^{3}}{36 c} + \frac {b d^{2} x^{2}}{5 c^{2}} + \frac {5 b d^{2} x}{12 c^{3}} + \frac {2 b d^{2} \log {\left (x - \frac {1}{c} \right )}}{5 c^{4}} - \frac {b d^{2} \operatorname {atanh}{\left (c x \right )}}{60 c^{4}} & \text {for}\: c \neq 0 \\\frac {a d^{2} x^{4}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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