Optimal. Leaf size=178 \[ \frac {1}{6} c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac {3}{5} c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {3}{4} c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{3} d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {37 b d^3 \log (1-c x)}{40 c^3}+\frac {b d^3 \log (c x+1)}{120 c^3}+\frac {1}{30} b c^2 d^3 x^5+\frac {11 b d^3 x}{12 c^2}+\frac {3}{20} b c d^3 x^4+\frac {7 b d^3 x^2}{15 c}+\frac {11}{36} b d^3 x^3 \]
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Rubi [A] time = 0.18, antiderivative size = 178, 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} \[ \frac {1}{6} c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac {3}{5} c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {3}{4} c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{3} d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{30} b c^2 d^3 x^5+\frac {11 b d^3 x}{12 c^2}+\frac {37 b d^3 \log (1-c x)}{40 c^3}+\frac {b d^3 \log (c x+1)}{120 c^3}+\frac {3}{20} b c d^3 x^4+\frac {7 b d^3 x^2}{15 c}+\frac {11}{36} b d^3 x^3 \]
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^2 (d+c d x)^3 \left (a+b \tanh ^{-1}(c x)\right ) \, dx &=\frac {1}{3} d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {3}{4} c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {3}{5} c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{6} c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )-(b c) \int \frac {d^3 x^3 \left (20+45 c x+36 c^2 x^2+10 c^3 x^3\right )}{60 \left (1-c^2 x^2\right )} \, dx\\ &=\frac {1}{3} d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {3}{4} c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {3}{5} c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{6} c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )-\frac {1}{60} \left (b c d^3\right ) \int \frac {x^3 \left (20+45 c x+36 c^2 x^2+10 c^3 x^3\right )}{1-c^2 x^2} \, dx\\ &=\frac {1}{3} d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {3}{4} c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {3}{5} c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{6} c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )-\frac {1}{60} \left (b c d^3\right ) \int \left (-\frac {55}{c^3}-\frac {56 x}{c^2}-\frac {55 x^2}{c}-36 x^3-10 c x^4+\frac {55+56 c x}{c^3 \left (1-c^2 x^2\right )}\right ) \, dx\\ &=\frac {11 b d^3 x}{12 c^2}+\frac {7 b d^3 x^2}{15 c}+\frac {11}{36} b d^3 x^3+\frac {3}{20} b c d^3 x^4+\frac {1}{30} b c^2 d^3 x^5+\frac {1}{3} d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {3}{4} c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {3}{5} c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{6} c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )-\frac {\left (b d^3\right ) \int \frac {55+56 c x}{1-c^2 x^2} \, dx}{60 c^2}\\ &=\frac {11 b d^3 x}{12 c^2}+\frac {7 b d^3 x^2}{15 c}+\frac {11}{36} b d^3 x^3+\frac {3}{20} b c d^3 x^4+\frac {1}{30} b c^2 d^3 x^5+\frac {1}{3} d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {3}{4} c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {3}{5} c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{6} c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )-\frac {\left (b d^3\right ) \int \frac {1}{-c-c^2 x} \, dx}{120 c}-\frac {\left (37 b d^3\right ) \int \frac {1}{c-c^2 x} \, dx}{40 c}\\ &=\frac {11 b d^3 x}{12 c^2}+\frac {7 b d^3 x^2}{15 c}+\frac {11}{36} b d^3 x^3+\frac {3}{20} b c d^3 x^4+\frac {1}{30} b c^2 d^3 x^5+\frac {1}{3} d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {3}{4} c d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {3}{5} c^2 d^3 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{6} c^3 d^3 x^6 \left (a+b \tanh ^{-1}(c x)\right )+\frac {37 b d^3 \log (1-c x)}{40 c^3}+\frac {b d^3 \log (1+c x)}{120 c^3}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 142, normalized size = 0.80 \[ \frac {d^3 \left (60 a c^6 x^6+216 a c^5 x^5+270 a c^4 x^4+120 a c^3 x^3+12 b c^5 x^5+54 b c^4 x^4+110 b c^3 x^3+168 b c^2 x^2+6 b c^3 x^3 \left (10 c^3 x^3+36 c^2 x^2+45 c x+20\right ) \tanh ^{-1}(c x)+330 b c x+333 b \log (1-c x)+3 b \log (c x+1)\right )}{360 c^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 178, normalized size = 1.00 \[ \frac {60 \, a c^{6} d^{3} x^{6} + 12 \, {\left (18 \, a + b\right )} c^{5} d^{3} x^{5} + 54 \, {\left (5 \, a + b\right )} c^{4} d^{3} x^{4} + 10 \, {\left (12 \, a + 11 \, b\right )} c^{3} d^{3} x^{3} + 168 \, b c^{2} d^{3} x^{2} + 330 \, b c d^{3} x + 3 \, b d^{3} \log \left (c x + 1\right ) + 333 \, b d^{3} \log \left (c x - 1\right ) + 3 \, {\left (10 \, b c^{6} d^{3} x^{6} + 36 \, b c^{5} d^{3} x^{5} + 45 \, b c^{4} d^{3} x^{4} + 20 \, b c^{3} d^{3} x^{3}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{360 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 621, normalized size = 3.49 \[ -\frac {1}{45} \, c {\left (\frac {42 \, b d^{3} \log \left (-\frac {c x + 1}{c x - 1} + 1\right )}{c^{4}} - \frac {6 \, {\left (\frac {60 \, {\left (c x + 1\right )}^{5} b d^{3}}{{\left (c x - 1\right )}^{5}} - \frac {90 \, {\left (c x + 1\right )}^{4} b d^{3}}{{\left (c x - 1\right )}^{4}} + \frac {140 \, {\left (c x + 1\right )}^{3} b d^{3}}{{\left (c x - 1\right )}^{3}} - \frac {105 \, {\left (c x + 1\right )}^{2} b d^{3}}{{\left (c x - 1\right )}^{2}} + \frac {42 \, {\left (c x + 1\right )} b d^{3}}{c x - 1} - 7 \, b d^{3}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{\frac {{\left (c x + 1\right )}^{6} c^{4}}{{\left (c x - 1\right )}^{6}} - \frac {6 \, {\left (c x + 1\right )}^{5} c^{4}}{{\left (c x - 1\right )}^{5}} + \frac {15 \, {\left (c x + 1\right )}^{4} c^{4}}{{\left (c x - 1\right )}^{4}} - \frac {20 \, {\left (c x + 1\right )}^{3} c^{4}}{{\left (c x - 1\right )}^{3}} + \frac {15 \, {\left (c x + 1\right )}^{2} c^{4}}{{\left (c x - 1\right )}^{2}} - \frac {6 \, {\left (c x + 1\right )} c^{4}}{c x - 1} + c^{4}} - \frac {42 \, b d^{3} \log \left (-\frac {c x + 1}{c x - 1}\right )}{c^{4}} - \frac {\frac {720 \, {\left (c x + 1\right )}^{5} a d^{3}}{{\left (c x - 1\right )}^{5}} - \frac {1080 \, {\left (c x + 1\right )}^{4} a d^{3}}{{\left (c x - 1\right )}^{4}} + \frac {1680 \, {\left (c x + 1\right )}^{3} a d^{3}}{{\left (c x - 1\right )}^{3}} - \frac {1260 \, {\left (c x + 1\right )}^{2} a d^{3}}{{\left (c x - 1\right )}^{2}} + \frac {504 \, {\left (c x + 1\right )} a d^{3}}{c x - 1} - 84 \, a d^{3} + \frac {318 \, {\left (c x + 1\right )}^{5} b d^{3}}{{\left (c x - 1\right )}^{5}} - \frac {1119 \, {\left (c x + 1\right )}^{4} b d^{3}}{{\left (c x - 1\right )}^{4}} + \frac {1742 \, {\left (c x + 1\right )}^{3} b d^{3}}{{\left (c x - 1\right )}^{3}} - \frac {1464 \, {\left (c x + 1\right )}^{2} b d^{3}}{{\left (c x - 1\right )}^{2}} + \frac {636 \, {\left (c x + 1\right )} b d^{3}}{c x - 1} - 113 \, b d^{3}}{\frac {{\left (c x + 1\right )}^{6} c^{4}}{{\left (c x - 1\right )}^{6}} - \frac {6 \, {\left (c x + 1\right )}^{5} c^{4}}{{\left (c x - 1\right )}^{5}} + \frac {15 \, {\left (c x + 1\right )}^{4} c^{4}}{{\left (c x - 1\right )}^{4}} - \frac {20 \, {\left (c x + 1\right )}^{3} c^{4}}{{\left (c x - 1\right )}^{3}} + \frac {15 \, {\left (c x + 1\right )}^{2} c^{4}}{{\left (c x - 1\right )}^{2}} - \frac {6 \, {\left (c x + 1\right )} c^{4}}{c x - 1} + c^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 187, normalized size = 1.05 \[ \frac {c^{3} d^{3} a \,x^{6}}{6}+\frac {3 c^{2} d^{3} a \,x^{5}}{5}+\frac {3 c \,d^{3} a \,x^{4}}{4}+\frac {d^{3} a \,x^{3}}{3}+\frac {c^{3} d^{3} b \arctanh \left (c x \right ) x^{6}}{6}+\frac {3 c^{2} d^{3} b \arctanh \left (c x \right ) x^{5}}{5}+\frac {3 c \,d^{3} b \arctanh \left (c x \right ) x^{4}}{4}+\frac {d^{3} b \arctanh \left (c x \right ) x^{3}}{3}+\frac {b \,c^{2} d^{3} x^{5}}{30}+\frac {3 b c \,d^{3} x^{4}}{20}+\frac {11 b \,d^{3} x^{3}}{36}+\frac {7 b \,d^{3} x^{2}}{15 c}+\frac {11 b \,d^{3} x}{12 c^{2}}+\frac {37 d^{3} b \ln \left (c x -1\right )}{40 c^{3}}+\frac {b \,d^{3} \ln \left (c x +1\right )}{120 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 265, normalized size = 1.49 \[ \frac {1}{6} \, a c^{3} d^{3} x^{6} + \frac {3}{5} \, a c^{2} d^{3} x^{5} + \frac {3}{4} \, a c d^{3} x^{4} + \frac {1}{180} \, {\left (30 \, x^{6} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac {15 \, \log \left (c x + 1\right )}{c^{7}} + \frac {15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} b c^{3} d^{3} + \frac {3}{20} \, {\left (4 \, x^{5} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} b c^{2} d^{3} + \frac {1}{3} \, a d^{3} x^{3} + \frac {1}{8} \, {\left (6 \, x^{4} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac {3 \, \log \left (c x + 1\right )}{c^{5}} + \frac {3 \, \log \left (c x - 1\right )}{c^{5}}\right )}\right )} b c d^{3} + \frac {1}{6} \, {\left (2 \, x^{3} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {x^{2}}{c^{2}} + \frac {\log \left (c^{2} x^{2} - 1\right )}{c^{4}}\right )}\right )} b d^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.03, size = 165, normalized size = 0.93 \[ \frac {\frac {7\,b\,c^2\,d^3\,x^2}{15}-\frac {d^3\,\left (165\,b\,\mathrm {atanh}\left (c\,x\right )-84\,b\,\ln \left (c^2\,x^2-1\right )\right )}{180}+\frac {11\,b\,c\,d^3\,x}{12}}{c^3}+\frac {d^3\,\left (60\,a\,x^3+55\,b\,x^3+60\,b\,x^3\,\mathrm {atanh}\left (c\,x\right )\right )}{180}+\frac {c^3\,d^3\,\left (30\,a\,x^6+30\,b\,x^6\,\mathrm {atanh}\left (c\,x\right )\right )}{180}+\frac {c\,d^3\,\left (135\,a\,x^4+27\,b\,x^4+135\,b\,x^4\,\mathrm {atanh}\left (c\,x\right )\right )}{180}+\frac {c^2\,d^3\,\left (108\,a\,x^5+6\,b\,x^5+108\,b\,x^5\,\mathrm {atanh}\left (c\,x\right )\right )}{180} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.29, size = 235, normalized size = 1.32 \[ \begin {cases} \frac {a c^{3} d^{3} x^{6}}{6} + \frac {3 a c^{2} d^{3} x^{5}}{5} + \frac {3 a c d^{3} x^{4}}{4} + \frac {a d^{3} x^{3}}{3} + \frac {b c^{3} d^{3} x^{6} \operatorname {atanh}{\left (c x \right )}}{6} + \frac {3 b c^{2} d^{3} x^{5} \operatorname {atanh}{\left (c x \right )}}{5} + \frac {b c^{2} d^{3} x^{5}}{30} + \frac {3 b c d^{3} x^{4} \operatorname {atanh}{\left (c x \right )}}{4} + \frac {3 b c d^{3} x^{4}}{20} + \frac {b d^{3} x^{3} \operatorname {atanh}{\left (c x \right )}}{3} + \frac {11 b d^{3} x^{3}}{36} + \frac {7 b d^{3} x^{2}}{15 c} + \frac {11 b d^{3} x}{12 c^{2}} + \frac {14 b d^{3} \log {\left (x - \frac {1}{c} \right )}}{15 c^{3}} + \frac {b d^{3} \operatorname {atanh}{\left (c x \right )}}{60 c^{3}} & \text {for}\: c \neq 0 \\\frac {a d^{3} x^{3}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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