Optimal. Leaf size=286 \[ \frac {1}{10} b c^2 d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^3 (c x+1)^5 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 c^2}-\frac {d^3 (c x+1)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c^2}-\frac {12 b d^3 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac {1}{2} b c d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {6}{5} b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {5 a b d^3 x}{2 c}-\frac {6 b^2 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{5 c^2}+\frac {3 b^2 d^3 \log \left (1-c^2 x^2\right )}{2 c^2}-\frac {13 b^2 d^3 \tanh ^{-1}(c x)}{10 c^2}+\frac {1}{30} b^2 c d^3 x^3+\frac {13 b^2 d^3 x}{10 c}+\frac {5 b^2 d^3 x \tanh ^{-1}(c x)}{2 c}+\frac {1}{4} b^2 d^3 x^2 \]
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Rubi [A] time = 0.60, antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 38, number of rules used = 14, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5940, 5928, 5910, 260, 5916, 321, 206, 266, 43, 1586, 5918, 2402, 2315, 302} \[ -\frac {6 b^2 d^3 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{5 c^2}+\frac {1}{10} b c^2 d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^3 (c x+1)^5 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 c^2}-\frac {d^3 (c x+1)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c^2}-\frac {12 b d^3 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac {1}{2} b c d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {6}{5} b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {5 a b d^3 x}{2 c}+\frac {3 b^2 d^3 \log \left (1-c^2 x^2\right )}{2 c^2}-\frac {13 b^2 d^3 \tanh ^{-1}(c x)}{10 c^2}+\frac {1}{30} b^2 c d^3 x^3+\frac {13 b^2 d^3 x}{10 c}+\frac {5 b^2 d^3 x \tanh ^{-1}(c x)}{2 c}+\frac {1}{4} b^2 d^3 x^2 \]
Antiderivative was successfully verified.
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Rule 43
Rule 206
Rule 260
Rule 266
Rule 302
Rule 321
Rule 1586
Rule 2315
Rule 2402
Rule 5910
Rule 5916
Rule 5918
Rule 5928
Rule 5940
Rubi steps
\begin {align*} \int x (d+c d x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx &=\int \left (-\frac {(d+c d x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {(d+c d x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{c d}\right ) \, dx\\ &=-\frac {\int (d+c d x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{c}+\frac {\int (d+c d x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{c d}\\ &=-\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c^2}+\frac {d^3 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 c^2}-\frac {(2 b) \int \left (-15 d^5 \left (a+b \tanh ^{-1}(c x)\right )-11 c d^5 x \left (a+b \tanh ^{-1}(c x)\right )-5 c^2 d^5 x^2 \left (a+b \tanh ^{-1}(c x)\right )-c^3 d^5 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {16 \left (d^5+c d^5 x\right ) \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2}\right ) \, dx}{5 c d^2}+\frac {b \int \left (-7 d^4 \left (a+b \tanh ^{-1}(c x)\right )-4 c d^4 x \left (a+b \tanh ^{-1}(c x)\right )-c^2 d^4 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {8 \left (d^4+c d^4 x\right ) \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2}\right ) \, dx}{2 c d}\\ &=-\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c^2}+\frac {d^3 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 c^2}-\frac {(32 b) \int \frac {\left (d^5+c d^5 x\right ) \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{5 c d^2}+\frac {(4 b) \int \frac {\left (d^4+c d^4 x\right ) \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{c d}-\left (2 b d^3\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx+\frac {1}{5} \left (22 b d^3\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac {\left (7 b d^3\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{2 c}+\frac {\left (6 b d^3\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c}-\frac {1}{2} \left (b c d^3\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx+\left (2 b c d^3\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx+\frac {1}{5} \left (2 b c^2 d^3\right ) \int x^3 \left (a+b \tanh ^{-1}(c x)\right ) \, dx\\ &=\frac {5 a b d^3 x}{2 c}+\frac {6}{5} b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{2} b c d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{10} b c^2 d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c^2}+\frac {d^3 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 c^2}-\frac {(32 b) \int \frac {a+b \tanh ^{-1}(c x)}{\frac {1}{d^5}-\frac {c x}{d^5}} \, dx}{5 c d^2}+\frac {(4 b) \int \frac {a+b \tanh ^{-1}(c x)}{\frac {1}{d^4}-\frac {c x}{d^4}} \, dx}{c d}-\frac {\left (7 b^2 d^3\right ) \int \tanh ^{-1}(c x) \, dx}{2 c}+\frac {\left (6 b^2 d^3\right ) \int \tanh ^{-1}(c x) \, dx}{c}+\left (b^2 c d^3\right ) \int \frac {x^2}{1-c^2 x^2} \, dx-\frac {1}{5} \left (11 b^2 c d^3\right ) \int \frac {x^2}{1-c^2 x^2} \, dx+\frac {1}{6} \left (b^2 c^2 d^3\right ) \int \frac {x^3}{1-c^2 x^2} \, dx-\frac {1}{3} \left (2 b^2 c^2 d^3\right ) \int \frac {x^3}{1-c^2 x^2} \, dx-\frac {1}{10} \left (b^2 c^3 d^3\right ) \int \frac {x^4}{1-c^2 x^2} \, dx\\ &=\frac {5 a b d^3 x}{2 c}+\frac {6 b^2 d^3 x}{5 c}+\frac {5 b^2 d^3 x \tanh ^{-1}(c x)}{2 c}+\frac {6}{5} b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{2} b c d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{10} b c^2 d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c^2}+\frac {d^3 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 c^2}-\frac {12 b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{5 c^2}+\frac {1}{2} \left (7 b^2 d^3\right ) \int \frac {x}{1-c^2 x^2} \, dx-\left (6 b^2 d^3\right ) \int \frac {x}{1-c^2 x^2} \, dx+\frac {\left (b^2 d^3\right ) \int \frac {1}{1-c^2 x^2} \, dx}{c}-\frac {\left (11 b^2 d^3\right ) \int \frac {1}{1-c^2 x^2} \, dx}{5 c}-\frac {\left (4 b^2 d^3\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{c}+\frac {\left (32 b^2 d^3\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{5 c}+\frac {1}{12} \left (b^2 c^2 d^3\right ) \operatorname {Subst}\left (\int \frac {x}{1-c^2 x} \, dx,x,x^2\right )-\frac {1}{3} \left (b^2 c^2 d^3\right ) \operatorname {Subst}\left (\int \frac {x}{1-c^2 x} \, dx,x,x^2\right )-\frac {1}{10} \left (b^2 c^3 d^3\right ) \int \left (-\frac {1}{c^4}-\frac {x^2}{c^2}+\frac {1}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx\\ &=\frac {5 a b d^3 x}{2 c}+\frac {13 b^2 d^3 x}{10 c}+\frac {1}{30} b^2 c d^3 x^3-\frac {6 b^2 d^3 \tanh ^{-1}(c x)}{5 c^2}+\frac {5 b^2 d^3 x \tanh ^{-1}(c x)}{2 c}+\frac {6}{5} b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{2} b c d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{10} b c^2 d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c^2}+\frac {d^3 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 c^2}-\frac {12 b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{5 c^2}+\frac {5 b^2 d^3 \log \left (1-c^2 x^2\right )}{4 c^2}+\frac {\left (4 b^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{c^2}-\frac {\left (32 b^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{5 c^2}-\frac {\left (b^2 d^3\right ) \int \frac {1}{1-c^2 x^2} \, dx}{10 c}+\frac {1}{12} \left (b^2 c^2 d^3\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{c^2}-\frac {1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )-\frac {1}{3} \left (b^2 c^2 d^3\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{c^2}-\frac {1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {5 a b d^3 x}{2 c}+\frac {13 b^2 d^3 x}{10 c}+\frac {1}{4} b^2 d^3 x^2+\frac {1}{30} b^2 c d^3 x^3-\frac {13 b^2 d^3 \tanh ^{-1}(c x)}{10 c^2}+\frac {5 b^2 d^3 x \tanh ^{-1}(c x)}{2 c}+\frac {6}{5} b d^3 x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{2} b c d^3 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{10} b c^2 d^3 x^4 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d^3 (1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c^2}+\frac {d^3 (1+c x)^5 \left (a+b \tanh ^{-1}(c x)\right )^2}{5 c^2}-\frac {12 b d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{5 c^2}+\frac {3 b^2 d^3 \log \left (1-c^2 x^2\right )}{2 c^2}-\frac {6 b^2 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{5 c^2}\\ \end {align*}
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Mathematica [A] time = 1.21, size = 325, normalized size = 1.14 \[ \frac {d^3 \left (12 a^2 c^5 x^5+45 a^2 c^4 x^4+60 a^2 c^3 x^3+30 a^2 c^2 x^2+6 a b c^4 x^4+30 a b c^3 x^3+72 a b c^2 x^2+72 a b \log \left (c^2 x^2-1\right )+6 b \tanh ^{-1}(c x) \left (a c^2 x^2 \left (4 c^3 x^3+15 c^2 x^2+20 c x+10\right )+b \left (c^4 x^4+5 c^3 x^3+12 c^2 x^2+25 c x-13\right )-24 b \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )+150 a b c x+75 a b \log (1-c x)-75 a b \log (c x+1)-18 a b+2 b^2 c^3 x^3+15 b^2 c^2 x^2+90 b^2 \log \left (1-c^2 x^2\right )+3 b^2 \left (4 c^5 x^5+15 c^4 x^4+20 c^3 x^3+10 c^2 x^2-49\right ) \tanh ^{-1}(c x)^2+72 b^2 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )+78 b^2 c x-15 b^2\right )}{60 c^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (a^{2} c^{3} d^{3} x^{4} + 3 \, a^{2} c^{2} d^{3} x^{3} + 3 \, a^{2} c d^{3} x^{2} + a^{2} d^{3} x + {\left (b^{2} c^{3} d^{3} x^{4} + 3 \, b^{2} c^{2} d^{3} x^{3} + 3 \, b^{2} c d^{3} x^{2} + b^{2} d^{3} x\right )} \operatorname {artanh}\left (c x\right )^{2} + 2 \, {\left (a b c^{3} d^{3} x^{4} + 3 \, a b c^{2} d^{3} x^{3} + 3 \, a b c d^{3} x^{2} + a b d^{3} x\right )} \operatorname {artanh}\left (c x\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c d x + d\right )}^{3} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 570, normalized size = 1.99 \[ \frac {5 a b \,d^{3} x}{2 c}+\frac {d^{3} b^{2} \arctanh \left (c x \right )^{2} x^{2}}{2}+\frac {6 d^{3} b^{2} \arctanh \left (c x \right ) x^{2}}{5}-\frac {6 d^{3} b^{2} \dilog \left (\frac {1}{2}+\frac {c x}{2}\right )}{5 c^{2}}+\frac {49 d^{3} b^{2} \ln \left (c x -1\right )^{2}}{80 c^{2}}+\frac {17 d^{3} b^{2} \ln \left (c x +1\right )}{20 c^{2}}+\frac {d^{3} b^{2} \ln \left (c x +1\right )^{2}}{80 c^{2}}+\frac {c^{3} d^{3} a^{2} x^{5}}{5}+\frac {3 c^{2} d^{3} a^{2} x^{4}}{4}+c \,d^{3} a^{2} x^{3}+\frac {6 d^{3} a b \,x^{2}}{5}+\frac {43 d^{3} b^{2} \ln \left (c x -1\right )}{20 c^{2}}+\frac {3 c^{2} d^{3} a b \arctanh \left (c x \right ) x^{4}}{2}+\frac {2 c^{3} d^{3} a b \arctanh \left (c x \right ) x^{5}}{5}+\frac {b^{2} d^{3} x^{2}}{4}+\frac {3 c^{2} d^{3} b^{2} \arctanh \left (c x \right )^{2} x^{4}}{4}+\frac {c \,d^{3} b^{2} \arctanh \left (c x \right ) x^{3}}{2}+\frac {c^{3} d^{3} b^{2} \arctanh \left (c x \right )^{2} x^{5}}{5}+\frac {c^{2} d^{3} b^{2} \arctanh \left (c x \right ) x^{4}}{10}+\frac {49 d^{3} b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{20 c^{2}}-\frac {d^{3} b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{20 c^{2}}+d^{3} a b \arctanh \left (c x \right ) x^{2}+c \,d^{3} b^{2} \arctanh \left (c x \right )^{2} x^{3}+\frac {5 b^{2} d^{3} x \arctanh \left (c x \right )}{2 c}-\frac {49 d^{3} b^{2} \ln \left (c x -1\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{40 c^{2}}-\frac {d^{3} b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{40 c^{2}}+\frac {d^{3} b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{40 c^{2}}+\frac {c^{2} d^{3} a b \,x^{4}}{10}+\frac {c \,d^{3} a b \,x^{3}}{2}-\frac {d^{3} a b \ln \left (c x +1\right )}{20 c^{2}}+\frac {49 d^{3} a b \ln \left (c x -1\right )}{20 c^{2}}+\frac {d^{3} a^{2} x^{2}}{2}+\frac {13 b^{2} d^{3} x}{10 c}+\frac {b^{2} c \,d^{3} x^{3}}{30}+2 c \,d^{3} a b \arctanh \left (c x \right ) x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.70, size = 780, normalized size = 2.73 \[ \frac {1}{5} \, a^{2} c^{3} d^{3} x^{5} + \frac {3}{4} \, a^{2} c^{2} d^{3} x^{4} + \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 )} a b c^{3} d^{3} + a^{2} c d^{3} x^{3} + \frac {1}{2} \, b^{2} d^{3} x^{2} \operatorname {artanh}\left (c x\right )^{2} + \frac {1}{4} \, {\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 )} a b c^{2} d^{3} + {\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 )} a b c d^{3} + \frac {1}{2} \, a^{2} d^{3} x^{2} + \frac {1}{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 )} a b d^{3} + \frac {1}{8} \, {\left (4 \, 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 )} \operatorname {artanh}\left (c x\right ) - \frac {2 \, {\left (\log \left (c x - 1\right ) - 2\right )} \log \left (c x + 1\right ) - \log \left (c x + 1\right )^{2} - \log \left (c x - 1\right )^{2} - 4 \, \log \left (c x - 1\right )}{c^{2}}\right )} b^{2} d^{3} + \frac {6 \, {\left (\log \left (c x + 1\right ) \log \left (-\frac {1}{2} \, c x + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c x + \frac {1}{2}\right )\right )} b^{2} d^{3}}{5 \, c^{2}} + \frac {7 \, b^{2} d^{3} \log \left (c x + 1\right )}{20 \, c^{2}} + \frac {33 \, b^{2} d^{3} \log \left (c x - 1\right )}{20 \, c^{2}} + \frac {8 \, b^{2} c^{3} d^{3} x^{3} + 60 \, b^{2} c^{2} d^{3} x^{2} + 312 \, b^{2} c d^{3} x + 3 \, {\left (4 \, b^{2} c^{5} d^{3} x^{5} + 15 \, b^{2} c^{4} d^{3} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 9 \, b^{2} d^{3}\right )} \log \left (c x + 1\right )^{2} + 3 \, {\left (4 \, b^{2} c^{5} d^{3} x^{5} + 15 \, b^{2} c^{4} d^{3} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} - 39 \, b^{2} d^{3}\right )} \log \left (-c x + 1\right )^{2} + 12 \, {\left (b^{2} c^{4} d^{3} x^{4} + 5 \, b^{2} c^{3} d^{3} x^{3} + 12 \, b^{2} c^{2} d^{3} x^{2} + 15 \, b^{2} c d^{3} x\right )} \log \left (c x + 1\right ) - 6 \, {\left (2 \, b^{2} c^{4} d^{3} x^{4} + 10 \, b^{2} c^{3} d^{3} x^{3} + 24 \, b^{2} c^{2} d^{3} x^{2} + 30 \, b^{2} c d^{3} x + {\left (4 \, b^{2} c^{5} d^{3} x^{5} + 15 \, b^{2} c^{4} d^{3} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 9 \, b^{2} d^{3}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{240 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ d^{3} \left (\int a^{2} x\, dx + \int 3 a^{2} c x^{2}\, dx + \int 3 a^{2} c^{2} x^{3}\, dx + \int a^{2} c^{3} x^{4}\, dx + \int b^{2} x \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b x \operatorname {atanh}{\left (c x \right )}\, dx + \int 3 b^{2} c x^{2} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 3 b^{2} c^{2} x^{3} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{3} x^{4} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 6 a b c x^{2} \operatorname {atanh}{\left (c x \right )}\, dx + \int 6 a b c^{2} x^{3} \operatorname {atanh}{\left (c x \right )}\, dx + \int 2 a b c^{3} x^{4} \operatorname {atanh}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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