Optimal. Leaf size=236 \[ \frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{12 c^3}-\frac {2 b d \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3}+\frac {a b d x}{2 c^2}+\frac {1}{4} c d x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{6} b d x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {b d x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}-\frac {b^2 d \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{3 c^3}-\frac {b^2 d \tanh ^{-1}(c x)}{3 c^3}+\frac {b^2 d x}{3 c^2}+\frac {b^2 d x \tanh ^{-1}(c x)}{2 c^2}+\frac {b^2 d \log \left (1-c^2 x^2\right )}{3 c^3}+\frac {b^2 d x^2}{12 c} \]
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Rubi [A] time = 0.53, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 14, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5940, 5916, 5980, 321, 206, 5984, 5918, 2402, 2315, 266, 43, 5910, 260, 5948} \[ -\frac {b^2 d \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{3 c^3}+\frac {a b d x}{2 c^2}+\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{12 c^3}-\frac {2 b d \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3}+\frac {1}{4} c d x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{6} b d x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {b d x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}+\frac {b^2 d \log \left (1-c^2 x^2\right )}{3 c^3}+\frac {b^2 d x}{3 c^2}+\frac {b^2 d x \tanh ^{-1}(c x)}{2 c^2}-\frac {b^2 d \tanh ^{-1}(c x)}{3 c^3}+\frac {b^2 d x^2}{12 c} \]
Antiderivative was successfully verified.
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Rule 43
Rule 206
Rule 260
Rule 266
Rule 321
Rule 2315
Rule 2402
Rule 5910
Rule 5916
Rule 5918
Rule 5940
Rule 5948
Rule 5980
Rule 5984
Rubi steps
\begin {align*} \int x^2 (d+c d x) \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx &=\int \left (d x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+c d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2\right ) \, dx\\ &=d \int x^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+(c d) \int x^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx\\ &=\frac {1}{3} d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} c d x^4 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {1}{3} (2 b c d) \int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\frac {1}{2} \left (b c^2 d\right ) \int \frac {x^4 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac {1}{3} d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} c d x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{2} (b d) \int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac {1}{2} (b d) \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx+\frac {(2 b d) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{3 c}-\frac {(2 b d) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 c}\\ &=\frac {b d x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}+\frac {1}{6} b d x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^3}+\frac {1}{3} d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} c d x^4 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {1}{3} \left (b^2 d\right ) \int \frac {x^2}{1-c^2 x^2} \, dx+\frac {(b d) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{2 c^2}-\frac {(b d) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{2 c^2}-\frac {(2 b d) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{3 c^2}-\frac {1}{6} \left (b^2 c d\right ) \int \frac {x^3}{1-c^2 x^2} \, dx\\ &=\frac {a b d x}{2 c^2}+\frac {b^2 d x}{3 c^2}+\frac {b d x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}+\frac {1}{6} b d x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{12 c^3}+\frac {1}{3} d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} c d x^4 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c^3}-\frac {\left (b^2 d\right ) \int \frac {1}{1-c^2 x^2} \, dx}{3 c^2}+\frac {\left (b^2 d\right ) \int \tanh ^{-1}(c x) \, dx}{2 c^2}+\frac {\left (2 b^2 d\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{3 c^2}-\frac {1}{12} \left (b^2 c d\right ) \operatorname {Subst}\left (\int \frac {x}{1-c^2 x} \, dx,x,x^2\right )\\ &=\frac {a b d x}{2 c^2}+\frac {b^2 d x}{3 c^2}-\frac {b^2 d \tanh ^{-1}(c x)}{3 c^3}+\frac {b^2 d x \tanh ^{-1}(c x)}{2 c^2}+\frac {b d x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}+\frac {1}{6} b d x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{12 c^3}+\frac {1}{3} d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} c d x^4 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c^3}-\frac {\left (2 b^2 d\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{3 c^3}-\frac {\left (b^2 d\right ) \int \frac {x}{1-c^2 x^2} \, dx}{2 c}-\frac {1}{12} \left (b^2 c d\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 {a b d x}{2 c^2}+\frac {b^2 d x}{3 c^2}+\frac {b^2 d x^2}{12 c}-\frac {b^2 d \tanh ^{-1}(c x)}{3 c^3}+\frac {b^2 d x \tanh ^{-1}(c x)}{2 c^2}+\frac {b d x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}+\frac {1}{6} b d x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{12 c^3}+\frac {1}{3} d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} c d x^4 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c^3}+\frac {b^2 d \log \left (1-c^2 x^2\right )}{3 c^3}-\frac {b^2 d \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{3 c^3}\\ \end {align*}
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Mathematica [A] time = 0.57, size = 234, normalized size = 0.99 \[ \frac {d \left (3 a^2 c^4 x^4+4 a^2 c^3 x^3+2 a b c^3 x^3+4 a b c^2 x^2+4 a b \log \left (c^2 x^2-1\right )+2 b \tanh ^{-1}(c x) \left (a c^3 x^3 (3 c x+4)+b \left (c^3 x^3+2 c^2 x^2+3 c x-2\right )-4 b \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )+6 a b c x+3 a b \log (1-c x)-3 a b \log (c x+1)+b^2 c^2 x^2+4 b^2 \log \left (1-c^2 x^2\right )+b^2 \left (3 c^4 x^4+4 c^3 x^3-7\right ) \tanh ^{-1}(c x)^2+4 b^2 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )+4 b^2 c x-b^2\right )}{12 c^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (a^{2} c d x^{3} + a^{2} d x^{2} + {\left (b^{2} c d x^{3} + b^{2} d x^{2}\right )} \operatorname {artanh}\left (c x\right )^{2} + 2 \, {\left (a b c d x^{3} + a b d x^{2}\right )} \operatorname {artanh}\left (c x\right ), x\right ) \]
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 \[ \int {\left (c d x + d\right )} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 383, normalized size = 1.62 \[ \frac {c d a b \arctanh \left (c x \right ) x^{4}}{2}-\frac {d \,b^{2} \dilog \left (\frac {1}{2}+\frac {c x}{2}\right )}{3 c^{3}}+\frac {d \,b^{2} \arctanh \left (c x \right ) x^{3}}{6}+\frac {d \,b^{2} \arctanh \left (c x \right )^{2} x^{3}}{3}+\frac {d a b \,x^{3}}{6}+\frac {7 d \,b^{2} \ln \left (c x -1\right )^{2}}{48 c^{3}}+\frac {d \,b^{2} \ln \left (c x +1\right )}{6 c^{3}}+\frac {d \,b^{2} \ln \left (c x -1\right )}{2 c^{3}}-\frac {d \,b^{2} \ln \left (c x +1\right )^{2}}{48 c^{3}}+\frac {c \,a^{2} d \,x^{4}}{4}+\frac {a b d x}{2 c^{2}}+\frac {b^{2} d x}{3 c^{2}}+\frac {b^{2} d \,x^{2}}{12 c}-\frac {7 d \,b^{2} \ln \left (c x -1\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{24 c^{3}}+\frac {b^{2} d x \arctanh \left (c x \right )}{2 c^{2}}+\frac {d \,b^{2} \arctanh \left (c x \right ) x^{2}}{3 c}+\frac {2 d a b \arctanh \left (c x \right ) x^{3}}{3}+\frac {7 d \,b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{12 c^{3}}+\frac {d \,b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{12 c^{3}}+\frac {c d \,b^{2} \arctanh \left (c x \right )^{2} x^{4}}{4}+\frac {d a b \,x^{2}}{3 c}+\frac {7 d a b \ln \left (c x -1\right )}{12 c^{3}}+\frac {d a b \ln \left (c x +1\right )}{12 c^{3}}-\frac {d \,b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{24 c^{3}}+\frac {d \,b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{24 c^{3}}+\frac {a^{2} d \,x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 402, normalized size = 1.70 \[ \frac {1}{4} \, a^{2} c d x^{4} + \frac {1}{3} \, a^{2} d x^{3} + \frac {1}{12} \, {\left (6 \, x^{4} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac {3 \, \log \left (c x + 1\right )}{c^{5}} + \frac {3 \, \log \left (c x - 1\right )}{c^{5}}\right )}\right )} a b c d + \frac {1}{3} \, {\left (2 \, x^{3} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {x^{2}}{c^{2}} + \frac {\log \left (c^{2} x^{2} - 1\right )}{c^{4}}\right )}\right )} a b d + \frac {{\left (\log \left (c x + 1\right ) \log \left (-\frac {1}{2} \, c x + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c x + \frac {1}{2}\right )\right )} b^{2} d}{3 \, c^{3}} + \frac {b^{2} d \log \left (c x + 1\right )}{6 \, c^{3}} + \frac {b^{2} d \log \left (c x - 1\right )}{2 \, c^{3}} + \frac {4 \, b^{2} c^{2} d x^{2} + 16 \, b^{2} c d x + {\left (3 \, b^{2} c^{4} d x^{4} + 4 \, b^{2} c^{3} d x^{3} + b^{2} d\right )} \log \left (c x + 1\right )^{2} + {\left (3 \, b^{2} c^{4} d x^{4} + 4 \, b^{2} c^{3} d x^{3} - 7 \, b^{2} d\right )} \log \left (-c x + 1\right )^{2} + 4 \, {\left (b^{2} c^{3} d x^{3} + 2 \, b^{2} c^{2} d x^{2} + 3 \, b^{2} c d x\right )} \log \left (c x + 1\right ) - 2 \, {\left (2 \, b^{2} c^{3} d x^{3} + 4 \, b^{2} c^{2} d x^{2} + 6 \, b^{2} c d x + {\left (3 \, b^{2} c^{4} d x^{4} + 4 \, b^{2} c^{3} d x^{3} + b^{2} d\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{48 \, c^{3}} \]
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 \[ \int x^2\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,\left (d+c\,d\,x\right ) \,d x \]
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 \[ d \left (\int a^{2} x^{2}\, dx + \int a^{2} c x^{3}\, dx + \int b^{2} x^{2} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b x^{2} \operatorname {atanh}{\left (c x \right )}\, dx + \int b^{2} c x^{3} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b c x^{3} \operatorname {atanh}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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