Optimal. Leaf size=312 \[ \frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{30 c^3}-\frac {16 b d^2 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{15 c^3}+\frac {1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {a b d^2 x}{c^2}+\frac {1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{10} b c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} b d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {8 b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{15 c}-\frac {8 b^2 d^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{15 c^3}-\frac {19 b^2 d^2 \tanh ^{-1}(c x)}{30 c^3}+\frac {19 b^2 d^2 x}{30 c^2}+\frac {b^2 d^2 x \tanh ^{-1}(c x)}{c^2}+\frac {2 b^2 d^2 \log \left (1-c^2 x^2\right )}{3 c^3}+\frac {b^2 d^2 x^2}{6 c}+\frac {1}{30} b^2 d^2 x^3 \]
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Rubi [A] time = 0.88, antiderivative size = 312, normalized size of antiderivative = 1.00, number of steps used = 36, number of rules used = 15, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {5940, 5916, 5980, 321, 206, 5984, 5918, 2402, 2315, 266, 43, 5910, 260, 5948, 302} \[ -\frac {8 b^2 d^2 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{15 c^3}+\frac {1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {a b d^2 x}{c^2}+\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{30 c^3}-\frac {16 b d^2 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{15 c^3}+\frac {1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{10} b c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} b d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {8 b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{15 c}+\frac {2 b^2 d^2 \log \left (1-c^2 x^2\right )}{3 c^3}+\frac {19 b^2 d^2 x}{30 c^2}+\frac {b^2 d^2 x \tanh ^{-1}(c x)}{c^2}-\frac {19 b^2 d^2 \tanh ^{-1}(c x)}{30 c^3}+\frac {b^2 d^2 x^2}{6 c}+\frac {1}{30} b^2 d^2 x^3 \]
Antiderivative was successfully verified.
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Rule 43
Rule 206
Rule 260
Rule 266
Rule 302
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)^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx &=\int \left (d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+2 c d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+c^2 d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^2 \int x^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+\left (2 c d^2\right ) \int x^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+\left (c^2 d^2\right ) \int x^4 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx\\ &=\frac {1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {1}{3} \left (2 b c d^2\right ) \int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\left (b c^2 d^2\right ) \int \frac {x^4 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\frac {1}{5} \left (2 b c^3 d^2\right ) \int \frac {x^5 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac {1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\left (b d^2\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\left (b d^2\right ) \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx+\frac {\left (2 b d^2\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{3 c}-\frac {\left (2 b d^2\right ) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 c}+\frac {1}{5} \left (2 b c d^2\right ) \int x^3 \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac {1}{5} \left (2 b c d^2\right ) \int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac {b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}+\frac {1}{3} b d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{10} b c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^3}+\frac {1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {1}{3} \left (b^2 d^2\right ) \int \frac {x^2}{1-c^2 x^2} \, dx-\frac {\left (2 b d^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{3 c^2}+\frac {\left (b d^2\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c^2}-\frac {\left (b d^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{c^2}+\frac {\left (2 b d^2\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{5 c}-\frac {\left (2 b d^2\right ) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{5 c}-\frac {1}{3} \left (b^2 c d^2\right ) \int \frac {x^3}{1-c^2 x^2} \, dx-\frac {1}{10} \left (b^2 c^2 d^2\right ) \int \frac {x^4}{1-c^2 x^2} \, dx\\ &=\frac {a b d^2 x}{c^2}+\frac {b^2 d^2 x}{3 c^2}+\frac {8 b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{15 c}+\frac {1}{3} b d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{10} b c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{30 c^3}+\frac {1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {2 b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c^3}-\frac {1}{5} \left (b^2 d^2\right ) \int \frac {x^2}{1-c^2 x^2} \, dx-\frac {\left (2 b d^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{5 c^2}-\frac {\left (b^2 d^2\right ) \int \frac {1}{1-c^2 x^2} \, dx}{3 c^2}+\frac {\left (2 b^2 d^2\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{3 c^2}+\frac {\left (b^2 d^2\right ) \int \tanh ^{-1}(c x) \, dx}{c^2}-\frac {1}{6} \left (b^2 c d^2\right ) \operatorname {Subst}\left (\int \frac {x}{1-c^2 x} \, dx,x,x^2\right )-\frac {1}{10} \left (b^2 c^2 d^2\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 {a b d^2 x}{c^2}+\frac {19 b^2 d^2 x}{30 c^2}+\frac {1}{30} b^2 d^2 x^3-\frac {b^2 d^2 \tanh ^{-1}(c x)}{3 c^3}+\frac {b^2 d^2 x \tanh ^{-1}(c x)}{c^2}+\frac {8 b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{15 c}+\frac {1}{3} b d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{10} b c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{30 c^3}+\frac {1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {16 b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{15 c^3}-\frac {\left (2 b^2 d^2\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^2\right ) \int \frac {1}{1-c^2 x^2} \, dx}{10 c^2}-\frac {\left (b^2 d^2\right ) \int \frac {1}{1-c^2 x^2} \, dx}{5 c^2}+\frac {\left (2 b^2 d^2\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{5 c^2}-\frac {\left (b^2 d^2\right ) \int \frac {x}{1-c^2 x^2} \, dx}{c}-\frac {1}{6} \left (b^2 c d^2\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^2 x}{c^2}+\frac {19 b^2 d^2 x}{30 c^2}+\frac {b^2 d^2 x^2}{6 c}+\frac {1}{30} b^2 d^2 x^3-\frac {19 b^2 d^2 \tanh ^{-1}(c x)}{30 c^3}+\frac {b^2 d^2 x \tanh ^{-1}(c x)}{c^2}+\frac {8 b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{15 c}+\frac {1}{3} b d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{10} b c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{30 c^3}+\frac {1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {16 b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{15 c^3}+\frac {2 b^2 d^2 \log \left (1-c^2 x^2\right )}{3 c^3}-\frac {b^2 d^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{3 c^3}-\frac {\left (2 b^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{5 c^3}\\ &=\frac {a b d^2 x}{c^2}+\frac {19 b^2 d^2 x}{30 c^2}+\frac {b^2 d^2 x^2}{6 c}+\frac {1}{30} b^2 d^2 x^3-\frac {19 b^2 d^2 \tanh ^{-1}(c x)}{30 c^3}+\frac {b^2 d^2 x \tanh ^{-1}(c x)}{c^2}+\frac {8 b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{15 c}+\frac {1}{3} b d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {1}{10} b c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{30 c^3}+\frac {1}{3} d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{2} c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{5} c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {16 b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{15 c^3}+\frac {2 b^2 d^2 \log \left (1-c^2 x^2\right )}{3 c^3}-\frac {8 b^2 d^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{15 c^3}\\ \end {align*}
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Mathematica [A] time = 1.03, size = 297, normalized size = 0.95 \[ \frac {d^2 \left (6 a^2 c^5 x^5+15 a^2 c^4 x^4+10 a^2 c^3 x^3+3 a b c^4 x^4+10 a b c^3 x^3+16 a b c^2 x^2+16 a b \log \left (c^2 x^2-1\right )+b \tanh ^{-1}(c x) \left (2 a c^3 x^3 \left (6 c^2 x^2+15 c x+10\right )+b \left (3 c^4 x^4+10 c^3 x^3+16 c^2 x^2+30 c x-19\right )-32 b \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )+30 a b c x+15 a b \log (1-c x)-15 a b \log (c x+1)-9 a b+b^2 c^3 x^3+5 b^2 c^2 x^2+20 b^2 \log \left (1-c^2 x^2\right )+b^2 \left (6 c^5 x^5+15 c^4 x^4+10 c^3 x^3-31\right ) \tanh ^{-1}(c x)^2+16 b^2 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )+19 b^2 c x-5 b^2\right )}{30 c^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (a^{2} c^{2} d^{2} x^{4} + 2 \, a^{2} c d^{2} x^{3} + a^{2} d^{2} x^{2} + {\left (b^{2} c^{2} d^{2} x^{4} + 2 \, b^{2} c d^{2} x^{3} + b^{2} d^{2} x^{2}\right )} \operatorname {artanh}\left (c x\right )^{2} + 2 \, {\left (a b c^{2} d^{2} x^{4} + 2 \, a b c d^{2} x^{3} + a b d^{2} x^{2}\right )} \operatorname {artanh}\left (c x\right ), x\right ) \]
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 \[ \int {\left (c d x + d\right )}^{2} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 521, normalized size = 1.67 \[ \frac {d^{2} b^{2} \arctanh \left (c x \right ) x^{3}}{3}+\frac {c \,d^{2} a^{2} x^{4}}{2}+\frac {d^{2} a b \,x^{3}}{3}+\frac {d^{2} b^{2} \arctanh \left (c x \right )^{2} x^{3}}{3}-\frac {8 d^{2} b^{2} \dilog \left (\frac {1}{2}+\frac {c x}{2}\right )}{15 c^{3}}+\frac {31 d^{2} b^{2} \ln \left (c x -1\right )^{2}}{120 c^{3}}+\frac {7 d^{2} b^{2} \ln \left (c x +1\right )}{20 c^{3}}+\frac {59 d^{2} b^{2} \ln \left (c x -1\right )}{60 c^{3}}+\frac {c^{2} d^{2} a^{2} x^{5}}{5}-\frac {d^{2} b^{2} \ln \left (c x +1\right )^{2}}{120 c^{3}}+c \,d^{2} a b \arctanh \left (c x \right ) x^{4}+\frac {2 c^{2} d^{2} a b \arctanh \left (c x \right ) x^{5}}{5}+\frac {b^{2} d^{2} x^{3}}{30}+\frac {19 b^{2} d^{2} x}{30 c^{2}}+\frac {b^{2} d^{2} x^{2}}{6 c}+\frac {c \,d^{2} b^{2} \arctanh \left (c x \right )^{2} x^{4}}{2}+\frac {d^{2} b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{30 c^{3}}+\frac {8 d^{2} b^{2} \arctanh \left (c x \right ) x^{2}}{15 c}-\frac {31 d^{2} b^{2} \ln \left (c x -1\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{60 c^{3}}+\frac {d^{2} a b \ln \left (c x +1\right )}{30 c^{3}}+\frac {31 d^{2} a b \ln \left (c x -1\right )}{30 c^{3}}+\frac {c \,d^{2} a b \,x^{4}}{10}+\frac {d^{2} b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{60 c^{3}}-\frac {d^{2} b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{60 c^{3}}+\frac {a b \,d^{2} x}{c^{2}}+\frac {d^{2} a^{2} x^{3}}{3}+\frac {8 d^{2} a b \,x^{2}}{15 c}+\frac {c \,d^{2} b^{2} \arctanh \left (c x \right ) x^{4}}{10}+\frac {c^{2} d^{2} b^{2} \arctanh \left (c x \right )^{2} x^{5}}{5}+\frac {2 d^{2} a b \arctanh \left (c x \right ) x^{3}}{3}+\frac {31 d^{2} b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{30 c^{3}}+\frac {b^{2} d^{2} x \arctanh \left (c x \right )}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.66, size = 604, normalized size = 1.94 \[ \frac {1}{5} \, a^{2} c^{2} d^{2} x^{5} + \frac {1}{2} \, a^{2} c d^{2} 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^{2} d^{2} + \frac {1}{3} \, a^{2} d^{2} x^{3} + \frac {1}{6} \, {\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^{2} + \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^{2} + \frac {8 \, {\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^{2}}{15 \, c^{3}} + \frac {7 \, b^{2} d^{2} \log \left (c x + 1\right )}{20 \, c^{3}} + \frac {59 \, b^{2} d^{2} \log \left (c x - 1\right )}{60 \, c^{3}} + \frac {4 \, b^{2} c^{3} d^{2} x^{3} + 20 \, b^{2} c^{2} d^{2} x^{2} + 76 \, b^{2} c d^{2} x + {\left (6 \, b^{2} c^{5} d^{2} x^{5} + 15 \, b^{2} c^{4} d^{2} x^{4} + 10 \, b^{2} c^{3} d^{2} x^{3} + b^{2} d^{2}\right )} \log \left (c x + 1\right )^{2} + {\left (6 \, b^{2} c^{5} d^{2} x^{5} + 15 \, b^{2} c^{4} d^{2} x^{4} + 10 \, b^{2} c^{3} d^{2} x^{3} - 31 \, b^{2} d^{2}\right )} \log \left (-c x + 1\right )^{2} + 2 \, {\left (3 \, b^{2} c^{4} d^{2} x^{4} + 10 \, b^{2} c^{3} d^{2} x^{3} + 16 \, b^{2} c^{2} d^{2} x^{2} + 30 \, b^{2} c d^{2} x\right )} \log \left (c x + 1\right ) - 2 \, {\left (3 \, b^{2} c^{4} d^{2} x^{4} + 10 \, b^{2} c^{3} d^{2} x^{3} + 16 \, b^{2} c^{2} d^{2} x^{2} + 30 \, b^{2} c d^{2} x + {\left (6 \, b^{2} c^{5} d^{2} x^{5} + 15 \, b^{2} c^{4} d^{2} x^{4} + 10 \, b^{2} c^{3} d^{2} x^{3} + b^{2} d^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{120 \, 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 \[ \int x^2\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^2 \,d x \]
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 \[ d^{2} \left (\int a^{2} x^{2}\, dx + \int 2 a^{2} c x^{3}\, dx + \int a^{2} c^{2} x^{4}\, dx + \int b^{2} x^{2} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b x^{2} \operatorname {atanh}{\left (c x \right )}\, dx + \int 2 b^{2} c x^{3} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{2} x^{4} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 4 a b c x^{3} \operatorname {atanh}{\left (c x \right )}\, dx + \int 2 a b c^{2} x^{4} \operatorname {atanh}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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