Optimal. Leaf size=196 \[ -\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^2}-\frac {2 b d \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{3 c^2}+\frac {1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} b d x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {a b d x}{c}-\frac {b^2 d \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{3 c^2}+\frac {b^2 d \log \left (1-c^2 x^2\right )}{2 c^2}-\frac {b^2 d \tanh ^{-1}(c x)}{3 c^2}+\frac {b^2 d x}{3 c}+\frac {b^2 d x \tanh ^{-1}(c x)}{c} \]
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Rubi [A] time = 0.39, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 12, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5940, 5916, 5980, 5910, 260, 5948, 321, 206, 5984, 5918, 2402, 2315} \[ -\frac {b^2 d \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{3 c^2}-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^2}-\frac {2 b d \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{3 c^2}+\frac {1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} b d x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {a b d x}{c}+\frac {b^2 d \log \left (1-c^2 x^2\right )}{2 c^2}-\frac {b^2 d \tanh ^{-1}(c x)}{3 c^2}+\frac {b^2 d x}{3 c}+\frac {b^2 d x \tanh ^{-1}(c x)}{c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 260
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 (d+c d x) \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx &=\int \left (d x \left (a+b \tanh ^{-1}(c x)\right )^2+c d x^2 \left (a+b \tanh ^{-1}(c x)\right )^2\right ) \, dx\\ &=d \int x \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+(c d) \int x^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx\\ &=\frac {1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2-(b c d) \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\frac {1}{3} \left (2 b c^2 d\right ) \int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac {1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} (2 b d) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac {1}{3} (2 b d) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx+\frac {(b d) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c}-\frac {(b d) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{c}\\ &=\frac {a b d x}{c}+\frac {1}{3} b d x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^2}+\frac {1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {(2 b d) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{3 c}+\frac {\left (b^2 d\right ) \int \tanh ^{-1}(c x) \, dx}{c}-\frac {1}{3} \left (b^2 c d\right ) \int \frac {x^2}{1-c^2 x^2} \, dx\\ &=\frac {a b d x}{c}+\frac {b^2 d x}{3 c}+\frac {b^2 d x \tanh ^{-1}(c x)}{c}+\frac {1}{3} b d x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^2}+\frac {1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} c d x^3 \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^2}-\left (b^2 d\right ) \int \frac {x}{1-c^2 x^2} \, dx-\frac {\left (b^2 d\right ) \int \frac {1}{1-c^2 x^2} \, dx}{3 c}+\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}\\ &=\frac {a b d x}{c}+\frac {b^2 d x}{3 c}-\frac {b^2 d \tanh ^{-1}(c x)}{3 c^2}+\frac {b^2 d x \tanh ^{-1}(c x)}{c}+\frac {1}{3} b d x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^2}+\frac {1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} c d x^3 \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^2}+\frac {b^2 d \log \left (1-c^2 x^2\right )}{2 c^2}-\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^2}\\ &=\frac {a b d x}{c}+\frac {b^2 d x}{3 c}-\frac {b^2 d \tanh ^{-1}(c x)}{3 c^2}+\frac {b^2 d x \tanh ^{-1}(c x)}{c}+\frac {1}{3} b d x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^2}+\frac {1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} c d x^3 \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^2}+\frac {b^2 d \log \left (1-c^2 x^2\right )}{2 c^2}-\frac {b^2 d \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{3 c^2}\\ \end {align*}
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Mathematica [A] time = 0.51, size = 201, normalized size = 1.03 \[ \frac {d \left (2 a^2 c^3 x^3+3 a^2 c^2 x^2+2 a b c^2 x^2+2 a b \log \left (c^2 x^2-1\right )+2 b \tanh ^{-1}(c x) \left (a c^2 x^2 (2 c x+3)+b \left (c^2 x^2+3 c x-1\right )-2 b \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )+6 a b c x+3 a b \log (1-c x)-3 a b \log (c x+1)+3 b^2 \log \left (1-c^2 x^2\right )+b^2 \left (2 c^3 x^3+3 c^2 x^2-5\right ) \tanh ^{-1}(c x)^2+2 b^2 \text {Li}_2\left (-e^{-2 \tanh ^{-1}(c x)}\right )+2 b^2 c x\right )}{6 c^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (a^{2} c d x^{2} + a^{2} d x + {\left (b^{2} c d x^{2} + b^{2} d x\right )} \operatorname {artanh}\left (c x\right )^{2} + 2 \, {\left (a b c d x^{2} + a b d x\right )} \operatorname {artanh}\left (c x\right ), x\right ) \]
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 \[ \int {\left (c d x + d\right )} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 341, normalized size = 1.74 \[ \frac {c \,a^{2} d \,x^{3}}{3}+\frac {a^{2} d \,x^{2}}{2}+\frac {c d \,b^{2} \arctanh \left (c x \right )^{2} x^{3}}{3}+\frac {d \,b^{2} \arctanh \left (c x \right )^{2} x^{2}}{2}+\frac {d \,b^{2} \arctanh \left (c x \right ) x^{2}}{3}+\frac {b^{2} d x \arctanh \left (c x \right )}{c}+\frac {5 d \,b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{6 c^{2}}-\frac {d \,b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{6 c^{2}}+\frac {5 d \,b^{2} \ln \left (c x -1\right )^{2}}{24 c^{2}}-\frac {d \,b^{2} \dilog \left (\frac {1}{2}+\frac {c x}{2}\right )}{3 c^{2}}-\frac {5 d \,b^{2} \ln \left (c x -1\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{12 c^{2}}+\frac {d \,b^{2} \ln \left (c x +1\right )^{2}}{24 c^{2}}-\frac {d \,b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{12 c^{2}}+\frac {d \,b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{12 c^{2}}+\frac {b^{2} d x}{3 c}+\frac {2 d \,b^{2} \ln \left (c x -1\right )}{3 c^{2}}+\frac {d \,b^{2} \ln \left (c x +1\right )}{3 c^{2}}+\frac {2 c d a b \arctanh \left (c x \right ) x^{3}}{3}+d a b \arctanh \left (c x \right ) x^{2}+\frac {d a b \,x^{2}}{3}+\frac {a b d x}{c}+\frac {5 d a b \ln \left (c x -1\right )}{6 c^{2}}-\frac {d a b \ln \left (c x +1\right )}{6 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, a^{2} c d x^{3} + \frac {1}{2} \, b^{2} d x^{2} \operatorname {artanh}\left (c x\right )^{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 c d - \frac {1}{216} \, {\left (2 \, c^{4} {\left (\frac {2 \, {\left (c^{2} x^{3} + 3 \, x\right )}}{c^{6}} - \frac {3 \, \log \left (c x + 1\right )}{c^{7}} + \frac {3 \, \log \left (c x - 1\right )}{c^{7}}\right )} - 3 \, c^{3} {\left (\frac {x^{2}}{c^{4}} + \frac {\log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )} - 648 \, c^{3} \int \frac {x^{3} \log \left (c x + 1\right )}{9 \, {\left (c^{4} x^{2} - c^{2}\right )}}\,{d x} + 9 \, c^{2} {\left (\frac {2 \, x}{c^{4}} - \frac {\log \left (c x + 1\right )}{c^{5}} + \frac {\log \left (c x - 1\right )}{c^{5}}\right )} - 324 \, c \int \frac {x \log \left (c x + 1\right )}{9 \, {\left (c^{4} x^{2} - c^{2}\right )}}\,{d x} - \frac {6 \, {\left (3 \, c^{3} x^{3} \log \left (c x + 1\right )^{2} + {\left (2 \, c^{3} x^{3} - 3 \, c^{2} x^{2} + 6 \, c x - 6 \, {\left (c^{3} x^{3} + 1\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )\right )}}{c^{3}} - \frac {2 \, {\left (c x - 1\right )}^{3} {\left (9 \, \log \left (-c x + 1\right )^{2} - 6 \, \log \left (-c x + 1\right ) + 2\right )} + 27 \, {\left (c x - 1\right )}^{2} {\left (2 \, \log \left (-c x + 1\right )^{2} - 2 \, \log \left (-c x + 1\right ) + 1\right )} + 54 \, {\left (c x - 1\right )} {\left (\log \left (-c x + 1\right )^{2} - 2 \, \log \left (-c x + 1\right ) + 2\right )}}{c^{3}} + \frac {18 \, \log \left (9 \, c^{4} x^{2} - 9 \, c^{2}\right )}{c^{3}} - 324 \, \int \frac {\log \left (c x + 1\right )}{9 \, {\left (c^{4} x^{2} - c^{2}\right )}}\,{d x}\right )} b^{2} c d + \frac {1}{2} \, a^{2} d 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 + \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 \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,\left (d+c\,d\,x\right ) \,d x \]
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 \[ d \left (\int a^{2} x\, dx + \int a^{2} c x^{2}\, dx + \int b^{2} x \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b x \operatorname {atanh}{\left (c x \right )}\, dx + \int b^{2} c x^{2} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b c x^{2} \operatorname {atanh}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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