Optimal. Leaf size=236 \[ \frac {a b d x}{2 c^2}+\frac {b^2 d x}{3 c^2}+\frac {b^2 d x^2}{12 c}-\frac {b^2 d \tanh ^{-1}(c x)}{3 c^3}+\frac {b^2 d x \tanh ^{-1}(c x)}{2 c^2}+\frac {b d x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}+\frac {1}{6} b d x^3 \left (a+b \tanh ^{-1}(c x)\right )+\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{12 c^3}+\frac {1}{3} d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} c d x^4 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c^3}+\frac {b^2 d \log \left (1-c^2 x^2\right )}{3 c^3}-\frac {b^2 d \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{3 c^3} \]
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Rubi [A]
time = 0.39, 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 = {6087, 6037, 6127, 327, 212, 6131, 6055, 2449, 2352, 272, 45, 6021, 266, 6095}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 212
Rule 266
Rule 272
Rule 327
Rule 2352
Rule 2449
Rule 6021
Rule 6037
Rule 6055
Rule 6087
Rule 6095
Rule 6127
Rule 6131
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 ) \text {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 ) \text {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 ) \text {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 = 0.37, size = 234, normalized size = 0.99 \begin {gather*} \frac {d \left (-b^2+6 a b c x+4 b^2 c x+4 a b c^2 x^2+b^2 c^2 x^2+4 a^2 c^3 x^3+2 a b c^3 x^3+3 a^2 c^4 x^4+b^2 \left (-7+4 c^3 x^3+3 c^4 x^4\right ) \tanh ^{-1}(c x)^2+2 b \tanh ^{-1}(c x) \left (a c^3 x^3 (4+3 c x)+b \left (-2+3 c x+2 c^2 x^2+c^3 x^3\right )-4 b \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )\right )+3 a b \log (1-c x)-3 a b \log (1+c x)+4 b^2 \log \left (1-c^2 x^2\right )+4 a b \log \left (-1+c^2 x^2\right )+4 b^2 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )\right )}{12 c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.31, size = 364, normalized size = 1.54
method | result | size |
derivativedivides | \(\frac {d \,a^{2} \left (\frac {1}{4} c^{4} x^{4}+\frac {1}{3} x^{3} c^{3}\right )+\frac {d a b \arctanh \left (c x \right ) c^{4} x^{4}}{2}+\frac {2 d a b \arctanh \left (c x \right ) c^{3} x^{3}}{3}+\frac {7 d a b \ln \left (c x -1\right )}{12}+\frac {d \,b^{2} \arctanh \left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {d \,b^{2} \arctanh \left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {d \,b^{2} \arctanh \left (c x \right ) c^{3} x^{3}}{6}+\frac {d \,b^{2} \arctanh \left (c x \right ) c x}{2}+\frac {d \,b^{2} \arctanh \left (c x \right ) c^{2} x^{2}}{3}+\frac {7 d \,b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{12}+\frac {d \,b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{12}+\frac {d a b \ln \left (c x +1\right )}{12}-\frac {7 d \,b^{2} \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{24}+\frac {d \,b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{24}-\frac {d \,b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{24}+\frac {7 d \,b^{2} \ln \left (c x -1\right )^{2}}{48}-\frac {d \,b^{2} \ln \left (c x +1\right )^{2}}{48}+\frac {d \,b^{2} \ln \left (c x -1\right )}{2}+\frac {d \,b^{2} \ln \left (c x +1\right )}{6}-\frac {d \,b^{2} \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )}{3}+\frac {d \,b^{2} c^{2} x^{2}}{12}+\frac {d \,b^{2} c x}{3}+\frac {d a b \,c^{3} x^{3}}{6}+\frac {d a b \,c^{2} x^{2}}{3}+\frac {d a b c x}{2}}{c^{3}}\) | \(364\) |
default | \(\frac {d \,a^{2} \left (\frac {1}{4} c^{4} x^{4}+\frac {1}{3} x^{3} c^{3}\right )+\frac {d a b \arctanh \left (c x \right ) c^{4} x^{4}}{2}+\frac {2 d a b \arctanh \left (c x \right ) c^{3} x^{3}}{3}+\frac {7 d a b \ln \left (c x -1\right )}{12}+\frac {d \,b^{2} \arctanh \left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {d \,b^{2} \arctanh \left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {d \,b^{2} \arctanh \left (c x \right ) c^{3} x^{3}}{6}+\frac {d \,b^{2} \arctanh \left (c x \right ) c x}{2}+\frac {d \,b^{2} \arctanh \left (c x \right ) c^{2} x^{2}}{3}+\frac {7 d \,b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{12}+\frac {d \,b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{12}+\frac {d a b \ln \left (c x +1\right )}{12}-\frac {7 d \,b^{2} \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{24}+\frac {d \,b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{24}-\frac {d \,b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{24}+\frac {7 d \,b^{2} \ln \left (c x -1\right )^{2}}{48}-\frac {d \,b^{2} \ln \left (c x +1\right )^{2}}{48}+\frac {d \,b^{2} \ln \left (c x -1\right )}{2}+\frac {d \,b^{2} \ln \left (c x +1\right )}{6}-\frac {d \,b^{2} \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )}{3}+\frac {d \,b^{2} c^{2} x^{2}}{12}+\frac {d \,b^{2} c x}{3}+\frac {d a b \,c^{3} x^{3}}{6}+\frac {d a b \,c^{2} x^{2}}{3}+\frac {d a b c x}{2}}{c^{3}}\) | \(364\) |
risch | \(-\frac {d c a b \ln \left (-c x +1\right ) 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 {d c \,b^{2} \ln \left (-c x +1\right ) x^{4}}{32}-\frac {7 d \,b^{2} \ln \left (-c x +1\right ) x^{2}}{48 c}-\frac {7 d \,b^{2} \ln \left (-c x +1\right ) x}{24 c^{2}}+\frac {7 d a b \ln \left (-c x +1\right )}{12 c^{3}}+\frac {d c \,b^{2} \ln \left (-c x +1\right )^{2} x^{4}}{16}+\frac {d \,b^{2} \left (3 c^{4} x^{4}+4 x^{3} c^{3}+1\right ) \ln \left (c x +1\right )^{2}}{48 c^{3}}-\frac {d a b \ln \left (-c x +1\right ) x^{3}}{3}-\frac {5 d \,b^{2} \ln \left (-c x +1\right ) \left (-c x +1\right )^{3}}{36 c^{3}}+\frac {5 d \,b^{2} \ln \left (-c x +1\right ) \left (-c x +1\right )^{2}}{24 c^{3}}-\frac {d \,b^{2} \ln \left (-c x +1\right ) \left (-c x +1\right )}{6 c^{3}}+\frac {d \,b^{2} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{3 c^{3}}-\frac {d \,b^{2} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{3 c^{3}}+\frac {d b \ln \left (-c x -1\right ) a}{12 c^{3}}+\frac {d \,b^{2} \ln \left (-c x +1\right ) \left (-c x +1\right )^{4}}{32 c^{3}}+\frac {d b a \,x^{2}}{3 c}+\frac {d \,b^{2} \dilog \left (-\frac {c x}{2}+\frac {1}{2}\right )}{3 c^{3}}+\frac {d c \,x^{4} a^{2}}{4}-\frac {7 d \,b^{2} \ln \left (-c x +1\right ) x^{3}}{72}-\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 )^{2} x^{3}}{12}-\frac {7 d \,a^{2}}{12 c^{3}}+\frac {d \,x^{3} a^{2}}{3}+\left (-\frac {d \,b^{2} x^{3} \left (3 c x +4\right ) \ln \left (-c x +1\right )}{24}+\frac {d b \left (6 c^{4} x^{4} a +8 c^{3} x^{3} a +2 b \,c^{3} x^{3}+4 b \,c^{2} x^{2}+6 b c x +7 b \ln \left (-c x +1\right )\right )}{24 c^{3}}\right ) \ln \left (c x +1\right )+\frac {d b a \,x^{3}}{6}+\frac {d \,b^{2} \ln \left (-c x -1\right )}{6 c^{3}}-\frac {d b a}{c^{3}}-\frac {5 d \,b^{2}}{12 c^{3}}+\frac {163 d \,b^{2} \ln \left (-c x +1\right )}{288 c^{3}}\) | \(594\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 402, normalized size = 1.70 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} 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 ) \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^2\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,\left (d+c\,d\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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