Optimal. Leaf size=130 \[ \frac {b^2 x}{3 c^2}-\frac {b^2 \tanh ^{-1}(c x)}{3 c^3}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^3}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {2 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c^3}-\frac {b^2 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{3 c^3} \]
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Rubi [A]
time = 0.15, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6037, 6127,
327, 212, 6131, 6055, 2449, 2352} \begin {gather*} \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^3}-\frac {2 b \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{3 c^3}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}-\frac {b^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{3 c^3}-\frac {b^2 \tanh ^{-1}(c x)}{3 c^3}+\frac {b^2 x}{3 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 327
Rule 2352
Rule 2449
Rule 6037
Rule 6055
Rule 6127
Rule 6131
Rubi steps
\begin {align*} \int x^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx &=\frac {1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {1}{3} (2 b c) \int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac {1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(2 b) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{3 c}-\frac {(2 b) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 c}\\ &=\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^3}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {1}{3} b^2 \int \frac {x^2}{1-c^2 x^2} \, dx-\frac {(2 b) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{3 c^2}\\ &=\frac {b^2 x}{3 c^2}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^3}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {2 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c^3}-\frac {b^2 \int \frac {1}{1-c^2 x^2} \, dx}{3 c^2}+\frac {\left (2 b^2\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{3 c^2}\\ &=\frac {b^2 x}{3 c^2}-\frac {b^2 \tanh ^{-1}(c x)}{3 c^3}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^3}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {2 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c^3}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{3 c^3}\\ &=\frac {b^2 x}{3 c^2}-\frac {b^2 \tanh ^{-1}(c x)}{3 c^3}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{3 c^3}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {2 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c^3}-\frac {b^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{3 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 122, normalized size = 0.94 \begin {gather*} \frac {b^2 c x+a b c^2 x^2+a^2 c^3 x^3+b^2 \left (-1+c^3 x^3\right ) \tanh ^{-1}(c x)^2+b \tanh ^{-1}(c x) \left (-b+b c^2 x^2+2 a c^3 x^3-2 b \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )\right )+a b \log \left (-1+c^2 x^2\right )+b^2 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )}{3 c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(244\) vs.
\(2(116)=232\).
time = 0.06, size = 245, normalized size = 1.88
method | result | size |
derivativedivides | \(\frac {\frac {c^{3} x^{3} a^{2}}{3}+\frac {b^{2} c^{3} x^{3} \arctanh \left (c x \right )^{2}}{3}+\frac {b^{2} \arctanh \left (c x \right ) c^{2} x^{2}}{3}+\frac {b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{3}+\frac {b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{3}+\frac {b^{2} c x}{3}+\frac {b^{2} \ln \left (c x -1\right )}{6}-\frac {b^{2} \ln \left (c x +1\right )}{6}-\frac {b^{2} \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )}{3}-\frac {b^{2} \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{6}+\frac {b^{2} \ln \left (c x -1\right )^{2}}{12}-\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{6}+\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{6}-\frac {b^{2} \ln \left (c x +1\right )^{2}}{12}+\frac {2 a b \,c^{3} x^{3} \arctanh \left (c x \right )}{3}+\frac {a b \,c^{2} x^{2}}{3}+\frac {a b \ln \left (c x -1\right )}{3}+\frac {a b \ln \left (c x +1\right )}{3}}{c^{3}}\) | \(245\) |
default | \(\frac {\frac {c^{3} x^{3} a^{2}}{3}+\frac {b^{2} c^{3} x^{3} \arctanh \left (c x \right )^{2}}{3}+\frac {b^{2} \arctanh \left (c x \right ) c^{2} x^{2}}{3}+\frac {b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{3}+\frac {b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{3}+\frac {b^{2} c x}{3}+\frac {b^{2} \ln \left (c x -1\right )}{6}-\frac {b^{2} \ln \left (c x +1\right )}{6}-\frac {b^{2} \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )}{3}-\frac {b^{2} \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{6}+\frac {b^{2} \ln \left (c x -1\right )^{2}}{12}-\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{6}+\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{6}-\frac {b^{2} \ln \left (c x +1\right )^{2}}{12}+\frac {2 a b \,c^{3} x^{3} \arctanh \left (c x \right )}{3}+\frac {a b \,c^{2} x^{2}}{3}+\frac {a b \ln \left (c x -1\right )}{3}+\frac {a b \ln \left (c x +1\right )}{3}}{c^{3}}\) | \(245\) |
risch | \(\frac {b^{2} x}{3 c^{2}}+\frac {a^{2} x^{3}}{3}+\frac {b^{2} \ln \left (c x +1\right )^{2} x^{3}}{12}+\frac {b^{2} \ln \left (c x +1\right )^{2}}{12 c^{3}}-\frac {b^{2} \ln \left (c x +1\right )}{6 c^{3}}-\frac {11 a b}{9 c^{3}}+\frac {b^{2} \ln \left (-c x +1\right )^{2} x^{3}}{12}-\frac {b^{2} \ln \left (-c x +1\right )^{2}}{12 c^{3}}+\frac {11 b^{2} \ln \left (-c x +1\right )}{18 c^{3}}+\frac {b^{2} \ln \left (c x +1\right ) x^{2}}{6 c}+\frac {a b \ln \left (-c x +1\right )}{3 c^{3}}-\frac {a b \ln \left (-c x +1\right ) x^{3}}{3}-\frac {b^{2} \ln \left (-c x +1\right ) x^{2}}{6 c}+\frac {a b \,x^{2}}{3 c}-\frac {a^{2}}{3 c^{3}}-\frac {17 b^{2}}{54 c^{3}}+\frac {b a \ln \left (c x +1\right ) x^{3}}{3}+\frac {b a \ln \left (c x +1\right )}{3 c^{3}}-\frac {b^{2} \ln \left (-c x +1\right ) \ln \left (c x +1\right ) x^{3}}{6}-\frac {b^{2} \ln \left (-c x +1\right ) \ln \left (c x +1\right )}{6 c^{3}}+\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{3 c^{3}}-\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{3 c^{3}}-\frac {b^{2} \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )}{3 c^{3}}-\frac {4 b^{2} \ln \left (c x -1\right )}{9 c^{3}}\) | \(350\) |
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 \begin {gather*} \text {Failed to integrate} \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*} \int x^{2} \left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{2}\, dx \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 = -0.01 \begin {gather*} \int x^2\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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