Optimal. Leaf size=123 \[ \frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2}+\frac {3 b x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c^2}+\frac {1}{2} x^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^2}-\frac {3 b^3 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{2 c^2} \]
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Rubi [A]
time = 0.18, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6037, 6127,
6021, 6131, 6055, 2449, 2352, 6095} \begin {gather*} -\frac {3 b^2 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^2}+\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c^2}+\frac {1}{2} x^2 \left (a+b \tanh ^{-1}(c x)\right )^3+\frac {3 b x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}-\frac {3 b^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{2 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2352
Rule 2449
Rule 6021
Rule 6037
Rule 6055
Rule 6095
Rule 6127
Rule 6131
Rubi steps
\begin {align*} \int x \left (a+b \tanh ^{-1}(c x)\right )^3 \, dx &=\frac {1}{2} x^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {1}{2} (3 b c) \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx\\ &=\frac {1}{2} x^2 \left (a+b \tanh ^{-1}(c x)\right )^3+\frac {(3 b) \int \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{2 c}-\frac {(3 b) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx}{2 c}\\ &=\frac {3 b x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c^2}+\frac {1}{2} x^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\left (3 b^2\right ) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2}+\frac {3 b x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c^2}+\frac {1}{2} x^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {\left (3 b^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{c}\\ &=\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2}+\frac {3 b x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c^2}+\frac {1}{2} x^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^2}+\frac {\left (3 b^3\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{c}\\ &=\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2}+\frac {3 b x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c^2}+\frac {1}{2} x^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^2}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{c^2}\\ &=\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2}+\frac {3 b x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{2 c^2}+\frac {1}{2} x^2 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^2}-\frac {3 b^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{2 c^2}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 161, normalized size = 1.31 \begin {gather*} \frac {6 b^2 (-1+c x) (a+b+a c x) \tanh ^{-1}(c x)^2+2 b^3 \left (-1+c^2 x^2\right ) \tanh ^{-1}(c x)^3+6 b \tanh ^{-1}(c x) \left (a c x (2 b+a c x)-2 b^2 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )\right )+a \left (6 a b c x+2 a^2 c^2 x^2+3 a b \log (1-c x)-3 a b \log (1+c x)+6 b^2 \log \left (1-c^2 x^2\right )\right )+6 b^3 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )}{4 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.49, size = 5831, normalized size = 47.41
method | result | size |
risch | \(\frac {3 b^{2} \left (-c x +1\right )^{2} \ln \left (-c x +1\right ) a}{8 c^{2}}+\frac {3 a^{2} b x}{2 c}+\frac {3 b^{3} \ln \left (-c x +1\right )^{2} x}{16 c}-\frac {9 b^{3} \ln \left (-c x +1\right ) x}{16 c}-\frac {3 a \,b^{2} \ln \left (-c x +1\right )^{2}}{8 c^{2}}+\frac {9 a \,b^{2} \ln \left (-c x +1\right )}{8 c^{2}}+\frac {3 a^{2} b \ln \left (-c x +1\right )}{4 c^{2}}-\frac {3 a \,b^{2} \ln \left (-c x +1\right ) x^{2}}{8}-\frac {3 a \,b^{2} \ln \left (-c x +1\right ) x}{4 c}-\frac {3 b^{3} \left (-c x +1\right )^{2} \ln \left (-c x +1\right )^{2}}{32 c^{2}}+\frac {3 b^{3} \left (-c x +1\right )^{2} \ln \left (-c x +1\right )}{32 c^{2}}-\frac {3 a^{2} b \ln \left (-c x -1\right )}{4 c^{2}}+\frac {3 a \,b^{2} \ln \left (-c x -1\right )}{2 c^{2}}-\frac {3 b^{3} \left (-c x +1\right ) \ln \left (-c x +1\right )}{4 c^{2}}+\frac {3 b^{3} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{2 c^{2}}-\frac {3 b^{3} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{2 c^{2}}+\frac {b^{3} \left (c^{2} x^{2}-1\right ) \ln \left (c x +1\right )^{3}}{16 c^{2}}+\frac {3 b^{2} \left (-b \,x^{2} \ln \left (-c x +1\right ) c^{2}+2 a \,c^{2} x^{2}+2 b c x +b \ln \left (-c x +1\right )-2 a +2 b \right ) \ln \left (c x +1\right )^{2}}{16 c^{2}}+\left (\frac {3 b^{3} \left (c^{2} x^{2}-1\right ) \ln \left (-c x +1\right )^{2}}{16 c^{2}}-\frac {3 b^{2} x \left (c x a +b \right ) \ln \left (-c x +1\right )}{4 c}-\frac {3 b \left (-c^{2} x^{2} a^{2}-2 a b c x -b \ln \left (-c x +1\right ) a -b^{2} \ln \left (-c x +1\right )\right )}{4 c^{2}}\right ) \ln \left (c x +1\right )-\frac {a^{3}}{2 c^{2}}+\frac {3 b^{3} \dilog \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2 c^{2}}-\frac {3 a^{2} b}{2 c^{2}}+\frac {3 b^{3} \ln \left (-c x +1\right )^{2} x^{2}}{32}+\frac {b^{3} \ln \left (-c x +1\right )^{3}}{16 c^{2}}-\frac {9 b^{3} \ln \left (-c x +1\right )^{2}}{32 c^{2}}-\frac {\ln \left (-c x +1\right )^{3} b^{3} x^{2}}{16}+\frac {21 b^{3} \ln \left (-c x +1\right )}{32 c^{2}}-\frac {3 x^{2} b^{3} \ln \left (-c x +1\right )}{32}+\frac {3 \ln \left (-c x +1\right )^{2} a \,b^{2} x^{2}}{8}-\frac {3 \ln \left (-c x +1\right ) a^{2} b \,x^{2}}{4}+\frac {a^{3} x^{2}}{2}\) | \(660\) |
derivativedivides | \(\text {Expression too large to display}\) | \(5831\) |
default | \(\text {Expression too large to display}\) | \(5831\) |
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 \left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3}\, 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\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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