Optimal. Leaf size=197 \[ \frac {a b^2 x}{c^2}+\frac {b^3 x \tanh ^{-1}(c x)}{c^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c^3}+\frac {b^3 \log \left (1-c^2 x^2\right )}{2 c^3}-\frac {b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^3}+\frac {b^3 \text {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 c^3} \]
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Rubi [A]
time = 0.36, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6037, 6127,
6021, 266, 6095, 6131, 6055, 6205, 6745} \begin {gather*} -\frac {b^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3}+\frac {a b^2 x}{c^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}-\frac {b \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^3+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {b^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 c^3}+\frac {b^3 x \tanh ^{-1}(c x)}{c^2}+\frac {b^3 \log \left (1-c^2 x^2\right )}{2 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 6021
Rule 6037
Rule 6055
Rule 6095
Rule 6127
Rule 6131
Rule 6205
Rule 6745
Rubi steps
\begin {align*} \int x^2 \left (a+b \tanh ^{-1}(c x)\right )^3 \, dx &=\frac {1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^3-(b c) \int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx\\ &=\frac {1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^3+\frac {b \int x \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{c}-\frac {b \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx}{c}\\ &=\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^3-b^2 \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\frac {b \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{1-c x} \, dx}{c^2}\\ &=\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c^3}+\frac {b^2 \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c^2}-\frac {b^2 \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{c^2}+\frac {\left (2 b^2\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{c^2}\\ &=\frac {a b^2 x}{c^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c^3}-\frac {b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c^3}+\frac {b^3 \int \tanh ^{-1}(c x) \, dx}{c^2}+\frac {b^3 \int \frac {\text {Li}_2\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{c^2}\\ &=\frac {a b^2 x}{c^2}+\frac {b^3 x \tanh ^{-1}(c x)}{c^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c^3}-\frac {b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c^3}+\frac {b^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 c^3}-\frac {b^3 \int \frac {x}{1-c^2 x^2} \, dx}{c}\\ &=\frac {a b^2 x}{c^2}+\frac {b^3 x \tanh ^{-1}(c x)}{c^2}-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}+\frac {1}{3} x^3 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c^3}+\frac {b^3 \log \left (1-c^2 x^2\right )}{2 c^3}-\frac {b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c^3}+\frac {b^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.37, size = 250, normalized size = 1.27 \begin {gather*} \frac {3 a^2 b c^2 x^2+2 a^3 c^3 x^3+6 a^2 b c^3 x^3 \tanh ^{-1}(c x)+3 a^2 b \log \left (1-c^2 x^2\right )+6 a b^2 \left (c x+\left (-1+c^3 x^3\right ) \tanh ^{-1}(c x)^2+\tanh ^{-1}(c x) \left (-1+c^2 x^2-2 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )\right )+\text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )\right )+b^3 \left (6 c x \tanh ^{-1}(c x)-3 \tanh ^{-1}(c x)^2+3 c^2 x^2 \tanh ^{-1}(c x)^2-2 \tanh ^{-1}(c x)^3+2 c^3 x^3 \tanh ^{-1}(c x)^3-6 \tanh ^{-1}(c x)^2 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )+3 \log \left (1-c^2 x^2\right )+6 \tanh ^{-1}(c x) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+3 \text {PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )\right )}{6 c^3} \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 = 1.21, size = 1093, normalized size = 5.55
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1093\) |
default | \(\text {Expression too large to display}\) | \(1093\) |
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 )^{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^2\,{\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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