Optimal. Leaf size=191 \[ \frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {3}{2} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^2 \left (a+b \tanh ^{-1}(c x)\right )^3}{2 c}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c}-\frac {3 b^3 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{2 c}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c}+\frac {3 b^3 \text {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 c} \]
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Rubi [A]
time = 0.23, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 10, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6065, 6021,
6131, 6055, 2449, 2352, 1600, 6095, 6205, 6745} \begin {gather*} -\frac {3 b^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c}-\frac {3 b^2 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c}+\frac {3}{2} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {(c x+1)^2 \left (a+b \tanh ^{-1}(c x)\right )^3}{2 c}-\frac {3 b \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c}-\frac {3 b^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{2 c}+\frac {3 b^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 1600
Rule 2352
Rule 2449
Rule 6021
Rule 6055
Rule 6065
Rule 6095
Rule 6131
Rule 6205
Rule 6745
Rubi steps
\begin {align*} \int (1+c x) \left (a+b \tanh ^{-1}(c x)\right )^3 \, dx &=\frac {(1+c x)^2 \left (a+b \tanh ^{-1}(c x)\right )^3}{2 c}-\frac {1}{2} (3 b) \int \left (-\left (a+b \tanh ^{-1}(c x)\right )^2+\frac {2 (1+c x) \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2}\right ) \, dx\\ &=\frac {(1+c x)^2 \left (a+b \tanh ^{-1}(c x)\right )^3}{2 c}+\frac {1}{2} (3 b) \int \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx-(3 b) \int \frac {(1+c x) \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx\\ &=\frac {3}{2} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^2 \left (a+b \tanh ^{-1}(c x)\right )^3}{2 c}-(3 b) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{1-c x} \, dx-\left (3 b^2 c\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}+\frac {3}{2} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^2 \left (a+b \tanh ^{-1}(c x)\right )^3}{2 c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c}-\left (3 b^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx+\left (6 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\\ &=\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {3}{2} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^2 \left (a+b \tanh ^{-1}(c x)\right )^3}{2 c}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c}+\left (3 b^3\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx+\left (3 b^3\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {3}{2} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^2 \left (a+b \tanh ^{-1}(c x)\right )^3}{2 c}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c}+\frac {3 b^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 c}-\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}\\ &=\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {3}{2} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^2 \left (a+b \tanh ^{-1}(c x)\right )^3}{2 c}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}-\frac {3 b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c}-\frac {3 b^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{2 c}-\frac {3 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c}+\frac {3 b^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 c}\\ \end {align*}
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Mathematica [A]
time = 0.48, size = 334, normalized size = 1.75 \begin {gather*} \frac {4 a^3 c x+6 a^2 b c x+2 a^3 c^2 x^2+12 a^2 b c x \tanh ^{-1}(c x)+12 a b^2 c x \tanh ^{-1}(c x)+6 a^2 b c^2 x^2 \tanh ^{-1}(c x)-18 a b^2 \tanh ^{-1}(c x)^2-6 b^3 \tanh ^{-1}(c x)^2+12 a b^2 c x \tanh ^{-1}(c x)^2+6 b^3 c x \tanh ^{-1}(c x)^2+6 a b^2 c^2 x^2 \tanh ^{-1}(c x)^2-6 b^3 \tanh ^{-1}(c x)^3+4 b^3 c x \tanh ^{-1}(c x)^3+2 b^3 c^2 x^2 \tanh ^{-1}(c x)^3-24 a b^2 \tanh ^{-1}(c x) \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )-12 b^3 \tanh ^{-1}(c x) \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )-12 b^3 \tanh ^{-1}(c x)^2 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )+9 a^2 b \log (1-c x)+3 a^2 b \log (1+c x)+6 a b^2 \log \left (1-c^2 x^2\right )+6 b^2 \left (2 a+b+2 b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+6 b^3 \text {PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )}{4 c} \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 = 10.72, size = 6152, normalized size = 32.21
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(6152\) |
default | \(\text {Expression too large to display}\) | \(6152\) |
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 \left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3} \left (c x + 1\right )\, 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 {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3\,\left (c\,x+1\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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