Optimal. Leaf size=244 \[ \frac {3 b e \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2}+\frac {3 b e x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {d \left (a+b \tanh ^{-1}(c x)\right )^3}{c}-\frac {\left (d^2+\frac {e^2}{c^2}\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{2 e}+\frac {(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )^3}{2 e}-\frac {3 b^2 e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^2}-\frac {3 b d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c}-\frac {3 b^3 e \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{2 c^2}-\frac {3 b^2 d \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c}+\frac {3 b^3 d \text {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 c} \]
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Rubi [A]
time = 0.43, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps
used = 14, 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, 6195, 6095, 6205, 6745} \begin {gather*} -\frac {3 b^2 e \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^2}-\frac {3 b^2 d \text {Li}_2\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c}-\frac {\left (\frac {e^2}{c^2}+d^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{2 e}+\frac {3 b e \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2}+\frac {(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )^3}{2 e}+\frac {d \left (a+b \tanh ^{-1}(c x)\right )^3}{c}-\frac {3 b d \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {3 b e x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}-\frac {3 b^3 e \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{2 c^2}+\frac {3 b^3 d \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 2352
Rule 2449
Rule 6021
Rule 6055
Rule 6065
Rule 6095
Rule 6131
Rule 6195
Rule 6205
Rule 6745
Rubi steps
\begin {align*} \int (d+e x) \left (a+b \tanh ^{-1}(c x)\right )^3 \, dx &=\frac {(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )^3}{2 e}-\frac {(3 b c) \int \left (-\frac {e^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2}+\frac {\left (c^2 d^2+e^2+2 c^2 d e x\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 \left (1-c^2 x^2\right )}\right ) \, dx}{2 e}\\ &=\frac {(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )^3}{2 e}-\frac {(3 b) \int \frac {\left (c^2 d^2+e^2+2 c^2 d e x\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx}{2 c e}+\frac {(3 b e) \int \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{2 c}\\ &=\frac {3 b e x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )^3}{2 e}-\frac {(3 b) \int \left (\frac {c^2 d^2 \left (1+\frac {e^2}{c^2 d^2}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2}+\frac {2 c^2 d e x \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2}\right ) \, dx}{2 c e}-\left (3 b^2 e\right ) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac {3 b e \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2}+\frac {3 b e x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )^3}{2 e}-(3 b c d) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx-\frac {\left (3 b^2 e\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{c}-\frac {\left (3 b \left (c^2 d^2+e^2\right )\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx}{2 c e}\\ &=\frac {3 b e \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2}+\frac {3 b e x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {d \left (a+b \tanh ^{-1}(c x)\right )^3}{c}-\frac {\left (d^2+\frac {e^2}{c^2}\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{2 e}+\frac {(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )^3}{2 e}-\frac {3 b^2 e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^2}-(3 b d) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{1-c x} \, dx+\frac {\left (3 b^3 e\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{c}\\ &=\frac {3 b e \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2}+\frac {3 b e x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {d \left (a+b \tanh ^{-1}(c x)\right )^3}{c}-\frac {\left (d^2+\frac {e^2}{c^2}\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{2 e}+\frac {(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )^3}{2 e}-\frac {3 b^2 e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^2}-\frac {3 b d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c}+\left (6 b^2 d\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 {\left (3 b^3 e\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 e \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2}+\frac {3 b e x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {d \left (a+b \tanh ^{-1}(c x)\right )^3}{c}-\frac {\left (d^2+\frac {e^2}{c^2}\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{2 e}+\frac {(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )^3}{2 e}-\frac {3 b^2 e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^2}-\frac {3 b d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c}-\frac {3 b^3 e \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{2 c^2}-\frac {3 b^2 d \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c}+\left (3 b^3 d\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=\frac {3 b e \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^2}+\frac {3 b e x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {d \left (a+b \tanh ^{-1}(c x)\right )^3}{c}-\frac {\left (d^2+\frac {e^2}{c^2}\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{2 e}+\frac {(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )^3}{2 e}-\frac {3 b^2 e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^2}-\frac {3 b d \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c}-\frac {3 b^3 e \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{2 c^2}-\frac {3 b^2 d \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c}+\frac {3 b^3 d \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 c}\\ \end {align*}
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Mathematica [A]
time = 0.44, size = 331, normalized size = 1.36 \begin {gather*} \frac {2 a^2 c (2 a c d+3 b e) x+2 a^3 c^2 e x^2+6 a^2 b c^2 x (2 d+e x) \tanh ^{-1}(c x)+3 a^2 b (2 c d+e) \log (1-c x)+3 a^2 b (2 c d-e) \log (1+c x)+6 a b^2 e \left (2 c x \tanh ^{-1}(c x)+\left (-1+c^2 x^2\right ) \tanh ^{-1}(c x)^2+\log \left (1-c^2 x^2\right )\right )-2 b^3 e \left (\tanh ^{-1}(c x) \left ((3-3 c x) \tanh ^{-1}(c x)+\left (1-c^2 x^2\right ) \tanh ^{-1}(c x)^2+6 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )\right )-3 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )\right )+12 a b^2 c d \left (\tanh ^{-1}(c x) \left ((-1+c x) \tanh ^{-1}(c x)-2 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )\right )+\text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )\right )+4 b^3 c d \left (\tanh ^{-1}(c x)^2 \left ((-1+c x) \tanh ^{-1}(c x)-3 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )\right )+3 \tanh ^{-1}(c x) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+\frac {3}{2} \text {PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )\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 = 6.66, size = 12134, normalized size = 49.73
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(12134\) |
default | \(\text {Expression too large to display}\) | \(12134\) |
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 (d + e x\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 = -0.00 \begin {gather*} \int {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3\,\left (d+e\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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