Optimal. Leaf size=387 \[ \frac {a b^2 e^2 x}{c^2}+\frac {b^3 e^2 x \tanh ^{-1}(c x)}{c^2}+\frac {3 b d e \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2}-\frac {b e^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac {3 b d e x \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {b e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {\left (3 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}-\frac {d \left (d^2+\frac {3 e^2}{c^2}\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}+\frac {(d+e x)^3 \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}-\frac {6 b^2 d e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^2}-\frac {b \left (3 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c^3}+\frac {b^3 e^2 \log \left (1-c^2 x^2\right )}{2 c^3}-\frac {3 b^3 d e \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^2}-\frac {b^2 \left (3 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c^3}+\frac {b^3 \left (3 c^2 d^2+e^2\right ) \text {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{2 c^3} \]
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Rubi [A]
time = 0.55, antiderivative size = 387, normalized size of antiderivative = 1.00, number
of steps used = 20, number of rules used = 13, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules
used = {6065, 6021, 6131, 6055, 2449, 2352, 6037, 6127, 266, 6095, 6195, 6205, 6745}
\begin {gather*} -\frac {6 b^2 d e \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^2}+\frac {a b^2 e^2 x}{c^2}-\frac {b^2 \left (3 c^2 d^2+e^2\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c^3}-\frac {b e^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}-\frac {d \left (\frac {3 e^2}{c^2}+d^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}+\frac {3 b d e \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2}+\frac {\left (3 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}-\frac {b \left (3 c^2 d^2+e^2\right ) \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^3}+\frac {3 b d e x \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {(d+e x)^3 \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}+\frac {b e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}-\frac {3 b^3 d e \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c^2}+\frac {b^3 e^2 x \tanh ^{-1}(c x)}{c^2}+\frac {b^3 \left (3 c^2 d^2+e^2\right ) \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 c^3}+\frac {b^3 e^2 \log \left (1-c^2 x^2\right )}{2 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 2352
Rule 2449
Rule 6021
Rule 6037
Rule 6055
Rule 6065
Rule 6095
Rule 6127
Rule 6131
Rule 6195
Rule 6205
Rule 6745
Rubi steps
\begin {align*} \int (d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )^3 \, dx &=\frac {(d+e x)^3 \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}-\frac {(b c) \int \left (-\frac {3 d e^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2}-\frac {e^3 x \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2}+\frac {\left (c^2 d^3+3 d e^2+e \left (3 c^2 d^2+e^2\right ) x\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2 \left (1-c^2 x^2\right )}\right ) \, dx}{e}\\ &=\frac {(d+e x)^3 \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}-\frac {b \int \frac {\left (c^2 d^3+3 d e^2+e \left (3 c^2 d^2+e^2\right ) x\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx}{c e}+\frac {(3 b d e) \int \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{c}+\frac {\left (b e^2\right ) \int x \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{c}\\ &=\frac {3 b d e x \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {b e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {(d+e x)^3 \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}-\frac {b \int \left (\frac {c^2 d^3 \left (1+\frac {3 e^2}{c^2 d^2}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2}+\frac {e \left (3 c^2 d^2+e^2\right ) x \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2}\right ) \, dx}{c e}-\left (6 b^2 d e\right ) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\left (b^2 e^2\right ) \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac {3 b d e \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2}+\frac {3 b d e x \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {b e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {(d+e x)^3 \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}-\frac {\left (6 b^2 d e\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{c}+\frac {\left (b^2 e^2\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c^2}-\frac {\left (b^2 e^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{c^2}-\left (b d \left (\frac {c d^2}{e}+\frac {3 e}{c}\right )\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx-\frac {\left (b \left (3 c^2 d^2+e^2\right )\right ) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx}{c}\\ &=\frac {a b^2 e^2 x}{c^2}+\frac {3 b d e \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2}-\frac {b e^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac {3 b d e x \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {b e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {\left (3 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}-\frac {d \left (d^2+\frac {3 e^2}{c^2}\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}+\frac {(d+e x)^3 \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}-\frac {6 b^2 d e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^2}+\frac {\left (6 b^3 d e\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{c}+\frac {\left (b^3 e^2\right ) \int \tanh ^{-1}(c x) \, dx}{c^2}-\frac {\left (b \left (3 c^2 d^2+e^2\right )\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{1-c x} \, dx}{c^2}\\ &=\frac {a b^2 e^2 x}{c^2}+\frac {b^3 e^2 x \tanh ^{-1}(c x)}{c^2}+\frac {3 b d e \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2}-\frac {b e^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac {3 b d e x \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {b e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {\left (3 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}-\frac {d \left (d^2+\frac {3 e^2}{c^2}\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}+\frac {(d+e x)^3 \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}-\frac {6 b^2 d e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^2}-\frac {b \left (3 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c^3}-\frac {\left (6 b^3 d e\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{c^2}-\frac {\left (b^3 e^2\right ) \int \frac {x}{1-c^2 x^2} \, dx}{c}+\frac {\left (2 b^2 \left (3 c^2 d^2+e^2\right )\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 e^2 x}{c^2}+\frac {b^3 e^2 x \tanh ^{-1}(c x)}{c^2}+\frac {3 b d e \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2}-\frac {b e^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac {3 b d e x \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {b e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {\left (3 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}-\frac {d \left (d^2+\frac {3 e^2}{c^2}\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}+\frac {(d+e x)^3 \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}-\frac {6 b^2 d e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^2}-\frac {b \left (3 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c^3}+\frac {b^3 e^2 \log \left (1-c^2 x^2\right )}{2 c^3}-\frac {3 b^3 d e \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c^2}-\frac {b^2 \left (3 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c^3}+\frac {\left (b^3 \left (3 c^2 d^2+e^2\right )\right ) \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 e^2 x}{c^2}+\frac {b^3 e^2 x \tanh ^{-1}(c x)}{c^2}+\frac {3 b d e \left (a+b \tanh ^{-1}(c x)\right )^2}{c^2}-\frac {b e^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^3}+\frac {3 b d e x \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {b e^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c}+\frac {\left (3 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{3 c^3}-\frac {d \left (d^2+\frac {3 e^2}{c^2}\right ) \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}+\frac {(d+e x)^3 \left (a+b \tanh ^{-1}(c x)\right )^3}{3 e}-\frac {6 b^2 d e \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c^2}-\frac {b \left (3 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c^3}+\frac {b^3 e^2 \log \left (1-c^2 x^2\right )}{2 c^3}-\frac {3 b^3 d e \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c^2}-\frac {b^2 \left (3 c^2 d^2+e^2\right ) \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c^3}+\frac {b^3 \left (3 c^2 d^2+e^2\right ) \text {Li}_3\left (1-\frac {2}{1-c x}\right )}{2 c^3}\\ \end {align*}
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Mathematica [A]
time = 1.28, size = 591, normalized size = 1.53 \begin {gather*} \frac {6 a^2 c^2 d (a c d+3 b e) x+3 a^2 c^2 e (2 a c d+b e) x^2+2 a^3 c^3 e^2 x^3+6 a^2 b c^3 x \left (3 d^2+3 d e x+e^2 x^2\right ) \tanh ^{-1}(c x)+3 a^2 b \left (3 c^2 d^2+3 c d e+e^2\right ) \log (1-c x)+3 a^2 b \left (3 c^2 d^2-3 c d e+e^2\right ) \log (1+c x)+18 a b^2 c d 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 )-6 b^3 c d 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 )+6 a b^2 e^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 )+18 a b^2 c^2 d^2 \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 )+6 b^3 c^2 d^2 \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 )+b^3 e^2 \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 = 16.24, size = 4520, normalized size = 11.68
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(4520\) |
default | \(\text {Expression too large to display}\) | \(4520\) |
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 )^{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 = -0.00 \begin {gather*} \int {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3\,{\left (d+e\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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