Optimal. Leaf size=306 \[ 3 a b^2 x+\frac {b^3 x}{4}-\frac {b^3 \tanh ^{-1}(c x)}{4 c}+3 b^3 x \tanh ^{-1}(c x)+\frac {1}{4} b^2 c x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4 b \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {21}{4} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} b c x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} b c^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}-\frac {11 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}-\frac {6 b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c}+\frac {3 b^3 \log \left (1-c^2 x^2\right )}{2 c}-\frac {11 b^3 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{2 c}-\frac {6 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c}+\frac {3 b^3 \text {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{c} \]
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Rubi [A]
time = 0.48, antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps
used = 26, number of rules used = 15, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used =
{6065, 6021, 6131, 6055, 2449, 2352, 6037, 6127, 266, 6095, 327, 212, 1600, 6205, 6745}
\begin {gather*} -\frac {6 b^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c}+\frac {1}{4} b^2 c x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac {11 b^2 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c}+3 a b^2 x+\frac {1}{4} b c^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} b c x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(c x+1)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}+\frac {21}{4} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {4 b \left (a+b \tanh ^{-1}(c x)\right )^2}{c}-\frac {6 b \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {3 b^3 \log \left (1-c^2 x^2\right )}{2 c}-\frac {11 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 )}{c}+3 b^3 x \tanh ^{-1}(c x)-\frac {b^3 \tanh ^{-1}(c x)}{4 c}+\frac {b^3 x}{4} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 266
Rule 327
Rule 1600
Rule 2352
Rule 2449
Rule 6021
Rule 6037
Rule 6055
Rule 6065
Rule 6095
Rule 6127
Rule 6131
Rule 6205
Rule 6745
Rubi steps
\begin {align*} \int (1+c x)^3 \left (a+b \tanh ^{-1}(c x)\right )^3 \, dx &=\frac {(1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}-\frac {1}{4} (3 b) \int \left (-7 \left (a+b \tanh ^{-1}(c x)\right )^2-4 c x \left (a+b \tanh ^{-1}(c x)\right )^2-c^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {8 (1+c x) \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2}\right ) \, dx\\ &=\frac {(1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}+\frac {1}{4} (21 b) \int \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx-(6 b) \int \frac {(1+c x) \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx+(3 b c) \int x \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+\frac {1}{4} \left (3 b c^2\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx\\ &=\frac {21}{4} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} b c x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} b c^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}-(6 b) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{1-c x} \, dx-\frac {1}{2} \left (21 b^2 c\right ) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\left (3 b^2 c^2\right ) \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\frac {1}{2} \left (b^2 c^3\right ) \int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac {21 b \left (a+b \tanh ^{-1}(c x)\right )^2}{4 c}+\frac {21}{4} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} b c x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} b c^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}-\frac {6 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 \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\left (3 b^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx-\frac {1}{2} \left (21 b^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx+\left (12 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 {1}{2} \left (b^2 c\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac {1}{2} \left (b^2 c\right ) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=3 a b^2 x+\frac {1}{4} b^2 c x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4 b \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {21}{4} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} b c x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} b c^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}-\frac {21 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{2 c}-\frac {6 b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c}-\frac {6 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c}-\frac {1}{2} b^2 \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx+\left (3 b^3\right ) \int \tanh ^{-1}(c x) \, dx+\left (6 b^3\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx+\frac {1}{2} \left (21 b^3\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\frac {1}{4} \left (b^3 c^2\right ) \int \frac {x^2}{1-c^2 x^2} \, dx\\ &=3 a b^2 x+\frac {b^3 x}{4}+3 b^3 x \tanh ^{-1}(c x)+\frac {1}{4} b^2 c x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4 b \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {21}{4} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} b c x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} b c^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}-\frac {11 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}-\frac {6 b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c}-\frac {6 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 )}{c}-\frac {1}{4} b^3 \int \frac {1}{1-c^2 x^2} \, dx+\frac {1}{2} b^3 \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\frac {\left (21 b^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{2 c}-\left (3 b^3 c\right ) \int \frac {x}{1-c^2 x^2} \, dx\\ &=3 a b^2 x+\frac {b^3 x}{4}-\frac {b^3 \tanh ^{-1}(c x)}{4 c}+3 b^3 x \tanh ^{-1}(c x)+\frac {1}{4} b^2 c x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4 b \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {21}{4} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} b c x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} b c^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}-\frac {11 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}-\frac {6 b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c}+\frac {3 b^3 \log \left (1-c^2 x^2\right )}{2 c}-\frac {21 b^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{4 c}-\frac {6 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 )}{c}-\frac {b^3 \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{2 c}\\ &=3 a b^2 x+\frac {b^3 x}{4}-\frac {b^3 \tanh ^{-1}(c x)}{4 c}+3 b^3 x \tanh ^{-1}(c x)+\frac {1}{4} b^2 c x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {4 b \left (a+b \tanh ^{-1}(c x)\right )^2}{c}+\frac {21}{4} b x \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {3}{2} b c x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{4} b c^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(1+c x)^4 \left (a+b \tanh ^{-1}(c x)\right )^3}{4 c}-\frac {11 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}-\frac {6 b \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (\frac {2}{1-c x}\right )}{c}+\frac {3 b^3 \log \left (1-c^2 x^2\right )}{2 c}-\frac {11 b^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{2 c}-\frac {6 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 )}{c}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(644\) vs. \(2(306)=612\).
time = 0.89, size = 644, normalized size = 2.10 \begin {gather*} \frac {-2 a b^2+8 a^3 c x+42 a^2 b c x+24 a b^2 c x+2 b^3 c x+12 a^3 c^2 x^2+12 a^2 b c^2 x^2+2 a b^2 c^2 x^2+8 a^3 c^3 x^3+2 a^2 b c^3 x^3+2 a^3 c^4 x^4-24 a b^2 \tanh ^{-1}(c x)-2 b^3 \tanh ^{-1}(c x)+24 a^2 b c x \tanh ^{-1}(c x)+84 a b^2 c x \tanh ^{-1}(c x)+24 b^3 c x \tanh ^{-1}(c x)+36 a^2 b c^2 x^2 \tanh ^{-1}(c x)+24 a b^2 c^2 x^2 \tanh ^{-1}(c x)+2 b^3 c^2 x^2 \tanh ^{-1}(c x)+24 a^2 b c^3 x^3 \tanh ^{-1}(c x)+4 a b^2 c^3 x^3 \tanh ^{-1}(c x)+6 a^2 b c^4 x^4 \tanh ^{-1}(c x)-90 a b^2 \tanh ^{-1}(c x)^2-56 b^3 \tanh ^{-1}(c x)^2+24 a b^2 c x \tanh ^{-1}(c x)^2+42 b^3 c x \tanh ^{-1}(c x)^2+36 a b^2 c^2 x^2 \tanh ^{-1}(c x)^2+12 b^3 c^2 x^2 \tanh ^{-1}(c x)^2+24 a b^2 c^3 x^3 \tanh ^{-1}(c x)^2+2 b^3 c^3 x^3 \tanh ^{-1}(c x)^2+6 a b^2 c^4 x^4 \tanh ^{-1}(c x)^2-30 b^3 \tanh ^{-1}(c x)^3+8 b^3 c x \tanh ^{-1}(c x)^3+12 b^3 c^2 x^2 \tanh ^{-1}(c x)^3+8 b^3 c^3 x^3 \tanh ^{-1}(c x)^3+2 b^3 c^4 x^4 \tanh ^{-1}(c x)^3-96 a b^2 \tanh ^{-1}(c x) \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )-88 b^3 \tanh ^{-1}(c x) \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )-48 b^3 \tanh ^{-1}(c x)^2 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )+45 a^2 b \log (1-c x)+3 a^2 b \log (1+c x)+44 a b^2 \log \left (1-c^2 x^2\right )+12 b^3 \log \left (1-c^2 x^2\right )+4 b^2 \left (12 a+11 b+12 b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+24 b^3 \text {PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )}{8 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 = 5.49, size = 883, normalized size = 2.89
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(883\) |
default | \(\text {Expression too large to display}\) | \(883\) |
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 )^{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 = -0.00 \begin {gather*} \int {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3\,{\left (c\,x+1\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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