Optimal. Leaf size=247 \[ \frac {19 b^3 x}{60 c^5}+\frac {b^3 x^3}{60 c^3}-\frac {19 b^3 \tanh ^{-1}(c x)}{60 c^6}+\frac {4 b^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{15 c^4}+\frac {b^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )}{20 c^2}+\frac {23 b \left (a+b \tanh ^{-1}(c x)\right )^2}{30 c^6}+\frac {b x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^5}+\frac {b x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^3}+\frac {b x^5 \left (a+b \tanh ^{-1}(c x)\right )^2}{10 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{6 c^6}+\frac {1}{6} x^6 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {23 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{15 c^6}-\frac {23 b^3 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{30 c^6} \]
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Rubi [A]
time = 0.67, antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps
used = 33, number of rules used = 11, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {6037, 6127,
308, 212, 327, 6131, 6055, 2449, 2352, 6021, 6095} \begin {gather*} -\frac {23 b^2 \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{15 c^6}+\frac {4 b^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{15 c^4}+\frac {b^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )}{20 c^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{6 c^6}+\frac {23 b \left (a+b \tanh ^{-1}(c x)\right )^2}{30 c^6}+\frac {b x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^5}+\frac {b x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^3}+\frac {1}{6} x^6 \left (a+b \tanh ^{-1}(c x)\right )^3+\frac {b x^5 \left (a+b \tanh ^{-1}(c x)\right )^2}{10 c}-\frac {23 b^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{30 c^6}-\frac {19 b^3 \tanh ^{-1}(c x)}{60 c^6}+\frac {19 b^3 x}{60 c^5}+\frac {b^3 x^3}{60 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 308
Rule 327
Rule 2352
Rule 2449
Rule 6021
Rule 6037
Rule 6055
Rule 6095
Rule 6127
Rule 6131
Rubi steps
\begin {align*} \int x^5 \left (a+b \tanh ^{-1}(c x)\right )^3 \, dx &=\frac {1}{6} x^6 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {1}{2} (b c) \int \frac {x^6 \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx\\ &=\frac {1}{6} x^6 \left (a+b \tanh ^{-1}(c x)\right )^3+\frac {b \int x^4 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{2 c}-\frac {b \int \frac {x^4 \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx}{2 c}\\ &=\frac {b x^5 \left (a+b \tanh ^{-1}(c x)\right )^2}{10 c}+\frac {1}{6} x^6 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {1}{5} b^2 \int \frac {x^5 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx+\frac {b \int x^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{2 c^3}-\frac {b \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx}{2 c^3}\\ &=\frac {b x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^3}+\frac {b x^5 \left (a+b \tanh ^{-1}(c x)\right )^2}{10 c}+\frac {1}{6} x^6 \left (a+b \tanh ^{-1}(c x)\right )^3+\frac {b \int \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx}{2 c^5}-\frac {b \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{1-c^2 x^2} \, dx}{2 c^5}+\frac {b^2 \int x^3 \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{5 c^2}-\frac {b^2 \int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{5 c^2}-\frac {b^2 \int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 c^2}\\ &=\frac {b^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )}{20 c^2}+\frac {b x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^5}+\frac {b x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^3}+\frac {b x^5 \left (a+b \tanh ^{-1}(c x)\right )^2}{10 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{6 c^6}+\frac {1}{6} x^6 \left (a+b \tanh ^{-1}(c x)\right )^3+\frac {b^2 \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{5 c^4}-\frac {b^2 \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{5 c^4}+\frac {b^2 \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{3 c^4}-\frac {b^2 \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 c^4}-\frac {b^2 \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{c^4}-\frac {b^3 \int \frac {x^4}{1-c^2 x^2} \, dx}{20 c}\\ &=\frac {4 b^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{15 c^4}+\frac {b^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )}{20 c^2}+\frac {23 b \left (a+b \tanh ^{-1}(c x)\right )^2}{30 c^6}+\frac {b x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^5}+\frac {b x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^3}+\frac {b x^5 \left (a+b \tanh ^{-1}(c x)\right )^2}{10 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{6 c^6}+\frac {1}{6} x^6 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {b^2 \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{5 c^5}-\frac {b^2 \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{3 c^5}-\frac {b^2 \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{c^5}-\frac {b^3 \int \frac {x^2}{1-c^2 x^2} \, dx}{10 c^3}-\frac {b^3 \int \frac {x^2}{1-c^2 x^2} \, dx}{6 c^3}-\frac {b^3 \int \left (-\frac {1}{c^4}-\frac {x^2}{c^2}+\frac {1}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx}{20 c}\\ &=\frac {19 b^3 x}{60 c^5}+\frac {b^3 x^3}{60 c^3}+\frac {4 b^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{15 c^4}+\frac {b^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )}{20 c^2}+\frac {23 b \left (a+b \tanh ^{-1}(c x)\right )^2}{30 c^6}+\frac {b x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^5}+\frac {b x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^3}+\frac {b x^5 \left (a+b \tanh ^{-1}(c x)\right )^2}{10 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{6 c^6}+\frac {1}{6} x^6 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {23 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{15 c^6}-\frac {b^3 \int \frac {1}{1-c^2 x^2} \, dx}{20 c^5}-\frac {b^3 \int \frac {1}{1-c^2 x^2} \, dx}{10 c^5}-\frac {b^3 \int \frac {1}{1-c^2 x^2} \, dx}{6 c^5}+\frac {b^3 \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{5 c^5}+\frac {b^3 \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{3 c^5}+\frac {b^3 \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{c^5}\\ &=\frac {19 b^3 x}{60 c^5}+\frac {b^3 x^3}{60 c^3}-\frac {19 b^3 \tanh ^{-1}(c x)}{60 c^6}+\frac {4 b^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{15 c^4}+\frac {b^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )}{20 c^2}+\frac {23 b \left (a+b \tanh ^{-1}(c x)\right )^2}{30 c^6}+\frac {b x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^5}+\frac {b x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^3}+\frac {b x^5 \left (a+b \tanh ^{-1}(c x)\right )^2}{10 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{6 c^6}+\frac {1}{6} x^6 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {23 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{15 c^6}-\frac {b^3 \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{5 c^6}-\frac {b^3 \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{3 c^6}-\frac {b^3 \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{c^6}\\ &=\frac {19 b^3 x}{60 c^5}+\frac {b^3 x^3}{60 c^3}-\frac {19 b^3 \tanh ^{-1}(c x)}{60 c^6}+\frac {4 b^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{15 c^4}+\frac {b^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )}{20 c^2}+\frac {23 b \left (a+b \tanh ^{-1}(c x)\right )^2}{30 c^6}+\frac {b x \left (a+b \tanh ^{-1}(c x)\right )^2}{2 c^5}+\frac {b x^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^3}+\frac {b x^5 \left (a+b \tanh ^{-1}(c x)\right )^2}{10 c}-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{6 c^6}+\frac {1}{6} x^6 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {23 b^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{15 c^6}-\frac {23 b^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{30 c^6}\\ \end {align*}
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Mathematica [A]
time = 0.51, size = 305, normalized size = 1.23 \begin {gather*} \frac {-19 a b^2+30 a^2 b c x+19 b^3 c x+16 a b^2 c^2 x^2+10 a^2 b c^3 x^3+b^3 c^3 x^3+3 a b^2 c^4 x^4+6 a^2 b c^5 x^5+10 a^3 c^6 x^6+2 b^2 \left (b \left (-23+15 c x+5 c^3 x^3+3 c^5 x^5\right )+15 a \left (-1+c^6 x^6\right )\right ) \tanh ^{-1}(c x)^2+10 b^3 \left (-1+c^6 x^6\right ) \tanh ^{-1}(c x)^3+b \tanh ^{-1}(c x) \left (30 a^2 c^6 x^6+4 a b c x \left (15+5 c^2 x^2+3 c^4 x^4\right )+b^2 \left (-19+16 c^2 x^2+3 c^4 x^4\right )-92 b^2 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )\right )+15 a^2 b \log (1-c x)-15 a^2 b \log (1+c x)+46 a b^2 \log \left (1-c^2 x^2\right )+46 b^3 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )}{60 c^6} \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 = 2.72, size = 1242, normalized size = 5.03
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1242\) |
default | \(\text {Expression too large to display}\) | \(1242\) |
risch | \(\text {Expression too large to display}\) | \(1362\) |
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^{5} \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 = -0.00 \begin {gather*} \int x^5\,{\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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