Optimal. Leaf size=162 \[ \frac {3 b^2 x}{10 c^4}+\frac {b^2 x^3}{30 c^2}-\frac {3 b^2 \tanh ^{-1}(c x)}{10 c^5}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^3}+\frac {b x^4 \left (a+b \tanh ^{-1}(c x)\right )}{10 c}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{5 c^5}+\frac {1}{5} x^5 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {2 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{5 c^5}-\frac {b^2 \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{5 c^5} \]
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Rubi [A]
time = 0.21, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6037, 6127,
308, 212, 327, 6131, 6055, 2449, 2352} \begin {gather*} \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{5 c^5}-\frac {2 b \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{5 c^5}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^3}+\frac {1}{5} x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {b x^4 \left (a+b \tanh ^{-1}(c x)\right )}{10 c}-\frac {b^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{5 c^5}-\frac {3 b^2 \tanh ^{-1}(c x)}{10 c^5}+\frac {3 b^2 x}{10 c^4}+\frac {b^2 x^3}{30 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 308
Rule 327
Rule 2352
Rule 2449
Rule 6037
Rule 6055
Rule 6127
Rule 6131
Rubi steps
\begin {align*} \int x^4 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx &=\frac {1}{5} x^5 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {1}{5} (2 b c) \int \frac {x^5 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac {1}{5} x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {(2 b) \int x^3 \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{5 c}-\frac {(2 b) \int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{5 c}\\ &=\frac {b x^4 \left (a+b \tanh ^{-1}(c x)\right )}{10 c}+\frac {1}{5} x^5 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {1}{10} b^2 \int \frac {x^4}{1-c^2 x^2} \, dx+\frac {(2 b) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{5 c^3}-\frac {(2 b) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{5 c^3}\\ &=\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^3}+\frac {b x^4 \left (a+b \tanh ^{-1}(c x)\right )}{10 c}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{5 c^5}+\frac {1}{5} x^5 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {1}{10} b^2 \int \left (-\frac {1}{c^4}-\frac {x^2}{c^2}+\frac {1}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx-\frac {(2 b) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{5 c^4}-\frac {b^2 \int \frac {x^2}{1-c^2 x^2} \, dx}{5 c^2}\\ &=\frac {3 b^2 x}{10 c^4}+\frac {b^2 x^3}{30 c^2}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^3}+\frac {b x^4 \left (a+b \tanh ^{-1}(c x)\right )}{10 c}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{5 c^5}+\frac {1}{5} x^5 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {2 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{5 c^5}-\frac {b^2 \int \frac {1}{1-c^2 x^2} \, dx}{10 c^4}-\frac {b^2 \int \frac {1}{1-c^2 x^2} \, dx}{5 c^4}+\frac {\left (2 b^2\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{5 c^4}\\ &=\frac {3 b^2 x}{10 c^4}+\frac {b^2 x^3}{30 c^2}-\frac {3 b^2 \tanh ^{-1}(c x)}{10 c^5}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^3}+\frac {b x^4 \left (a+b \tanh ^{-1}(c x)\right )}{10 c}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{5 c^5}+\frac {1}{5} x^5 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {2 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{5 c^5}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{5 c^5}\\ &=\frac {3 b^2 x}{10 c^4}+\frac {b^2 x^3}{30 c^2}-\frac {3 b^2 \tanh ^{-1}(c x)}{10 c^5}+\frac {b x^2 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^3}+\frac {b x^4 \left (a+b \tanh ^{-1}(c x)\right )}{10 c}+\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{5 c^5}+\frac {1}{5} x^5 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {2 b \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{5 c^5}-\frac {b^2 \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{5 c^5}\\ \end {align*}
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Mathematica [A]
time = 0.33, size = 161, normalized size = 0.99 \begin {gather*} \frac {-9 a b+9 b^2 c x+6 a b c^2 x^2+b^2 c^3 x^3+3 a b c^4 x^4+6 a^2 c^5 x^5+6 b^2 \left (-1+c^5 x^5\right ) \tanh ^{-1}(c x)^2+3 b \tanh ^{-1}(c x) \left (4 a c^5 x^5+b \left (-3+2 c^2 x^2+c^4 x^4\right )-4 b \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )\right )+6 a b \log \left (-1+c^2 x^2\right )+6 b^2 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )}{30 c^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 281, normalized size = 1.73
method | result | size |
derivativedivides | \(\frac {\frac {c^{5} x^{5} a^{2}}{5}+\frac {c^{5} x^{5} b^{2} \arctanh \left (c x \right )^{2}}{5}+\frac {b^{2} \arctanh \left (c x \right ) c^{4} x^{4}}{10}+\frac {b^{2} \arctanh \left (c x \right ) c^{2} x^{2}}{5}+\frac {b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{5}+\frac {b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{5}+\frac {b^{2} c^{3} x^{3}}{30}+\frac {3 b^{2} c x}{10}+\frac {3 b^{2} \ln \left (c x -1\right )}{20}-\frac {3 b^{2} \ln \left (c x +1\right )}{20}-\frac {b^{2} \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )}{5}-\frac {b^{2} \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{10}+\frac {b^{2} \ln \left (c x -1\right )^{2}}{20}-\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{10}+\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{10}-\frac {b^{2} \ln \left (c x +1\right )^{2}}{20}+\frac {2 c^{5} x^{5} a b \arctanh \left (c x \right )}{5}+\frac {c^{4} x^{4} a b}{10}+\frac {a b \,c^{2} x^{2}}{5}+\frac {a b \ln \left (c x -1\right )}{5}+\frac {a b \ln \left (c x +1\right )}{5}}{c^{5}}\) | \(281\) |
default | \(\frac {\frac {c^{5} x^{5} a^{2}}{5}+\frac {c^{5} x^{5} b^{2} \arctanh \left (c x \right )^{2}}{5}+\frac {b^{2} \arctanh \left (c x \right ) c^{4} x^{4}}{10}+\frac {b^{2} \arctanh \left (c x \right ) c^{2} x^{2}}{5}+\frac {b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{5}+\frac {b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{5}+\frac {b^{2} c^{3} x^{3}}{30}+\frac {3 b^{2} c x}{10}+\frac {3 b^{2} \ln \left (c x -1\right )}{20}-\frac {3 b^{2} \ln \left (c x +1\right )}{20}-\frac {b^{2} \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )}{5}-\frac {b^{2} \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{10}+\frac {b^{2} \ln \left (c x -1\right )^{2}}{20}-\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{10}+\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{10}-\frac {b^{2} \ln \left (c x +1\right )^{2}}{20}+\frac {2 c^{5} x^{5} a b \arctanh \left (c x \right )}{5}+\frac {c^{4} x^{4} a b}{10}+\frac {a b \,c^{2} x^{2}}{5}+\frac {a b \ln \left (c x -1\right )}{5}+\frac {a b \ln \left (c x +1\right )}{5}}{c^{5}}\) | \(281\) |
risch | \(\frac {3 b^{2} x}{10 c^{4}}+\frac {b^{2} x^{3}}{30 c^{2}}+\frac {b^{2} \ln \left (-c x +1\right )^{2} x^{5}}{20}-\frac {b^{2} \ln \left (-c x +1\right )^{2}}{20 c^{5}}+\frac {137 b^{2} \ln \left (-c x +1\right )}{300 c^{5}}-\frac {137 a b}{150 c^{5}}-\frac {23 b^{2} \ln \left (c x -1\right )}{75 c^{5}}+\frac {b^{2} \ln \left (c x +1\right ) x^{2}}{10 c^{3}}+\frac {b^{2} \ln \left (c x +1\right ) x^{4}}{20 c}+\frac {a b \ln \left (-c x +1\right )}{5 c^{5}}-\frac {a b \ln \left (-c x +1\right ) x^{5}}{5}+\frac {a b \,x^{4}}{10 c}-\frac {b^{2} \ln \left (-c x +1\right ) x^{4}}{20 c}-\frac {b^{2} \ln \left (-c x +1\right ) x^{2}}{10 c^{3}}+\frac {a b \,x^{2}}{5 c^{3}}-\frac {a^{2}}{5 c^{5}}-\frac {413 b^{2}}{2250 c^{5}}-\frac {b^{2} \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )}{5 c^{5}}+\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{5 c^{5}}-\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{5 c^{5}}-\frac {b^{2} \ln \left (-c x +1\right ) \ln \left (c x +1\right ) x^{5}}{10}+\frac {b a \ln \left (c x +1\right )}{5 c^{5}}+\frac {b a \ln \left (c x +1\right ) x^{5}}{5}-\frac {b^{2} \ln \left (-c x +1\right ) \ln \left (c x +1\right )}{10 c^{5}}+\frac {a^{2} x^{5}}{5}+\frac {b^{2} \ln \left (c x +1\right )^{2}}{20 c^{5}}-\frac {3 b^{2} \ln \left (c x +1\right )}{20 c^{5}}+\frac {b^{2} \ln \left (c x +1\right )^{2} x^{5}}{20}\) | \(406\) |
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^{4} \left (a + b \operatorname {atanh}{\left (c x \right )}\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 = -0.01 \begin {gather*} \int x^4\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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