Optimal. Leaf size=196 \[ \frac {a b d x}{c}+\frac {b^2 d x}{3 c}-\frac {b^2 d \tanh ^{-1}(c x)}{3 c^2}+\frac {b^2 d x \tanh ^{-1}(c x)}{c}+\frac {1}{3} b d x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^2}+\frac {1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c^2}+\frac {b^2 d \log \left (1-c^2 x^2\right )}{2 c^2}-\frac {b^2 d \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{3 c^2} \]
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Rubi [A]
time = 0.29, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 12, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6087, 6037,
6127, 6021, 266, 6095, 327, 212, 6131, 6055, 2449, 2352} \begin {gather*} -\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^2}-\frac {2 b d \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{3 c^2}+\frac {1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} b d x^2 \left (a+b \tanh ^{-1}(c x)\right )+\frac {a b d x}{c}-\frac {b^2 d \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{3 c^2}+\frac {b^2 d \log \left (1-c^2 x^2\right )}{2 c^2}-\frac {b^2 d \tanh ^{-1}(c x)}{3 c^2}+\frac {b^2 d x}{3 c}+\frac {b^2 d x \tanh ^{-1}(c x)}{c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 266
Rule 327
Rule 2352
Rule 2449
Rule 6021
Rule 6037
Rule 6055
Rule 6087
Rule 6095
Rule 6127
Rule 6131
Rubi steps
\begin {align*} \int x (d+c d x) \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx &=\int \left (d x \left (a+b \tanh ^{-1}(c x)\right )^2+c d x^2 \left (a+b \tanh ^{-1}(c x)\right )^2\right ) \, dx\\ &=d \int x \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+(c d) \int x^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx\\ &=\frac {1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2-(b c d) \int \frac {x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\frac {1}{3} \left (2 b c^2 d\right ) \int \frac {x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac {1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} (2 b d) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac {1}{3} (2 b d) \int \frac {x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx+\frac {(b d) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{c}-\frac {(b d) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{c}\\ &=\frac {a b d x}{c}+\frac {1}{3} b d x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^2}+\frac {1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {(2 b d) \int \frac {a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{3 c}+\frac {\left (b^2 d\right ) \int \tanh ^{-1}(c x) \, dx}{c}-\frac {1}{3} \left (b^2 c d\right ) \int \frac {x^2}{1-c^2 x^2} \, dx\\ &=\frac {a b d x}{c}+\frac {b^2 d x}{3 c}+\frac {b^2 d x \tanh ^{-1}(c x)}{c}+\frac {1}{3} b d x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^2}+\frac {1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c^2}-\left (b^2 d\right ) \int \frac {x}{1-c^2 x^2} \, dx-\frac {\left (b^2 d\right ) \int \frac {1}{1-c^2 x^2} \, dx}{3 c}+\frac {\left (2 b^2 d\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx}{3 c}\\ &=\frac {a b d x}{c}+\frac {b^2 d x}{3 c}-\frac {b^2 d \tanh ^{-1}(c x)}{3 c^2}+\frac {b^2 d x \tanh ^{-1}(c x)}{c}+\frac {1}{3} b d x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^2}+\frac {1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c^2}+\frac {b^2 d \log \left (1-c^2 x^2\right )}{2 c^2}-\frac {\left (2 b^2 d\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )}{3 c^2}\\ &=\frac {a b d x}{c}+\frac {b^2 d x}{3 c}-\frac {b^2 d \tanh ^{-1}(c x)}{3 c^2}+\frac {b^2 d x \tanh ^{-1}(c x)}{c}+\frac {1}{3} b d x^2 \left (a+b \tanh ^{-1}(c x)\right )-\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{6 c^2}+\frac {1}{2} d x^2 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {1}{3} c d x^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{3 c^2}+\frac {b^2 d \log \left (1-c^2 x^2\right )}{2 c^2}-\frac {b^2 d \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{3 c^2}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 201, normalized size = 1.03 \begin {gather*} \frac {d \left (6 a b c x+2 b^2 c x+3 a^2 c^2 x^2+2 a b c^2 x^2+2 a^2 c^3 x^3+b^2 \left (-5+3 c^2 x^2+2 c^3 x^3\right ) \tanh ^{-1}(c x)^2+2 b \tanh ^{-1}(c x) \left (a c^2 x^2 (3+2 c x)+b \left (-1+3 c x+c^2 x^2\right )-2 b \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )\right )+3 a b \log (1-c x)-3 a b \log (1+c x)+3 b^2 \log \left (1-c^2 x^2\right )+2 a b \log \left (-1+c^2 x^2\right )+2 b^2 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )\right )}{6 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 322, normalized size = 1.64
method | result | size |
derivativedivides | \(\frac {d \,a^{2} \left (\frac {1}{3} x^{3} c^{3}+\frac {1}{2} c^{2} x^{2}\right )+\frac {d \,b^{2} \arctanh \left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {d \,b^{2} \arctanh \left (c x \right )^{2} c^{2} x^{2}}{2}+\frac {d \,b^{2} \arctanh \left (c x \right ) c^{2} x^{2}}{3}+d \,b^{2} \arctanh \left (c x \right ) c x +\frac {5 d \,b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{6}-\frac {d \,b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{6}+\frac {5 d \,b^{2} \ln \left (c x -1\right )^{2}}{24}-\frac {d \,b^{2} \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )}{3}-\frac {5 d \,b^{2} \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{12}-\frac {d \,b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{12}+\frac {d \,b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{12}+\frac {d \,b^{2} \ln \left (c x +1\right )^{2}}{24}+\frac {d \,b^{2} c x}{3}+\frac {2 d \,b^{2} \ln \left (c x -1\right )}{3}+\frac {d \,b^{2} \ln \left (c x +1\right )}{3}+\frac {2 d a b \arctanh \left (c x \right ) c^{3} x^{3}}{3}+d a b \arctanh \left (c x \right ) c^{2} x^{2}+\frac {d a b \,c^{2} x^{2}}{3}+d a b c x +\frac {5 d a b \ln \left (c x -1\right )}{6}-\frac {d a b \ln \left (c x +1\right )}{6}}{c^{2}}\) | \(322\) |
default | \(\frac {d \,a^{2} \left (\frac {1}{3} x^{3} c^{3}+\frac {1}{2} c^{2} x^{2}\right )+\frac {d \,b^{2} \arctanh \left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {d \,b^{2} \arctanh \left (c x \right )^{2} c^{2} x^{2}}{2}+\frac {d \,b^{2} \arctanh \left (c x \right ) c^{2} x^{2}}{3}+d \,b^{2} \arctanh \left (c x \right ) c x +\frac {5 d \,b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{6}-\frac {d \,b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{6}+\frac {5 d \,b^{2} \ln \left (c x -1\right )^{2}}{24}-\frac {d \,b^{2} \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )}{3}-\frac {5 d \,b^{2} \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{12}-\frac {d \,b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{12}+\frac {d \,b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{12}+\frac {d \,b^{2} \ln \left (c x +1\right )^{2}}{24}+\frac {d \,b^{2} c x}{3}+\frac {2 d \,b^{2} \ln \left (c x -1\right )}{3}+\frac {d \,b^{2} \ln \left (c x +1\right )}{3}+\frac {2 d a b \arctanh \left (c x \right ) c^{3} x^{3}}{3}+d a b \arctanh \left (c x \right ) c^{2} x^{2}+\frac {d a b \,c^{2} x^{2}}{3}+d a b c x +\frac {5 d a b \ln \left (c x -1\right )}{6}-\frac {d a b \ln \left (c x +1\right )}{6}}{c^{2}}\) | \(322\) |
risch | \(\frac {a b d x}{c}-\frac {d c a b \ln \left (-c x +1\right ) x^{3}}{3}+\frac {b^{2} d x}{3 c}+\frac {d \,b^{2} \left (2 x^{3} c^{3}+3 c^{2} x^{2}-1\right ) \ln \left (c x +1\right )^{2}}{24 c^{2}}-\frac {d a b \ln \left (-c x +1\right ) x^{2}}{2}+\frac {5 d a b \ln \left (-c x +1\right )}{6 c^{2}}+\frac {d c \,b^{2} \ln \left (-c x +1\right )^{2} x^{3}}{12}-\frac {d c \,b^{2} \ln \left (-c x +1\right ) x^{3}}{18}-\frac {5 d \,b^{2} \ln \left (-c x +1\right ) x}{12 c}-\frac {d b \ln \left (-c x -1\right ) a}{6 c^{2}}-\frac {d \,b^{2} \ln \left (-c x +1\right ) \left (-c x +1\right )^{3}}{18 c^{2}}+\frac {5 d \,b^{2} \ln \left (-c x +1\right ) \left (-c x +1\right )^{2}}{24 c^{2}}-\frac {d \,b^{2} \ln \left (-c x +1\right ) \left (-c x +1\right )}{6 c^{2}}-\frac {d \,b^{2} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{3 c^{2}}+\frac {d \,b^{2} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{3 c^{2}}+\frac {d \,b^{2} \dilog \left (-\frac {c x}{2}+\frac {1}{2}\right )}{3 c^{2}}+\left (-\frac {d \,b^{2} x^{2} \left (2 c x +3\right ) \ln \left (-c x +1\right )}{12}-\frac {d b \left (-4 c^{3} x^{3} a -6 a \,c^{2} x^{2}-2 b \,c^{2} x^{2}-6 b c x -5 b \ln \left (-c x +1\right )\right )}{12 c^{2}}\right ) \ln \left (c x +1\right )-\frac {5 d \,b^{2} \ln \left (-c x +1\right )^{2}}{24 c^{2}}+\frac {d c \,x^{3} a^{2}}{3}+\frac {d \,b^{2} \ln \left (-c x +1\right )^{2} x^{2}}{8}-\frac {5 d \,b^{2} \ln \left (-c x +1\right ) x^{2}}{24}-\frac {5 d \,a^{2}}{6 c^{2}}+\frac {d \,x^{2} a^{2}}{2}+\frac {d b a \,x^{2}}{3}-\frac {4 d b a}{3 c^{2}}-\frac {d \,b^{2}}{3 c^{2}}+\frac {49 d \,b^{2} \ln \left (-c x +1\right )}{72 c^{2}}+\frac {d \,b^{2} \ln \left (-c x -1\right )}{3 c^{2}}\) | \(518\) |
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*} d \left (\int a^{2} x\, dx + \int a^{2} c x^{2}\, dx + \int b^{2} x \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b x \operatorname {atanh}{\left (c x \right )}\, dx + \int b^{2} c x^{2} \operatorname {atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b c x^{2} \operatorname {atanh}{\left (c x \right )}\, dx\right ) \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\,{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2\,\left (d+c\,d\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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