Optimal. Leaf size=136 \[ \frac {(a+b x) \coth ^{-1}(a+b x)}{b^2}-\frac {a \coth ^{-1}(a+b x)^2}{b^2}-\frac {\left (1+a^2\right ) \coth ^{-1}(a+b x)^2}{2 b^2}+\frac {1}{2} x^2 \coth ^{-1}(a+b x)^2+\frac {2 a \coth ^{-1}(a+b x) \log \left (\frac {2}{1-a-b x}\right )}{b^2}+\frac {\log \left (1-(a+b x)^2\right )}{2 b^2}+\frac {a \text {PolyLog}\left (2,-\frac {1+a+b x}{1-a-b x}\right )}{b^2} \]
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Rubi [A]
time = 0.15, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6247, 6066,
6022, 266, 6196, 6096, 6132, 6056, 2449, 2352} \begin {gather*} -\frac {\left (a^2+1\right ) \coth ^{-1}(a+b x)^2}{2 b^2}+\frac {a \text {Li}_2\left (-\frac {a+b x+1}{-a-b x+1}\right )}{b^2}+\frac {\log \left (1-(a+b x)^2\right )}{2 b^2}-\frac {a \coth ^{-1}(a+b x)^2}{b^2}+\frac {(a+b x) \coth ^{-1}(a+b x)}{b^2}+\frac {2 a \log \left (\frac {2}{-a-b x+1}\right ) \coth ^{-1}(a+b x)}{b^2}+\frac {1}{2} x^2 \coth ^{-1}(a+b x)^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 2352
Rule 2449
Rule 6022
Rule 6056
Rule 6066
Rule 6096
Rule 6132
Rule 6196
Rule 6247
Rubi steps
\begin {align*} \int x \coth ^{-1}(a+b x)^2 \, dx &=\frac {\text {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right ) \coth ^{-1}(x)^2 \, dx,x,a+b x\right )}{b}\\ &=\frac {1}{2} x^2 \coth ^{-1}(a+b x)^2-\text {Subst}\left (\int \left (-\frac {\coth ^{-1}(x)}{b^2}+\frac {\left (1+a^2-2 a x\right ) \coth ^{-1}(x)}{b^2 \left (1-x^2\right )}\right ) \, dx,x,a+b x\right )\\ &=\frac {1}{2} x^2 \coth ^{-1}(a+b x)^2+\frac {\text {Subst}\left (\int \coth ^{-1}(x) \, dx,x,a+b x\right )}{b^2}-\frac {\text {Subst}\left (\int \frac {\left (1+a^2-2 a x\right ) \coth ^{-1}(x)}{1-x^2} \, dx,x,a+b x\right )}{b^2}\\ &=\frac {(a+b x) \coth ^{-1}(a+b x)}{b^2}+\frac {1}{2} x^2 \coth ^{-1}(a+b x)^2-\frac {\text {Subst}\left (\int \frac {x}{1-x^2} \, dx,x,a+b x\right )}{b^2}-\frac {\text {Subst}\left (\int \left (\frac {\left (1+a^2\right ) \coth ^{-1}(x)}{1-x^2}-\frac {2 a x \coth ^{-1}(x)}{1-x^2}\right ) \, dx,x,a+b x\right )}{b^2}\\ &=\frac {(a+b x) \coth ^{-1}(a+b x)}{b^2}+\frac {1}{2} x^2 \coth ^{-1}(a+b x)^2+\frac {\log \left (1-(a+b x)^2\right )}{2 b^2}+\frac {(2 a) \text {Subst}\left (\int \frac {x \coth ^{-1}(x)}{1-x^2} \, dx,x,a+b x\right )}{b^2}-\frac {\left (1+a^2\right ) \text {Subst}\left (\int \frac {\coth ^{-1}(x)}{1-x^2} \, dx,x,a+b x\right )}{b^2}\\ &=\frac {(a+b x) \coth ^{-1}(a+b x)}{b^2}-\frac {a \coth ^{-1}(a+b x)^2}{b^2}-\frac {\left (1+a^2\right ) \coth ^{-1}(a+b x)^2}{2 b^2}+\frac {1}{2} x^2 \coth ^{-1}(a+b x)^2+\frac {\log \left (1-(a+b x)^2\right )}{2 b^2}+\frac {(2 a) \text {Subst}\left (\int \frac {\coth ^{-1}(x)}{1-x} \, dx,x,a+b x\right )}{b^2}\\ &=\frac {(a+b x) \coth ^{-1}(a+b x)}{b^2}-\frac {a \coth ^{-1}(a+b x)^2}{b^2}-\frac {\left (1+a^2\right ) \coth ^{-1}(a+b x)^2}{2 b^2}+\frac {1}{2} x^2 \coth ^{-1}(a+b x)^2+\frac {2 a \coth ^{-1}(a+b x) \log \left (\frac {2}{1-a-b x}\right )}{b^2}+\frac {\log \left (1-(a+b x)^2\right )}{2 b^2}-\frac {(2 a) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,a+b x\right )}{b^2}\\ &=\frac {(a+b x) \coth ^{-1}(a+b x)}{b^2}-\frac {a \coth ^{-1}(a+b x)^2}{b^2}-\frac {\left (1+a^2\right ) \coth ^{-1}(a+b x)^2}{2 b^2}+\frac {1}{2} x^2 \coth ^{-1}(a+b x)^2+\frac {2 a \coth ^{-1}(a+b x) \log \left (\frac {2}{1-a-b x}\right )}{b^2}+\frac {\log \left (1-(a+b x)^2\right )}{2 b^2}+\frac {(2 a) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-a-b x}\right )}{b^2}\\ &=\frac {(a+b x) \coth ^{-1}(a+b x)}{b^2}-\frac {a \coth ^{-1}(a+b x)^2}{b^2}-\frac {\left (1+a^2\right ) \coth ^{-1}(a+b x)^2}{2 b^2}+\frac {1}{2} x^2 \coth ^{-1}(a+b x)^2+\frac {2 a \coth ^{-1}(a+b x) \log \left (\frac {2}{1-a-b x}\right )}{b^2}+\frac {\log \left (1-(a+b x)^2\right )}{2 b^2}+\frac {a \text {Li}_2\left (1-\frac {2}{1-a-b x}\right )}{b^2}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 106, normalized size = 0.78 \begin {gather*} \frac {\left (-1+2 a-a^2+b^2 x^2\right ) \coth ^{-1}(a+b x)^2+2 \coth ^{-1}(a+b x) \left (a+b x+2 a \log \left (1-e^{-2 \coth ^{-1}(a+b x)}\right )\right )-2 \log \left (\frac {1}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}\right )-2 a \text {PolyLog}\left (2,e^{-2 \coth ^{-1}(a+b x)}\right )}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(311\) vs.
\(2(132)=264\).
time = 0.22, size = 312, normalized size = 2.29
method | result | size |
derivativedivides | \(\frac {\frac {\mathrm {arccoth}\left (b x +a \right )^{2} \left (b x +a \right )^{2}}{2}-\mathrm {arccoth}\left (b x +a \right )^{2} a \left (b x +a \right )+\left (b x +a \right ) \mathrm {arccoth}\left (b x +a \right )-\mathrm {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a +\frac {\mathrm {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right )}{2}-\mathrm {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a -\frac {\mathrm {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right )}{2}+\frac {\ln \left (b x +a -1\right )}{2}+\frac {\ln \left (b x +a +1\right )}{2}+\dilog \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right ) a +\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right ) a}{2}-\frac {\ln \left (b x +a -1\right )^{2} a}{4}-\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (b x +a -1\right )^{2}}{8}+\frac {\ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right ) \ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right ) a}{2}-\frac {\ln \left (b x +a +1\right ) \ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right ) a}{2}+\frac {\ln \left (b x +a +1\right )^{2} a}{4}+\frac {\ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right ) \ln \left (b x +a +1\right )}{4}+\frac {\ln \left (b x +a +1\right )^{2}}{8}}{b^{2}}\) | \(312\) |
default | \(\frac {\frac {\mathrm {arccoth}\left (b x +a \right )^{2} \left (b x +a \right )^{2}}{2}-\mathrm {arccoth}\left (b x +a \right )^{2} a \left (b x +a \right )+\left (b x +a \right ) \mathrm {arccoth}\left (b x +a \right )-\mathrm {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right ) a +\frac {\mathrm {arccoth}\left (b x +a \right ) \ln \left (b x +a -1\right )}{2}-\mathrm {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right ) a -\frac {\mathrm {arccoth}\left (b x +a \right ) \ln \left (b x +a +1\right )}{2}+\frac {\ln \left (b x +a -1\right )}{2}+\frac {\ln \left (b x +a +1\right )}{2}+\dilog \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right ) a +\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right ) a}{2}-\frac {\ln \left (b x +a -1\right )^{2} a}{4}-\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (b x +a -1\right )^{2}}{8}+\frac {\ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right ) \ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right ) a}{2}-\frac {\ln \left (b x +a +1\right ) \ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right ) a}{2}+\frac {\ln \left (b x +a +1\right )^{2} a}{4}+\frac {\ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right ) \ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (-\frac {b x}{2}-\frac {a}{2}+\frac {1}{2}\right ) \ln \left (b x +a +1\right )}{4}+\frac {\ln \left (b x +a +1\right )^{2}}{8}}{b^{2}}\) | \(312\) |
risch | \(-\frac {\left (-b^{2} x^{2}+a^{2}+2 a +1\right ) \ln \left (b x +a +1\right )^{2}}{8 b^{2}}+\left (-\frac {x^{2} \ln \left (b x +a -1\right )}{4}+\frac {\ln \left (b x +a -1\right ) a^{2}-2 \ln \left (b x +a -1\right ) a +2 b x +\ln \left (b x +a -1\right )}{4 b^{2}}\right ) \ln \left (b x +a +1\right )-\frac {x^{2} \ln \left (b x +a -1\right )}{8}+\frac {x^{2} \ln \left (b x +a -1\right )^{2}}{8}-\frac {\ln \left (b x +a -1\right ) \left (b x +a -1\right ) a}{2 b^{2}}+\frac {\ln \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right ) \ln \left (b x +a -1\right ) a}{b^{2}}+\frac {\ln \left (b x +a -1\right ) x a}{4 b}+\frac {\ln \left (b x +a +1\right ) a}{2 b^{2}}+\frac {\ln \left (b x +a -1\right ) \left (b x +a -1\right )^{2}}{8 b^{2}}+\frac {\dilog \left (\frac {b x}{2}+\frac {a}{2}+\frac {1}{2}\right ) a}{b^{2}}+\frac {\ln \left (b x +a -1\right )^{2} a}{4 b^{2}}+\frac {3 \ln \left (b x +a -1\right ) a^{2}}{8 b^{2}}-\frac {3 \ln \left (b x +a -1\right ) a}{4 b^{2}}-\frac {\ln \left (b x +a -1\right ) x}{4 b}-\frac {\ln \left (b x +a -1\right )^{2} a^{2}}{8 b^{2}}+\frac {\ln \left (b x +a +1\right )}{2 b^{2}}-\frac {\ln \left (b x +a -1\right )^{2}}{8 b^{2}}+\frac {3 \ln \left (b x +a -1\right )}{8 b^{2}}\) | \(331\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 202, normalized size = 1.49 \begin {gather*} \frac {1}{2} \, x^{2} \operatorname {arcoth}\left (b x + a\right )^{2} + \frac {1}{8} \, b^{2} {\left (\frac {8 \, {\left (\log \left (b x + a - 1\right ) \log \left (\frac {1}{2} \, b x + \frac {1}{2} \, a + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, b x - \frac {1}{2} \, a + \frac {1}{2}\right )\right )} a}{b^{4}} + \frac {4 \, {\left (a + 1\right )} \log \left (b x + a + 1\right )}{b^{4}} + \frac {{\left (a^{2} + 2 \, a + 1\right )} \log \left (b x + a + 1\right )^{2} - 2 \, {\left (a^{2} + 2 \, a + 1\right )} \log \left (b x + a + 1\right ) \log \left (b x + a - 1\right ) + {\left (a^{2} - 2 \, a + 1\right )} \log \left (b x + a - 1\right )^{2} - 4 \, {\left (a - 1\right )} \log \left (b x + a - 1\right )}{b^{4}}\right )} + \frac {1}{2} \, b {\left (\frac {2 \, x}{b^{2}} - \frac {{\left (a^{2} + 2 \, a + 1\right )} \log \left (b x + a + 1\right )}{b^{3}} + \frac {{\left (a^{2} - 2 \, a + 1\right )} \log \left (b x + a - 1\right )}{b^{3}}\right )} \operatorname {arcoth}\left (b x + a\right ) \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 \operatorname {acoth}^{2}{\left (a + b x \right )}\, 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\,{\mathrm {acoth}\left (a+b\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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