Integrand size = 14, antiderivative size = 54 \[ \int (a+b x)^2 \coth ^{-1}(a+b x) \, dx=\frac {(a+b x)^2}{6 b}+\frac {(a+b x)^3 \coth ^{-1}(a+b x)}{3 b}+\frac {\log \left (1-(a+b x)^2\right )}{6 b} \]
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Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6243, 6038, 272, 45} \[ \int (a+b x)^2 \coth ^{-1}(a+b x) \, dx=\frac {(a+b x)^2}{6 b}+\frac {\log \left (1-(a+b x)^2\right )}{6 b}+\frac {(a+b x)^3 \coth ^{-1}(a+b x)}{3 b} \]
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Rule 45
Rule 272
Rule 6038
Rule 6243
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^2 \coth ^{-1}(x) \, dx,x,a+b x\right )}{b} \\ & = \frac {(a+b x)^3 \coth ^{-1}(a+b x)}{3 b}-\frac {\text {Subst}\left (\int \frac {x^3}{1-x^2} \, dx,x,a+b x\right )}{3 b} \\ & = \frac {(a+b x)^3 \coth ^{-1}(a+b x)}{3 b}-\frac {\text {Subst}\left (\int \frac {x}{1-x} \, dx,x,(a+b x)^2\right )}{6 b} \\ & = \frac {(a+b x)^3 \coth ^{-1}(a+b x)}{3 b}-\frac {\text {Subst}\left (\int \left (-1+\frac {1}{1-x}\right ) \, dx,x,(a+b x)^2\right )}{6 b} \\ & = \frac {(a+b x)^2}{6 b}+\frac {(a+b x)^3 \coth ^{-1}(a+b x)}{3 b}+\frac {\log \left (1-(a+b x)^2\right )}{6 b} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int (a+b x)^2 \coth ^{-1}(a+b x) \, dx=\frac {(a+b x)^2+2 (a+b x)^3 \coth ^{-1}(a+b x)+\log \left (1-(a+b x)^2\right )}{6 b} \]
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Time = 0.45 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {\frac {\operatorname {arccoth}\left (b x +a \right ) \left (b x +a \right )^{3}}{3}+\frac {\left (b x +a \right )^{2}}{6}+\frac {\ln \left (b x +a -1\right )}{6}+\frac {\ln \left (b x +a +1\right )}{6}}{b}\) | \(48\) |
default | \(\frac {\frac {\operatorname {arccoth}\left (b x +a \right ) \left (b x +a \right )^{3}}{3}+\frac {\left (b x +a \right )^{2}}{6}+\frac {\ln \left (b x +a -1\right )}{6}+\frac {\ln \left (b x +a +1\right )}{6}}{b}\) | \(48\) |
parts | \(\frac {\operatorname {arccoth}\left (b x +a \right ) b^{2} x^{3}}{3}+\operatorname {arccoth}\left (b x +a \right ) b a \,x^{2}+\operatorname {arccoth}\left (b x +a \right ) a^{2} x +\frac {\operatorname {arccoth}\left (b x +a \right ) a^{3}}{3 b}+\frac {b \,x^{2}}{6}+\frac {a x}{3}+\frac {\ln \left (b x +a -1\right )}{6 b}+\frac {\ln \left (b x +a +1\right )}{6 b}\) | \(87\) |
parallelrisch | \(-\frac {-2 b^{4} \operatorname {arccoth}\left (b x +a \right ) x^{3}-6 x^{2} \operatorname {arccoth}\left (b x +a \right ) a \,b^{3}-6 x \,\operatorname {arccoth}\left (b x +a \right ) a^{2} b^{2}-b^{3} x^{2}-2 \,\operatorname {arccoth}\left (b x +a \right ) a^{3} b -2 a \,b^{2} x +5 b \,a^{2}-2 b \ln \left (b x +a -1\right )-2 \,\operatorname {arccoth}\left (b x +a \right ) b -b}{6 b^{2}}\) | \(106\) |
risch | \(\frac {\left (b x +a \right )^{3} \ln \left (b x +a +1\right )}{6 b}-\frac {b^{2} \ln \left (b x +a -1\right ) x^{3}}{6}-\frac {b \ln \left (b x +a -1\right ) x^{2} a}{2}-\frac {a^{2} x \ln \left (b x +a -1\right )}{2}-\frac {\ln \left (b x +a -1\right ) a^{3}}{6 b}+\frac {b \,x^{2}}{6}+\frac {a x}{3}+\frac {\ln \left (b^{2} x^{2}+2 a b x +a^{2}-1\right )}{6 b}\) | \(111\) |
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Time = 0.26 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.59 \[ \int (a+b x)^2 \coth ^{-1}(a+b x) \, dx=\frac {b^{2} x^{2} + 2 \, a b x + {\left (a^{3} + 1\right )} \log \left (b x + a + 1\right ) - {\left (a^{3} - 1\right )} \log \left (b x + a - 1\right ) + {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x\right )} \log \left (\frac {b x + a + 1}{b x + a - 1}\right )}{6 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (39) = 78\).
Time = 0.31 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.80 \[ \int (a+b x)^2 \coth ^{-1}(a+b x) \, dx=\begin {cases} \frac {a^{3} \operatorname {acoth}{\left (a + b x \right )}}{3 b} + a^{2} x \operatorname {acoth}{\left (a + b x \right )} + a b x^{2} \operatorname {acoth}{\left (a + b x \right )} + \frac {a x}{3} + \frac {b^{2} x^{3} \operatorname {acoth}{\left (a + b x \right )}}{3} + \frac {b x^{2}}{6} + \frac {\log {\left (\frac {a}{b} + x + \frac {1}{b} \right )}}{3 b} - \frac {\operatorname {acoth}{\left (a + b x \right )}}{3 b} & \text {for}\: b \neq 0 \\a^{2} x \operatorname {acoth}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.50 \[ \int (a+b x)^2 \coth ^{-1}(a+b x) \, dx=\frac {1}{6} \, b {\left (\frac {b x^{2} + 2 \, a x}{b} + \frac {{\left (a^{3} + 1\right )} \log \left (b x + a + 1\right )}{b^{2}} - \frac {{\left (a^{3} - 1\right )} \log \left (b x + a - 1\right )}{b^{2}}\right )} + \frac {1}{3} \, {\left (b^{2} x^{3} + 3 \, a b x^{2} + 3 \, a^{2} x\right )} \operatorname {arcoth}\left (b x + a\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (48) = 96\).
Time = 0.28 (sec) , antiderivative size = 255, normalized size of antiderivative = 4.72 \[ \int (a+b x)^2 \coth ^{-1}(a+b x) \, dx=\frac {1}{6} \, {\left ({\left (a + 1\right )} b - {\left (a - 1\right )} b\right )} {\left (\frac {\log \left (\frac {{\left | b x + a + 1 \right |}}{{\left | b x + a - 1 \right |}}\right )}{b^{2}} - \frac {\log \left ({\left | \frac {b x + a + 1}{b x + a - 1} - 1 \right |}\right )}{b^{2}} + \frac {{\left (\frac {3 \, {\left (b x + a + 1\right )}^{2}}{{\left (b x + a - 1\right )}^{2}} + 1\right )} \log \left (-\frac {\frac {1}{a - \frac {{\left (\frac {{\left (b x + a + 1\right )} {\left (a - 1\right )}}{b x + a - 1} - a - 1\right )} b}{\frac {{\left (b x + a + 1\right )} b}{b x + a - 1} - b}} + 1}{\frac {1}{a - \frac {{\left (\frac {{\left (b x + a + 1\right )} {\left (a - 1\right )}}{b x + a - 1} - a - 1\right )} b}{\frac {{\left (b x + a + 1\right )} b}{b x + a - 1} - b}} - 1}\right )}{b^{2} {\left (\frac {b x + a + 1}{b x + a - 1} - 1\right )}^{3}} + \frac {2 \, {\left (b x + a + 1\right )}}{{\left (b x + a - 1\right )} b^{2} {\left (\frac {b x + a + 1}{b x + a - 1} - 1\right )}^{2}}\right )} \]
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Time = 4.38 (sec) , antiderivative size = 114, normalized size of antiderivative = 2.11 \[ \int (a+b x)^2 \coth ^{-1}(a+b x) \, dx=\frac {a\,x}{3}+\ln \left (\frac {1}{a+b\,x}+1\right )\,\left (\frac {a^2\,x}{2}+\frac {a\,b\,x^2}{2}+\frac {b^2\,x^3}{6}\right )+\frac {b\,x^2}{6}-\ln \left (1-\frac {1}{a+b\,x}\right )\,\left (\frac {a^2\,x}{2}+\frac {a\,b\,x^2}{2}+\frac {b^2\,x^3}{6}\right )-\frac {\ln \left (a+b\,x-1\right )\,\left (a^3-1\right )}{6\,b}+\frac {\ln \left (a+b\,x+1\right )\,\left (a^3+1\right )}{6\,b} \]
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