Integrand size = 10, antiderivative size = 78 \[ \int x^2 \coth ^{-1}(a+b x) \, dx=-\frac {a x}{b^2}+\frac {(a+b x)^2}{6 b^3}+\frac {1}{3} x^3 \coth ^{-1}(a+b x)+\frac {(1-a)^3 \log (1-a-b x)}{6 b^3}+\frac {(1+a)^3 \log (1+a+b x)}{6 b^3} \]
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Time = 0.09 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6247, 6064, 716, 647, 31} \[ \int x^2 \coth ^{-1}(a+b x) \, dx=\frac {(a+b x)^2}{6 b^3}+\frac {(1-a)^3 \log (-a-b x+1)}{6 b^3}+\frac {(a+1)^3 \log (a+b x+1)}{6 b^3}-\frac {a x}{b^2}+\frac {1}{3} x^3 \coth ^{-1}(a+b x) \]
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Rule 31
Rule 647
Rule 716
Rule 6064
Rule 6247
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right )^2 \coth ^{-1}(x) \, dx,x,a+b x\right )}{b} \\ & = \frac {1}{3} x^3 \coth ^{-1}(a+b x)-\frac {1}{3} \text {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^3}{1-x^2} \, dx,x,a+b x\right ) \\ & = \frac {1}{3} x^3 \coth ^{-1}(a+b x)-\frac {1}{3} \text {Subst}\left (\int \left (\frac {3 a}{b^3}-\frac {x}{b^3}-\frac {a \left (3+a^2\right )-\left (1+3 a^2\right ) x}{b^3 \left (1-x^2\right )}\right ) \, dx,x,a+b x\right ) \\ & = -\frac {a x}{b^2}+\frac {(a+b x)^2}{6 b^3}+\frac {1}{3} x^3 \coth ^{-1}(a+b x)+\frac {\text {Subst}\left (\int \frac {a \left (3+a^2\right )-\left (1+3 a^2\right ) x}{1-x^2} \, dx,x,a+b x\right )}{3 b^3} \\ & = -\frac {a x}{b^2}+\frac {(a+b x)^2}{6 b^3}+\frac {1}{3} x^3 \coth ^{-1}(a+b x)-\frac {(1-a)^3 \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,a+b x\right )}{6 b^3}-\frac {(1+a)^3 \text {Subst}\left (\int \frac {1}{-1-x} \, dx,x,a+b x\right )}{6 b^3} \\ & = -\frac {a x}{b^2}+\frac {(a+b x)^2}{6 b^3}+\frac {1}{3} x^3 \coth ^{-1}(a+b x)+\frac {(1-a)^3 \log (1-a-b x)}{6 b^3}+\frac {(1+a)^3 \log (1+a+b x)}{6 b^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.18 \[ \int x^2 \coth ^{-1}(a+b x) \, dx=-\frac {2 a x}{3 b^2}+\frac {x^2}{6 b}+\frac {1}{3} x^3 \coth ^{-1}(a+b x)+\frac {\left (1-3 a+3 a^2-a^3\right ) \log (1-a-b x)}{6 b^3}+\frac {\left (1+3 a+3 a^2+a^3\right ) \log (1+a+b x)}{6 b^3} \]
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Time = 0.10 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.09
method | result | size |
parts | \(\frac {x^{3} \operatorname {arccoth}\left (b x +a \right )}{3}+\frac {b \left (-\frac {-\frac {1}{2} b \,x^{2}+2 a x}{b^{3}}+\frac {\left (-a^{3}+3 a^{2}-3 a +1\right ) \ln \left (b x +a -1\right )}{2 b^{4}}+\frac {\left (a^{3}+3 a^{2}+3 a +1\right ) \ln \left (b x +a +1\right )}{2 b^{4}}\right )}{3}\) | \(85\) |
parallelrisch | \(\frac {2 \,\operatorname {arccoth}\left (b x +a \right ) x^{3} b^{3}+1+b^{2} x^{2}+2 \,\operatorname {arccoth}\left (b x +a \right ) a^{3}+6 \ln \left (b x +a -1\right ) a^{2}-4 a b x +6 \,\operatorname {arccoth}\left (b x +a \right ) a^{2}+6 \,\operatorname {arccoth}\left (b x +a \right ) a +7 a^{2}+2 \ln \left (b x +a -1\right )+2 \,\operatorname {arccoth}\left (b x +a \right )}{6 b^{3}}\) | \(99\) |
derivativedivides | \(\frac {-\frac {\operatorname {arccoth}\left (b x +a \right ) a^{3}}{3}+\operatorname {arccoth}\left (b x +a \right ) a^{2} \left (b x +a \right )-\operatorname {arccoth}\left (b x +a \right ) a \left (b x +a \right )^{2}+\frac {\operatorname {arccoth}\left (b x +a \right ) \left (b x +a \right )^{3}}{3}-\left (b x +a \right ) a +\frac {\left (b x +a \right )^{2}}{6}-\frac {\left (a^{3}-3 a^{2}+3 a -1\right ) \ln \left (b x +a -1\right )}{6}+\frac {\left (a^{3}+3 a^{2}+3 a +1\right ) \ln \left (b x +a +1\right )}{6}}{b^{3}}\) | \(124\) |
default | \(\frac {-\frac {\operatorname {arccoth}\left (b x +a \right ) a^{3}}{3}+\operatorname {arccoth}\left (b x +a \right ) a^{2} \left (b x +a \right )-\operatorname {arccoth}\left (b x +a \right ) a \left (b x +a \right )^{2}+\frac {\operatorname {arccoth}\left (b x +a \right ) \left (b x +a \right )^{3}}{3}-\left (b x +a \right ) a +\frac {\left (b x +a \right )^{2}}{6}-\frac {\left (a^{3}-3 a^{2}+3 a -1\right ) \ln \left (b x +a -1\right )}{6}+\frac {\left (a^{3}+3 a^{2}+3 a +1\right ) \ln \left (b x +a +1\right )}{6}}{b^{3}}\) | \(124\) |
risch | \(\frac {x^{3} \ln \left (b x +a +1\right )}{6}-\frac {\ln \left (b x +a -1\right ) x^{3}}{6}+\frac {\ln \left (-b x -a -1\right ) a^{3}}{6 b^{3}}-\frac {\ln \left (b x +a -1\right ) a^{3}}{6 b^{3}}+\frac {x^{2}}{6 b}+\frac {\ln \left (-b x -a -1\right ) a^{2}}{2 b^{3}}+\frac {\ln \left (b x +a -1\right ) a^{2}}{2 b^{3}}-\frac {2 x a}{3 b^{2}}+\frac {\ln \left (-b x -a -1\right ) a}{2 b^{3}}-\frac {\ln \left (b x +a -1\right ) a}{2 b^{3}}+\frac {\ln \left (-b x -a -1\right )}{6 b^{3}}+\frac {\ln \left (b x +a -1\right )}{6 b^{3}}\) | \(163\) |
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Time = 0.27 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.08 \[ \int x^2 \coth ^{-1}(a+b x) \, dx=\frac {b^{3} x^{3} \log \left (\frac {b x + a + 1}{b x + a - 1}\right ) + b^{2} x^{2} - 4 \, a b x + {\left (a^{3} + 3 \, a^{2} + 3 \, a + 1\right )} \log \left (b x + a + 1\right ) - {\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )} \log \left (b x + a - 1\right )}{6 \, b^{3}} \]
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Time = 0.38 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.50 \[ \int x^2 \coth ^{-1}(a+b x) \, dx=\begin {cases} \frac {a^{3} \operatorname {acoth}{\left (a + b x \right )}}{3 b^{3}} + \frac {a^{2} \log {\left (a + b x + 1 \right )}}{b^{3}} - \frac {a^{2} \operatorname {acoth}{\left (a + b x \right )}}{b^{3}} - \frac {2 a x}{3 b^{2}} + \frac {a \operatorname {acoth}{\left (a + b x \right )}}{b^{3}} + \frac {x^{3} \operatorname {acoth}{\left (a + b x \right )}}{3} + \frac {x^{2}}{6 b} + \frac {\log {\left (a + b x + 1 \right )}}{3 b^{3}} - \frac {\operatorname {acoth}{\left (a + b x \right )}}{3 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \operatorname {acoth}{\left (a \right )}}{3} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.01 \[ \int x^2 \coth ^{-1}(a+b x) \, dx=\frac {1}{3} \, x^{3} \operatorname {arcoth}\left (b x + a\right ) + \frac {1}{6} \, b {\left (\frac {b x^{2} - 4 \, a x}{b^{3}} + \frac {{\left (a^{3} + 3 \, a^{2} + 3 \, a + 1\right )} \log \left (b x + a + 1\right )}{b^{4}} - \frac {{\left (a^{3} - 3 \, a^{2} + 3 \, a - 1\right )} \log \left (b x + a - 1\right )}{b^{4}}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (68) = 136\).
Time = 0.29 (sec) , antiderivative size = 360, normalized size of antiderivative = 4.62 \[ \int x^2 \coth ^{-1}(a+b x) \, dx=\frac {1}{6} \, {\left ({\left (a + 1\right )} b - {\left (a - 1\right )} b\right )} {\left (\frac {{\left (3 \, a^{2} + 1\right )} \log \left (\frac {{\left | b x + a + 1 \right |}}{{\left | b x + a - 1 \right |}}\right )}{b^{4}} - \frac {{\left (3 \, a^{2} + 1\right )} \log \left ({\left | \frac {b x + a + 1}{b x + a - 1} - 1 \right |}\right )}{b^{4}} - \frac {2 \, {\left (\frac {{\left (b x + a + 1\right )} {\left (3 \, a - 1\right )}}{b x + a - 1} - 3 \, a\right )}}{b^{4} {\left (\frac {b x + a + 1}{b x + a - 1} - 1\right )}^{2}} + \frac {{\left (\frac {3 \, {\left (b x + a + 1\right )}^{2} a^{2}}{{\left (b x + a - 1\right )}^{2}} - \frac {6 \, {\left (b x + a + 1\right )} a^{2}}{b x + a - 1} + 3 \, a^{2} - \frac {6 \, {\left (b x + a + 1\right )}^{2} a}{{\left (b x + a - 1\right )}^{2}} + \frac {6 \, {\left (b x + a + 1\right )} a}{b x + a - 1} + \frac {3 \, {\left (b x + a + 1\right )}^{2}}{{\left (b x + a - 1\right )}^{2}} + 1\right )} \log \left (-\frac {\frac {1}{a - \frac {{\left (\frac {{\left (b x + a + 1\right )} {\left (a - 1\right )}}{b x + a - 1} - a - 1\right )} b}{\frac {{\left (b x + a + 1\right )} b}{b x + a - 1} - b}} + 1}{\frac {1}{a - \frac {{\left (\frac {{\left (b x + a + 1\right )} {\left (a - 1\right )}}{b x + a - 1} - a - 1\right )} b}{\frac {{\left (b x + a + 1\right )} b}{b x + a - 1} - b}} - 1}\right )}{b^{4} {\left (\frac {b x + a + 1}{b x + a - 1} - 1\right )}^{3}}\right )} \]
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Time = 4.65 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.26 \[ \int x^2 \coth ^{-1}(a+b x) \, dx=\frac {x^3\,\ln \left (\frac {1}{a+b\,x}+1\right )}{6}-\frac {x^3\,\ln \left (1-\frac {1}{a+b\,x}\right )}{6}+\frac {x^2}{6\,b}-\frac {\ln \left (a+b\,x-1\right )\,\left (a^3-3\,a^2+3\,a-1\right )}{6\,b^3}+\frac {\ln \left (a+b\,x+1\right )\,\left (a^3+3\,a^2+3\,a+1\right )}{6\,b^3}-\frac {2\,a\,x}{3\,b^2} \]
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