Integrand size = 12, antiderivative size = 57 \[ \int \left (c+d x^2\right ) \coth ^{-1}(a x) \, dx=\frac {d x^2}{6 a}+c x \coth ^{-1}(a x)+\frac {1}{3} d x^3 \coth ^{-1}(a x)+\frac {\left (3 a^2 c+d\right ) \log \left (1-a^2 x^2\right )}{6 a^3} \]
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Time = 0.05 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6124, 1607, 455, 45} \[ \int \left (c+d x^2\right ) \coth ^{-1}(a x) \, dx=\frac {\left (3 a^2 c+d\right ) \log \left (1-a^2 x^2\right )}{6 a^3}+c x \coth ^{-1}(a x)+\frac {1}{3} d x^3 \coth ^{-1}(a x)+\frac {d x^2}{6 a} \]
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Rule 45
Rule 455
Rule 1607
Rule 6124
Rubi steps \begin{align*} \text {integral}& = c x \coth ^{-1}(a x)+\frac {1}{3} d x^3 \coth ^{-1}(a x)-a \int \frac {c x+\frac {d x^3}{3}}{1-a^2 x^2} \, dx \\ & = c x \coth ^{-1}(a x)+\frac {1}{3} d x^3 \coth ^{-1}(a x)-a \int \frac {x \left (c+\frac {d x^2}{3}\right )}{1-a^2 x^2} \, dx \\ & = c x \coth ^{-1}(a x)+\frac {1}{3} d x^3 \coth ^{-1}(a x)-\frac {1}{2} a \text {Subst}\left (\int \frac {c+\frac {d x}{3}}{1-a^2 x} \, dx,x,x^2\right ) \\ & = c x \coth ^{-1}(a x)+\frac {1}{3} d x^3 \coth ^{-1}(a x)-\frac {1}{2} a \text {Subst}\left (\int \left (-\frac {d}{3 a^2}+\frac {-3 a^2 c-d}{3 a^2 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right ) \\ & = \frac {d x^2}{6 a}+c x \coth ^{-1}(a x)+\frac {1}{3} d x^3 \coth ^{-1}(a x)+\frac {\left (3 a^2 c+d\right ) \log \left (1-a^2 x^2\right )}{6 a^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.21 \[ \int \left (c+d x^2\right ) \coth ^{-1}(a x) \, dx=\frac {d x^2}{6 a}+c x \coth ^{-1}(a x)+\frac {1}{3} d x^3 \coth ^{-1}(a x)+\frac {c \log \left (1-a^2 x^2\right )}{2 a}+\frac {d \log \left (1-a^2 x^2\right )}{6 a^3} \]
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Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.96
method | result | size |
parts | \(\frac {d \,x^{3} \operatorname {arccoth}\left (a x \right )}{3}+c x \,\operatorname {arccoth}\left (a x \right )+\frac {a \left (\frac {d \,x^{2}}{2 a^{2}}+\frac {\left (3 a^{2} c +d \right ) \ln \left (a^{2} x^{2}-1\right )}{2 a^{4}}\right )}{3}\) | \(55\) |
derivativedivides | \(\frac {\operatorname {arccoth}\left (a x \right ) c a x +\frac {a \,\operatorname {arccoth}\left (a x \right ) d \,x^{3}}{3}+\frac {\frac {a^{2} d \,x^{2}}{2}+\frac {\left (3 a^{2} c +d \right ) \ln \left (a x -1\right )}{2}-\frac {\left (-3 a^{2} c -d \right ) \ln \left (a x +1\right )}{2}}{3 a^{2}}}{a}\) | \(74\) |
default | \(\frac {\operatorname {arccoth}\left (a x \right ) c a x +\frac {a \,\operatorname {arccoth}\left (a x \right ) d \,x^{3}}{3}+\frac {\frac {a^{2} d \,x^{2}}{2}+\frac {\left (3 a^{2} c +d \right ) \ln \left (a x -1\right )}{2}-\frac {\left (-3 a^{2} c -d \right ) \ln \left (a x +1\right )}{2}}{3 a^{2}}}{a}\) | \(74\) |
parallelrisch | \(-\frac {-2 x^{3} \operatorname {arccoth}\left (a x \right ) a^{3} d -6 c \,\operatorname {arccoth}\left (a x \right ) x \,a^{3}-a^{2} d \,x^{2}-6 \ln \left (a x -1\right ) a^{2} c -6 \,\operatorname {arccoth}\left (a x \right ) a^{2} c -2 \ln \left (a x -1\right ) d -2 \,\operatorname {arccoth}\left (a x \right ) d}{6 a^{3}}\) | \(78\) |
risch | \(\left (\frac {1}{6} d \,x^{3}+\frac {1}{2} c x \right ) \ln \left (a x +1\right )-\frac {d \,x^{3} \ln \left (a x -1\right )}{6}-\frac {c x \ln \left (a x -1\right )}{2}+\frac {d \,x^{2}}{6 a}+\frac {\ln \left (a^{2} x^{2}-1\right ) c}{2 a}+\frac {\ln \left (a^{2} x^{2}-1\right ) d}{6 a^{3}}\) | \(83\) |
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Time = 0.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.12 \[ \int \left (c+d x^2\right ) \coth ^{-1}(a x) \, dx=\frac {a^{2} d x^{2} + {\left (3 \, a^{2} c + d\right )} \log \left (a^{2} x^{2} - 1\right ) + {\left (a^{3} d x^{3} + 3 \, a^{3} c x\right )} \log \left (\frac {a x + 1}{a x - 1}\right )}{6 \, a^{3}} \]
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.53 \[ \int \left (c+d x^2\right ) \coth ^{-1}(a x) \, dx=\begin {cases} c x \operatorname {acoth}{\left (a x \right )} + \frac {d x^{3} \operatorname {acoth}{\left (a x \right )}}{3} + \frac {c \log {\left (x - \frac {1}{a} \right )}}{a} + \frac {c \operatorname {acoth}{\left (a x \right )}}{a} + \frac {d x^{2}}{6 a} + \frac {d \log {\left (x - \frac {1}{a} \right )}}{3 a^{3}} + \frac {d \operatorname {acoth}{\left (a x \right )}}{3 a^{3}} & \text {for}\: a \neq 0 \\\frac {i \pi \left (c x + \frac {d x^{3}}{3}\right )}{2} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.14 \[ \int \left (c+d x^2\right ) \coth ^{-1}(a x) \, dx=\frac {1}{6} \, a {\left (\frac {d x^{2}}{a^{2}} + \frac {{\left (3 \, a^{2} c + d\right )} \log \left (a x + 1\right )}{a^{4}} + \frac {{\left (3 \, a^{2} c + d\right )} \log \left (a x - 1\right )}{a^{4}}\right )} + \frac {1}{3} \, {\left (d x^{3} + 3 \, c x\right )} \operatorname {arcoth}\left (a x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (51) = 102\).
Time = 0.28 (sec) , antiderivative size = 268, normalized size of antiderivative = 4.70 \[ \int \left (c+d x^2\right ) \coth ^{-1}(a x) \, dx=\frac {1}{3} \, a {\left (\frac {{\left (3 \, a^{2} c + d\right )} \log \left (\frac {{\left | a x + 1 \right |}}{{\left | a x - 1 \right |}}\right )}{a^{4}} - \frac {{\left (3 \, a^{2} c + d\right )} \log \left ({\left | \frac {a x + 1}{a x - 1} - 1 \right |}\right )}{a^{4}} + \frac {2 \, {\left (a x + 1\right )} d}{{\left (a x - 1\right )} a^{4} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{2}} + \frac {{\left (\frac {3 \, {\left (a x + 1\right )}^{2} a^{2} c}{{\left (a x - 1\right )}^{2}} - \frac {6 \, {\left (a x + 1\right )} a^{2} c}{a x - 1} + 3 \, a^{2} c + \frac {3 \, {\left (a x + 1\right )}^{2} d}{{\left (a x - 1\right )}^{2}} + d\right )} \log \left (-\frac {\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} + 1}{\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} - 1}\right )}{a^{4} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{3}}\right )} \]
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Time = 4.25 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.05 \[ \int \left (c+d x^2\right ) \coth ^{-1}(a x) \, dx=\frac {\frac {d\,\ln \left (a^2\,x^2-1\right )}{6}+a^2\,\left (\frac {c\,\ln \left (a^2\,x^2-1\right )}{2}+\frac {d\,x^2}{6}\right )}{a^3}+\frac {d\,x^3\,\mathrm {acoth}\left (a\,x\right )}{3}+c\,x\,\mathrm {acoth}\left (a\,x\right ) \]
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