Integrand size = 14, antiderivative size = 110 \[ \int \left (c+d x^2\right )^2 \coth ^{-1}(a x) \, dx=\frac {d \left (10 a^2 c+3 d\right ) x^2}{30 a^3}+\frac {d^2 x^4}{20 a}+c^2 x \coth ^{-1}(a x)+\frac {2}{3} c d x^3 \coth ^{-1}(a x)+\frac {1}{5} d^2 x^5 \coth ^{-1}(a x)+\frac {\left (15 a^4 c^2+10 a^2 c d+3 d^2\right ) \log \left (1-a^2 x^2\right )}{30 a^5} \]
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Time = 0.10 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {200, 6124, 1608, 1261, 712} \[ \int \left (c+d x^2\right )^2 \coth ^{-1}(a x) \, dx=\frac {d x^2 \left (10 a^2 c+3 d\right )}{30 a^3}+\frac {\left (15 a^4 c^2+10 a^2 c d+3 d^2\right ) \log \left (1-a^2 x^2\right )}{30 a^5}+c^2 x \coth ^{-1}(a x)+\frac {2}{3} c d x^3 \coth ^{-1}(a x)+\frac {1}{5} d^2 x^5 \coth ^{-1}(a x)+\frac {d^2 x^4}{20 a} \]
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Rule 200
Rule 712
Rule 1261
Rule 1608
Rule 6124
Rubi steps \begin{align*} \text {integral}& = c^2 x \coth ^{-1}(a x)+\frac {2}{3} c d x^3 \coth ^{-1}(a x)+\frac {1}{5} d^2 x^5 \coth ^{-1}(a x)-a \int \frac {c^2 x+\frac {2}{3} c d x^3+\frac {d^2 x^5}{5}}{1-a^2 x^2} \, dx \\ & = c^2 x \coth ^{-1}(a x)+\frac {2}{3} c d x^3 \coth ^{-1}(a x)+\frac {1}{5} d^2 x^5 \coth ^{-1}(a x)-a \int \frac {x \left (c^2+\frac {2}{3} c d x^2+\frac {d^2 x^4}{5}\right )}{1-a^2 x^2} \, dx \\ & = c^2 x \coth ^{-1}(a x)+\frac {2}{3} c d x^3 \coth ^{-1}(a x)+\frac {1}{5} d^2 x^5 \coth ^{-1}(a x)-\frac {1}{2} a \text {Subst}\left (\int \frac {c^2+\frac {2 c d x}{3}+\frac {d^2 x^2}{5}}{1-a^2 x} \, dx,x,x^2\right ) \\ & = c^2 x \coth ^{-1}(a x)+\frac {2}{3} c d x^3 \coth ^{-1}(a x)+\frac {1}{5} d^2 x^5 \coth ^{-1}(a x)-\frac {1}{2} a \text {Subst}\left (\int \left (-\frac {d \left (10 a^2 c+3 d\right )}{15 a^4}-\frac {d^2 x}{5 a^2}+\frac {-15 a^4 c^2-10 a^2 c d-3 d^2}{15 a^4 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right ) \\ & = \frac {d \left (10 a^2 c+3 d\right ) x^2}{30 a^3}+\frac {d^2 x^4}{20 a}+c^2 x \coth ^{-1}(a x)+\frac {2}{3} c d x^3 \coth ^{-1}(a x)+\frac {1}{5} d^2 x^5 \coth ^{-1}(a x)+\frac {\left (15 a^4 c^2+10 a^2 c d+3 d^2\right ) \log \left (1-a^2 x^2\right )}{30 a^5} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.89 \[ \int \left (c+d x^2\right )^2 \coth ^{-1}(a x) \, dx=\frac {a^2 d x^2 \left (6 d+a^2 \left (20 c+3 d x^2\right )\right )+4 a^5 x \left (15 c^2+10 c d x^2+3 d^2 x^4\right ) \coth ^{-1}(a x)+\left (30 a^4 c^2+20 a^2 c d+6 d^2\right ) \log \left (1-a^2 x^2\right )}{60 a^5} \]
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Time = 0.48 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.95
method | result | size |
parts | \(\frac {d^{2} x^{5} \operatorname {arccoth}\left (a x \right )}{5}+\frac {2 c d \,x^{3} \operatorname {arccoth}\left (a x \right )}{3}+c^{2} x \,\operatorname {arccoth}\left (a x \right )+\frac {a \left (\frac {d \left (\frac {3}{2} a^{2} d \,x^{4}+10 a^{2} c \,x^{2}+3 d \,x^{2}\right )}{2 a^{4}}+\frac {\left (15 a^{4} c^{2}+10 a^{2} c d +3 d^{2}\right ) \ln \left (a^{2} x^{2}-1\right )}{2 a^{6}}\right )}{15}\) | \(105\) |
derivativedivides | \(\frac {\operatorname {arccoth}\left (a x \right ) c^{2} a x +\frac {2 a \,\operatorname {arccoth}\left (a x \right ) c d \,x^{3}}{3}+\frac {a \,\operatorname {arccoth}\left (a x \right ) d^{2} x^{5}}{5}+\frac {5 c \,a^{4} d \,x^{2}+\frac {3 d^{2} a^{4} x^{4}}{4}+\frac {3 a^{2} d^{2} x^{2}}{2}+\frac {\left (15 a^{4} c^{2}+10 a^{2} c d +3 d^{2}\right ) \ln \left (a x -1\right )}{2}-\frac {\left (-15 a^{4} c^{2}-10 a^{2} c d -3 d^{2}\right ) \ln \left (a x +1\right )}{2}}{15 a^{4}}}{a}\) | \(137\) |
default | \(\frac {\operatorname {arccoth}\left (a x \right ) c^{2} a x +\frac {2 a \,\operatorname {arccoth}\left (a x \right ) c d \,x^{3}}{3}+\frac {a \,\operatorname {arccoth}\left (a x \right ) d^{2} x^{5}}{5}+\frac {5 c \,a^{4} d \,x^{2}+\frac {3 d^{2} a^{4} x^{4}}{4}+\frac {3 a^{2} d^{2} x^{2}}{2}+\frac {\left (15 a^{4} c^{2}+10 a^{2} c d +3 d^{2}\right ) \ln \left (a x -1\right )}{2}-\frac {\left (-15 a^{4} c^{2}-10 a^{2} c d -3 d^{2}\right ) \ln \left (a x +1\right )}{2}}{15 a^{4}}}{a}\) | \(137\) |
parallelrisch | \(-\frac {-12 x^{5} \operatorname {arccoth}\left (a x \right ) a^{5} d^{2}-40 x^{3} \operatorname {arccoth}\left (a x \right ) a^{5} c d -3 d^{2} a^{4} x^{4}-60 c^{2} x \,\operatorname {arccoth}\left (a x \right ) a^{5}-20 c \,a^{4} d \,x^{2}-60 \ln \left (a x -1\right ) a^{4} c^{2}-60 \,\operatorname {arccoth}\left (a x \right ) a^{4} c^{2}-6 a^{2} d^{2} x^{2}-40 \ln \left (a x -1\right ) a^{2} c d -40 \,\operatorname {arccoth}\left (a x \right ) a^{2} c d -20 a^{2} c d -12 \ln \left (a x -1\right ) d^{2}-12 \,\operatorname {arccoth}\left (a x \right ) d^{2}-6 d^{2}}{60 a^{5}}\) | \(163\) |
risch | \(\left (\frac {1}{10} d^{2} x^{5}+\frac {1}{3} c d \,x^{3}+\frac {1}{2} c^{2} x \right ) \ln \left (a x +1\right )-\frac {d^{2} x^{5} \ln \left (a x -1\right )}{10}-\frac {c d \,x^{3} \ln \left (a x -1\right )}{3}+\frac {d^{2} x^{4}}{20 a}-\frac {c^{2} x \ln \left (a x -1\right )}{2}+\frac {c d \,x^{2}}{3 a}+\frac {\ln \left (a^{2} x^{2}-1\right ) c^{2}}{2 a}+\frac {5 c^{2}}{9 a}+\frac {d^{2} x^{2}}{10 a^{3}}+\frac {\ln \left (a^{2} x^{2}-1\right ) c d}{3 a^{3}}+\frac {c d}{3 a^{3}}+\frac {\ln \left (a^{2} x^{2}-1\right ) d^{2}}{10 a^{5}}+\frac {d^{2}}{20 a^{5}}\) | \(178\) |
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Time = 0.25 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.07 \[ \int \left (c+d x^2\right )^2 \coth ^{-1}(a x) \, dx=\frac {3 \, a^{4} d^{2} x^{4} + 2 \, {\left (10 \, a^{4} c d + 3 \, a^{2} d^{2}\right )} x^{2} + 2 \, {\left (15 \, a^{4} c^{2} + 10 \, a^{2} c d + 3 \, d^{2}\right )} \log \left (a^{2} x^{2} - 1\right ) + 2 \, {\left (3 \, a^{5} d^{2} x^{5} + 10 \, a^{5} c d x^{3} + 15 \, a^{5} c^{2} x\right )} \log \left (\frac {a x + 1}{a x - 1}\right )}{60 \, a^{5}} \]
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Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.65 \[ \int \left (c+d x^2\right )^2 \coth ^{-1}(a x) \, dx=\begin {cases} c^{2} x \operatorname {acoth}{\left (a x \right )} + \frac {2 c d x^{3} \operatorname {acoth}{\left (a x \right )}}{3} + \frac {d^{2} x^{5} \operatorname {acoth}{\left (a x \right )}}{5} + \frac {c^{2} \log {\left (x - \frac {1}{a} \right )}}{a} + \frac {c^{2} \operatorname {acoth}{\left (a x \right )}}{a} + \frac {c d x^{2}}{3 a} + \frac {d^{2} x^{4}}{20 a} + \frac {2 c d \log {\left (x - \frac {1}{a} \right )}}{3 a^{3}} + \frac {2 c d \operatorname {acoth}{\left (a x \right )}}{3 a^{3}} + \frac {d^{2} x^{2}}{10 a^{3}} + \frac {d^{2} \log {\left (x - \frac {1}{a} \right )}}{5 a^{5}} + \frac {d^{2} \operatorname {acoth}{\left (a x \right )}}{5 a^{5}} & \text {for}\: a \neq 0 \\\frac {i \pi \left (c^{2} x + \frac {2 c d x^{3}}{3} + \frac {d^{2} x^{5}}{5}\right )}{2} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.19 \[ \int \left (c+d x^2\right )^2 \coth ^{-1}(a x) \, dx=\frac {1}{60} \, a {\left (\frac {3 \, a^{2} d^{2} x^{4} + 2 \, {\left (10 \, a^{2} c d + 3 \, d^{2}\right )} x^{2}}{a^{4}} + \frac {2 \, {\left (15 \, a^{4} c^{2} + 10 \, a^{2} c d + 3 \, d^{2}\right )} \log \left (a x + 1\right )}{a^{6}} + \frac {2 \, {\left (15 \, a^{4} c^{2} + 10 \, a^{2} c d + 3 \, d^{2}\right )} \log \left (a x - 1\right )}{a^{6}}\right )} + \frac {1}{15} \, {\left (3 \, d^{2} x^{5} + 10 \, c d x^{3} + 15 \, c^{2} x\right )} \operatorname {arcoth}\left (a x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 529 vs. \(2 (100) = 200\).
Time = 0.28 (sec) , antiderivative size = 529, normalized size of antiderivative = 4.81 \[ \int \left (c+d x^2\right )^2 \coth ^{-1}(a x) \, dx=\frac {1}{15} \, a {\left (\frac {{\left (15 \, a^{4} c^{2} + 10 \, a^{2} c d + 3 \, d^{2}\right )} \log \left (\frac {{\left | a x + 1 \right |}}{{\left | a x - 1 \right |}}\right )}{a^{6}} - \frac {{\left (15 \, a^{4} c^{2} + 10 \, a^{2} c d + 3 \, d^{2}\right )} \log \left ({\left | \frac {a x + 1}{a x - 1} - 1 \right |}\right )}{a^{6}} + \frac {4 \, {\left (\frac {{\left (5 \, a^{2} c d + 3 \, d^{2}\right )} {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} - \frac {{\left (10 \, a^{2} c d + 3 \, d^{2}\right )} {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {{\left (5 \, a^{2} c d + 3 \, d^{2}\right )} {\left (a x + 1\right )}}{a x - 1}\right )}}{a^{6} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{4}} + \frac {{\left (\frac {15 \, {\left (a x + 1\right )}^{4} a^{4} c^{2}}{{\left (a x - 1\right )}^{4}} - \frac {60 \, {\left (a x + 1\right )}^{3} a^{4} c^{2}}{{\left (a x - 1\right )}^{3}} + \frac {90 \, {\left (a x + 1\right )}^{2} a^{4} c^{2}}{{\left (a x - 1\right )}^{2}} - \frac {60 \, {\left (a x + 1\right )} a^{4} c^{2}}{a x - 1} + 15 \, a^{4} c^{2} + \frac {30 \, {\left (a x + 1\right )}^{4} a^{2} c d}{{\left (a x - 1\right )}^{4}} - \frac {60 \, {\left (a x + 1\right )}^{3} a^{2} c d}{{\left (a x - 1\right )}^{3}} + \frac {40 \, {\left (a x + 1\right )}^{2} a^{2} c d}{{\left (a x - 1\right )}^{2}} - \frac {20 \, {\left (a x + 1\right )} a^{2} c d}{a x - 1} + 10 \, a^{2} c d + \frac {15 \, {\left (a x + 1\right )}^{4} d^{2}}{{\left (a x - 1\right )}^{4}} + \frac {30 \, {\left (a x + 1\right )}^{2} d^{2}}{{\left (a x - 1\right )}^{2}} + 3 \, d^{2}\right )} \log \left (-\frac {\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} + 1}{\frac {\frac {{\left (a x + 1\right )} a}{a x - 1} - a}{a {\left (\frac {a x + 1}{a x - 1} + 1\right )}} - 1}\right )}{a^{6} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{5}}\right )} \]
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Time = 4.40 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.05 \[ \int \left (c+d x^2\right )^2 \coth ^{-1}(a x) \, dx=\frac {a^4\,\left (\frac {c^2\,\ln \left (a^2\,x^2-1\right )}{2}+\frac {d^2\,x^4}{20}+\frac {c\,d\,x^2}{3}\right )+a^2\,\left (\frac {d^2\,x^2}{10}+\frac {c\,d\,\ln \left (a^2\,x^2-1\right )}{3}\right )+\frac {d^2\,\ln \left (a^2\,x^2-1\right )}{10}}{a^5}+c^2\,x\,\mathrm {acoth}\left (a\,x\right )+\frac {d^2\,x^5\,\mathrm {acoth}\left (a\,x\right )}{5}+\frac {2\,c\,d\,x^3\,\mathrm {acoth}\left (a\,x\right )}{3} \]
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