Integrand size = 10, antiderivative size = 105 \[ \int x^5 \coth ^{-1}(a x)^2 \, dx=\frac {4 x^2}{45 a^4}+\frac {x^4}{60 a^2}+\frac {x \coth ^{-1}(a x)}{3 a^5}+\frac {x^3 \coth ^{-1}(a x)}{9 a^3}+\frac {x^5 \coth ^{-1}(a x)}{15 a}-\frac {\coth ^{-1}(a x)^2}{6 a^6}+\frac {1}{6} x^6 \coth ^{-1}(a x)^2+\frac {23 \log \left (1-a^2 x^2\right )}{90 a^6} \]
[Out]
Time = 0.19 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6038, 6128, 272, 45, 6022, 266, 6096} \[ \int x^5 \coth ^{-1}(a x)^2 \, dx=-\frac {\coth ^{-1}(a x)^2}{6 a^6}+\frac {x \coth ^{-1}(a x)}{3 a^5}+\frac {4 x^2}{45 a^4}+\frac {x^3 \coth ^{-1}(a x)}{9 a^3}+\frac {x^4}{60 a^2}+\frac {23 \log \left (1-a^2 x^2\right )}{90 a^6}+\frac {1}{6} x^6 \coth ^{-1}(a x)^2+\frac {x^5 \coth ^{-1}(a x)}{15 a} \]
[In]
[Out]
Rule 45
Rule 266
Rule 272
Rule 6022
Rule 6038
Rule 6096
Rule 6128
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x^6 \coth ^{-1}(a x)^2-\frac {1}{3} a \int \frac {x^6 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx \\ & = \frac {1}{6} x^6 \coth ^{-1}(a x)^2+\frac {\int x^4 \coth ^{-1}(a x) \, dx}{3 a}-\frac {\int \frac {x^4 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{3 a} \\ & = \frac {x^5 \coth ^{-1}(a x)}{15 a}+\frac {1}{6} x^6 \coth ^{-1}(a x)^2-\frac {1}{15} \int \frac {x^5}{1-a^2 x^2} \, dx+\frac {\int x^2 \coth ^{-1}(a x) \, dx}{3 a^3}-\frac {\int \frac {x^2 \coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{3 a^3} \\ & = \frac {x^3 \coth ^{-1}(a x)}{9 a^3}+\frac {x^5 \coth ^{-1}(a x)}{15 a}+\frac {1}{6} x^6 \coth ^{-1}(a x)^2-\frac {1}{30} \text {Subst}\left (\int \frac {x^2}{1-a^2 x} \, dx,x,x^2\right )+\frac {\int \coth ^{-1}(a x) \, dx}{3 a^5}-\frac {\int \frac {\coth ^{-1}(a x)}{1-a^2 x^2} \, dx}{3 a^5}-\frac {\int \frac {x^3}{1-a^2 x^2} \, dx}{9 a^2} \\ & = \frac {x \coth ^{-1}(a x)}{3 a^5}+\frac {x^3 \coth ^{-1}(a x)}{9 a^3}+\frac {x^5 \coth ^{-1}(a x)}{15 a}-\frac {\coth ^{-1}(a x)^2}{6 a^6}+\frac {1}{6} x^6 \coth ^{-1}(a x)^2-\frac {1}{30} \text {Subst}\left (\int \left (-\frac {1}{a^4}-\frac {x}{a^2}-\frac {1}{a^4 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right )-\frac {\int \frac {x}{1-a^2 x^2} \, dx}{3 a^4}-\frac {\text {Subst}\left (\int \frac {x}{1-a^2 x} \, dx,x,x^2\right )}{18 a^2} \\ & = \frac {x^2}{30 a^4}+\frac {x^4}{60 a^2}+\frac {x \coth ^{-1}(a x)}{3 a^5}+\frac {x^3 \coth ^{-1}(a x)}{9 a^3}+\frac {x^5 \coth ^{-1}(a x)}{15 a}-\frac {\coth ^{-1}(a x)^2}{6 a^6}+\frac {1}{6} x^6 \coth ^{-1}(a x)^2+\frac {\log \left (1-a^2 x^2\right )}{5 a^6}-\frac {\text {Subst}\left (\int \left (-\frac {1}{a^2}-\frac {1}{a^2 \left (-1+a^2 x\right )}\right ) \, dx,x,x^2\right )}{18 a^2} \\ & = \frac {4 x^2}{45 a^4}+\frac {x^4}{60 a^2}+\frac {x \coth ^{-1}(a x)}{3 a^5}+\frac {x^3 \coth ^{-1}(a x)}{9 a^3}+\frac {x^5 \coth ^{-1}(a x)}{15 a}-\frac {\coth ^{-1}(a x)^2}{6 a^6}+\frac {1}{6} x^6 \coth ^{-1}(a x)^2+\frac {23 \log \left (1-a^2 x^2\right )}{90 a^6} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.76 \[ \int x^5 \coth ^{-1}(a x)^2 \, dx=\frac {16 a^2 x^2+3 a^4 x^4+4 a x \left (15+5 a^2 x^2+3 a^4 x^4\right ) \coth ^{-1}(a x)+30 \left (-1+a^6 x^6\right ) \coth ^{-1}(a x)^2+46 \log \left (1-a^2 x^2\right )}{180 a^6} \]
[In]
[Out]
Time = 0.30 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.88
method | result | size |
parallelrisch | \(-\frac {-30 a^{6} x^{6} \operatorname {arccoth}\left (a x \right )^{2}-12 a^{5} x^{5} \operatorname {arccoth}\left (a x \right )-16-3 a^{4} x^{4}-20 a^{3} x^{3} \operatorname {arccoth}\left (a x \right )-16 a^{2} x^{2}-60 a x \,\operatorname {arccoth}\left (a x \right )+30 \operatorname {arccoth}\left (a x \right )^{2}-92 \ln \left (a x -1\right )-92 \,\operatorname {arccoth}\left (a x \right )}{180 a^{6}}\) | \(92\) |
parts | \(\frac {x^{6} \operatorname {arccoth}\left (a x \right )^{2}}{6}+\frac {\frac {a^{5} x^{5} \operatorname {arccoth}\left (a x \right )}{5}+\frac {a^{3} x^{3} \operatorname {arccoth}\left (a x \right )}{3}+a x \,\operatorname {arccoth}\left (a x \right )+\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{2}-\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{2}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {\ln \left (a x -1\right )^{2}}{8}+\frac {\ln \left (a x +1\right )^{2}}{8}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{4}+\frac {a^{4} x^{4}}{20}+\frac {4 a^{2} x^{2}}{15}+\frac {23 \ln \left (a x -1\right )}{30}+\frac {23 \ln \left (a x +1\right )}{30}}{3 a^{6}}\) | \(166\) |
derivativedivides | \(\frac {\frac {a^{6} x^{6} \operatorname {arccoth}\left (a x \right )^{2}}{6}+\frac {a^{5} x^{5} \operatorname {arccoth}\left (a x \right )}{15}+\frac {a^{3} x^{3} \operatorname {arccoth}\left (a x \right )}{9}+\frac {a x \,\operatorname {arccoth}\left (a x \right )}{3}+\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{6}-\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{6}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{12}+\frac {\ln \left (a x -1\right )^{2}}{24}+\frac {\ln \left (a x +1\right )^{2}}{24}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{12}+\frac {a^{4} x^{4}}{60}+\frac {4 a^{2} x^{2}}{45}+\frac {23 \ln \left (a x -1\right )}{90}+\frac {23 \ln \left (a x +1\right )}{90}}{a^{6}}\) | \(168\) |
default | \(\frac {\frac {a^{6} x^{6} \operatorname {arccoth}\left (a x \right )^{2}}{6}+\frac {a^{5} x^{5} \operatorname {arccoth}\left (a x \right )}{15}+\frac {a^{3} x^{3} \operatorname {arccoth}\left (a x \right )}{9}+\frac {a x \,\operatorname {arccoth}\left (a x \right )}{3}+\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x -1\right )}{6}-\frac {\operatorname {arccoth}\left (a x \right ) \ln \left (a x +1\right )}{6}-\frac {\ln \left (a x -1\right ) \ln \left (\frac {a x}{2}+\frac {1}{2}\right )}{12}+\frac {\ln \left (a x -1\right )^{2}}{24}+\frac {\ln \left (a x +1\right )^{2}}{24}-\frac {\left (\ln \left (a x +1\right )-\ln \left (\frac {a x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {a x}{2}+\frac {1}{2}\right )}{12}+\frac {a^{4} x^{4}}{60}+\frac {4 a^{2} x^{2}}{45}+\frac {23 \ln \left (a x -1\right )}{90}+\frac {23 \ln \left (a x +1\right )}{90}}{a^{6}}\) | \(168\) |
risch | \(\frac {\left (a^{6} x^{6}-1\right ) \ln \left (a x +1\right )^{2}}{24 a^{6}}-\frac {\left (15 \ln \left (a x -1\right ) x^{6} a^{6}-6 a^{5} x^{5}-10 a^{3} x^{3}-30 a x -15 \ln \left (a x -1\right )\right ) \ln \left (a x +1\right )}{180 a^{6}}+\frac {x^{6} \ln \left (a x -1\right )^{2}}{24}-\frac {\ln \left (a x -1\right ) x^{5}}{30 a}+\frac {x^{4}}{60 a^{2}}-\frac {\ln \left (a x -1\right ) x^{3}}{18 a^{3}}+\frac {4 x^{2}}{45 a^{4}}-\frac {\ln \left (a x -1\right ) x}{6 a^{5}}-\frac {\ln \left (a x -1\right )^{2}}{24 a^{6}}+\frac {23 \ln \left (a^{2} x^{2}-1\right )}{90 a^{6}}+\frac {16}{135 a^{6}}\) | \(180\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.93 \[ \int x^5 \coth ^{-1}(a x)^2 \, dx=\frac {6 \, a^{4} x^{4} + 32 \, a^{2} x^{2} + 15 \, {\left (a^{6} x^{6} - 1\right )} \log \left (\frac {a x + 1}{a x - 1}\right )^{2} + 4 \, {\left (3 \, a^{5} x^{5} + 5 \, a^{3} x^{3} + 15 \, a x\right )} \log \left (\frac {a x + 1}{a x - 1}\right ) + 92 \, \log \left (a^{2} x^{2} - 1\right )}{360 \, a^{6}} \]
[In]
[Out]
Time = 0.40 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.09 \[ \int x^5 \coth ^{-1}(a x)^2 \, dx=\begin {cases} \frac {x^{6} \operatorname {acoth}^{2}{\left (a x \right )}}{6} + \frac {x^{5} \operatorname {acoth}{\left (a x \right )}}{15 a} + \frac {x^{4}}{60 a^{2}} + \frac {x^{3} \operatorname {acoth}{\left (a x \right )}}{9 a^{3}} + \frac {4 x^{2}}{45 a^{4}} + \frac {x \operatorname {acoth}{\left (a x \right )}}{3 a^{5}} + \frac {23 \log {\left (a x + 1 \right )}}{45 a^{6}} - \frac {\operatorname {acoth}^{2}{\left (a x \right )}}{6 a^{6}} - \frac {23 \operatorname {acoth}{\left (a x \right )}}{45 a^{6}} & \text {for}\: a \neq 0 \\- \frac {\pi ^{2} x^{6}}{24} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.29 \[ \int x^5 \coth ^{-1}(a x)^2 \, dx=\frac {1}{6} \, x^{6} \operatorname {arcoth}\left (a x\right )^{2} + \frac {1}{90} \, a {\left (\frac {2 \, {\left (3 \, a^{4} x^{5} + 5 \, a^{2} x^{3} + 15 \, x\right )}}{a^{6}} - \frac {15 \, \log \left (a x + 1\right )}{a^{7}} + \frac {15 \, \log \left (a x - 1\right )}{a^{7}}\right )} \operatorname {arcoth}\left (a x\right ) + \frac {6 \, a^{4} x^{4} + 32 \, a^{2} x^{2} - 2 \, {\left (15 \, \log \left (a x - 1\right ) - 46\right )} \log \left (a x + 1\right ) + 15 \, \log \left (a x + 1\right )^{2} + 15 \, \log \left (a x - 1\right )^{2} + 92 \, \log \left (a x - 1\right )}{360 \, a^{6}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 534 vs. \(2 (89) = 178\).
Time = 0.28 (sec) , antiderivative size = 534, normalized size of antiderivative = 5.09 \[ \int x^5 \coth ^{-1}(a x)^2 \, dx=\frac {1}{90} \, {\left (\frac {15 \, {\left (\frac {3 \, {\left (a x + 1\right )}^{5}}{{\left (a x - 1\right )}^{5}} + \frac {10 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {3 \, {\left (a x + 1\right )}}{a x - 1}\right )} \log \left (\frac {a x + 1}{a x - 1}\right )^{2}}{\frac {{\left (a x + 1\right )}^{6} a^{7}}{{\left (a x - 1\right )}^{6}} - \frac {6 \, {\left (a x + 1\right )}^{5} a^{7}}{{\left (a x - 1\right )}^{5}} + \frac {15 \, {\left (a x + 1\right )}^{4} a^{7}}{{\left (a x - 1\right )}^{4}} - \frac {20 \, {\left (a x + 1\right )}^{3} a^{7}}{{\left (a x - 1\right )}^{3}} + \frac {15 \, {\left (a x + 1\right )}^{2} a^{7}}{{\left (a x - 1\right )}^{2}} - \frac {6 \, {\left (a x + 1\right )} a^{7}}{a x - 1} + a^{7}} + \frac {2 \, {\left (\frac {45 \, {\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} - \frac {90 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {140 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} - \frac {70 \, {\left (a x + 1\right )}}{a x - 1} + 23\right )} \log \left (\frac {a x + 1}{a x - 1}\right )}{\frac {{\left (a x + 1\right )}^{5} a^{7}}{{\left (a x - 1\right )}^{5}} - \frac {5 \, {\left (a x + 1\right )}^{4} a^{7}}{{\left (a x - 1\right )}^{4}} + \frac {10 \, {\left (a x + 1\right )}^{3} a^{7}}{{\left (a x - 1\right )}^{3}} - \frac {10 \, {\left (a x + 1\right )}^{2} a^{7}}{{\left (a x - 1\right )}^{2}} + \frac {5 \, {\left (a x + 1\right )} a^{7}}{a x - 1} - a^{7}} + \frac {4 \, {\left (\frac {11 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} - \frac {16 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {11 \, {\left (a x + 1\right )}}{a x - 1}\right )}}{\frac {{\left (a x + 1\right )}^{4} a^{7}}{{\left (a x - 1\right )}^{4}} - \frac {4 \, {\left (a x + 1\right )}^{3} a^{7}}{{\left (a x - 1\right )}^{3}} + \frac {6 \, {\left (a x + 1\right )}^{2} a^{7}}{{\left (a x - 1\right )}^{2}} - \frac {4 \, {\left (a x + 1\right )} a^{7}}{a x - 1} + a^{7}} - \frac {46 \, \log \left (\frac {a x + 1}{a x - 1} - 1\right )}{a^{7}} + \frac {46 \, \log \left (\frac {a x + 1}{a x - 1}\right )}{a^{7}}\right )} a \]
[In]
[Out]
Time = 4.63 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.81 \[ \int x^5 \coth ^{-1}(a x)^2 \, dx=\frac {x^6\,{\mathrm {acoth}\left (a\,x\right )}^2}{6}+\frac {\frac {23\,\ln \left (a^2\,x^2-1\right )}{90}+\frac {4\,a^2\,x^2}{45}+\frac {a^4\,x^4}{60}-\frac {{\mathrm {acoth}\left (a\,x\right )}^2}{6}+\frac {a^3\,x^3\,\mathrm {acoth}\left (a\,x\right )}{9}+\frac {a^5\,x^5\,\mathrm {acoth}\left (a\,x\right )}{15}+\frac {a\,x\,\mathrm {acoth}\left (a\,x\right )}{3}}{a^6} \]
[In]
[Out]