Integrand size = 12, antiderivative size = 59 \[ \int x^5 (a+b \text {arctanh}(c x)) \, dx=\frac {b x}{6 c^5}+\frac {b x^3}{18 c^3}+\frac {b x^5}{30 c}-\frac {b \text {arctanh}(c x)}{6 c^6}+\frac {1}{6} x^6 (a+b \text {arctanh}(c x)) \]
Time = 0.02 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.37 \[ \int x^5 (a+b \text {arctanh}(c x)) \, dx=\frac {b x}{6 c^5}+\frac {b x^3}{18 c^3}+\frac {b x^5}{30 c}+\frac {a x^6}{6}+\frac {1}{6} b x^6 \text {arctanh}(c x)+\frac {b \log (1-c x)}{12 c^6}-\frac {b \log (1+c x)}{12 c^6} \]
(b*x)/(6*c^5) + (b*x^3)/(18*c^3) + (b*x^5)/(30*c) + (a*x^6)/6 + (b*x^6*Arc Tanh[c*x])/6 + (b*Log[1 - c*x])/(12*c^6) - (b*Log[1 + c*x])/(12*c^6)
Time = 0.23 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6452, 254, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^5 (a+b \text {arctanh}(c x)) \, dx\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))-\frac {1}{6} b c \int \frac {x^6}{1-c^2 x^2}dx\) |
\(\Big \downarrow \) 254 |
\(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))-\frac {1}{6} b c \int \left (-\frac {x^4}{c^2}-\frac {x^2}{c^4}+\frac {1}{c^6 \left (1-c^2 x^2\right )}-\frac {1}{c^6}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{6} x^6 (a+b \text {arctanh}(c x))-\frac {1}{6} b c \left (\frac {\text {arctanh}(c x)}{c^7}-\frac {x}{c^6}-\frac {x^3}{3 c^4}-\frac {x^5}{5 c^2}\right )\) |
(x^6*(a + b*ArcTanh[c*x]))/6 - (b*c*(-(x/c^6) - x^3/(3*c^4) - x^5/(5*c^2) + ArcTanh[c*x]/c^7))/6
3.1.1.3.1 Defintions of rubi rules used
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : > Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x ], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 ] && IntegerQ[m])) && NeQ[m, -1]
Time = 0.20 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00
method | result | size |
parallelrisch | \(-\frac {-15 b \,\operatorname {arctanh}\left (c x \right ) x^{6} c^{6}-15 a \,c^{6} x^{6}-3 b \,c^{5} x^{5}-5 b \,c^{3} x^{3}-15 b c x +15 b \,\operatorname {arctanh}\left (c x \right )}{90 c^{6}}\) | \(59\) |
parts | \(\frac {a \,x^{6}}{6}+\frac {b \left (\frac {c^{6} x^{6} \operatorname {arctanh}\left (c x \right )}{6}+\frac {c^{5} x^{5}}{30}+\frac {c^{3} x^{3}}{18}+\frac {c x}{6}+\frac {\ln \left (c x -1\right )}{12}-\frac {\ln \left (c x +1\right )}{12}\right )}{c^{6}}\) | \(62\) |
derivativedivides | \(\frac {\frac {a \,c^{6} x^{6}}{6}+b \left (\frac {c^{6} x^{6} \operatorname {arctanh}\left (c x \right )}{6}+\frac {c^{5} x^{5}}{30}+\frac {c^{3} x^{3}}{18}+\frac {c x}{6}+\frac {\ln \left (c x -1\right )}{12}-\frac {\ln \left (c x +1\right )}{12}\right )}{c^{6}}\) | \(66\) |
default | \(\frac {\frac {a \,c^{6} x^{6}}{6}+b \left (\frac {c^{6} x^{6} \operatorname {arctanh}\left (c x \right )}{6}+\frac {c^{5} x^{5}}{30}+\frac {c^{3} x^{3}}{18}+\frac {c x}{6}+\frac {\ln \left (c x -1\right )}{12}-\frac {\ln \left (c x +1\right )}{12}\right )}{c^{6}}\) | \(66\) |
risch | \(\frac {b \,x^{6} \ln \left (c x +1\right )}{12}-\frac {b \,x^{6} \ln \left (-c x +1\right )}{12}+\frac {a \,x^{6}}{6}+\frac {b \,x^{5}}{30 c}+\frac {b \,x^{3}}{18 c^{3}}+\frac {b x}{6 c^{5}}-\frac {b \ln \left (c x +1\right )}{12 c^{6}}+\frac {b \ln \left (-c x +1\right )}{12 c^{6}}\) | \(83\) |
-1/90*(-15*b*arctanh(c*x)*x^6*c^6-15*a*c^6*x^6-3*b*c^5*x^5-5*b*c^3*x^3-15* b*c*x+15*b*arctanh(c*x))/c^6
Time = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.14 \[ \int x^5 (a+b \text {arctanh}(c x)) \, dx=\frac {30 \, a c^{6} x^{6} + 6 \, b c^{5} x^{5} + 10 \, b c^{3} x^{3} + 30 \, b c x + 15 \, {\left (b c^{6} x^{6} - b\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{180 \, c^{6}} \]
1/180*(30*a*c^6*x^6 + 6*b*c^5*x^5 + 10*b*c^3*x^3 + 30*b*c*x + 15*(b*c^6*x^ 6 - b)*log(-(c*x + 1)/(c*x - 1)))/c^6
Time = 0.37 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.07 \[ \int x^5 (a+b \text {arctanh}(c x)) \, dx=\begin {cases} \frac {a x^{6}}{6} + \frac {b x^{6} \operatorname {atanh}{\left (c x \right )}}{6} + \frac {b x^{5}}{30 c} + \frac {b x^{3}}{18 c^{3}} + \frac {b x}{6 c^{5}} - \frac {b \operatorname {atanh}{\left (c x \right )}}{6 c^{6}} & \text {for}\: c \neq 0 \\\frac {a x^{6}}{6} & \text {otherwise} \end {cases} \]
Piecewise((a*x**6/6 + b*x**6*atanh(c*x)/6 + b*x**5/(30*c) + b*x**3/(18*c** 3) + b*x/(6*c**5) - b*atanh(c*x)/(6*c**6), Ne(c, 0)), (a*x**6/6, True))
Time = 0.21 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.19 \[ \int x^5 (a+b \text {arctanh}(c x)) \, dx=\frac {1}{6} \, a x^{6} + \frac {1}{180} \, {\left (30 \, x^{6} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac {15 \, \log \left (c x + 1\right )}{c^{7}} + \frac {15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} b \]
1/6*a*x^6 + 1/180*(30*x^6*arctanh(c*x) + c*(2*(3*c^4*x^5 + 5*c^2*x^3 + 15* x)/c^6 - 15*log(c*x + 1)/c^7 + 15*log(c*x - 1)/c^7))*b
Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (49) = 98\).
Time = 0.30 (sec) , antiderivative size = 442, normalized size of antiderivative = 7.49 \[ \int x^5 (a+b \text {arctanh}(c x)) \, dx=\frac {1}{45} \, c {\left (\frac {15 \, {\left (\frac {3 \, {\left (c x + 1\right )}^{5} b}{{\left (c x - 1\right )}^{5}} + \frac {10 \, {\left (c x + 1\right )}^{3} b}{{\left (c x - 1\right )}^{3}} + \frac {3 \, {\left (c x + 1\right )} b}{c x - 1}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{\frac {{\left (c x + 1\right )}^{6} c^{7}}{{\left (c x - 1\right )}^{6}} - \frac {6 \, {\left (c x + 1\right )}^{5} c^{7}}{{\left (c x - 1\right )}^{5}} + \frac {15 \, {\left (c x + 1\right )}^{4} c^{7}}{{\left (c x - 1\right )}^{4}} - \frac {20 \, {\left (c x + 1\right )}^{3} c^{7}}{{\left (c x - 1\right )}^{3}} + \frac {15 \, {\left (c x + 1\right )}^{2} c^{7}}{{\left (c x - 1\right )}^{2}} - \frac {6 \, {\left (c x + 1\right )} c^{7}}{c x - 1} + c^{7}} + \frac {\frac {90 \, {\left (c x + 1\right )}^{5} a}{{\left (c x - 1\right )}^{5}} + \frac {300 \, {\left (c x + 1\right )}^{3} a}{{\left (c x - 1\right )}^{3}} + \frac {90 \, {\left (c x + 1\right )} a}{c x - 1} + \frac {45 \, {\left (c x + 1\right )}^{5} b}{{\left (c x - 1\right )}^{5}} - \frac {135 \, {\left (c x + 1\right )}^{4} b}{{\left (c x - 1\right )}^{4}} + \frac {230 \, {\left (c x + 1\right )}^{3} b}{{\left (c x - 1\right )}^{3}} - \frac {210 \, {\left (c x + 1\right )}^{2} b}{{\left (c x - 1\right )}^{2}} + \frac {93 \, {\left (c x + 1\right )} b}{c x - 1} - 23 \, b}{\frac {{\left (c x + 1\right )}^{6} c^{7}}{{\left (c x - 1\right )}^{6}} - \frac {6 \, {\left (c x + 1\right )}^{5} c^{7}}{{\left (c x - 1\right )}^{5}} + \frac {15 \, {\left (c x + 1\right )}^{4} c^{7}}{{\left (c x - 1\right )}^{4}} - \frac {20 \, {\left (c x + 1\right )}^{3} c^{7}}{{\left (c x - 1\right )}^{3}} + \frac {15 \, {\left (c x + 1\right )}^{2} c^{7}}{{\left (c x - 1\right )}^{2}} - \frac {6 \, {\left (c x + 1\right )} c^{7}}{c x - 1} + c^{7}}\right )} \]
1/45*c*(15*(3*(c*x + 1)^5*b/(c*x - 1)^5 + 10*(c*x + 1)^3*b/(c*x - 1)^3 + 3 *(c*x + 1)*b/(c*x - 1))*log(-(c*x + 1)/(c*x - 1))/((c*x + 1)^6*c^7/(c*x - 1)^6 - 6*(c*x + 1)^5*c^7/(c*x - 1)^5 + 15*(c*x + 1)^4*c^7/(c*x - 1)^4 - 20 *(c*x + 1)^3*c^7/(c*x - 1)^3 + 15*(c*x + 1)^2*c^7/(c*x - 1)^2 - 6*(c*x + 1 )*c^7/(c*x - 1) + c^7) + (90*(c*x + 1)^5*a/(c*x - 1)^5 + 300*(c*x + 1)^3*a /(c*x - 1)^3 + 90*(c*x + 1)*a/(c*x - 1) + 45*(c*x + 1)^5*b/(c*x - 1)^5 - 1 35*(c*x + 1)^4*b/(c*x - 1)^4 + 230*(c*x + 1)^3*b/(c*x - 1)^3 - 210*(c*x + 1)^2*b/(c*x - 1)^2 + 93*(c*x + 1)*b/(c*x - 1) - 23*b)/((c*x + 1)^6*c^7/(c* x - 1)^6 - 6*(c*x + 1)^5*c^7/(c*x - 1)^5 + 15*(c*x + 1)^4*c^7/(c*x - 1)^4 - 20*(c*x + 1)^3*c^7/(c*x - 1)^3 + 15*(c*x + 1)^2*c^7/(c*x - 1)^2 - 6*(c*x + 1)*c^7/(c*x - 1) + c^7))
Time = 3.32 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.88 \[ \int x^5 (a+b \text {arctanh}(c x)) \, dx=\frac {\frac {b\,c^3\,x^3}{18}-\frac {b\,\mathrm {atanh}\left (c\,x\right )}{6}+\frac {b\,c^5\,x^5}{30}+\frac {b\,c\,x}{6}}{c^6}+\frac {a\,x^6}{6}+\frac {b\,x^6\,\mathrm {atanh}\left (c\,x\right )}{6} \]