Integrand size = 20, antiderivative size = 87 \[ \int x^3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x) \, dx=\frac {x}{24 a^3}+\frac {x^3}{72 a}-\frac {a x^5}{24}+\frac {a^3 x^7}{56}-\frac {\text {arctanh}(a x)}{24 a^4}+\frac {1}{4} x^4 \text {arctanh}(a x)-\frac {1}{3} a^2 x^6 \text {arctanh}(a x)+\frac {1}{8} a^4 x^8 \text {arctanh}(a x) \]
1/24*x/a^3+1/72*x^3/a-1/24*a*x^5+1/56*a^3*x^7-1/24*arctanh(a*x)/a^4+1/4*x^ 4*arctanh(a*x)-1/3*a^2*x^6*arctanh(a*x)+1/8*a^4*x^8*arctanh(a*x)
Time = 0.03 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.18 \[ \int x^3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x) \, dx=\frac {x}{24 a^3}+\frac {x^3}{72 a}-\frac {a x^5}{24}+\frac {a^3 x^7}{56}+\frac {1}{4} x^4 \text {arctanh}(a x)-\frac {1}{3} a^2 x^6 \text {arctanh}(a x)+\frac {1}{8} a^4 x^8 \text {arctanh}(a x)+\frac {\log (1-a x)}{48 a^4}-\frac {\log (1+a x)}{48 a^4} \]
x/(24*a^3) + x^3/(72*a) - (a*x^5)/24 + (a^3*x^7)/56 + (x^4*ArcTanh[a*x])/4 - (a^2*x^6*ArcTanh[a*x])/3 + (a^4*x^8*ArcTanh[a*x])/8 + Log[1 - a*x]/(48* a^4) - Log[1 + a*x]/(48*a^4)
Time = 0.34 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6574, 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.
\(\displaystyle \int x^3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x) \, dx\) |
\(\Big \downarrow \) 6574 |
\(\displaystyle \int \left (a^4 x^7 \text {arctanh}(a x)-2 a^2 x^5 \text {arctanh}(a x)+x^3 \text {arctanh}(a x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{8} a^4 x^8 \text {arctanh}(a x)-\frac {\text {arctanh}(a x)}{24 a^4}+\frac {a^3 x^7}{56}+\frac {x}{24 a^3}-\frac {1}{3} a^2 x^6 \text {arctanh}(a x)+\frac {1}{4} x^4 \text {arctanh}(a x)-\frac {a x^5}{24}+\frac {x^3}{72 a}\) |
x/(24*a^3) + x^3/(72*a) - (a*x^5)/24 + (a^3*x^7)/56 - ArcTanh[a*x]/(24*a^4 ) + (x^4*ArcTanh[a*x])/4 - (a^2*x^6*ArcTanh[a*x])/3 + (a^4*x^8*ArcTanh[a*x ])/8
3.2.93.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(q_), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0] && IGtQ[q, 1]
Time = 0.42 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.89
method | result | size |
parallelrisch | \(-\frac {-63 \,\operatorname {arctanh}\left (a x \right ) a^{8} x^{8}-9 a^{7} x^{7}+168 \,\operatorname {arctanh}\left (a x \right ) a^{6} x^{6}+21 a^{5} x^{5}-126 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )-7 a^{3} x^{3}-21 a x +21 \,\operatorname {arctanh}\left (a x \right )}{504 a^{4}}\) | \(77\) |
derivativedivides | \(\frac {\frac {\operatorname {arctanh}\left (a x \right ) a^{8} x^{8}}{8}-\frac {\operatorname {arctanh}\left (a x \right ) a^{6} x^{6}}{3}+\frac {a^{4} x^{4} \operatorname {arctanh}\left (a x \right )}{4}+\frac {a^{7} x^{7}}{56}-\frac {a^{5} x^{5}}{24}+\frac {a^{3} x^{3}}{72}+\frac {a x}{24}+\frac {\ln \left (a x -1\right )}{48}-\frac {\ln \left (a x +1\right )}{48}}{a^{4}}\) | \(86\) |
default | \(\frac {\frac {\operatorname {arctanh}\left (a x \right ) a^{8} x^{8}}{8}-\frac {\operatorname {arctanh}\left (a x \right ) a^{6} x^{6}}{3}+\frac {a^{4} x^{4} \operatorname {arctanh}\left (a x \right )}{4}+\frac {a^{7} x^{7}}{56}-\frac {a^{5} x^{5}}{24}+\frac {a^{3} x^{3}}{72}+\frac {a x}{24}+\frac {\ln \left (a x -1\right )}{48}-\frac {\ln \left (a x +1\right )}{48}}{a^{4}}\) | \(86\) |
parts | \(\frac {a^{4} x^{8} \operatorname {arctanh}\left (a x \right )}{8}-\frac {a^{2} x^{6} \operatorname {arctanh}\left (a x \right )}{3}+\frac {x^{4} \operatorname {arctanh}\left (a x \right )}{4}-\frac {a \left (-\frac {\frac {3}{7} a^{6} x^{7}-a^{4} x^{5}+\frac {1}{3} a^{2} x^{3}+x}{a^{4}}+\frac {\ln \left (a x +1\right )}{2 a^{5}}-\frac {\ln \left (a x -1\right )}{2 a^{5}}\right )}{24}\) | \(92\) |
risch | \(\left (\frac {1}{16} a^{4} x^{8}-\frac {1}{6} a^{2} x^{6}+\frac {1}{8} x^{4}\right ) \ln \left (a x +1\right )-\frac {a^{4} x^{8} \ln \left (-a x +1\right )}{16}+\frac {a^{3} x^{7}}{56}+\frac {a^{2} x^{6} \ln \left (-a x +1\right )}{6}-\frac {a \,x^{5}}{24}-\frac {x^{4} \ln \left (-a x +1\right )}{8}+\frac {x^{3}}{72 a}+\frac {x}{24 a^{3}}+\frac {\ln \left (-a x +1\right )}{48 a^{4}}-\frac {\ln \left (a x +1\right )}{48 a^{4}}\) | \(124\) |
meijerg | \(-\frac {i \left (\frac {i x a \left (45 a^{6} x^{6}+63 a^{4} x^{4}+105 a^{2} x^{2}+315\right )}{630}+\frac {i x a \left (-9 a^{8} x^{8}+9\right ) \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{36 \sqrt {a^{2} x^{2}}}\right )}{4 a^{4}}-\frac {i \left (-\frac {2 i x a \left (21 a^{4} x^{4}+35 a^{2} x^{2}+105\right )}{315}-\frac {i x a \left (-7 a^{6} x^{6}+7\right ) \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{21 \sqrt {a^{2} x^{2}}}\right )}{2 a^{4}}-\frac {i \left (\frac {i x a \left (5 a^{2} x^{2}+15\right )}{15}+\frac {i x a \left (-5 a^{4} x^{4}+5\right ) \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{10 \sqrt {a^{2} x^{2}}}\right )}{4 a^{4}}\) | \(251\) |
-1/504*(-63*arctanh(a*x)*a^8*x^8-9*a^7*x^7+168*arctanh(a*x)*a^6*x^6+21*a^5 *x^5-126*a^4*x^4*arctanh(a*x)-7*a^3*x^3-21*a*x+21*arctanh(a*x))/a^4
Time = 0.25 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.89 \[ \int x^3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x) \, dx=\frac {18 \, a^{7} x^{7} - 42 \, a^{5} x^{5} + 14 \, a^{3} x^{3} + 42 \, a x + 21 \, {\left (3 \, a^{8} x^{8} - 8 \, a^{6} x^{6} + 6 \, a^{4} x^{4} - 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{1008 \, a^{4}} \]
1/1008*(18*a^7*x^7 - 42*a^5*x^5 + 14*a^3*x^3 + 42*a*x + 21*(3*a^8*x^8 - 8* a^6*x^6 + 6*a^4*x^4 - 1)*log(-(a*x + 1)/(a*x - 1)))/a^4
Time = 0.56 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.87 \[ \int x^3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x) \, dx=\begin {cases} \frac {a^{4} x^{8} \operatorname {atanh}{\left (a x \right )}}{8} + \frac {a^{3} x^{7}}{56} - \frac {a^{2} x^{6} \operatorname {atanh}{\left (a x \right )}}{3} - \frac {a x^{5}}{24} + \frac {x^{4} \operatorname {atanh}{\left (a x \right )}}{4} + \frac {x^{3}}{72 a} + \frac {x}{24 a^{3}} - \frac {\operatorname {atanh}{\left (a x \right )}}{24 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((a**4*x**8*atanh(a*x)/8 + a**3*x**7/56 - a**2*x**6*atanh(a*x)/3 - a*x**5/24 + x**4*atanh(a*x)/4 + x**3/(72*a) + x/(24*a**3) - atanh(a*x)/( 24*a**4), Ne(a, 0)), (0, True))
Time = 0.20 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.01 \[ \int x^3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x) \, dx=\frac {1}{1008} \, a {\left (\frac {2 \, {\left (9 \, a^{6} x^{7} - 21 \, a^{4} x^{5} + 7 \, a^{2} x^{3} + 21 \, x\right )}}{a^{4}} - \frac {21 \, \log \left (a x + 1\right )}{a^{5}} + \frac {21 \, \log \left (a x - 1\right )}{a^{5}}\right )} + \frac {1}{24} \, {\left (3 \, a^{4} x^{8} - 8 \, a^{2} x^{6} + 6 \, x^{4}\right )} \operatorname {artanh}\left (a x\right ) \]
1/1008*a*(2*(9*a^6*x^7 - 21*a^4*x^5 + 7*a^2*x^3 + 21*x)/a^4 - 21*log(a*x + 1)/a^5 + 21*log(a*x - 1)/a^5) + 1/24*(3*a^4*x^8 - 8*a^2*x^6 + 6*x^4)*arct anh(a*x)
Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (71) = 142\).
Time = 0.29 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.76 \[ \int x^3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x) \, dx=\frac {4}{63} \, a {\left (\frac {\frac {28 \, {\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} - \frac {7 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {21 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} - \frac {7 \, {\left (a x + 1\right )}}{a x - 1} + 1}{a^{5} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{7}} + \frac {84 \, {\left (\frac {{\left (a x + 1\right )}^{5}}{{\left (a x - 1\right )}^{5}} + \frac {{\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} + \frac {{\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}}\right )} \log \left (-\frac {\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} + 1}{\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} - 1}\right )}{a^{5} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{8}}\right )} \]
4/63*a*((28*(a*x + 1)^4/(a*x - 1)^4 - 7*(a*x + 1)^3/(a*x - 1)^3 + 21*(a*x + 1)^2/(a*x - 1)^2 - 7*(a*x + 1)/(a*x - 1) + 1)/(a^5*((a*x + 1)/(a*x - 1) - 1)^7) + 84*((a*x + 1)^5/(a*x - 1)^5 + (a*x + 1)^4/(a*x - 1)^4 + (a*x + 1 )^3/(a*x - 1)^3)*log(-(a*((a*x + 1)/(a*x - 1) + 1)/((a*x + 1)*a/(a*x - 1) - a) + 1)/(a*((a*x + 1)/(a*x - 1) + 1)/((a*x + 1)*a/(a*x - 1) - a) - 1))/( a^5*((a*x + 1)/(a*x - 1) - 1)^8))
Time = 4.15 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.16 \[ \int x^3 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x) \, dx=\frac {x}{24\,a^3}-\frac {a\,x^5}{24}+\ln \left (a\,x+1\right )\,\left (\frac {a^4\,x^8}{16}-\frac {a^2\,x^6}{6}+\frac {x^4}{8}\right )-\ln \left (1-a\,x\right )\,\left (\frac {a^4\,x^8}{16}-\frac {a^2\,x^6}{6}+\frac {x^4}{8}\right )+\frac {x^3}{72\,a}+\frac {a^3\,x^7}{56}+\frac {\mathrm {atan}\left (a\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{24\,a^4} \]