Integrand size = 20, antiderivative size = 96 \[ \int x^4 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x) \, dx=\frac {4 x^2}{315 a^3}+\frac {2 x^4}{315 a}-\frac {11 a x^6}{378}+\frac {a^3 x^8}{72}+\frac {1}{5} x^5 \text {arctanh}(a x)-\frac {2}{7} a^2 x^7 \text {arctanh}(a x)+\frac {1}{9} a^4 x^9 \text {arctanh}(a x)+\frac {4 \log \left (1-a^2 x^2\right )}{315 a^5} \] Output:
4/315*x^2/a^3+2/315*x^4/a-11/378*a*x^6+1/72*a^3*x^8+1/5*x^5*arctanh(a*x)-2 /7*a^2*x^7*arctanh(a*x)+1/9*a^4*x^9*arctanh(a*x)+4/315*ln(-a^2*x^2+1)/a^5
Time = 0.04 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00 \[ \int x^4 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x) \, dx=\frac {4 x^2}{315 a^3}+\frac {2 x^4}{315 a}-\frac {11 a x^6}{378}+\frac {a^3 x^8}{72}+\frac {1}{5} x^5 \text {arctanh}(a x)-\frac {2}{7} a^2 x^7 \text {arctanh}(a x)+\frac {1}{9} a^4 x^9 \text {arctanh}(a x)+\frac {4 \log \left (1-a^2 x^2\right )}{315 a^5} \] Input:
Integrate[x^4*(1 - a^2*x^2)^2*ArcTanh[a*x],x]
Output:
(4*x^2)/(315*a^3) + (2*x^4)/(315*a) - (11*a*x^6)/378 + (a^3*x^8)/72 + (x^5 *ArcTanh[a*x])/5 - (2*a^2*x^7*ArcTanh[a*x])/7 + (a^4*x^9*ArcTanh[a*x])/9 + (4*Log[1 - a^2*x^2])/(315*a^5)
Time = 0.40 (sec) , antiderivative size = 96, 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.
\(\displaystyle \int x^4 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x) \, dx\) |
\(\Big \downarrow \) 6574 |
\(\displaystyle \int \left (a^4 x^8 \text {arctanh}(a x)-2 a^2 x^6 \text {arctanh}(a x)+x^4 \text {arctanh}(a x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{9} a^4 x^9 \text {arctanh}(a x)+\frac {a^3 x^8}{72}+\frac {4 x^2}{315 a^3}-\frac {2}{7} a^2 x^7 \text {arctanh}(a x)+\frac {4 \log \left (1-a^2 x^2\right )}{315 a^5}+\frac {1}{5} x^5 \text {arctanh}(a x)-\frac {11 a x^6}{378}+\frac {2 x^4}{315 a}\) |
Input:
Int[x^4*(1 - a^2*x^2)^2*ArcTanh[a*x],x]
Output:
(4*x^2)/(315*a^3) + (2*x^4)/(315*a) - (11*a*x^6)/378 + (a^3*x^8)/72 + (x^5 *ArcTanh[a*x])/5 - (2*a^2*x^7*ArcTanh[a*x])/7 + (a^4*x^9*ArcTanh[a*x])/9 + (4*Log[1 - a^2*x^2])/(315*a^5)
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 = 0.64 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.93
method | result | size |
parts | \(\frac {a^{4} x^{9} \operatorname {arctanh}\left (a x \right )}{9}-\frac {2 a^{2} x^{7} \operatorname {arctanh}\left (a x \right )}{7}+\frac {x^{5} \operatorname {arctanh}\left (a x \right )}{5}-\frac {a \left (-\frac {\frac {35}{4} a^{6} x^{8}-\frac {55}{3} a^{4} x^{6}+4 a^{2} x^{4}+8 x^{2}}{2 a^{4}}-\frac {4 \ln \left (a^{2} x^{2}-1\right )}{a^{6}}\right )}{315}\) | \(89\) |
derivativedivides | \(\frac {\frac {\operatorname {arctanh}\left (a x \right ) a^{9} x^{9}}{9}-\frac {2 \,\operatorname {arctanh}\left (a x \right ) a^{7} x^{7}}{7}+\frac {\operatorname {arctanh}\left (a x \right ) a^{5} x^{5}}{5}+\frac {a^{8} x^{8}}{72}-\frac {11 a^{6} x^{6}}{378}+\frac {2 a^{4} x^{4}}{315}+\frac {4 a^{2} x^{2}}{315}+\frac {4 \ln \left (a x -1\right )}{315}+\frac {4 \ln \left (a x +1\right )}{315}}{a^{5}}\) | \(90\) |
default | \(\frac {\frac {\operatorname {arctanh}\left (a x \right ) a^{9} x^{9}}{9}-\frac {2 \,\operatorname {arctanh}\left (a x \right ) a^{7} x^{7}}{7}+\frac {\operatorname {arctanh}\left (a x \right ) a^{5} x^{5}}{5}+\frac {a^{8} x^{8}}{72}-\frac {11 a^{6} x^{6}}{378}+\frac {2 a^{4} x^{4}}{315}+\frac {4 a^{2} x^{2}}{315}+\frac {4 \ln \left (a x -1\right )}{315}+\frac {4 \ln \left (a x +1\right )}{315}}{a^{5}}\) | \(90\) |
parallelrisch | \(-\frac {-840 \,\operatorname {arctanh}\left (a x \right ) a^{9} x^{9}-105 a^{8} x^{8}+2160 \,\operatorname {arctanh}\left (a x \right ) a^{7} x^{7}+220 a^{6} x^{6}-1512 \,\operatorname {arctanh}\left (a x \right ) a^{5} x^{5}-48 a^{4} x^{4}-96-96 a^{2} x^{2}-192 \ln \left (a x -1\right )-192 \,\operatorname {arctanh}\left (a x \right )}{7560 a^{5}}\) | \(90\) |
risch | \(\left (\frac {1}{18} a^{4} x^{9}-\frac {1}{7} a^{2} x^{7}+\frac {1}{10} x^{5}\right ) \ln \left (a x +1\right )-\frac {a^{4} x^{9} \ln \left (-a x +1\right )}{18}+\frac {a^{3} x^{8}}{72}+\frac {a^{2} x^{7} \ln \left (-a x +1\right )}{7}-\frac {11 a \,x^{6}}{378}-\frac {x^{5} \ln \left (-a x +1\right )}{10}+\frac {2 x^{4}}{315 a}+\frac {4 x^{2}}{315 a^{3}}+\frac {4 \ln \left (a^{2} x^{2}-1\right )}{315 a^{5}}\) | \(118\) |
meijerg | \(-\frac {-\frac {x^{2} a^{2} \left (15 a^{6} x^{6}+20 a^{4} x^{4}+30 a^{2} x^{2}+60\right )}{270}+\frac {2 x^{10} a^{10} \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{9 \sqrt {a^{2} x^{2}}}-\frac {2 \ln \left (-a^{2} x^{2}+1\right )}{9}}{4 a^{5}}-\frac {\frac {x^{2} a^{2} \left (4 a^{4} x^{4}+6 a^{2} x^{2}+12\right )}{42}-\frac {2 x^{8} a^{8} \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{7 \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (-a^{2} x^{2}+1\right )}{7}}{2 a^{5}}-\frac {-\frac {a^{2} x^{2} \left (3 a^{2} x^{2}+6\right )}{15}+\frac {2 a^{6} x^{6} \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{5 \sqrt {a^{2} x^{2}}}-\frac {2 \ln \left (-a^{2} x^{2}+1\right )}{5}}{4 a^{5}}\) | \(275\) |
Input:
int(x^4*(-a^2*x^2+1)^2*arctanh(a*x),x,method=_RETURNVERBOSE)
Output:
1/9*a^4*x^9*arctanh(a*x)-2/7*a^2*x^7*arctanh(a*x)+1/5*x^5*arctanh(a*x)-1/3 15*a*(-1/2/a^4*(35/4*a^6*x^8-55/3*a^4*x^6+4*a^2*x^4+8*x^2)-4/a^6*ln(a^2*x^ 2-1))
Time = 0.08 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.96 \[ \int x^4 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x) \, dx=\frac {105 \, a^{8} x^{8} - 220 \, a^{6} x^{6} + 48 \, a^{4} x^{4} + 96 \, a^{2} x^{2} + 12 \, {\left (35 \, a^{9} x^{9} - 90 \, a^{7} x^{7} + 63 \, a^{5} x^{5}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) + 96 \, \log \left (a^{2} x^{2} - 1\right )}{7560 \, a^{5}} \] Input:
integrate(x^4*(-a^2*x^2+1)^2*arctanh(a*x),x, algorithm="fricas")
Output:
1/7560*(105*a^8*x^8 - 220*a^6*x^6 + 48*a^4*x^4 + 96*a^2*x^2 + 12*(35*a^9*x ^9 - 90*a^7*x^7 + 63*a^5*x^5)*log(-(a*x + 1)/(a*x - 1)) + 96*log(a^2*x^2 - 1))/a^5
Time = 0.66 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.04 \[ \int x^4 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x) \, dx=\begin {cases} \frac {a^{4} x^{9} \operatorname {atanh}{\left (a x \right )}}{9} + \frac {a^{3} x^{8}}{72} - \frac {2 a^{2} x^{7} \operatorname {atanh}{\left (a x \right )}}{7} - \frac {11 a x^{6}}{378} + \frac {x^{5} \operatorname {atanh}{\left (a x \right )}}{5} + \frac {2 x^{4}}{315 a} + \frac {4 x^{2}}{315 a^{3}} + \frac {8 \log {\left (x - \frac {1}{a} \right )}}{315 a^{5}} + \frac {8 \operatorname {atanh}{\left (a x \right )}}{315 a^{5}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate(x**4*(-a**2*x**2+1)**2*atanh(a*x),x)
Output:
Piecewise((a**4*x**9*atanh(a*x)/9 + a**3*x**8/72 - 2*a**2*x**7*atanh(a*x)/ 7 - 11*a*x**6/378 + x**5*atanh(a*x)/5 + 2*x**4/(315*a) + 4*x**2/(315*a**3) + 8*log(x - 1/a)/(315*a**5) + 8*atanh(a*x)/(315*a**5), Ne(a, 0)), (0, Tru e))
Time = 0.03 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.93 \[ \int x^4 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x) \, dx=\frac {1}{7560} \, a {\left (\frac {105 \, a^{6} x^{8} - 220 \, a^{4} x^{6} + 48 \, a^{2} x^{4} + 96 \, x^{2}}{a^{4}} + \frac {96 \, \log \left (a x + 1\right )}{a^{6}} + \frac {96 \, \log \left (a x - 1\right )}{a^{6}}\right )} + \frac {1}{315} \, {\left (35 \, a^{4} x^{9} - 90 \, a^{2} x^{7} + 63 \, x^{5}\right )} \operatorname {artanh}\left (a x\right ) \] Input:
integrate(x^4*(-a^2*x^2+1)^2*arctanh(a*x),x, algorithm="maxima")
Output:
1/7560*a*((105*a^6*x^8 - 220*a^4*x^6 + 48*a^2*x^4 + 96*x^2)/a^4 + 96*log(a *x + 1)/a^6 + 96*log(a*x - 1)/a^6) + 1/315*(35*a^4*x^9 - 90*a^2*x^7 + 63*x ^5)*arctanh(a*x)
Leaf count of result is larger than twice the leaf count of optimal. 383 vs. \(2 (80) = 160\).
Time = 0.13 (sec) , antiderivative size = 383, normalized size of antiderivative = 3.99 \[ \int x^4 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x) \, dx=\frac {4}{945} \, a {\left (\frac {6 \, \log \left (\frac {{\left | -a x - 1 \right |}}{{\left | a x - 1 \right |}}\right )}{a^{6}} - \frac {6 \, \log \left ({\left | -\frac {a x + 1}{a x - 1} + 1 \right |}\right )}{a^{6}} - \frac {\frac {6 \, {\left (a x + 1\right )}^{7}}{{\left (a x - 1\right )}^{7}} - \frac {45 \, {\left (a x + 1\right )}^{6}}{{\left (a x - 1\right )}^{6}} - \frac {274 \, {\left (a x + 1\right )}^{5}}{{\left (a x - 1\right )}^{5}} - \frac {214 \, {\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} - \frac {274 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} - \frac {45 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} + \frac {6 \, {\left (a x + 1\right )}}{a x - 1}}{a^{6} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{8}} + \frac {6 \, {\left (\frac {210 \, {\left (a x + 1\right )}^{6}}{{\left (a x - 1\right )}^{6}} + \frac {315 \, {\left (a x + 1\right )}^{5}}{{\left (a x - 1\right )}^{5}} + \frac {441 \, {\left (a x + 1\right )}^{4}}{{\left (a x - 1\right )}^{4}} + \frac {126 \, {\left (a x + 1\right )}^{3}}{{\left (a x - 1\right )}^{3}} + \frac {36 \, {\left (a x + 1\right )}^{2}}{{\left (a x - 1\right )}^{2}} - \frac {9 \, {\left (a x + 1\right )}}{a x - 1} + 1\right )} \log \left (-\frac {\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} + 1}{\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} - 1}\right )}{a^{6} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{9}}\right )} \] Input:
integrate(x^4*(-a^2*x^2+1)^2*arctanh(a*x),x, algorithm="giac")
Output:
4/945*a*(6*log(abs(-a*x - 1)/abs(a*x - 1))/a^6 - 6*log(abs(-(a*x + 1)/(a*x - 1) + 1))/a^6 - (6*(a*x + 1)^7/(a*x - 1)^7 - 45*(a*x + 1)^6/(a*x - 1)^6 - 274*(a*x + 1)^5/(a*x - 1)^5 - 214*(a*x + 1)^4/(a*x - 1)^4 - 274*(a*x + 1 )^3/(a*x - 1)^3 - 45*(a*x + 1)^2/(a*x - 1)^2 + 6*(a*x + 1)/(a*x - 1))/(a^6 *((a*x + 1)/(a*x - 1) - 1)^8) + 6*(210*(a*x + 1)^6/(a*x - 1)^6 + 315*(a*x + 1)^5/(a*x - 1)^5 + 441*(a*x + 1)^4/(a*x - 1)^4 + 126*(a*x + 1)^3/(a*x - 1)^3 + 36*(a*x + 1)^2/(a*x - 1)^2 - 9*(a*x + 1)/(a*x - 1) + 1)*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^6*((a*x + 1)/(a*x - 1) - 1 )^9))
Time = 3.85 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.10 \[ \int x^4 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x) \, dx=\frac {4\,\ln \left (a^2\,x^2-1\right )}{315\,a^5}-\frac {11\,a\,x^6}{378}+\ln \left (a\,x+1\right )\,\left (\frac {a^4\,x^9}{18}-\frac {a^2\,x^7}{7}+\frac {x^5}{10}\right )-\ln \left (1-a\,x\right )\,\left (\frac {a^4\,x^9}{18}-\frac {a^2\,x^7}{7}+\frac {x^5}{10}\right )+\frac {2\,x^4}{315\,a}+\frac {4\,x^2}{315\,a^3}+\frac {a^3\,x^8}{72} \] Input:
int(x^4*atanh(a*x)*(a^2*x^2 - 1)^2,x)
Output:
(4*log(a^2*x^2 - 1))/(315*a^5) - (11*a*x^6)/378 + log(a*x + 1)*(x^5/10 - ( a^2*x^7)/7 + (a^4*x^9)/18) - log(1 - a*x)*(x^5/10 - (a^2*x^7)/7 + (a^4*x^9 )/18) + (2*x^4)/(315*a) + (4*x^2)/(315*a^3) + (a^3*x^8)/72
Time = 0.16 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.96 \[ \int x^4 \left (1-a^2 x^2\right )^2 \text {arctanh}(a x) \, dx=\frac {840 \mathit {atanh} \left (a x \right ) a^{9} x^{9}-2160 \mathit {atanh} \left (a x \right ) a^{7} x^{7}+1512 \mathit {atanh} \left (a x \right ) a^{5} x^{5}+192 \mathit {atanh} \left (a x \right )+192 \,\mathrm {log}\left (a^{2} x -a \right )+105 a^{8} x^{8}-220 a^{6} x^{6}+48 a^{4} x^{4}+96 a^{2} x^{2}}{7560 a^{5}} \] Input:
int(x^4*(-a^2*x^2+1)^2*atanh(a*x),x)
Output:
(840*atanh(a*x)*a**9*x**9 - 2160*atanh(a*x)*a**7*x**7 + 1512*atanh(a*x)*a* *5*x**5 + 192*atanh(a*x) + 192*log(a**2*x - a) + 105*a**8*x**8 - 220*a**6* x**6 + 48*a**4*x**4 + 96*a**2*x**2)/(7560*a**5)