Integrand size = 20, antiderivative size = 77 \[ \int \frac {x^3 \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3} \, dx=-\frac {x^3}{16 a \left (1-a^2 x^2\right )^2}+\frac {3 x}{32 a^3 \left (1-a^2 x^2\right )}-\frac {3 \text {arctanh}(a x)}{32 a^4}+\frac {x^4 \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2} \] Output:
-1/16*x^3/a/(-a^2*x^2+1)^2+3/32*x/a^3/(-a^2*x^2+1)-3/32*arctanh(a*x)/a^4+1 /4*x^4*arctanh(a*x)/(-a^2*x^2+1)^2
Time = 0.10 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.27 \[ \int \frac {x^3 \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3} \, dx=-\frac {x}{16 a^3 \left (-1+a^2 x^2\right )^2}-\frac {5 x}{32 a^3 \left (-1+a^2 x^2\right )}+\frac {\left (-1+2 a^2 x^2\right ) \text {arctanh}(a x)}{4 a^4 \left (-1+a^2 x^2\right )^2}-\frac {5 \log (1-a x)}{64 a^4}+\frac {5 \log (1+a x)}{64 a^4} \] Input:
Integrate[(x^3*ArcTanh[a*x])/(1 - a^2*x^2)^3,x]
Output:
-1/16*x/(a^3*(-1 + a^2*x^2)^2) - (5*x)/(32*a^3*(-1 + a^2*x^2)) + ((-1 + 2* a^2*x^2)*ArcTanh[a*x])/(4*a^4*(-1 + a^2*x^2)^2) - (5*Log[1 - a*x])/(64*a^4 ) + (5*Log[1 + a*x])/(64*a^4)
Time = 0.28 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.18, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6570, 252, 252, 219}
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.
\(\displaystyle \int \frac {x^3 \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 6570 |
\(\displaystyle \frac {x^4 \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{4} a \int \frac {x^4}{\left (1-a^2 x^2\right )^3}dx\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {x^4 \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{4} a \left (\frac {x^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \int \frac {x^2}{\left (1-a^2 x^2\right )^2}dx}{4 a^2}\right )\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {x^4 \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{4} a \left (\frac {x^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {x}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\int \frac {1}{1-a^2 x^2}dx}{2 a^2}\right )}{4 a^2}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {x^4 \text {arctanh}(a x)}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{4} a \left (\frac {x^3}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac {3 \left (\frac {x}{2 a^2 \left (1-a^2 x^2\right )}-\frac {\text {arctanh}(a x)}{2 a^3}\right )}{4 a^2}\right )\) |
Input:
Int[(x^3*ArcTanh[a*x])/(1 - a^2*x^2)^3,x]
Output:
(x^4*ArcTanh[a*x])/(4*(1 - a^2*x^2)^2) - (a*(x^3/(4*a^2*(1 - a^2*x^2)^2) - (3*(x/(2*a^2*(1 - a^2*x^2)) - ArcTanh[a*x]/(2*a^3)))/(4*a^2)))/4
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e _.)*(x_)^2)^(q_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])^p/(d*(m + 1))), x] - Simp[b*c*(p/(m + 1)) Int[(f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTanh[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[c^2*d + e, 0] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]
Time = 0.44 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.78
method | result | size |
parallelrisch | \(-\frac {-5 a^{4} x^{4} \operatorname {arctanh}\left (a x \right )+5 a^{3} x^{3}-6 a^{2} x^{2} \operatorname {arctanh}\left (a x \right )-3 a x +3 \,\operatorname {arctanh}\left (a x \right )}{32 \left (a^{2} x^{2}-1\right )^{2} a^{4}}\) | \(60\) |
derivativedivides | \(\frac {\frac {\operatorname {arctanh}\left (a x \right )}{16 \left (a x +1\right )^{2}}-\frac {3 \,\operatorname {arctanh}\left (a x \right )}{16 \left (a x +1\right )}+\frac {\operatorname {arctanh}\left (a x \right )}{16 \left (a x -1\right )^{2}}+\frac {3 \,\operatorname {arctanh}\left (a x \right )}{16 \left (a x -1\right )}+\frac {1}{64 \left (a x +1\right )^{2}}-\frac {5}{64 \left (a x +1\right )}+\frac {5 \ln \left (a x +1\right )}{64}-\frac {1}{64 \left (a x -1\right )^{2}}-\frac {5}{64 \left (a x -1\right )}-\frac {5 \ln \left (a x -1\right )}{64}}{a^{4}}\) | \(110\) |
default | \(\frac {\frac {\operatorname {arctanh}\left (a x \right )}{16 \left (a x +1\right )^{2}}-\frac {3 \,\operatorname {arctanh}\left (a x \right )}{16 \left (a x +1\right )}+\frac {\operatorname {arctanh}\left (a x \right )}{16 \left (a x -1\right )^{2}}+\frac {3 \,\operatorname {arctanh}\left (a x \right )}{16 \left (a x -1\right )}+\frac {1}{64 \left (a x +1\right )^{2}}-\frac {5}{64 \left (a x +1\right )}+\frac {5 \ln \left (a x +1\right )}{64}-\frac {1}{64 \left (a x -1\right )^{2}}-\frac {5}{64 \left (a x -1\right )}-\frac {5 \ln \left (a x -1\right )}{64}}{a^{4}}\) | \(110\) |
parts | \(\frac {\operatorname {arctanh}\left (a x \right )}{16 a^{4} \left (a x +1\right )^{2}}-\frac {3 \,\operatorname {arctanh}\left (a x \right )}{16 a^{4} \left (a x +1\right )}+\frac {\operatorname {arctanh}\left (a x \right )}{16 a^{4} \left (a x -1\right )^{2}}+\frac {3 \,\operatorname {arctanh}\left (a x \right )}{16 a^{4} \left (a x -1\right )}-\frac {-\frac {1}{16 a \left (a x +1\right )^{2}}+\frac {5}{16 a \left (a x +1\right )}-\frac {5 \ln \left (a x +1\right )}{16 a}+\frac {1}{16 a \left (a x -1\right )^{2}}+\frac {5}{16 a \left (a x -1\right )}+\frac {5 \ln \left (a x -1\right )}{16 a}}{4 a^{3}}\) | \(142\) |
orering | \(-\frac {\left (a x -1\right ) \left (a x +1\right ) \left (5 a^{4} x^{4}+3 a^{2} x^{2}-3\right ) \operatorname {arctanh}\left (a x \right )}{8 a^{4} \left (-a^{2} x^{2}+1\right )^{3}}-\frac {\left (5 a^{2} x^{2}-3\right ) \left (a x +1\right )^{2} \left (a x -1\right )^{2} \left (\frac {3 x^{2} \operatorname {arctanh}\left (a x \right )}{\left (-a^{2} x^{2}+1\right )^{3}}+\frac {x^{3} a}{\left (-a^{2} x^{2}+1\right )^{4}}+\frac {6 x^{4} \operatorname {arctanh}\left (a x \right ) a^{2}}{\left (-a^{2} x^{2}+1\right )^{4}}\right )}{32 x^{2} a^{4}}\) | \(146\) |
risch | \(\frac {\left (2 a^{2} x^{2}-1\right ) \ln \left (a x +1\right )}{8 a^{4} \left (a^{2} x^{2}-1\right )^{2}}+\frac {5 \ln \left (-a x -1\right ) a^{4} x^{4}-5 \ln \left (a x -1\right ) x^{4} a^{4}-10 a^{3} x^{3}-10 \ln \left (-a x -1\right ) a^{2} x^{2}+10 \ln \left (a x -1\right ) a^{2} x^{2}-16 x^{2} \ln \left (-a x +1\right ) a^{2}+6 a x +5 \ln \left (-a x -1\right )-5 \ln \left (a x -1\right )+8 \ln \left (-a x +1\right )}{64 a^{4} \left (a x -1\right ) \left (a x +1\right ) \left (a^{2} x^{2}-1\right )}\) | \(176\) |
Input:
int(x^3*arctanh(a*x)/(-a^2*x^2+1)^3,x,method=_RETURNVERBOSE)
Output:
-1/32*(-5*a^4*x^4*arctanh(a*x)+5*a^3*x^3-6*a^2*x^2*arctanh(a*x)-3*a*x+3*ar ctanh(a*x))/(a^2*x^2-1)^2/a^4
Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.92 \[ \int \frac {x^3 \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3} \, dx=-\frac {10 \, a^{3} x^{3} - 6 \, a x - {\left (5 \, a^{4} x^{4} + 6 \, a^{2} x^{2} - 3\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{64 \, {\left (a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}\right )}} \] Input:
integrate(x^3*arctanh(a*x)/(-a^2*x^2+1)^3,x, algorithm="fricas")
Output:
-1/64*(10*a^3*x^3 - 6*a*x - (5*a^4*x^4 + 6*a^2*x^2 - 3)*log(-(a*x + 1)/(a* x - 1)))/(a^8*x^4 - 2*a^6*x^2 + a^4)
Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (65) = 130\).
Time = 0.78 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.05 \[ \int \frac {x^3 \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3} \, dx=\begin {cases} \frac {5 a^{4} x^{4} \operatorname {atanh}{\left (a x \right )}}{32 a^{8} x^{4} - 64 a^{6} x^{2} + 32 a^{4}} - \frac {5 a^{3} x^{3}}{32 a^{8} x^{4} - 64 a^{6} x^{2} + 32 a^{4}} + \frac {6 a^{2} x^{2} \operatorname {atanh}{\left (a x \right )}}{32 a^{8} x^{4} - 64 a^{6} x^{2} + 32 a^{4}} + \frac {3 a x}{32 a^{8} x^{4} - 64 a^{6} x^{2} + 32 a^{4}} - \frac {3 \operatorname {atanh}{\left (a x \right )}}{32 a^{8} x^{4} - 64 a^{6} x^{2} + 32 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate(x**3*atanh(a*x)/(-a**2*x**2+1)**3,x)
Output:
Piecewise((5*a**4*x**4*atanh(a*x)/(32*a**8*x**4 - 64*a**6*x**2 + 32*a**4) - 5*a**3*x**3/(32*a**8*x**4 - 64*a**6*x**2 + 32*a**4) + 6*a**2*x**2*atanh( a*x)/(32*a**8*x**4 - 64*a**6*x**2 + 32*a**4) + 3*a*x/(32*a**8*x**4 - 64*a* *6*x**2 + 32*a**4) - 3*atanh(a*x)/(32*a**8*x**4 - 64*a**6*x**2 + 32*a**4), Ne(a, 0)), (0, True))
Time = 0.02 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.29 \[ \int \frac {x^3 \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3} \, dx=-\frac {1}{64} \, a {\left (\frac {2 \, {\left (5 \, a^{2} x^{3} - 3 \, x\right )}}{a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}} - \frac {5 \, \log \left (a x + 1\right )}{a^{5}} + \frac {5 \, \log \left (a x - 1\right )}{a^{5}}\right )} + \frac {{\left (2 \, a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )}{4 \, {\left (a^{8} x^{4} - 2 \, a^{6} x^{2} + a^{4}\right )}} \] Input:
integrate(x^3*arctanh(a*x)/(-a^2*x^2+1)^3,x, algorithm="maxima")
Output:
-1/64*a*(2*(5*a^2*x^3 - 3*x)/(a^8*x^4 - 2*a^6*x^2 + a^4) - 5*log(a*x + 1)/ a^5 + 5*log(a*x - 1)/a^5) + 1/4*(2*a^2*x^2 - 1)*arctanh(a*x)/(a^8*x^4 - 2* a^6*x^2 + a^4)
Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (66) = 132\).
Time = 0.13 (sec) , antiderivative size = 239, normalized size of antiderivative = 3.10 \[ \int \frac {x^3 \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3} \, dx=\frac {1}{256} \, {\left (2 \, {\left (\frac {{\left (a x - 1\right )}^{2} {\left (\frac {4 \, {\left (a x + 1\right )}}{a x - 1} + 1\right )}}{{\left (a x + 1\right )}^{2} a^{5}} + \frac {\frac {{\left (a x + 1\right )}^{2} a^{5}}{{\left (a x - 1\right )}^{2}} + \frac {4 \, {\left (a x + 1\right )} a^{5}}{a x - 1}}{a^{10}}\right )} \log \left (-\frac {\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} + 1}{\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} - 1}\right ) + \frac {{\left (a x - 1\right )}^{2} {\left (\frac {8 \, {\left (a x + 1\right )}}{a x - 1} + 1\right )}}{{\left (a x + 1\right )}^{2} a^{5}} - \frac {\frac {{\left (a x + 1\right )}^{2} a^{5}}{{\left (a x - 1\right )}^{2}} + \frac {8 \, {\left (a x + 1\right )} a^{5}}{a x - 1}}{a^{10}}\right )} a \] Input:
integrate(x^3*arctanh(a*x)/(-a^2*x^2+1)^3,x, algorithm="giac")
Output:
1/256*(2*((a*x - 1)^2*(4*(a*x + 1)/(a*x - 1) + 1)/((a*x + 1)^2*a^5) + ((a* x + 1)^2*a^5/(a*x - 1)^2 + 4*(a*x + 1)*a^5/(a*x - 1))/a^10)*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*x - 1)^2*(8*(a*x + 1)/(a*x - 1) + 1)/((a*x + 1)^2*a^5) - ((a*x + 1)^2*a^5/(a*x - 1)^2 + 8*(a*x + 1)*a ^5/(a*x - 1))/a^10)*a
Time = 4.25 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.08 \[ \int \frac {x^3 \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3} \, dx=\frac {5\,\mathrm {atanh}\left (a\,x\right )}{32\,a^4}+\frac {\frac {\ln \left (1-a\,x\right )}{8}-\frac {\ln \left (a\,x+1\right )}{8}+\frac {3\,a\,x}{32}+x^2\,\left (\frac {a^2\,\ln \left (a\,x+1\right )}{4}-\frac {a^2\,\ln \left (1-a\,x\right )}{4}\right )-\frac {5\,a^3\,x^3}{32}}{a^4\,{\left (a^2\,x^2-1\right )}^2} \] Input:
int(-(x^3*atanh(a*x))/(a^2*x^2 - 1)^3,x)
Output:
(5*atanh(a*x))/(32*a^4) + (log(1 - a*x)/8 - log(a*x + 1)/8 + (3*a*x)/32 + x^2*((a^2*log(a*x + 1))/4 - (a^2*log(1 - a*x))/4) - (5*a^3*x^3)/32)/(a^4*( a^2*x^2 - 1)^2)
Time = 0.18 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.81 \[ \int \frac {x^3 \text {arctanh}(a x)}{\left (1-a^2 x^2\right )^3} \, dx=\frac {16 \mathit {atanh} \left (a x \right ) a^{4} x^{4}+3 \,\mathrm {log}\left (a^{2} x -a \right ) a^{4} x^{4}-6 \,\mathrm {log}\left (a^{2} x -a \right ) a^{2} x^{2}+3 \,\mathrm {log}\left (a^{2} x -a \right )-3 \,\mathrm {log}\left (a^{2} x +a \right ) a^{4} x^{4}+6 \,\mathrm {log}\left (a^{2} x +a \right ) a^{2} x^{2}-3 \,\mathrm {log}\left (a^{2} x +a \right )-10 a^{3} x^{3}+6 a x}{64 a^{4} \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right )} \] Input:
int(x^3*atanh(a*x)/(-a^2*x^2+1)^3,x)
Output:
(16*atanh(a*x)*a**4*x**4 + 3*log(a**2*x - a)*a**4*x**4 - 6*log(a**2*x - a) *a**2*x**2 + 3*log(a**2*x - a) - 3*log(a**2*x + a)*a**4*x**4 + 6*log(a**2* x + a)*a**2*x**2 - 3*log(a**2*x + a) - 10*a**3*x**3 + 6*a*x)/(64*a**4*(a** 4*x**4 - 2*a**2*x**2 + 1))