Integrand size = 22, antiderivative size = 111 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {8 \sqrt {1-\frac {1}{a^2 x^2}} x}{3 c^4}-\frac {a x}{3 c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )}-\frac {\left (4 a+\frac {3}{x}\right ) x}{3 a c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^4} \]
arctanh((1-1/a^2/x^2)^(1/2))/a/c^4-1/3*a*x/c^4/(a-1/x)/(1-1/a^2/x^2)^(1/2) -1/3*(4*a+3/x)*x/a/c^4/(1-1/a^2/x^2)^(1/2)+8/3*x*(1-1/a^2/x^2)^(1/2)/c^4
Time = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.85 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {8-5 a x-7 a^2 x^2+3 a^3 x^3+3 a \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x) \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{3 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)} \]
(8 - 5*a*x - 7*a^2*x^2 + 3*a^3*x^3 + 3*a*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x )*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(3*a^2*c^4*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x))
Time = 0.36 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.95, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {6731, 27, 569, 25, 532, 25, 534, 243, 73, 221}
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.
| \(\displaystyle \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx\) |
| \(\Big \downarrow \) 6731 |
| \(\displaystyle -\frac {\int \frac {a x^2}{c \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (a-\frac {1}{x}\right )}d\frac {1}{x}}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \int \frac {x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (a-\frac {1}{x}\right )}d\frac {1}{x}}{c^4}\) |
| \(\Big \downarrow \) 569 |
| \(\displaystyle -\frac {a \left (\frac {x}{3 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )}-\frac {\int -\frac {\left (4 a+\frac {3}{x}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{3 a^2}\right )}{c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a \left (\frac {\int \frac {\left (4 a+\frac {3}{x}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{3 a^2}+\frac {x}{3 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )}\right )}{c^4}\) |
| \(\Big \downarrow \) 532 |
| \(\displaystyle -\frac {a \left (\frac {\frac {3 a+\frac {4}{x}}{a \sqrt {1-\frac {1}{a^2 x^2}}}-\int -\frac {\left (4 a+\frac {3}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{3 a^2}+\frac {x}{3 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )}\right )}{c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a \left (\frac {\int \frac {\left (4 a+\frac {3}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {3 a+\frac {4}{x}}{a \sqrt {1-\frac {1}{a^2 x^2}}}}{3 a^2}+\frac {x}{3 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )}\right )}{c^4}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle -\frac {a \left (\frac {3 \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {3 a+\frac {4}{x}}{a \sqrt {1-\frac {1}{a^2 x^2}}}-4 a x \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^2}+\frac {x}{3 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )}\right )}{c^4}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {a \left (\frac {\frac {3}{2} \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}+\frac {3 a+\frac {4}{x}}{a \sqrt {1-\frac {1}{a^2 x^2}}}-4 a x \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^2}+\frac {x}{3 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )}\right )}{c^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {a \left (\frac {-3 a^2 \int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}-4 a x \sqrt {1-\frac {1}{a^2 x^2}}+\frac {3 a+\frac {4}{x}}{a \sqrt {1-\frac {1}{a^2 x^2}}}}{3 a^2}+\frac {x}{3 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )}\right )}{c^4}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {a \left (\frac {-3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {3 a+\frac {4}{x}}{a \sqrt {1-\frac {1}{a^2 x^2}}}-4 a x \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^2}+\frac {x}{3 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )}\right )}{c^4}\) |
-((a*(x/(3*Sqrt[1 - 1/(a^2*x^2)]*(a - x^(-1))) + ((3*a + 4/x)/(a*Sqrt[1 - 1/(a^2*x^2)]) - 4*a*Sqrt[1 - 1/(a^2*x^2)]*x - 3*ArcTanh[Sqrt[1 - 1/(a^2*x^ 2)]])/(3*a^2)))/c^4)
3.5.36.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) *((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[x^m *(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(Qx/x^m) + e*((2*p + 3)/x^m), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d Int[ x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
Int[((x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_))/((c_) + (d_.)*(x_)), x_Symbol] : > Simp[(-x^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*p*(c + d*x))), x] + Simp[1/(2 *c^2*p) Int[x^m*(a + b*x^2)^p*(c*(m + 2*p + 1) - d*(m + 2*p + 2)*x), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && ILtQ[m + 2*p, 0]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> S imp[-c^n Subst[Int[(c + d*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^2), x], x, 1/ x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(217\) vs. \(2(97)=194\).
Time = 0.24 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.96
| method | result | size |
| risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{4}}+\frac {\left (\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{4} \sqrt {a^{2}}}+\frac {\sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{4 a^{6} \left (x +\frac {1}{a}\right )}-\frac {19 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{12 a^{6} \left (x -\frac {1}{a}\right )}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{6 a^{7} \left (x -\frac {1}{a}\right )^{2}}\right ) a^{4} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{4} \left (a x -1\right )}\) | \(218\) |
| default | \(\frac {\left (24 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}+45 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{5} x^{5}-24 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}-21 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}-45 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{4} x^{4}-48 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-11 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}-90 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}+48 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+5 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x +90 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}+24 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x +19 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+45 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x -24 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )-45 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{24 a \sqrt {a^{2}}\, c^{4} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )^{4}}\) | \(523\) |
1/a*(a*x+1)/c^4*((a*x-1)/(a*x+1))^(1/2)+(1/a^4*ln(a^2*x/(a^2)^(1/2)+(a^2*x ^2-1)^(1/2))/(a^2)^(1/2)+1/4/a^6/(x+1/a)*(a^2*(x+1/a)^2-2*a*(x+1/a))^(1/2) -19/12/a^6/(x-1/a)*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1/2)-1/6/a^7/(x-1/a)^2*((x -1/a)^2*a^2+2*(x-1/a)*a)^(1/2))*a^4/c^4*((a*x-1)/(a*x+1))^(1/2)/(a*x-1)*(( a*x-1)*(a*x+1))^(1/2)
Time = 0.25 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.21 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {3 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 3 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (3 \, a^{3} x^{3} - 7 \, a^{2} x^{2} - 5 \, a x + 8\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} \]
1/3*(3*(a^2*x^2 - 2*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 3*(a^2*x ^2 - 2*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (3*a^3*x^3 - 7*a^2*x^ 2 - 5*a*x + 8)*sqrt((a*x - 1)/(a*x + 1)))/(a^3*c^4*x^2 - 2*a^2*c^4*x + a*c ^4)
\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {a^{4} \left (\int \left (- \frac {x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{5} x^{5} - 3 a^{4} x^{4} + 2 a^{3} x^{3} + 2 a^{2} x^{2} - 3 a x + 1}\right )\, dx + \int \frac {a x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{5} x^{5} - 3 a^{4} x^{4} + 2 a^{3} x^{3} + 2 a^{2} x^{2} - 3 a x + 1}\, dx\right )}{c^{4}} \]
a**4*(Integral(-x**4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**5*x**5 - 3*a**4 *x**4 + 2*a**3*x**3 + 2*a**2*x**2 - 3*a*x + 1), x) + Integral(a*x**5*sqrt( a*x/(a*x + 1) - 1/(a*x + 1))/(a**5*x**5 - 3*a**4*x**4 + 2*a**3*x**3 + 2*a* *2*x**2 - 3*a*x + 1), x))/c**4
Time = 0.19 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.44 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {1}{12} \, a {\left (\frac {\frac {17 \, {\left (a x - 1\right )}}{a x + 1} - \frac {42 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}{a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} + \frac {12 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac {12 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{4}} + \frac {3 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{4}}\right )} \]
1/12*a*((17*(a*x - 1)/(a*x + 1) - 42*(a*x - 1)^2/(a*x + 1)^2 + 1)/(a^2*c^4 *((a*x - 1)/(a*x + 1))^(5/2) - a^2*c^4*((a*x - 1)/(a*x + 1))^(3/2)) + 12*l og(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^4) - 12*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^4) + 3*sqrt((a*x - 1)/(a*x + 1))/(a^2*c^4))
\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\int { \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{{\left (c - \frac {c}{a x}\right )}^{4}} \,d x } \]
Time = 3.89 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.15 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{4\,a\,c^4}-\frac {\frac {17\,\left (a\,x-1\right )}{3\,\left (a\,x+1\right )}-\frac {14\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {1}{3}}{4\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}-4\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}+\frac {2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^4} \]