Integrand size = 22, antiderivative size = 105 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=-\frac {2 \left (a+\frac {1}{x}\right )}{3 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {6 a+\frac {7}{x}}{3 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3}+\frac {2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^3} \]
-2/3*(a+1/x)/a^2/c^3/(1-1/a^2/x^2)^(3/2)+2*arctanh((1-1/a^2/x^2)^(1/2))/a/ c^3+1/3*(-6*a-7/x)/a^2/c^3/(1-1/a^2/x^2)^(1/2)+x*(1-1/a^2/x^2)^(1/2)/c^3
Time = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {10-4 a x-11 a^2 x^2+3 a^3 x^3+6 a \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x) \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{3 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)} \]
(10 - 4*a*x - 11*a^2*x^2 + 3*a^3*x^3 + 6*a*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a *x)*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(3*a^2*c^3*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x))
Time = 0.46 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.95, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6731, 27, 570, 532, 25, 2336, 27, 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.
| \(\displaystyle \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx\) |
| \(\Big \downarrow \) 6731 |
| \(\displaystyle -\frac {\int \frac {a^2 x^2}{c^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )^2}d\frac {1}{x}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^2 \int \frac {x^2}{\sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )^2}d\frac {1}{x}}{c^3}\) |
| \(\Big \downarrow \) 570 |
| \(\displaystyle -\frac {\int \frac {\left (a+\frac {1}{x}\right )^2 x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2}}d\frac {1}{x}}{a^2 c^3}\) |
| \(\Big \downarrow \) 532 |
| \(\displaystyle -\frac {\frac {2 \left (a+\frac {1}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {1}{3} \int -\frac {\left (3 a^2+\frac {6 a}{x}+\frac {4}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{a^2 c^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{3} \int \frac {\left (3 a^2+\frac {6 a}{x}+\frac {4}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}+\frac {2 \left (a+\frac {1}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}}{a^2 c^3}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle -\frac {\frac {1}{3} \left (\frac {6 a+\frac {7}{x}}{\sqrt {1-\frac {1}{a^2 x^2}}}-\int -\frac {3 a \left (a+\frac {2}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\right )+\frac {2 \left (a+\frac {1}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}}{a^2 c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{3} \left (3 a \int \frac {\left (a+\frac {2}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {6 a+\frac {7}{x}}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {2 \left (a+\frac {1}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}}{a^2 c^3}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle -\frac {\frac {1}{3} \left (3 a \left (2 \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {6 a+\frac {7}{x}}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {2 \left (a+\frac {1}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}}{a^2 c^3}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {\frac {1}{3} \left (3 a \left (\int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {6 a+\frac {7}{x}}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {2 \left (a+\frac {1}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}}{a^2 c^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\frac {1}{3} \left (3 a \left (-2 a^2 \int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {6 a+\frac {7}{x}}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {2 \left (a+\frac {1}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}}{a^2 c^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {1}{3} \left (3 a \left (-2 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {6 a+\frac {7}{x}}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {2 \left (a+\frac {1}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}}{a^2 c^3}\) |
-(((2*(a + x^(-1)))/(3*(1 - 1/(a^2*x^2))^(3/2)) + ((6*a + 7/x)/Sqrt[1 - 1/ (a^2*x^2)] + 3*a*(-(a*Sqrt[1 - 1/(a^2*x^2)]*x) - 2*ArcTanh[Sqrt[1 - 1/(a^2 *x^2)]]))/3)/(a^2*c^3))
3.5.19.3.1 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[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c^(2*n)/a^n Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I LtQ[n, -1] && !(IGtQ[m, 0] && ILtQ[m + n, 0] && !GtQ[p, 1])
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {Q = PolynomialQuotient[(c*x)^m*Pq, a + b*x^2, x], f = Coeff[PolynomialRema inder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(c*x) ^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a* b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 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]
Time = 0.23 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.76
| method | result | size |
| risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{3}}+\frac {\left (\frac {2 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{3} \sqrt {a^{2}}}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{3 a^{6} \left (x -\frac {1}{a}\right )^{2}}-\frac {8 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{3 a^{5} \left (x -\frac {1}{a}\right )}\right ) a^{3} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{3} \left (a x -1\right )}\) | \(185\) |
| default | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \left (-27 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}-24 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+15 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x +81 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}+72 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-13 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-81 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x -72 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x +27 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}+24 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )\right )}{12 a \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, c^{3} \sqrt {a^{2}}\, \left (a x -1\right )^{3}}\) | \(344\) |
1/a*(a*x+1)/c^3*((a*x-1)/(a*x+1))^(1/2)+(2/a^3*ln(a^2*x/(a^2)^(1/2)+(a^2*x ^2-1)^(1/2))/(a^2)^(1/2)-1/3/a^6/(x-1/a)^2*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1/ 2)-8/3/a^5/(x-1/a)*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1/2))*a^3/c^3*((a*x-1)/(a* x+1))^(1/2)/(a*x-1)*((a*x-1)*(a*x+1))^(1/2)
Time = 0.26 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.28 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {6 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 6 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (3 \, a^{3} x^{3} - 11 \, a^{2} x^{2} - 4 \, a x + 10\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\right )}} \]
1/3*(6*(a^2*x^2 - 2*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 6*(a^2*x ^2 - 2*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (3*a^3*x^3 - 11*a^2*x ^2 - 4*a*x + 10)*sqrt((a*x - 1)/(a*x + 1)))/(a^3*c^3*x^2 - 2*a^2*c^3*x + a *c^3)
\[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {a^{3} \int \frac {x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{3} x^{3} - 3 a^{2} x^{2} + 3 a x - 1}\, dx}{c^{3}} \]
a**3*Integral(x**3*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**3*x**3 - 3*a**2*x **2 + 3*a*x - 1), x)/c**3
Time = 0.19 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.30 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {1}{6} \, a {\left (\frac {\frac {14 \, {\left (a x - 1\right )}}{a x + 1} - \frac {27 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}{a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} + \frac {12 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{3}} - \frac {12 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{3}}\right )} \]
1/6*a*((14*(a*x - 1)/(a*x + 1) - 27*(a*x - 1)^2/(a*x + 1)^2 + 1)/(a^2*c^3* ((a*x - 1)/(a*x + 1))^(5/2) - a^2*c^3*((a*x - 1)/(a*x + 1))^(3/2)) + 12*lo g(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^3) - 12*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^3))
Exception generated. \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Time = 3.87 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^3}-\frac {\frac {14\,\left (a\,x-1\right )}{3\,\left (a\,x+1\right )}-\frac {9\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {1}{3}}{2\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}-2\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}} \]