Integrand size = 22, antiderivative size = 215 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=-\frac {13 a^4 \sqrt {1-\frac {1}{a^2 x^2}}}{9 c^4 \left (a-\frac {1}{x}\right )^5}-\frac {115 a^3 \sqrt {1-\frac {1}{a^2 x^2}}}{63 c^4 \left (a-\frac {1}{x}\right )^4}-\frac {262 a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{105 c^4 \left (a-\frac {1}{x}\right )^3}-\frac {1259 a \sqrt {1-\frac {1}{a^2 x^2}}}{315 c^4 \left (a-\frac {1}{x}\right )^2}-\frac {3464 \sqrt {1-\frac {1}{a^2 x^2}}}{315 c^4 \left (a-\frac {1}{x}\right )}+\frac {a^5 \sqrt {1-\frac {1}{a^2 x^2}} x}{c^4 \left (a-\frac {1}{x}\right )^5}+\frac {7 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^4} \] Output:
-13/9*a^4*(1-1/a^2/x^2)^(1/2)/c^4/(a-1/x)^5-115/63*a^3*(1-1/a^2/x^2)^(1/2) /c^4/(a-1/x)^4-262/105*a^2*(1-1/a^2/x^2)^(1/2)/c^4/(a-1/x)^3-1259/315*a*(1 -1/a^2/x^2)^(1/2)/c^4/(a-1/x)^2-3464/315*(1-1/a^2/x^2)^(1/2)/c^4/(a-1/x)+a ^5*(1-1/a^2/x^2)^(1/2)*x/c^4/(a-1/x)^5+7*arctanh((1-1/a^2/x^2)^(1/2))/a/c^ 4
Time = 0.12 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.56 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {-3464+11651 a x-10232 a^2 x^2-5567 a^3 x^3+13241 a^4 x^4-6224 a^5 x^5+315 a^6 x^6+2205 a \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{315 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^4} \] Input:
Integrate[E^(3*ArcCoth[a*x])/(c - c/(a*x))^4,x]
Output:
(-3464 + 11651*a*x - 10232*a^2*x^2 - 5567*a^3*x^3 + 13241*a^4*x^4 - 6224*a ^5*x^5 + 315*a^6*x^6 + 2205*a*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^4*ArcTanh [Sqrt[1 - 1/(a^2*x^2)]])/(315*a^2*c^4*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^4 )
Time = 1.61 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.99, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.773, Rules used = {6731, 27, 570, 532, 25, 2336, 27, 2336, 25, 2336, 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.
| \(\displaystyle \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx\) |
| \(\Big \downarrow \) 6731 |
| \(\displaystyle -c^3 \int \frac {a^7 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^2}{c^7 \left (a-\frac {1}{x}\right )^7}d\frac {1}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^7 \int \frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^2}{\left (a-\frac {1}{x}\right )^7}d\frac {1}{x}}{c^4}\) |
| \(\Big \downarrow \) 570 |
| \(\displaystyle -\frac {\int \frac {\left (a+\frac {1}{x}\right )^7 x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{11/2}}d\frac {1}{x}}{a^7 c^4}\) |
| \(\Big \downarrow \) 532 |
| \(\displaystyle -\frac {\frac {64 a^5 \left (a+\frac {1}{x}\right )}{9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}-\frac {1}{9} \int -\frac {\left (9 a^7+\frac {63 a^6}{x}+\frac {134 a^5}{x^2}-\frac {198 a^4}{x^3}-\frac {63 a^3}{x^4}-\frac {9 a^2}{x^5}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{9/2}}d\frac {1}{x}}{a^7 c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{9} \int \frac {\left (9 a^7+\frac {63 a^6}{x}+\frac {134 a^5}{x^2}-\frac {198 a^4}{x^3}-\frac {63 a^3}{x^4}-\frac {9 a^2}{x^5}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{9/2}}d\frac {1}{x}+\frac {64 a^5 \left (a+\frac {1}{x}\right )}{9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}}{a^7 c^4}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle -\frac {\frac {1}{9} \left (-\frac {1}{7} \int -\frac {3 \left (21 a^7+\frac {147 a^6}{x}+\frac {307 a^5}{x^2}+\frac {21 a^4}{x^3}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{7/2}}d\frac {1}{x}-\frac {16 a^5 \left (9 a-\frac {5}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}\right )+\frac {64 a^5 \left (a+\frac {1}{x}\right )}{9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}}{a^7 c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{9} \left (\frac {3}{7} \int \frac {\left (21 a^7+\frac {147 a^6}{x}+\frac {307 a^5}{x^2}+\frac {21 a^4}{x^3}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{7/2}}d\frac {1}{x}-\frac {16 a^5 \left (9 a-\frac {5}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}\right )+\frac {64 a^5 \left (a+\frac {1}{x}\right )}{9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}}{a^7 c^4}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle -\frac {\frac {1}{9} \left (\frac {3}{7} \left (\frac {8 a^5 \left (21 a+\frac {41}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {1}{5} \int -\frac {\left (105 a^7+\frac {735 a^6}{x}+\frac {1312 a^5}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2}}d\frac {1}{x}\right )-\frac {16 a^5 \left (9 a-\frac {5}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}\right )+\frac {64 a^5 \left (a+\frac {1}{x}\right )}{9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}}{a^7 c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{9} \left (\frac {3}{7} \left (\frac {1}{5} \int \frac {\left (105 a^7+\frac {735 a^6}{x}+\frac {1312 a^5}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2}}d\frac {1}{x}+\frac {8 a^5 \left (21 a+\frac {41}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}\right )-\frac {16 a^5 \left (9 a-\frac {5}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}\right )+\frac {64 a^5 \left (a+\frac {1}{x}\right )}{9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}}{a^7 c^4}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle -\frac {\frac {1}{9} \left (\frac {3}{7} \left (\frac {1}{5} \left (\frac {a^5 \left (735 a+\frac {1417}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {1}{3} \int -\frac {\left (315 a^7+\frac {2205 a^6}{x}+\frac {2834 a^5}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}\right )+\frac {8 a^5 \left (21 a+\frac {41}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}\right )-\frac {16 a^5 \left (9 a-\frac {5}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}\right )+\frac {64 a^5 \left (a+\frac {1}{x}\right )}{9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}}{a^7 c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{9} \left (\frac {3}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {\left (315 a^7+\frac {2205 a^6}{x}+\frac {2834 a^5}{x^2}\right ) x^2}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}+\frac {a^5 \left (735 a+\frac {1417}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {8 a^5 \left (21 a+\frac {41}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}\right )-\frac {16 a^5 \left (9 a-\frac {5}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}\right )+\frac {64 a^5 \left (a+\frac {1}{x}\right )}{9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}}{a^7 c^4}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle -\frac {\frac {1}{9} \left (\frac {3}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {a^5 \left (2205 a+\frac {3149}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}-\int -\frac {315 a^6 \left (a+\frac {7}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\right )+\frac {a^5 \left (735 a+\frac {1417}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {8 a^5 \left (21 a+\frac {41}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}\right )-\frac {16 a^5 \left (9 a-\frac {5}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}\right )+\frac {64 a^5 \left (a+\frac {1}{x}\right )}{9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}}{a^7 c^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{9} \left (\frac {3}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (315 a^6 \int \frac {\left (a+\frac {7}{x}\right ) x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {a^5 \left (2205 a+\frac {3149}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {a^5 \left (735 a+\frac {1417}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {8 a^5 \left (21 a+\frac {41}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}\right )-\frac {16 a^5 \left (9 a-\frac {5}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}\right )+\frac {64 a^5 \left (a+\frac {1}{x}\right )}{9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}}{a^7 c^4}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle -\frac {\frac {1}{9} \left (\frac {3}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (315 a^6 \left (7 \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {a^5 \left (2205 a+\frac {3149}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {a^5 \left (735 a+\frac {1417}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {8 a^5 \left (21 a+\frac {41}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}\right )-\frac {16 a^5 \left (9 a-\frac {5}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}\right )+\frac {64 a^5 \left (a+\frac {1}{x}\right )}{9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}}{a^7 c^4}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {\frac {1}{9} \left (\frac {3}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (315 a^6 \left (\frac {7}{2} \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {a^5 \left (2205 a+\frac {3149}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {a^5 \left (735 a+\frac {1417}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {8 a^5 \left (21 a+\frac {41}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}\right )-\frac {16 a^5 \left (9 a-\frac {5}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}\right )+\frac {64 a^5 \left (a+\frac {1}{x}\right )}{9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}}{a^7 c^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\frac {1}{9} \left (\frac {3}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (315 a^6 \left (-7 a^2 \int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {a^5 \left (2205 a+\frac {3149}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {a^5 \left (735 a+\frac {1417}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}\right )+\frac {8 a^5 \left (21 a+\frac {41}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}\right )-\frac {16 a^5 \left (9 a-\frac {5}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}\right )+\frac {64 a^5 \left (a+\frac {1}{x}\right )}{9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}}{a^7 c^4}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {64 a^5 \left (a+\frac {1}{x}\right )}{9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}+\frac {1}{9} \left (\frac {3}{7} \left (\frac {8 a^5 \left (21 a+\frac {41}{x}\right )}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}+\frac {1}{5} \left (\frac {a^5 \left (735 a+\frac {1417}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {1}{3} \left (315 a^6 \left (-7 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-a x \sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {a^5 \left (2205 a+\frac {3149}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )\right )\right )-\frac {16 a^5 \left (9 a-\frac {5}{x}\right )}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}\right )}{a^7 c^4}\) |
Input:
Int[E^(3*ArcCoth[a*x])/(c - c/(a*x))^4,x]
Output:
-(((64*a^5*(a + x^(-1)))/(9*(1 - 1/(a^2*x^2))^(9/2)) + ((-16*a^5*(9*a - 5/ x))/(7*(1 - 1/(a^2*x^2))^(7/2)) + (3*((8*a^5*(21*a + 41/x))/(5*(1 - 1/(a^2 *x^2))^(5/2)) + ((a^5*(735*a + 1417/x))/(3*(1 - 1/(a^2*x^2))^(3/2)) + ((a^ 5*(2205*a + 3149/x))/Sqrt[1 - 1/(a^2*x^2)] + 315*a^6*(-(a*Sqrt[1 - 1/(a^2* x^2)]*x) - 7*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]))/3)/5))/7)/9)/(a^7*c^4))
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 = 0.15 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.42
| method | result | size |
| risch | \(\frac {a x -1}{a \,c^{4} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {7 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{4} \sqrt {a^{2}}}-\frac {4964 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{315 a^{6} \left (x -\frac {1}{a}\right )}-\frac {4 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{9 a^{10} \left (x -\frac {1}{a}\right )^{5}}-\frac {164 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{63 a^{9} \left (x -\frac {1}{a}\right )^{4}}-\frac {697 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{105 a^{8} \left (x -\frac {1}{a}\right )^{3}}-\frac {3226 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 a \left (x -\frac {1}{a}\right )}}{315 a^{7} \left (x -\frac {1}{a}\right )^{2}}\right ) a^{4} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c^{4} \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(305\) |
| default | \(-\frac {-2205 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{6} x^{6}-2205 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{7} x^{6}+1890 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{4} x^{4}+13230 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{5} x^{5}+13230 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}-6376 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{3} x^{3}-33075 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{4} x^{4}-33075 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+8646 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a^{2} x^{2}+44100 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}+44100 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-5349 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x -33075 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}-33075 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+1259 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+13230 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a x +13230 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x -2205 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}-2205 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )}{315 a \left (a x -1\right )^{4} \sqrt {a^{2}}\, c^{4} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(622\) |
Input:
int(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^4,x,method=_RETURNVERBOSE)
Output:
1/a*(a*x-1)/c^4/((a*x-1)/(a*x+1))^(1/2)+(7/a^4*ln(a^2*x/(a^2)^(1/2)+(a^2*x ^2-1)^(1/2))/(a^2)^(1/2)-4964/315/a^6/(x-1/a)*((x-1/a)^2*a^2+2*a*(x-1/a))^ (1/2)-4/9/a^10/(x-1/a)^5*((x-1/a)^2*a^2+2*a*(x-1/a))^(1/2)-164/63/a^9/(x-1 /a)^4*((x-1/a)^2*a^2+2*a*(x-1/a))^(1/2)-697/105/a^8/(x-1/a)^3*((x-1/a)^2*a ^2+2*a*(x-1/a))^(1/2)-3226/315/a^7/(x-1/a)^2*((x-1/a)^2*a^2+2*a*(x-1/a))^( 1/2))*a^4/c^4/(a*x+1)/((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1))^(1/2)
Time = 0.12 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.12 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {2205 \, {\left (a^{5} x^{5} - 5 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - 10 \, a^{2} x^{2} + 5 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 2205 \, {\left (a^{5} x^{5} - 5 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - 10 \, a^{2} x^{2} + 5 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (315 \, a^{6} x^{6} - 6224 \, a^{5} x^{5} + 13241 \, a^{4} x^{4} - 5567 \, a^{3} x^{3} - 10232 \, a^{2} x^{2} + 11651 \, a x - 3464\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{315 \, {\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right )}} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^4,x, algorithm="fricas")
Output:
1/315*(2205*(a^5*x^5 - 5*a^4*x^4 + 10*a^3*x^3 - 10*a^2*x^2 + 5*a*x - 1)*lo g(sqrt((a*x - 1)/(a*x + 1)) + 1) - 2205*(a^5*x^5 - 5*a^4*x^4 + 10*a^3*x^3 - 10*a^2*x^2 + 5*a*x - 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (315*a^6*x^ 6 - 6224*a^5*x^5 + 13241*a^4*x^4 - 5567*a^3*x^3 - 10232*a^2*x^2 + 11651*a* x - 3464)*sqrt((a*x - 1)/(a*x + 1)))/(a^6*c^4*x^5 - 5*a^5*c^4*x^4 + 10*a^4 *c^4*x^3 - 10*a^3*c^4*x^2 + 5*a^2*c^4*x - a*c^4)
\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {a^{4} \int \frac {x^{4}}{\frac {a^{5} x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {5 a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {10 a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {10 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {5 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx}{c^{4}} \] Input:
integrate(1/((a*x-1)/(a*x+1))**(3/2)/(c-c/a/x)**4,x)
Output:
a**4*Integral(x**4/(a**5*x**5*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - 5*a**4*x**4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) + 10*a**3*x**3*s qrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - 10*a**2*x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) + 5*a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1)), x)/c**4
Time = 0.04 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.86 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {1}{1260} \, a {\left (\frac {\frac {235 \, {\left (a x - 1\right )}}{a x + 1} + \frac {801 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {2289 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + \frac {11760 \, {\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} - \frac {17640 \, {\left (a x - 1\right )}^{5}}{{\left (a x + 1\right )}^{5}} + 35}{a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} - a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}}} + \frac {8820 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac {8820 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{4}}\right )} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^4,x, algorithm="maxima")
Output:
1/1260*a*((235*(a*x - 1)/(a*x + 1) + 801*(a*x - 1)^2/(a*x + 1)^2 + 2289*(a *x - 1)^3/(a*x + 1)^3 + 11760*(a*x - 1)^4/(a*x + 1)^4 - 17640*(a*x - 1)^5/ (a*x + 1)^5 + 35)/(a^2*c^4*((a*x - 1)/(a*x + 1))^(11/2) - a^2*c^4*((a*x - 1)/(a*x + 1))^(9/2)) + 8820*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c^4) - 8820*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/(a^2*c^4))
Time = 0.22 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.29 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=-\frac {7 \, \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{c^{4} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} + \frac {\sqrt {a^{2} x^{2} - 1}}{a c^{4} \mathrm {sgn}\left (a x + 1\right )} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^4,x, algorithm="giac")
Output:
-7*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))/(c^4*abs(a)*sgn(a*x + 1)) + sqr t(a^2*x^2 - 1)/(a*c^4*sgn(a*x + 1))
Time = 13.74 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.71 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {14\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^4}-\frac {\frac {89\,{\left (a\,x-1\right )}^2}{35\,{\left (a\,x+1\right )}^2}+\frac {109\,{\left (a\,x-1\right )}^3}{15\,{\left (a\,x+1\right )}^3}+\frac {112\,{\left (a\,x-1\right )}^4}{3\,{\left (a\,x+1\right )}^4}-\frac {56\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}+\frac {47\,\left (a\,x-1\right )}{63\,\left (a\,x+1\right )}+\frac {1}{9}}{4\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}-4\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}} \] Input:
int(1/((c - c/(a*x))^4*((a*x - 1)/(a*x + 1))^(3/2)),x)
Output:
(14*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(a*c^4) - ((89*(a*x - 1)^2)/(35*(a *x + 1)^2) + (109*(a*x - 1)^3)/(15*(a*x + 1)^3) + (112*(a*x - 1)^4)/(3*(a* x + 1)^4) - (56*(a*x - 1)^5)/(a*x + 1)^5 + (47*(a*x - 1))/(63*(a*x + 1)) + 1/9)/(4*a*c^4*((a*x - 1)/(a*x + 1))^(9/2) - 4*a*c^4*((a*x - 1)/(a*x + 1)) ^(11/2))
Time = 0.18 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.57 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {8820 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a^{4} x^{4}-35280 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a^{3} x^{3}+52920 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a^{2} x^{2}-35280 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a x +8820 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right )+7513 \sqrt {a x -1}\, a^{4} x^{4}-30052 \sqrt {a x -1}\, a^{3} x^{3}+45078 \sqrt {a x -1}\, a^{2} x^{2}-30052 \sqrt {a x -1}\, a x +7513 \sqrt {a x -1}+630 \sqrt {a x +1}\, a^{5} x^{5}-13078 \sqrt {a x +1}\, a^{4} x^{4}+39560 \sqrt {a x +1}\, a^{3} x^{3}-50694 \sqrt {a x +1}\, a^{2} x^{2}+30230 \sqrt {a x +1}\, a x -6928 \sqrt {a x +1}}{630 \sqrt {a x -1}\, a \,c^{4} \left (a^{4} x^{4}-4 a^{3} x^{3}+6 a^{2} x^{2}-4 a x +1\right )} \] Input:
int(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^4,x)
Output:
(8820*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a**4*x**4 - 35280*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a**3*x **3 + 52920*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a** 2*x**2 - 35280*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))* a*x + 8820*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2)) + 75 13*sqrt(a*x - 1)*a**4*x**4 - 30052*sqrt(a*x - 1)*a**3*x**3 + 45078*sqrt(a* x - 1)*a**2*x**2 - 30052*sqrt(a*x - 1)*a*x + 7513*sqrt(a*x - 1) + 630*sqrt (a*x + 1)*a**5*x**5 - 13078*sqrt(a*x + 1)*a**4*x**4 + 39560*sqrt(a*x + 1)* a**3*x**3 - 50694*sqrt(a*x + 1)*a**2*x**2 + 30230*sqrt(a*x + 1)*a*x - 6928 *sqrt(a*x + 1))/(630*sqrt(a*x - 1)*a*c**4*(a**4*x**4 - 4*a**3*x**3 + 6*a** 2*x**2 - 4*a*x + 1))