Integrand size = 22, antiderivative size = 235 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\frac {173 c^5 \sqrt {1-\frac {1}{a^2 x^2}}}{105 a \sqrt {c-\frac {c}{a x}}}+\frac {227 c^4 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}}}{105 a}+\frac {59 c^3 \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{3/2}}{35 a}+\frac {9 c^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{5/2}}{7 a}+c \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{7/2} x-\frac {7 c^{9/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a} \] Output:
173/105*c^5*(1-1/a^2/x^2)^(1/2)/a/(c-c/a/x)^(1/2)+227/105*c^4*(1-1/a^2/x^2 )^(1/2)*(c-c/a/x)^(1/2)/a+59/35*c^3*(1-1/a^2/x^2)^(1/2)*(c-c/a/x)^(3/2)/a+ 9/7*c^2*(1-1/a^2/x^2)^(1/2)*(c-c/a/x)^(5/2)/a+c*(1-1/a^2/x^2)^(1/2)*(c-c/a /x)^(7/2)*x-7*c^(9/2)*arctanh(c^(1/2)*(1-1/a^2/x^2)^(1/2)/(c-c/a/x)^(1/2)) /a
Time = 0.14 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.46 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\frac {c^4 \sqrt {c-\frac {c}{a x}} \left (\sqrt {1+\frac {1}{a x}} \left (-30+162 a x-356 a^2 x^2+292 a^3 x^3+105 a^4 x^4\right )-735 a^3 x^3 \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )\right )}{105 a^4 \sqrt {1-\frac {1}{a x}} x^3} \] Input:
Integrate[E^ArcCoth[a*x]*(c - c/(a*x))^(9/2),x]
Output:
(c^4*Sqrt[c - c/(a*x)]*(Sqrt[1 + 1/(a*x)]*(-30 + 162*a*x - 356*a^2*x^2 + 2 92*a^3*x^3 + 105*a^4*x^4) - 735*a^3*x^3*ArcTanh[Sqrt[1 + 1/(a*x)]]))/(105* a^4*Sqrt[1 - 1/(a*x)]*x^3)
Time = 0.70 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.78, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {6731, 585, 27, 108, 27, 170, 27, 170, 27, 164, 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.
| \(\displaystyle \int \left (c-\frac {c}{a x}\right )^{9/2} e^{\coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6731 |
| \(\displaystyle -c \int \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{7/2} x^2d\frac {1}{x}\) |
| \(\Big \downarrow \) 585 |
| \(\displaystyle -\frac {c^4 \sqrt {c-\frac {c}{a x}} \int \frac {\left (a-\frac {1}{x}\right )^4 \sqrt {1+\frac {1}{a x}} x^2}{a^4}d\frac {1}{x}}{\sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c^4 \sqrt {c-\frac {c}{a x}} \int \left (a-\frac {1}{x}\right )^4 \sqrt {1+\frac {1}{a x}} x^2d\frac {1}{x}}{a^4 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle -\frac {c^4 \sqrt {c-\frac {c}{a x}} \left (\int -\frac {\left (a-\frac {1}{x}\right )^3 \left (7 a+\frac {9}{x}\right ) x}{2 a \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-x \left (a-\frac {1}{x}\right )^4 \sqrt {\frac {1}{a x}+1}\right )}{a^4 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c^4 \sqrt {c-\frac {c}{a x}} \left (x \left (-\sqrt {\frac {1}{a x}+1}\right ) \left (a-\frac {1}{x}\right )^4-\frac {\int \frac {\left (a-\frac {1}{x}\right )^3 \left (7 a+\frac {9}{x}\right ) x}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{2 a}\right )}{a^4 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle -\frac {c^4 \sqrt {c-\frac {c}{a x}} \left (x \left (-\sqrt {\frac {1}{a x}+1}\right ) \left (a-\frac {1}{x}\right )^4-\frac {\frac {2}{7} a \int \frac {\left (a-\frac {1}{x}\right )^2 \left (49 a+\frac {59}{x}\right ) x}{2 \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {18}{7} a \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^3}{2 a}\right )}{a^4 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c^4 \sqrt {c-\frac {c}{a x}} \left (x \left (-\sqrt {\frac {1}{a x}+1}\right ) \left (a-\frac {1}{x}\right )^4-\frac {\frac {1}{7} a \int \frac {\left (a-\frac {1}{x}\right )^2 \left (49 a+\frac {59}{x}\right ) x}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {18}{7} a \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^3}{2 a}\right )}{a^4 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle -\frac {c^4 \sqrt {c-\frac {c}{a x}} \left (x \left (-\sqrt {\frac {1}{a x}+1}\right ) \left (a-\frac {1}{x}\right )^4-\frac {\frac {1}{7} a \left (\frac {2}{5} a \int \frac {\left (a-\frac {1}{x}\right ) \left (245 a+\frac {227}{x}\right ) x}{2 \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {118}{5} a \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^2\right )+\frac {18}{7} a \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^3}{2 a}\right )}{a^4 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c^4 \sqrt {c-\frac {c}{a x}} \left (x \left (-\sqrt {\frac {1}{a x}+1}\right ) \left (a-\frac {1}{x}\right )^4-\frac {\frac {1}{7} a \left (\frac {1}{5} a \int \frac {\left (a-\frac {1}{x}\right ) \left (245 a+\frac {227}{x}\right ) x}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {118}{5} a \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^2\right )+\frac {18}{7} a \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^3}{2 a}\right )}{a^4 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle -\frac {c^4 \sqrt {c-\frac {c}{a x}} \left (x \left (-\sqrt {\frac {1}{a x}+1}\right ) \left (a-\frac {1}{x}\right )^4-\frac {\frac {1}{7} a \left (\frac {1}{5} a \left (245 a^2 \int \frac {x}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+\frac {2}{3} a \left (400 a-\frac {227}{x}\right ) \sqrt {\frac {1}{a x}+1}\right )+\frac {118}{5} a \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^2\right )+\frac {18}{7} a \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^3}{2 a}\right )}{a^4 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {c^4 \sqrt {c-\frac {c}{a x}} \left (x \left (-\sqrt {\frac {1}{a x}+1}\right ) \left (a-\frac {1}{x}\right )^4-\frac {\frac {1}{7} a \left (\frac {1}{5} a \left (490 a^3 \int \frac {1}{\frac {a}{x^2}-a}d\sqrt {1+\frac {1}{a x}}+\frac {2}{3} a \left (400 a-\frac {227}{x}\right ) \sqrt {\frac {1}{a x}+1}\right )+\frac {118}{5} a \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^2\right )+\frac {18}{7} a \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^3}{2 a}\right )}{a^4 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {c^4 \left (x \left (-\sqrt {\frac {1}{a x}+1}\right ) \left (a-\frac {1}{x}\right )^4-\frac {\frac {1}{7} a \left (\frac {1}{5} a \left (\frac {2}{3} a \left (400 a-\frac {227}{x}\right ) \sqrt {\frac {1}{a x}+1}-490 a^2 \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right )\right )+\frac {118}{5} a \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^2\right )+\frac {18}{7} a \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^3}{2 a}\right ) \sqrt {c-\frac {c}{a x}}}{a^4 \sqrt {1-\frac {1}{a x}}}\) |
Input:
Int[E^ArcCoth[a*x]*(c - c/(a*x))^(9/2),x]
Output:
-((c^4*Sqrt[c - c/(a*x)]*(-((a - x^(-1))^4*Sqrt[1 + 1/(a*x)]*x) - ((18*a*( a - x^(-1))^3*Sqrt[1 + 1/(a*x)])/7 + (a*((118*a*(a - x^(-1))^2*Sqrt[1 + 1/ (a*x)])/5 + (a*((2*a*(400*a - 227/x)*Sqrt[1 + 1/(a*x)])/3 - 490*a^2*ArcTan h[Sqrt[1 + 1/(a*x)]]))/5))/7)/(2*a)))/(a^4*Sqrt[1 - 1/(a*x)]))
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ ))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h *(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)) Int[( a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]
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[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_) , x_Symbol] :> Simp[a^p*c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^F racPart[n]) Int[(e*x)^m*(1 - d*(x/c))^p*(1 + d*(x/c))^(n + p), x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[a, 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 = 0.10 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.71
| method | result | size |
| default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{4} \left (210 a^{\frac {9}{2}} \sqrt {x \left (a x +1\right )}\, x^{4}+584 a^{\frac {7}{2}} x^{3} \sqrt {x \left (a x +1\right )}-735 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{4} x^{4}-712 a^{\frac {5}{2}} x^{2} \sqrt {x \left (a x +1\right )}+324 a^{\frac {3}{2}} x \sqrt {x \left (a x +1\right )}-60 \sqrt {x \left (a x +1\right )}\, \sqrt {a}\right )}{210 \sqrt {\frac {a x -1}{a x +1}}\, x^{3} a^{\frac {9}{2}} \sqrt {x \left (a x +1\right )}}\) | \(166\) |
| risch | \(\frac {\left (105 a^{5} x^{5}+397 a^{4} x^{4}-64 a^{3} x^{3}-194 a^{2} x^{2}+132 a x -30\right ) c^{4} \sqrt {\frac {c \left (a x -1\right )}{a x}}}{105 x^{3} a^{4} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}-\frac {7 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right ) c^{4} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{2 \sqrt {a^{2} c}\, \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(184\) |
Input:
int(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x)^(9/2),x,method=_RETURNVERBOSE)
Output:
1/210/((a*x-1)/(a*x+1))^(1/2)*(c*(a*x-1)/a/x)^(1/2)*c^4*(210*a^(9/2)*(x*(a *x+1))^(1/2)*x^4+584*a^(7/2)*x^3*(x*(a*x+1))^(1/2)-735*ln(1/2*(2*(x*(a*x+1 ))^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2))*a^4*x^4-712*a^(5/2)*x^2*(x*(a*x+1))^(1/ 2)+324*a^(3/2)*x*(x*(a*x+1))^(1/2)-60*(x*(a*x+1))^(1/2)*a^(1/2))/x^3/a^(9/ 2)/(x*(a*x+1))^(1/2)
Time = 0.12 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.86 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\left [\frac {735 \, {\left (a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (105 \, a^{5} c^{4} x^{5} + 397 \, a^{4} c^{4} x^{4} - 64 \, a^{3} c^{4} x^{3} - 194 \, a^{2} c^{4} x^{2} + 132 \, a c^{4} x - 30 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{420 \, {\left (a^{5} x^{4} - a^{4} x^{3}\right )}}, \frac {735 \, {\left (a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (105 \, a^{5} c^{4} x^{5} + 397 \, a^{4} c^{4} x^{4} - 64 \, a^{3} c^{4} x^{3} - 194 \, a^{2} c^{4} x^{2} + 132 \, a c^{4} x - 30 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{210 \, {\left (a^{5} x^{4} - a^{4} x^{3}\right )}}\right ] \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x)^(9/2),x, algorithm="fricas")
Output:
[1/420*(735*(a^4*c^4*x^4 - a^3*c^4*x^3)*sqrt(c)*log(-(8*a^3*c*x^3 - 7*a*c* x - 4*(2*a^3*x^3 + 3*a^2*x^2 + a*x)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt ((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(105*a^5*c^4*x^5 + 397*a^4*c^4*x^4 - 64*a^3*c^4*x^3 - 194*a^2*c^4*x^2 + 132*a*c^4*x - 30*c^4)*sqrt((a*x - 1) /(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^5*x^4 - a^4*x^3), 1/210*(735*(a^4* c^4*x^4 - a^3*c^4*x^3)*sqrt(-c)*arctan(2*(a^2*x^2 + a*x)*sqrt(-c)*sqrt((a* x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) + 2*( 105*a^5*c^4*x^5 + 397*a^4*c^4*x^4 - 64*a^3*c^4*x^3 - 194*a^2*c^4*x^2 + 132 *a*c^4*x - 30*c^4)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^5 *x^4 - a^4*x^3)]
Timed out. \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\text {Timed out} \] Input:
integrate(1/((a*x-1)/(a*x+1))**(1/2)*(c-c/a/x)**(9/2),x)
Output:
Timed out
\[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\int { \frac {{\left (c - \frac {c}{a x}\right )}^{\frac {9}{2}}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x)^(9/2),x, algorithm="maxima")
Output:
integrate((c - c/(a*x))^(9/2)/sqrt((a*x - 1)/(a*x + 1)), x)
\[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\int { \frac {{\left (c - \frac {c}{a x}\right )}^{\frac {9}{2}}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x)^(9/2),x, algorithm="giac")
Output:
integrate((c - c/(a*x))^(9/2)/sqrt((a*x - 1)/(a*x + 1)), x)
Timed out. \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\int \frac {{\left (c-\frac {c}{a\,x}\right )}^{9/2}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \] Input:
int((c - c/(a*x))^(9/2)/((a*x - 1)/(a*x + 1))^(1/2),x)
Output:
int((c - c/(a*x))^(9/2)/((a*x - 1)/(a*x + 1))^(1/2), x)
Time = 0.18 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.52 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\frac {\sqrt {c}\, c^{4} \left (105 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{4} x^{4}+292 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{3} x^{3}-356 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{2} x^{2}+162 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a x -30 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}-735 \,\mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}\right ) a^{4} x^{4}-337 a^{4} x^{4}\right )}{105 a^{5} x^{4}} \] Input:
int(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x)^(9/2),x)
Output:
(sqrt(c)*c**4*(105*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a**4*x**4 + 292*sqrt(x)*s qrt(a)*sqrt(a*x + 1)*a**3*x**3 - 356*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a**2*x* *2 + 162*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a*x - 30*sqrt(x)*sqrt(a)*sqrt(a*x + 1) - 735*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a))*a**4*x**4 - 337*a**4*x**4)) /(105*a**5*x**4)