Integrand size = 22, antiderivative size = 213 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=-\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {11 \sqrt {1-\frac {1}{a^2 x^2}} x}{8 c \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {23 \sqrt {1-\frac {1}{a^2 x^2}} x}{8 c^2 \sqrt {c-\frac {c}{a x}}}+\frac {7 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a c^{5/2}}-\frac {79 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {2} \sqrt {c-\frac {c}{a x}}}\right )}{8 \sqrt {2} a c^{5/2}} \] Output:
-1/2*(1-1/a^2/x^2)^(1/2)*x/(c-c/a/x)^(5/2)-11/8*(1-1/a^2/x^2)^(1/2)*x/c/(c -c/a/x)^(3/2)+23/8*(1-1/a^2/x^2)^(1/2)*x/c^2/(c-c/a/x)^(1/2)+7*arctanh(c^( 1/2)*(1-1/a^2/x^2)^(1/2)/(c-c/a/x)^(1/2))/a/c^(5/2)-79/16*arctanh(1/2*c^(1 /2)*(1-1/a^2/x^2)^(1/2)*2^(1/2)/(c-c/a/x)^(1/2))*2^(1/2)/a/c^(5/2)
Time = 0.14 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.63 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\frac {\sqrt {1-\frac {1}{a x}} \left (2 a \sqrt {1+\frac {1}{a x}} x \left (23-35 a x+8 a^2 x^2\right )+112 (-1+a x)^2 \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )-79 \sqrt {2} (-1+a x)^2 \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )\right )}{16 a c^2 \sqrt {c-\frac {c}{a x}} (-1+a x)^2} \] Input:
Integrate[E^ArcCoth[a*x]/(c - c/(a*x))^(5/2),x]
Output:
(Sqrt[1 - 1/(a*x)]*(2*a*Sqrt[1 + 1/(a*x)]*x*(23 - 35*a*x + 8*a^2*x^2) + 11 2*(-1 + a*x)^2*ArcTanh[Sqrt[1 + 1/(a*x)]] - 79*Sqrt[2]*(-1 + a*x)^2*ArcTan h[Sqrt[1 + 1/(a*x)]/Sqrt[2]]))/(16*a*c^2*Sqrt[c - c/(a*x)]*(-1 + a*x)^2)
Time = 0.71 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.82, 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, 110, 27, 168, 27, 168, 25, 174, 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.
| \(\displaystyle \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6731 |
| \(\displaystyle -c \int \frac {\sqrt {1-\frac {1}{a^2 x^2}} x^2}{\left (c-\frac {c}{a x}\right )^{7/2}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 585 |
| \(\displaystyle -\frac {\sqrt {1-\frac {1}{a x}} \int \frac {a^3 \sqrt {1+\frac {1}{a x}} x^2}{\left (a-\frac {1}{x}\right )^3}d\frac {1}{x}}{c^2 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \sqrt {1-\frac {1}{a x}} \int \frac {\sqrt {1+\frac {1}{a x}} x^2}{\left (a-\frac {1}{x}\right )^3}d\frac {1}{x}}{c^2 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle -\frac {a^3 \sqrt {1-\frac {1}{a x}} \left (\frac {x \sqrt {\frac {1}{a x}+1}}{2 a \left (a-\frac {1}{x}\right )^2}-\frac {\int -\frac {\left (6 a+\frac {5}{x}\right ) x^2}{2 a \left (a-\frac {1}{x}\right )^2 \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{2 a}\right )}{c^2 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \sqrt {1-\frac {1}{a x}} \left (\frac {\int \frac {\left (6 a+\frac {5}{x}\right ) x^2}{\left (a-\frac {1}{x}\right )^2 \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{4 a^2}+\frac {x \sqrt {\frac {1}{a x}+1}}{2 a \left (a-\frac {1}{x}\right )^2}\right )}{c^2 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {a^3 \sqrt {1-\frac {1}{a x}} \left (\frac {\frac {11 x \sqrt {\frac {1}{a x}+1}}{2 \left (a-\frac {1}{x}\right )}-\frac {\int -\frac {\left (46 a+\frac {33}{x}\right ) x^2}{2 \left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{2 a}}{4 a^2}+\frac {x \sqrt {\frac {1}{a x}+1}}{2 a \left (a-\frac {1}{x}\right )^2}\right )}{c^2 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \sqrt {1-\frac {1}{a x}} \left (\frac {\frac {\int \frac {\left (46 a+\frac {33}{x}\right ) x^2}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{4 a}+\frac {11 x \sqrt {\frac {1}{a x}+1}}{2 \left (a-\frac {1}{x}\right )}}{4 a^2}+\frac {x \sqrt {\frac {1}{a x}+1}}{2 a \left (a-\frac {1}{x}\right )^2}\right )}{c^2 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {a^3 \sqrt {1-\frac {1}{a x}} \left (\frac {\frac {-\frac {\int -\frac {\left (56 a+\frac {23}{x}\right ) x}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{a}-46 x \sqrt {\frac {1}{a x}+1}}{4 a}+\frac {11 x \sqrt {\frac {1}{a x}+1}}{2 \left (a-\frac {1}{x}\right )}}{4 a^2}+\frac {x \sqrt {\frac {1}{a x}+1}}{2 a \left (a-\frac {1}{x}\right )^2}\right )}{c^2 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a^3 \sqrt {1-\frac {1}{a x}} \left (\frac {\frac {\frac {\int \frac {\left (56 a+\frac {23}{x}\right ) x}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{a}-46 x \sqrt {\frac {1}{a x}+1}}{4 a}+\frac {11 x \sqrt {\frac {1}{a x}+1}}{2 \left (a-\frac {1}{x}\right )}}{4 a^2}+\frac {x \sqrt {\frac {1}{a x}+1}}{2 a \left (a-\frac {1}{x}\right )^2}\right )}{c^2 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle -\frac {a^3 \sqrt {1-\frac {1}{a x}} \left (\frac {\frac {\frac {79 \int \frac {1}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+56 \int \frac {x}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{a}-46 x \sqrt {\frac {1}{a x}+1}}{4 a}+\frac {11 x \sqrt {\frac {1}{a x}+1}}{2 \left (a-\frac {1}{x}\right )}}{4 a^2}+\frac {x \sqrt {\frac {1}{a x}+1}}{2 a \left (a-\frac {1}{x}\right )^2}\right )}{c^2 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {a^3 \sqrt {1-\frac {1}{a x}} \left (\frac {\frac {\frac {158 a \int \frac {1}{2 a-\frac {a}{x^2}}d\sqrt {1+\frac {1}{a x}}+112 a \int \frac {1}{\frac {a}{x^2}-a}d\sqrt {1+\frac {1}{a x}}}{a}-46 x \sqrt {\frac {1}{a x}+1}}{4 a}+\frac {11 x \sqrt {\frac {1}{a x}+1}}{2 \left (a-\frac {1}{x}\right )}}{4 a^2}+\frac {x \sqrt {\frac {1}{a x}+1}}{2 a \left (a-\frac {1}{x}\right )^2}\right )}{c^2 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {a^3 \sqrt {1-\frac {1}{a x}} \left (\frac {\frac {\frac {79 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {\frac {1}{a x}+1}}{\sqrt {2}}\right )-112 \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right )}{a}-46 x \sqrt {\frac {1}{a x}+1}}{4 a}+\frac {11 x \sqrt {\frac {1}{a x}+1}}{2 \left (a-\frac {1}{x}\right )}}{4 a^2}+\frac {x \sqrt {\frac {1}{a x}+1}}{2 a \left (a-\frac {1}{x}\right )^2}\right )}{c^2 \sqrt {c-\frac {c}{a x}}}\) |
Input:
Int[E^ArcCoth[a*x]/(c - c/(a*x))^(5/2),x]
Output:
-((a^3*Sqrt[1 - 1/(a*x)]*((Sqrt[1 + 1/(a*x)]*x)/(2*a*(a - x^(-1))^2) + ((1 1*Sqrt[1 + 1/(a*x)]*x)/(2*(a - x^(-1))) + (-46*Sqrt[1 + 1/(a*x)]*x + (-112 *ArcTanh[Sqrt[1 + 1/(a*x)]] + 79*Sqrt[2]*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2] ])/a)/(4*a))/(4*a^2)))/(c^2*Sqrt[c - c/(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 + 1)/((m + 1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f)) Int[(a + b*x)^(m + 1) *(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt Q[n, 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_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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[((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 = 0.20 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.54
| method | result | size |
| risch | \(\frac {a x -1}{a \,c^{2} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}+\frac {\left (\frac {7 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{2 a^{3} \sqrt {a^{2} c}}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +3 \left (x -\frac {1}{a}\right ) a c +2 c}}{2 a^{6} c \left (x -\frac {1}{a}\right )^{2}}-\frac {19 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +3 \left (x -\frac {1}{a}\right ) a c +2 c}}{8 a^{5} c \left (x -\frac {1}{a}\right )}-\frac {79 \sqrt {2}\, \ln \left (\frac {4 c +3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +3 \left (x -\frac {1}{a}\right ) a c +2 c}}{x -\frac {1}{a}}\right )}{32 a^{4} \sqrt {c}}\right ) a^{2} \sqrt {\left (a x +1\right ) a c x}\, \left (a x -1\right )}{c^{2} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) x \sqrt {\frac {c \left (a x -1\right )}{a x}}}\) | \(328\) |
| default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (32 a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, x^{2}-79 a^{\frac {5}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, a +3 a x +1}{a x -1}\right ) x^{2}-140 a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, x +112 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{3} \sqrt {\frac {1}{a}}\, x^{2}+158 a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, a +3 a x +1}{a x -1}\right ) x +92 \sqrt {x \left (a x +1\right )}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}-224 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{2} \sqrt {\frac {1}{a}}\, x +112 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}-79 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, a +3 a x +1}{a x -1}\right ) \sqrt {a}\right )}{32 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x -1\right )^{2} a^{\frac {3}{2}} c^{3} \sqrt {x \left (a x +1\right )}\, \sqrt {\frac {1}{a}}}\) | \(366\) |
Input:
int(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/a/c^2/((a*x-1)/(a*x+1))^(1/2)/(c*(a*x-1)/a/x)^(1/2)*(a*x-1)+(7/2/a^3*ln( (1/2*a*c+a^2*c*x)/(a^2*c)^(1/2)+(a^2*c*x^2+a*c*x)^(1/2))/(a^2*c)^(1/2)-1/2 /a^6/c/(x-1/a)^2*((x-1/a)^2*a^2*c+3*(x-1/a)*a*c+2*c)^(1/2)-19/8/a^5/c/(x-1 /a)*((x-1/a)^2*a^2*c+3*(x-1/a)*a*c+2*c)^(1/2)-79/32/a^4/c^(1/2)*2^(1/2)*ln ((4*c+3*(x-1/a)*a*c+2*2^(1/2)*c^(1/2)*((x-1/a)^2*a^2*c+3*(x-1/a)*a*c+2*c)^ (1/2))/(x-1/a)))*a^2/c^2/((a*x-1)/(a*x+1))^(1/2)/(a*x+1)/x/(c*(a*x-1)/a/x) ^(1/2)*((a*x+1)*a*c*x)^(1/2)*(a*x-1)
Time = 0.15 (sec) , antiderivative size = 668, normalized size of antiderivative = 3.14 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\left [\frac {79 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt {2} {\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 112 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 8 \, {\left (8 \, a^{4} x^{4} - 27 \, a^{3} x^{3} - 12 \, a^{2} x^{2} + 23 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{64 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}}, \frac {79 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {2} {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - 112 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 4 \, {\left (8 \, a^{4} x^{4} - 27 \, a^{3} x^{3} - 12 \, a^{2} x^{2} + 23 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{32 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}}\right ] \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^(5/2),x, algorithm="fricas")
Output:
[1/64*(79*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(c)*log(-(17*a^3*c *x^3 - 3*a^2*c*x^2 - 13*a*c*x - 4*sqrt(2)*(3*a^3*x^3 + 4*a^2*x^2 + a*x)*sq rt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3* a^2*x^2 + 3*a*x - 1)) + 112*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*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)) + 8*(8*a^4*x^4 - 27*a^3*x^3 - 12*a^2*x^2 + 23*a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^4*c^3*x^3 - 3*a^3*c^3*x^2 + 3*a^2*c^3*x - a*c^3), 1/32*(79* sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(-c)*arctan(2*sqrt(2)*(a^2*x ^2 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(3*a^ 2*c*x^2 - 2*a*c*x - c)) - 112*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(-c)*a rctan(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)) + 4*(8*a^4*x^4 - 27*a^3*x^3 - 12*a^2*x ^2 + 23*a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^4*c^3*x ^3 - 3*a^3*c^3*x^2 + 3*a^2*c^3*x - a*c^3)]
Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(1/((a*x-1)/(a*x+1))**(1/2)/(c-c/a/x)**(5/2),x)
Output:
Timed out
\[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^(5/2),x, algorithm="maxima")
Output:
integrate(1/((c - c/(a*x))^(5/2)*sqrt((a*x - 1)/(a*x + 1))), x)
\[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}} \sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^(5/2),x, algorithm="giac")
Output:
integrate(1/((c - c/(a*x))^(5/2)*sqrt((a*x - 1)/(a*x + 1))), x)
Timed out. \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\int \frac {1}{{\left (c-\frac {c}{a\,x}\right )}^{5/2}\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \] Input:
int(1/((c - c/(a*x))^(5/2)*((a*x - 1)/(a*x + 1))^(1/2)),x)
Output:
int(1/((c - c/(a*x))^(5/2)*((a*x - 1)/(a*x + 1))^(1/2)), x)
Time = 0.17 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.97 \[ \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\frac {\sqrt {c}\, \left (96 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{2} x^{2}-420 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a x +276 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}+237 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}-1\right ) a^{2} x^{2}-474 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}-1\right ) a x +237 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}-1\right )-237 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}+1\right ) a^{2} x^{2}+474 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}+1\right ) a x -237 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}+1\right )-237 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}-1\right ) a^{2} x^{2}+474 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}-1\right ) a x -237 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}-1\right )+237 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}+1\right ) a^{2} x^{2}-474 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}+1\right ) a x +237 \sqrt {2}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}+1\right )+672 \,\mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}\right ) a^{2} x^{2}-1344 \,\mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}\right ) a x +672 \,\mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}\right )+172 a^{2} x^{2}-344 a x +172\right )}{96 a \,c^{3} \left (a^{2} x^{2}-2 a x +1\right )} \] Input:
int(1/((a*x-1)/(a*x+1))^(1/2)/(c-c/a/x)^(5/2),x)
Output:
(sqrt(c)*(96*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a**2*x**2 - 420*sqrt(x)*sqrt(a) *sqrt(a*x + 1)*a*x + 276*sqrt(x)*sqrt(a)*sqrt(a*x + 1) + 237*sqrt(2)*log(s qrt(a*x + 1) + sqrt(x)*sqrt(a) - sqrt(2) - 1)*a**2*x**2 - 474*sqrt(2)*log( sqrt(a*x + 1) + sqrt(x)*sqrt(a) - sqrt(2) - 1)*a*x + 237*sqrt(2)*log(sqrt( a*x + 1) + sqrt(x)*sqrt(a) - sqrt(2) - 1) - 237*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) - sqrt(2) + 1)*a**2*x**2 + 474*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) - sqrt(2) + 1)*a*x - 237*sqrt(2)*log(sqrt(a*x + 1) + sq rt(x)*sqrt(a) - sqrt(2) + 1) - 237*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqr t(a) + sqrt(2) - 1)*a**2*x**2 + 474*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sq rt(a) + sqrt(2) - 1)*a*x - 237*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + sqrt(2) - 1) + 237*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + sqrt(2 ) + 1)*a**2*x**2 - 474*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + sqrt( 2) + 1)*a*x + 237*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + sqrt(2) + 1) + 672*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a))*a**2*x**2 - 1344*log(sqrt(a* x + 1) + sqrt(x)*sqrt(a))*a*x + 672*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a)) + 172*a**2*x**2 - 344*a*x + 172))/(96*a*c**3*(a**2*x**2 - 2*a*x + 1))