Integrand size = 24, antiderivative size = 249 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=-\frac {5 c \sqrt {1-\frac {1}{a^2 x^2}}}{3 a \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {29 \sqrt {1-\frac {1}{a^2 x^2}}}{12 a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {73 \sqrt {1-\frac {1}{a^2 x^2}}}{16 a c \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x}{\left (c-\frac {c}{a x}\right )^{7/2}}+\frac {11 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a c^{5/2}}-\frac {249 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {2} \sqrt {c-\frac {c}{a x}}}\right )}{16 \sqrt {2} a c^{5/2}} \] Output:
-5/3*c*(1-1/a^2/x^2)^(1/2)/a/(c-c/a/x)^(7/2)-29/12*(1-1/a^2/x^2)^(1/2)/a/( c-c/a/x)^(5/2)-73/16*(1-1/a^2/x^2)^(1/2)/a/c/(c-c/a/x)^(3/2)+c*(1-1/a^2/x^ 2)^(1/2)*x/(c-c/a/x)^(7/2)+11*arctanh(c^(1/2)*(1-1/a^2/x^2)^(1/2)/(c-c/a/x )^(1/2))/a/c^(5/2)-249/32*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 = 0.18 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.57 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\frac {\sqrt {1-\frac {1}{a x}} \left (2 a \sqrt {1+\frac {1}{a x}} x \left (-219+554 a x-415 a^2 x^2+48 a^3 x^3\right )+1056 (-1+a x)^3 \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )-747 \sqrt {2} (-1+a x)^3 \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )\right )}{96 a c^2 \sqrt {c-\frac {c}{a x}} (-1+a x)^3} \] Input:
Integrate[E^(3*ArcCoth[a*x])/(c - c/(a*x))^(5/2),x]
Output:
(Sqrt[1 - 1/(a*x)]*(2*a*Sqrt[1 + 1/(a*x)]*x*(-219 + 554*a*x - 415*a^2*x^2 + 48*a^3*x^3) + 1056*(-1 + a*x)^3*ArcTanh[Sqrt[1 + 1/(a*x)]] - 747*Sqrt[2] *(-1 + a*x)^3*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]]))/(96*a*c^2*Sqrt[c - c/(a *x)]*(-1 + a*x)^3)
Time = 0.78 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.84, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6731, 585, 27, 109, 25, 27, 168, 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.
| \(\displaystyle \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6731 |
| \(\displaystyle -c^3 \int \frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^2}{\left (c-\frac {c}{a x}\right )^{11/2}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 585 |
| \(\displaystyle -\frac {\sqrt {1-\frac {1}{a x}} \int \frac {a^4 \left (1+\frac {1}{a x}\right )^{3/2} x^2}{\left (a-\frac {1}{x}\right )^4}d\frac {1}{x}}{c^2 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^4 \sqrt {1-\frac {1}{a x}} \int \frac {\left (1+\frac {1}{a x}\right )^{3/2} x^2}{\left (a-\frac {1}{x}\right )^4}d\frac {1}{x}}{c^2 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle -\frac {a^4 \sqrt {1-\frac {1}{a x}} \left (\frac {2 x \sqrt {\frac {1}{a x}+1}}{3 a \left (a-\frac {1}{x}\right )^3}-\frac {\int -\frac {\left (5 a+\frac {4}{x}\right ) x^2}{a \left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{3 a}\right )}{c^2 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a^4 \sqrt {1-\frac {1}{a x}} \left (\frac {\int \frac {\left (5 a+\frac {4}{x}\right ) x^2}{a \left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{3 a}+\frac {2 x \sqrt {\frac {1}{a x}+1}}{3 a \left (a-\frac {1}{x}\right )^3}\right )}{c^2 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^4 \sqrt {1-\frac {1}{a x}} \left (\frac {\int \frac {\left (5 a+\frac {4}{x}\right ) x^2}{\left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{3 a^2}+\frac {2 x \sqrt {\frac {1}{a x}+1}}{3 a \left (a-\frac {1}{x}\right )^3}\right )}{c^2 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {a^4 \sqrt {1-\frac {1}{a x}} \left (\frac {\frac {9 x \sqrt {\frac {1}{a x}+1}}{4 \left (a-\frac {1}{x}\right )^2}-\frac {\int -\frac {\left (58 a+\frac {45}{x}\right ) x^2}{2 \left (a-\frac {1}{x}\right )^2 \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{4 a}}{3 a^2}+\frac {2 x \sqrt {\frac {1}{a x}+1}}{3 a \left (a-\frac {1}{x}\right )^3}\right )}{c^2 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^4 \sqrt {1-\frac {1}{a x}} \left (\frac {\frac {\int \frac {\left (58 a+\frac {45}{x}\right ) x^2}{\left (a-\frac {1}{x}\right )^2 \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{8 a}+\frac {9 x \sqrt {\frac {1}{a x}+1}}{4 \left (a-\frac {1}{x}\right )^2}}{3 a^2}+\frac {2 x \sqrt {\frac {1}{a x}+1}}{3 a \left (a-\frac {1}{x}\right )^3}\right )}{c^2 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {a^4 \sqrt {1-\frac {1}{a x}} \left (\frac {\frac {\frac {103 x \sqrt {\frac {1}{a x}+1}}{2 \left (a-\frac {1}{x}\right )}-\frac {\int -\frac {3 \left (146 a+\frac {103}{x}\right ) x^2}{2 \left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{2 a}}{8 a}+\frac {9 x \sqrt {\frac {1}{a x}+1}}{4 \left (a-\frac {1}{x}\right )^2}}{3 a^2}+\frac {2 x \sqrt {\frac {1}{a x}+1}}{3 a \left (a-\frac {1}{x}\right )^3}\right )}{c^2 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^4 \sqrt {1-\frac {1}{a x}} \left (\frac {\frac {\frac {3 \int \frac {\left (146 a+\frac {103}{x}\right ) x^2}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{4 a}+\frac {103 x \sqrt {\frac {1}{a x}+1}}{2 \left (a-\frac {1}{x}\right )}}{8 a}+\frac {9 x \sqrt {\frac {1}{a x}+1}}{4 \left (a-\frac {1}{x}\right )^2}}{3 a^2}+\frac {2 x \sqrt {\frac {1}{a x}+1}}{3 a \left (a-\frac {1}{x}\right )^3}\right )}{c^2 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {a^4 \sqrt {1-\frac {1}{a x}} \left (\frac {\frac {\frac {3 \left (-\frac {\int -\frac {\left (176 a+\frac {73}{x}\right ) x}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{a}-146 x \sqrt {\frac {1}{a x}+1}\right )}{4 a}+\frac {103 x \sqrt {\frac {1}{a x}+1}}{2 \left (a-\frac {1}{x}\right )}}{8 a}+\frac {9 x \sqrt {\frac {1}{a x}+1}}{4 \left (a-\frac {1}{x}\right )^2}}{3 a^2}+\frac {2 x \sqrt {\frac {1}{a x}+1}}{3 a \left (a-\frac {1}{x}\right )^3}\right )}{c^2 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a^4 \sqrt {1-\frac {1}{a x}} \left (\frac {\frac {\frac {3 \left (\frac {\int \frac {\left (176 a+\frac {73}{x}\right ) x}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{a}-146 x \sqrt {\frac {1}{a x}+1}\right )}{4 a}+\frac {103 x \sqrt {\frac {1}{a x}+1}}{2 \left (a-\frac {1}{x}\right )}}{8 a}+\frac {9 x \sqrt {\frac {1}{a x}+1}}{4 \left (a-\frac {1}{x}\right )^2}}{3 a^2}+\frac {2 x \sqrt {\frac {1}{a x}+1}}{3 a \left (a-\frac {1}{x}\right )^3}\right )}{c^2 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle -\frac {a^4 \sqrt {1-\frac {1}{a x}} \left (\frac {\frac {\frac {3 \left (\frac {249 \int \frac {1}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}+176 \int \frac {x}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{a}-146 x \sqrt {\frac {1}{a x}+1}\right )}{4 a}+\frac {103 x \sqrt {\frac {1}{a x}+1}}{2 \left (a-\frac {1}{x}\right )}}{8 a}+\frac {9 x \sqrt {\frac {1}{a x}+1}}{4 \left (a-\frac {1}{x}\right )^2}}{3 a^2}+\frac {2 x \sqrt {\frac {1}{a x}+1}}{3 a \left (a-\frac {1}{x}\right )^3}\right )}{c^2 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {a^4 \sqrt {1-\frac {1}{a x}} \left (\frac {\frac {\frac {3 \left (\frac {498 a \int \frac {1}{2 a-\frac {a}{x^2}}d\sqrt {1+\frac {1}{a x}}+352 a \int \frac {1}{\frac {a}{x^2}-a}d\sqrt {1+\frac {1}{a x}}}{a}-146 x \sqrt {\frac {1}{a x}+1}\right )}{4 a}+\frac {103 x \sqrt {\frac {1}{a x}+1}}{2 \left (a-\frac {1}{x}\right )}}{8 a}+\frac {9 x \sqrt {\frac {1}{a x}+1}}{4 \left (a-\frac {1}{x}\right )^2}}{3 a^2}+\frac {2 x \sqrt {\frac {1}{a x}+1}}{3 a \left (a-\frac {1}{x}\right )^3}\right )}{c^2 \sqrt {c-\frac {c}{a x}}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {a^4 \sqrt {1-\frac {1}{a x}} \left (\frac {\frac {\frac {3 \left (\frac {249 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {\frac {1}{a x}+1}}{\sqrt {2}}\right )-352 \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right )}{a}-146 x \sqrt {\frac {1}{a x}+1}\right )}{4 a}+\frac {103 x \sqrt {\frac {1}{a x}+1}}{2 \left (a-\frac {1}{x}\right )}}{8 a}+\frac {9 x \sqrt {\frac {1}{a x}+1}}{4 \left (a-\frac {1}{x}\right )^2}}{3 a^2}+\frac {2 x \sqrt {\frac {1}{a x}+1}}{3 a \left (a-\frac {1}{x}\right )^3}\right )}{c^2 \sqrt {c-\frac {c}{a x}}}\) |
Input:
Int[E^(3*ArcCoth[a*x])/(c - c/(a*x))^(5/2),x]
Output:
-((a^4*Sqrt[1 - 1/(a*x)]*((2*Sqrt[1 + 1/(a*x)]*x)/(3*a*(a - x^(-1))^3) + ( (9*Sqrt[1 + 1/(a*x)]*x)/(4*(a - x^(-1))^2) + ((103*Sqrt[1 + 1/(a*x)]*x)/(2 *(a - x^(-1))) + (3*(-146*Sqrt[1 + 1/(a*x)]*x + (-352*ArcTanh[Sqrt[1 + 1/( a*x)]] + 249*Sqrt[2]*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/a))/(4*a))/(8*a)) /(3*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[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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 = 376, normalized size of antiderivative = 1.51
| method | result | size |
| risch | \(\frac {a x -1}{a \,c^{2} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}+\frac {\left (\frac {11 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{2 a^{3} \sqrt {a^{2} c}}-\frac {2 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +3 \left (x -\frac {1}{a}\right ) a c +2 c}}{3 a^{7} c \left (x -\frac {1}{a}\right )^{3}}-\frac {11 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +3 \left (x -\frac {1}{a}\right ) a c +2 c}}{4 a^{6} c \left (x -\frac {1}{a}\right )^{2}}-\frac {271 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +3 \left (x -\frac {1}{a}\right ) a c +2 c}}{48 a^{5} c \left (x -\frac {1}{a}\right )}-\frac {249 \sqrt {2}\, \ln \left (\frac {4 c +3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +3 \left (x -\frac {1}{a}\right ) a c +2 c}}{x -\frac {1}{a}}\right )}{64 a^{4} \sqrt {c}}\right ) a^{2} \sqrt {\left (a x +1\right ) a c x}\, \left (a x -1\right )}{c^{2} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) x \sqrt {\frac {c \left (a x -1\right )}{a x}}}\) | \(376\) |
| default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (192 \sqrt {x \left (a x +1\right )}\, a^{\frac {9}{2}} \sqrt {\frac {1}{a}}\, x^{3}-747 a^{\frac {7}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, a +3 a x +1}{a x -1}\right ) x^{3}-1660 a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, x^{2}+1056 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{4} \sqrt {\frac {1}{a}}\, x^{3}+2241 a^{\frac {5}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, a +3 a x +1}{a x -1}\right ) x^{2}+2216 a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, x -3168 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{3} \sqrt {\frac {1}{a}}\, x^{2}-2241 a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, a +3 a x +1}{a x -1}\right ) x -876 \sqrt {x \left (a x +1\right )}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}+3168 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{2} \sqrt {\frac {1}{a}}\, x -1056 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}+747 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, a +3 a x +1}{a x -1}\right ) \sqrt {a}\right )}{192 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x -1\right )^{2} \left (a x +1\right ) a^{\frac {3}{2}} c^{3} \sqrt {x \left (a x +1\right )}\, \sqrt {\frac {1}{a}}}\) | \(480\) |
Input:
int(1/((a*x-1)/(a*x+1))^(3/2)/(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)+(11/2/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)-2/ 3/a^7/c/(x-1/a)^3*((x-1/a)^2*a^2*c+3*(x-1/a)*a*c+2*c)^(1/2)-11/4/a^6/c/(x- 1/a)^2*((x-1/a)^2*a^2*c+3*(x-1/a)*a*c+2*c)^(1/2)-271/48/a^5/c/(x-1/a)*((x- 1/a)^2*a^2*c+3*(x-1/a)*a*c+2*c)^(1/2)-249/64/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 = 0.17 (sec) , antiderivative size = 736, normalized size of antiderivative = 2.96 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\left [\frac {747 \, \sqrt {2} {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, 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 ) + 1056 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, 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 (48 \, a^{5} x^{5} - 367 \, a^{4} x^{4} + 139 \, a^{3} x^{3} + 335 \, a^{2} x^{2} - 219 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{384 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}}, \frac {747 \, \sqrt {2} {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, 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 ) - 1056 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, 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 (48 \, a^{5} x^{5} - 367 \, a^{4} x^{4} + 139 \, a^{3} x^{3} + 335 \, a^{2} x^{2} - 219 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{192 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}}\right ] \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^(5/2),x, algorithm="fricas")
Output:
[1/384*(747*sqrt(2)*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*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)*sqrt(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)) + 1056*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^ 2 - 4*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*(48*a^5*x^5 - 367*a^4*x^4 + 139*a^3*x^3 + 335*a^2*x^2 - 219 *a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^5*c^3*x^4 - 4* a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*a^2*c^3*x + a*c^3), 1/192*(747*sqrt(2)*(a^ 4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*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)) - 1056*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*sqrt(-c)*arctan(2*(a^2*x^2 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*s qrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) + 4*(48*a^5*x^5 - 367*a^ 4*x^4 + 139*a^3*x^3 + 335*a^2*x^2 - 219*a*x)*sqrt((a*x - 1)/(a*x + 1))*sqr t((a*c*x - c)/(a*x)))/(a^5*c^3*x^4 - 4*a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*a^2 *c^3*x + a*c^3)]
Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(1/((a*x-1)/(a*x+1))**(3/2)/(c-c/a/x)**(5/2),x)
Output:
Timed out
\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^(5/2),x, algorithm="maxima")
Output:
integrate(1/((c - c/(a*x))^(5/2)*((a*x - 1)/(a*x + 1))^(3/2)), x)
\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^(5/2),x, algorithm="giac")
Output:
integrate(1/((c - c/(a*x))^(5/2)*((a*x - 1)/(a*x + 1))^(3/2)), x)
Timed out. \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\int \frac {1}{{\left (c-\frac {c}{a\,x}\right )}^{5/2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \] Input:
int(1/((c - c/(a*x))^(5/2)*((a*x - 1)/(a*x + 1))^(3/2)),x)
Output:
int(1/((c - c/(a*x))^(5/2)*((a*x - 1)/(a*x + 1))^(3/2)), x)
Time = 0.17 (sec) , antiderivative size = 583, normalized size of antiderivative = 2.34 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
int(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a/x)^(5/2),x)
Output:
(sqrt(c)*(576*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a**3*x**3 - 4980*sqrt(x)*sqrt( a)*sqrt(a*x + 1)*a**2*x**2 + 6648*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a*x - 2628 *sqrt(x)*sqrt(a)*sqrt(a*x + 1) + 2241*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)* sqrt(a) - sqrt(2) - 1)*a**3*x**3 - 6723*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x )*sqrt(a) - sqrt(2) - 1)*a**2*x**2 + 6723*sqrt(2)*log(sqrt(a*x + 1) + sqrt (x)*sqrt(a) - sqrt(2) - 1)*a*x - 2241*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)* sqrt(a) - sqrt(2) - 1) - 2241*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) - sqrt(2) + 1)*a**3*x**3 + 6723*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a ) - sqrt(2) + 1)*a**2*x**2 - 6723*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt (a) - sqrt(2) + 1)*a*x + 2241*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) - sqrt(2) + 1) - 2241*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + sqrt(2 ) - 1)*a**3*x**3 + 6723*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + sqrt (2) - 1)*a**2*x**2 - 6723*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + sq rt(2) - 1)*a*x + 2241*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + sqrt(2 ) - 1) + 2241*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + sqrt(2) + 1)*a **3*x**3 - 6723*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + sqrt(2) + 1) *a**2*x**2 + 6723*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + sqrt(2) + 1)*a*x - 2241*sqrt(2)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a) + sqrt(2) + 1) + 6336*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a))*a**3*x**3 - 19008*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a))*a**2*x**2 + 19008*log(sqrt(a*x + 1) + sqrt(x)*s...