Integrand size = 24, antiderivative size = 161 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=-\frac {\left (16 a+\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{5/2}}{3 a^2 \left (1-\frac {1}{a x}\right )^{5/2}}+\frac {\left (a-\frac {1}{x}\right )^2 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{5/2} x}{a^2 \left (1-\frac {1}{a x}\right )^{5/2}}-\frac {7 \left (c-\frac {c}{a x}\right )^{5/2} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \left (1-\frac {1}{a x}\right )^{5/2}} \]
-7*(c-c/a/x)^(5/2)*arctanh((1+1/a/x)^(1/2))/a/(1-1/a/x)^(5/2)-1/3*(16*a+1/ x)*(c-c/a/x)^(5/2)*(1+1/a/x)^(1/2)/a^2/(1-1/a/x)^(5/2)+(a-1/x)^2*(c-c/a/x) ^(5/2)*x*(1+1/a/x)^(1/2)/a^2/(1-1/a/x)^(5/2)
Time = 0.07 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.55 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\frac {c^2 \sqrt {c-\frac {c}{a x}} \left (\sqrt {1+\frac {1}{a x}} \left (2-22 a x+3 a^2 x^2\right )-21 a x \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )\right )}{3 a^2 \sqrt {1-\frac {1}{a x}} x} \]
(c^2*Sqrt[c - c/(a*x)]*(Sqrt[1 + 1/(a*x)]*(2 - 22*a*x + 3*a^2*x^2) - 21*a* x*ArcTanh[Sqrt[1 + 1/(a*x)]]))/(3*a^2*Sqrt[1 - 1/(a*x)]*x)
Time = 0.35 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.70, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6731, 585, 27, 109, 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.
| \(\displaystyle \int \left (c-\frac {c}{a x}\right )^{5/2} e^{-\coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6731 |
| \(\displaystyle -\frac {\int \frac {\left (c-\frac {c}{a x}\right )^{7/2} x^2}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}}{c}\) |
| \(\Big \downarrow \) 585 |
| \(\displaystyle -\frac {c^2 \sqrt {c-\frac {c}{a x}} \int \frac {\left (a-\frac {1}{x}\right )^3 x^2}{a^3 \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{\sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c^2 \sqrt {c-\frac {c}{a x}} \int \frac {\left (a-\frac {1}{x}\right )^3 x^2}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}}{a^3 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle -\frac {c^2 \sqrt {c-\frac {c}{a x}} \left (-\int \frac {\left (a-\frac {1}{x}\right ) \left (7 a+\frac {1}{x}\right ) x}{2 \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-a x \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^2\right )}{a^3 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c^2 \sqrt {c-\frac {c}{a x}} \left (-\frac {1}{2} \int \frac {\left (a-\frac {1}{x}\right ) \left (7 a+\frac {1}{x}\right ) x}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-a x \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^2\right )}{a^3 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle -\frac {c^2 \sqrt {c-\frac {c}{a x}} \left (\frac {1}{2} \left (\frac {2}{3} a \left (16 a+\frac {1}{x}\right ) \sqrt {\frac {1}{a x}+1}-7 a^2 \int \frac {x}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}\right )-a x \left (a-\frac {1}{x}\right )^2 \sqrt {\frac {1}{a x}+1}\right )}{a^3 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {c^2 \sqrt {c-\frac {c}{a x}} \left (\frac {1}{2} \left (\frac {2}{3} a \left (16 a+\frac {1}{x}\right ) \sqrt {\frac {1}{a x}+1}-14 a^3 \int \frac {1}{\frac {a}{x^2}-a}d\sqrt {1+\frac {1}{a x}}\right )-a x \left (a-\frac {1}{x}\right )^2 \sqrt {\frac {1}{a x}+1}\right )}{a^3 \sqrt {1-\frac {1}{a x}}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {c^2 \left (\frac {1}{2} \left (14 a^2 \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right )+\frac {2}{3} a \left (16 a+\frac {1}{x}\right ) \sqrt {\frac {1}{a x}+1}\right )-a x \left (a-\frac {1}{x}\right )^2 \sqrt {\frac {1}{a x}+1}\right ) \sqrt {c-\frac {c}{a x}}}{a^3 \sqrt {1-\frac {1}{a x}}}\) |
-((c^2*Sqrt[c - c/(a*x)]*(-(a*(a - x^(-1))^2*Sqrt[1 + 1/(a*x)]*x) + ((2*a* (16*a + x^(-1))*Sqrt[1 + 1/(a*x)])/3 + 14*a^2*ArcTanh[Sqrt[1 + 1/(a*x)]])/ 2))/(a^3*Sqrt[1 - 1/(a*x)]))
3.5.64.3.1 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_))^(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_ ))*((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_)^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 = 0.16 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.89
| method | result | size |
| default | \(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{2} \left (6 a^{\frac {5}{2}} x^{2} \sqrt {\left (a x +1\right ) x}-44 a^{\frac {3}{2}} x \sqrt {\left (a x +1\right ) x}-21 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{2} x^{2}+4 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}\right )}{6 x \,a^{\frac {5}{2}} \left (a x -1\right ) \sqrt {\left (a x +1\right ) x}}\) | \(144\) |
| risch | \(\frac {\left (3 a^{3} x^{3}-19 a^{2} x^{2}-20 a x +2\right ) c^{2} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{3 x \,a^{2} \left (a x -1\right )}-\frac {7 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right ) c^{2} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{2 \sqrt {a^{2} c}\, \left (a x -1\right )}\) | \(168\) |
1/6*((a*x-1)/(a*x+1))^(1/2)*(a*x+1)*(c*(a*x-1)/a/x)^(1/2)*c^2*(6*a^(5/2)*x ^2*((a*x+1)*x)^(1/2)-44*a^(3/2)*x*((a*x+1)*x)^(1/2)-21*ln(1/2*(2*((a*x+1)* x)^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2))*a^2*x^2+4*((a*x+1)*x)^(1/2)*a^(1/2))/x/ a^(5/2)/(a*x-1)/((a*x+1)*x)^(1/2)
Time = 0.28 (sec) , antiderivative size = 381, normalized size of antiderivative = 2.37 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\left [\frac {21 \, {\left (a^{2} c^{2} x^{2} - a c^{2} x\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (3 \, a^{3} c^{2} x^{3} - 19 \, a^{2} c^{2} x^{2} - 20 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{12 \, {\left (a^{3} x^{2} - a^{2} x\right )}}, \frac {21 \, {\left (a^{2} c^{2} x^{2} - a c^{2} x\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (3 \, a^{3} c^{2} x^{3} - 19 \, a^{2} c^{2} x^{2} - 20 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{6 \, {\left (a^{3} x^{2} - a^{2} x\right )}}\right ] \]
[1/12*(21*(a^2*c^2*x^2 - a*c^2*x)*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*(3*a^3*c^2*x^3 - 19*a^2*c^2*x^2 - 20*a*c ^2*x + 2*c^2)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2 - a^2*x), 1/6*(21*(a^2*c^2*x^2 - a*c^2*x)*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*(3*a^3*c^2*x^3 - 19*a^2*c^2*x^2 - 20*a*c^2*x + 2*c^2)*sq rt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2 - a^2*x)]
Timed out. \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\text {Timed out} \]
\[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\int { {\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}} \sqrt {\frac {a x - 1}{a x + 1}} \,d x } \]
Exception generated. \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\text {Exception raised: TypeError} \]
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
Timed out. \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\int {\left (c-\frac {c}{a\,x}\right )}^{5/2}\,\sqrt {\frac {a\,x-1}{a\,x+1}} \,d x \]