Integrand size = 24, antiderivative size = 225 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=-\frac {94 a^2 \left (c-\frac {c}{a x}\right )^{9/2} x^3 \sqrt {1+a x}}{21 (1-a x)^{5/2}}+\frac {6 a \left (c-\frac {c}{a x}\right )^{9/2} x^2 \sqrt {1+a x}}{5 (1-a x)^{3/2}}-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x \sqrt {1+a x}}{7 \sqrt {1-a x}}+\frac {a^3 \left (c-\frac {c}{a x}\right )^{9/2} x^4 \sqrt {1+a x} (2718+521 a x)}{105 (1-a x)^{9/2}}+\frac {11 a^{7/2} \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{9/2}} \]
11*a^(7/2)*(c-c/a/x)^(9/2)*x^(9/2)*arcsinh(a^(1/2)*x^(1/2))/(-a*x+1)^(9/2) -94/21*a^2*(c-c/a/x)^(9/2)*x^3*(a*x+1)^(1/2)/(-a*x+1)^(5/2)+6/5*a*(c-c/a/x )^(9/2)*x^2*(a*x+1)^(1/2)/(-a*x+1)^(3/2)+1/105*a^3*(c-c/a/x)^(9/2)*x^4*(52 1*a*x+2718)*(a*x+1)^(1/2)/(-a*x+1)^(9/2)-2/7*(c-c/a/x)^(9/2)*x*(a*x+1)^(1/ 2)/(-a*x+1)^(1/2)
Time = 2.35 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.48 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\frac {c^4 \sqrt {c-\frac {c}{a x}} \left (\sqrt {-a x (1+a x)} \left (30-246 a x+1028 a^2 x^2-4156 a^3 x^3+105 a^4 x^4\right )-1155 a^4 x^4 \arcsin \left (\sqrt {-a x}\right )\right )}{105 a^3 x^2 (-a x)^{3/2} \sqrt {1-a x}} \]
(c^4*Sqrt[c - c/(a*x)]*(Sqrt[-(a*x*(1 + a*x))]*(30 - 246*a*x + 1028*a^2*x^ 2 - 4156*a^3*x^3 + 105*a^4*x^4) - 1155*a^4*x^4*ArcSin[Sqrt[-(a*x)]]))/(105 *a^3*x^2*(-(a*x))^(3/2)*Sqrt[1 - a*x])
Time = 0.47 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.75, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {6684, 6679, 109, 27, 167, 27, 167, 27, 160, 63, 222}
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.
| \(\displaystyle \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx\) |
| \(\Big \downarrow \) 6684 |
| \(\displaystyle \frac {x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \int \frac {e^{-\text {arctanh}(a x)} (1-a x)^{9/2}}{x^{9/2}}dx}{(1-a x)^{9/2}}\) |
| \(\Big \downarrow \) 6679 |
| \(\displaystyle \frac {x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \int \frac {(1-a x)^5}{x^{9/2} \sqrt {a x+1}}dx}{(1-a x)^{9/2}}\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \left (-\frac {2}{7} \int \frac {a (21-5 a x) (1-a x)^3}{2 x^{7/2} \sqrt {a x+1}}dx-\frac {2 \sqrt {a x+1} (1-a x)^4}{7 x^{7/2}}\right )}{(1-a x)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \left (-\frac {1}{7} a \int \frac {(21-5 a x) (1-a x)^3}{x^{7/2} \sqrt {a x+1}}dx-\frac {2 \sqrt {a x+1} (1-a x)^4}{7 x^{7/2}}\right )}{(1-a x)^{9/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \left (-\frac {1}{7} a \left (\frac {2}{5} \int -\frac {a (1-a x)^2 (17 a x+235)}{2 x^{5/2} \sqrt {a x+1}}dx-\frac {42 (1-a x)^3 \sqrt {a x+1}}{5 x^{5/2}}\right )-\frac {2 \sqrt {a x+1} (1-a x)^4}{7 x^{7/2}}\right )}{(1-a x)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \left (-\frac {1}{7} a \left (-\frac {1}{5} a \int \frac {(1-a x)^2 (17 a x+235)}{x^{5/2} \sqrt {a x+1}}dx-\frac {42 \sqrt {a x+1} (1-a x)^3}{5 x^{5/2}}\right )-\frac {2 \sqrt {a x+1} (1-a x)^4}{7 x^{7/2}}\right )}{(1-a x)^{9/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \left (-\frac {1}{7} a \left (-\frac {1}{5} a \left (\frac {2}{3} \int -\frac {a (1-a x) (521 a x+1359)}{2 x^{3/2} \sqrt {a x+1}}dx-\frac {470 (1-a x)^2 \sqrt {a x+1}}{3 x^{3/2}}\right )-\frac {42 \sqrt {a x+1} (1-a x)^3}{5 x^{5/2}}\right )-\frac {2 \sqrt {a x+1} (1-a x)^4}{7 x^{7/2}}\right )}{(1-a x)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \left (-\frac {1}{7} a \left (-\frac {1}{5} a \left (-\frac {1}{3} a \int \frac {(1-a x) (521 a x+1359)}{x^{3/2} \sqrt {a x+1}}dx-\frac {470 \sqrt {a x+1} (1-a x)^2}{3 x^{3/2}}\right )-\frac {42 \sqrt {a x+1} (1-a x)^3}{5 x^{5/2}}\right )-\frac {2 \sqrt {a x+1} (1-a x)^4}{7 x^{7/2}}\right )}{(1-a x)^{9/2}}\) |
| \(\Big \downarrow \) 160 |
| \(\displaystyle \frac {x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \left (-\frac {1}{7} a \left (-\frac {1}{5} a \left (-\frac {1}{3} a \left (-\frac {1155}{2} a \int \frac {1}{\sqrt {x} \sqrt {a x+1}}dx-\frac {\sqrt {a x+1} (521 a x+2718)}{\sqrt {x}}\right )-\frac {470 \sqrt {a x+1} (1-a x)^2}{3 x^{3/2}}\right )-\frac {42 \sqrt {a x+1} (1-a x)^3}{5 x^{5/2}}\right )-\frac {2 \sqrt {a x+1} (1-a x)^4}{7 x^{7/2}}\right )}{(1-a x)^{9/2}}\) |
| \(\Big \downarrow \) 63 |
| \(\displaystyle \frac {x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \left (-\frac {1}{7} a \left (-\frac {1}{5} a \left (-\frac {1}{3} a \left (-1155 a \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}-\frac {\sqrt {a x+1} (521 a x+2718)}{\sqrt {x}}\right )-\frac {470 \sqrt {a x+1} (1-a x)^2}{3 x^{3/2}}\right )-\frac {42 \sqrt {a x+1} (1-a x)^3}{5 x^{5/2}}\right )-\frac {2 \sqrt {a x+1} (1-a x)^4}{7 x^{7/2}}\right )}{(1-a x)^{9/2}}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {x^{9/2} \left (-\frac {1}{7} a \left (-\frac {1}{5} a \left (-\frac {1}{3} a \left (-1155 \sqrt {a} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )-\frac {\sqrt {a x+1} (521 a x+2718)}{\sqrt {x}}\right )-\frac {470 \sqrt {a x+1} (1-a x)^2}{3 x^{3/2}}\right )-\frac {42 \sqrt {a x+1} (1-a x)^3}{5 x^{5/2}}\right )-\frac {2 \sqrt {a x+1} (1-a x)^4}{7 x^{7/2}}\right ) \left (c-\frac {c}{a x}\right )^{9/2}}{(1-a x)^{9/2}}\) |
((c - c/(a*x))^(9/2)*x^(9/2)*((-2*(1 - a*x)^4*Sqrt[1 + a*x])/(7*x^(7/2)) - (a*((-42*(1 - a*x)^3*Sqrt[1 + a*x])/(5*x^(5/2)) - (a*((-470*(1 - a*x)^2*S qrt[1 + a*x])/(3*x^(3/2)) - (a*(-((Sqrt[1 + a*x]*(2718 + 521*a*x))/Sqrt[x] ) - 1155*Sqrt[a]*ArcSinh[Sqrt[a]*Sqrt[x]]))/3))/5))/7))/(1 - a*x)^(9/2)
3.6.35.3.1 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[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2/b S ubst[Int[1/Sqrt[c + d*(x^2/b)], x], x, Sqrt[b*x]], x] /; FreeQ[{b, c, d}, x ] && GtQ[c, 0]
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[(b^2*d*e*g - a^2*d*f*h*m - a*b*(d*(f*g + e*h) - c*f*h*(m + 1)) + b*f*h*(b*c - a*d)*(m + 1)*x)*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d*(b*c - a*d)*(m + 1))), x] + Simp[(a*d*f*h*m + b*(d* (f*g + e*h) - c*f*h*(m + 2)))/(b^2*d) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1] && (SumSimplerQ[m, 1] || !SumSimplerQ[n, 1])
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*((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 - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol ] :> Simp[c^p Int[u*(1 + d*(x/c))^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] , x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ[p] || GtQ[c, 0])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] :> Simp[x^p*((c + d/x)^p/(1 + c*(x/d))^p) Int[u*(1 + c*(x/d))^p*(E^(n*Ar cTanh[a*x])/x^p), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && !IntegerQ[p]
Time = 0.11 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.76
| method | result | size |
| default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{4} \sqrt {-a^{2} x^{2}+1}\, \left (210 \sqrt {-\left (a x +1\right ) x}\, a^{\frac {9}{2}} x^{4}+1155 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) a^{4} x^{4}-8312 x^{3} \sqrt {-\left (a x +1\right ) x}\, a^{\frac {7}{2}}+2056 a^{\frac {5}{2}} x^{2} \sqrt {-\left (a x +1\right ) x}-492 a^{\frac {3}{2}} x \sqrt {-\left (a x +1\right ) x}+60 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}\right )}{210 x^{3} a^{\frac {9}{2}} \left (a x -1\right ) \sqrt {-\left (a x +1\right ) x}}\) | \(172\) |
| risch | \(-\frac {\left (105 a^{5} x^{5}-4051 a^{4} x^{4}-3128 a^{3} x^{3}+782 a^{2} x^{2}-216 a x +30\right ) c^{4} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{105 x^{3} \sqrt {-\left (a x +1\right ) a c x}\, \sqrt {-a^{2} x^{2}+1}\, a^{4}}+\frac {11 \arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-a c x}}\right ) c^{4} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{2 \sqrt {a^{2} c}\, \sqrt {-a^{2} x^{2}+1}}\) | \(208\) |
1/210*(c*(a*x-1)/a/x)^(1/2)/x^3*c^4/a^(9/2)*(-a^2*x^2+1)^(1/2)*(210*(-(a*x +1)*x)^(1/2)*a^(9/2)*x^4+1155*arctan(1/2/a^(1/2)*(2*a*x+1)/(-(a*x+1)*x)^(1 /2))*a^4*x^4-8312*x^3*(-(a*x+1)*x)^(1/2)*a^(7/2)+2056*a^(5/2)*x^2*(-(a*x+1 )*x)^(1/2)-492*a^(3/2)*x*(-(a*x+1)*x)^(1/2)+60*a^(1/2)*(-(a*x+1)*x)^(1/2)) /(a*x-1)/(-(a*x+1)*x)^(1/2)
Time = 0.28 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.72 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\left [\frac {1155 \, {\left (a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (105 \, a^{4} c^{4} x^{4} - 4156 \, a^{3} c^{4} x^{3} + 1028 \, a^{2} c^{4} x^{2} - 246 \, a c^{4} x + 30 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{420 \, {\left (a^{5} x^{4} - a^{4} x^{3}\right )}}, -\frac {1155 \, {\left (a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (105 \, a^{4} c^{4} x^{4} - 4156 \, a^{3} c^{4} x^{3} + 1028 \, a^{2} c^{4} x^{2} - 246 \, a c^{4} x + 30 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{210 \, {\left (a^{5} x^{4} - a^{4} x^{3}\right )}}\right ] \]
[1/420*(1155*(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^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a* x)) - c)/(a*x - 1)) + 4*(105*a^4*c^4*x^4 - 4156*a^3*c^4*x^3 + 1028*a^2*c^4 *x^2 - 246*a*c^4*x + 30*c^4)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/( a^5*x^4 - a^4*x^3), -1/210*(1155*(a^4*c^4*x^4 - a^3*c^4*x^3)*sqrt(c)*arcta n(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(105*a^4*c^4*x^4 - 4156*a^3*c^4*x^3 + 1028*a^2*c^4*x^2 - 2 46*a*c^4*x + 30*c^4)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^5*x^4 - a^4*x^3)]
\[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\int \frac {\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {9}{2}} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{a x + 1}\, dx \]
\[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} {\left (c - \frac {c}{a x}\right )}^{\frac {9}{2}}}{a x + 1} \,d x } \]
Exception generated. \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\int \frac {{\left (c-\frac {c}{a\,x}\right )}^{9/2}\,\sqrt {1-a^2\,x^2}}{a\,x+1} \,d x \]