Integrand size = 20, antiderivative size = 194 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\frac {16384 c^5 \sqrt {1-\frac {1}{a^2 x^2}} x}{693 \sqrt {c-a c x}}+\frac {4096}{693} c^4 \sqrt {1-\frac {1}{a^2 x^2}} x \sqrt {c-a c x}+\frac {512}{231} c^3 \sqrt {1-\frac {1}{a^2 x^2}} x (c-a c x)^{3/2}+\frac {640}{693} c^2 \sqrt {1-\frac {1}{a^2 x^2}} x (c-a c x)^{5/2}+\frac {40}{99} c \sqrt {1-\frac {1}{a^2 x^2}} x (c-a c x)^{7/2}+\frac {2}{11} \sqrt {1-\frac {1}{a^2 x^2}} x (c-a c x)^{9/2} \]
512/231*c^3*x*(-a*c*x+c)^(3/2)*(1-1/a^2/x^2)^(1/2)+640/693*c^2*x*(-a*c*x+c )^(5/2)*(1-1/a^2/x^2)^(1/2)+40/99*c*x*(-a*c*x+c)^(7/2)*(1-1/a^2/x^2)^(1/2) +2/11*x*(-a*c*x+c)^(9/2)*(1-1/a^2/x^2)^(1/2)+16384/693*c^5*x*(1-1/a^2/x^2) ^(1/2)/(-a*c*x+c)^(1/2)+4096/693*c^4*x*(1-1/a^2/x^2)^(1/2)*(-a*c*x+c)^(1/2 )
Time = 0.07 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.44 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\frac {2 c^4 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x} \left (-11531+5419 a x-3198 a^2 x^2+1510 a^3 x^3-455 a^4 x^4+63 a^5 x^5\right )}{693 a \sqrt {1-\frac {1}{a x}}} \]
(2*c^4*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x]*(-11531 + 5419*a*x - 3198*a^2*x^2 + 1510*a^3*x^3 - 455*a^4*x^4 + 63*a^5*x^5))/(693*a*Sqrt[1 - 1/(a*x)])
Time = 0.32 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.20, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6727, 27, 105, 105, 105, 100, 27, 87, 48}
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.
| \(\displaystyle \int (c-a c x)^{9/2} e^{-\coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6727 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \int \frac {\left (a-\frac {1}{x}\right )^5}{a^5 \sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{13/2}}d\frac {1}{x}}{\left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \int \frac {\left (a-\frac {1}{x}\right )^5}{\sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{13/2}}d\frac {1}{x}}{a^5 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \left (-\frac {20}{11} \int \frac {\left (a-\frac {1}{x}\right )^4}{\sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{11/2}}d\frac {1}{x}-\frac {2 \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^5}{11 \left (\frac {1}{x}\right )^{11/2}}\right )}{a^5 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \left (-\frac {20}{11} \left (-\frac {16}{9} \int \frac {\left (a-\frac {1}{x}\right )^3}{\sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{9/2}}d\frac {1}{x}-\frac {2 \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^4}{9 \left (\frac {1}{x}\right )^{9/2}}\right )-\frac {2 \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^5}{11 \left (\frac {1}{x}\right )^{11/2}}\right )}{a^5 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \left (-\frac {20}{11} \left (-\frac {16}{9} \left (-\frac {12}{7} \int \frac {\left (a-\frac {1}{x}\right )^2}{\sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}-\frac {2 \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^3}{7 \left (\frac {1}{x}\right )^{7/2}}\right )-\frac {2 \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^4}{9 \left (\frac {1}{x}\right )^{9/2}}\right )-\frac {2 \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^5}{11 \left (\frac {1}{x}\right )^{11/2}}\right )}{a^5 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \left (-\frac {20}{11} \left (-\frac {16}{9} \left (-\frac {12}{7} \left (\frac {2}{5} \int -\frac {14 a-\frac {5}{x}}{2 \sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{5/2}}d\frac {1}{x}-\frac {2 a^2 \sqrt {\frac {1}{a x}+1}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )-\frac {2 \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^3}{7 \left (\frac {1}{x}\right )^{7/2}}\right )-\frac {2 \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^4}{9 \left (\frac {1}{x}\right )^{9/2}}\right )-\frac {2 \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^5}{11 \left (\frac {1}{x}\right )^{11/2}}\right )}{a^5 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \left (-\frac {20}{11} \left (-\frac {16}{9} \left (-\frac {12}{7} \left (-\frac {1}{5} \int \frac {14 a-\frac {5}{x}}{\sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{5/2}}d\frac {1}{x}-\frac {2 a^2 \sqrt {\frac {1}{a x}+1}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )-\frac {2 \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^3}{7 \left (\frac {1}{x}\right )^{7/2}}\right )-\frac {2 \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^4}{9 \left (\frac {1}{x}\right )^{9/2}}\right )-\frac {2 \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^5}{11 \left (\frac {1}{x}\right )^{11/2}}\right )}{a^5 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2} \left (-\frac {20}{11} \left (-\frac {16}{9} \left (-\frac {12}{7} \left (\frac {1}{5} \left (\frac {43}{3} \int \frac {1}{\sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}+\frac {28 a \sqrt {\frac {1}{a x}+1}}{3 \left (\frac {1}{x}\right )^{3/2}}\right )-\frac {2 a^2 \sqrt {\frac {1}{a x}+1}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )-\frac {2 \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^3}{7 \left (\frac {1}{x}\right )^{7/2}}\right )-\frac {2 \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^4}{9 \left (\frac {1}{x}\right )^{9/2}}\right )-\frac {2 \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^5}{11 \left (\frac {1}{x}\right )^{11/2}}\right )}{a^5 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{9/2} \left (-\frac {20}{11} \left (-\frac {16}{9} \left (-\frac {12}{7} \left (\frac {1}{5} \left (\frac {28 a \sqrt {\frac {1}{a x}+1}}{3 \left (\frac {1}{x}\right )^{3/2}}-\frac {86 \sqrt {\frac {1}{a x}+1}}{3 \sqrt {\frac {1}{x}}}\right )-\frac {2 a^2 \sqrt {\frac {1}{a x}+1}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )-\frac {2 \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^3}{7 \left (\frac {1}{x}\right )^{7/2}}\right )-\frac {2 \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^4}{9 \left (\frac {1}{x}\right )^{9/2}}\right )-\frac {2 \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^5}{11 \left (\frac {1}{x}\right )^{11/2}}\right ) (c-a c x)^{9/2}}{a^5 \left (1-\frac {1}{a x}\right )^{9/2}}\) |
-((((-20*((-16*((-12*(((28*a*Sqrt[1 + 1/(a*x)])/(3*(x^(-1))^(3/2)) - (86*S qrt[1 + 1/(a*x)])/(3*Sqrt[x^(-1)]))/5 - (2*a^2*Sqrt[1 + 1/(a*x)])/(5*(x^(- 1))^(5/2))))/7 - (2*(a - x^(-1))^3*Sqrt[1 + 1/(a*x)])/(7*(x^(-1))^(7/2)))) /9 - (2*(a - x^(-1))^4*Sqrt[1 + 1/(a*x)])/(9*(x^(-1))^(9/2))))/11 - (2*(a - x^(-1))^5*Sqrt[1 + 1/(a*x)])/(11*(x^(-1))^(11/2)))*(x^(-1))^(9/2)*(c - a *c*x)^(9/2))/(a^5*(1 - 1/(a*x))^(9/2)))
3.3.52.3.1 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] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
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[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_), x_Symbol] :> Si mp[(-(1/x)^p)*((c + d*x)^p/(1 + c/(d*x))^p) Subst[Int[((1 + c*(x/d))^p*(( 1 + x/a)^(n/2)/x^(p + 2)))/(1 - x/a)^(n/2), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && !IntegerQ[p]
Time = 0.42 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.40
| method | result | size |
| risch | \(-\frac {2 c^{5} \sqrt {\frac {a x -1}{a x +1}}\, \left (63 a^{5} x^{5}-455 a^{4} x^{4}+1510 a^{3} x^{3}-3198 a^{2} x^{2}+5419 a x -11531\right ) \left (a x +1\right )}{693 \sqrt {-c \left (a x -1\right )}\, a}\) | \(77\) |
| gosper | \(\frac {2 \left (a x +1\right ) \left (63 a^{5} x^{5}-455 a^{4} x^{4}+1510 a^{3} x^{3}-3198 a^{2} x^{2}+5419 a x -11531\right ) \left (-a c x +c \right )^{\frac {9}{2}} \sqrt {\frac {a x -1}{a x +1}}}{693 a \left (a x -1\right )^{5}}\) | \(80\) |
| default | \(\frac {2 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, c^{4} \left (63 a^{5} x^{5}-455 a^{4} x^{4}+1510 a^{3} x^{3}-3198 a^{2} x^{2}+5419 a x -11531\right )}{693 \left (a x -1\right ) a}\) | \(84\) |
-2/693*c^5*((a*x-1)/(a*x+1))^(1/2)/(-c*(a*x-1))^(1/2)*(63*a^5*x^5-455*a^4* x^4+1510*a^3*x^3-3198*a^2*x^2+5419*a*x-11531)/a*(a*x+1)
Time = 0.25 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.54 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\frac {2 \, {\left (63 \, a^{6} c^{4} x^{6} - 392 \, a^{5} c^{4} x^{5} + 1055 \, a^{4} c^{4} x^{4} - 1688 \, a^{3} c^{4} x^{3} + 2221 \, a^{2} c^{4} x^{2} - 6112 \, a c^{4} x - 11531 \, c^{4}\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{693 \, {\left (a^{2} x - a\right )}} \]
2/693*(63*a^6*c^4*x^6 - 392*a^5*c^4*x^5 + 1055*a^4*c^4*x^4 - 1688*a^3*c^4* x^3 + 2221*a^2*c^4*x^2 - 6112*a*c^4*x - 11531*c^4)*sqrt(-a*c*x + c)*sqrt(( a*x - 1)/(a*x + 1))/(a^2*x - a)
Timed out. \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\text {Timed out} \]
Time = 0.24 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.66 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\frac {2 \, {\left (63 \, a^{6} \sqrt {-c} c^{4} x^{6} - 392 \, a^{5} \sqrt {-c} c^{4} x^{5} + 1055 \, a^{4} \sqrt {-c} c^{4} x^{4} - 1688 \, a^{3} \sqrt {-c} c^{4} x^{3} + 2221 \, a^{2} \sqrt {-c} c^{4} x^{2} - 6112 \, a \sqrt {-c} c^{4} x - 11531 \, \sqrt {-c} c^{4}\right )} {\left (a x - 1\right )}}{693 \, {\left (a^{2} x - a\right )} \sqrt {a x + 1}} \]
2/693*(63*a^6*sqrt(-c)*c^4*x^6 - 392*a^5*sqrt(-c)*c^4*x^5 + 1055*a^4*sqrt( -c)*c^4*x^4 - 1688*a^3*sqrt(-c)*c^4*x^3 + 2221*a^2*sqrt(-c)*c^4*x^2 - 6112 *a*sqrt(-c)*c^4*x - 11531*sqrt(-c)*c^4)*(a*x - 1)/((a^2*x - a)*sqrt(a*x + 1))
Exception generated. \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\text {Exception raised: TypeError} \]
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
Time = 4.32 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.57 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\frac {2\,c^4\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (63\,a^5\,x^5-329\,a^4\,x^4+726\,a^3\,x^3-962\,a^2\,x^2+1259\,a\,x-4853\right )}{693\,a}-\frac {32768\,c^4\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{693\,a\,\left (a\,x-1\right )} \]