Integrand size = 20, antiderivative size = 311 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {40 \left (a-\frac {1}{x}\right )^4 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}+\frac {11776 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x^4}+\frac {5120 (c-a c x)^{7/2}}{63 a^4 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x^3}-\frac {512 (c-a c x)^{7/2}}{63 a^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x^2}+\frac {128 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x}+\frac {2 \left (a-\frac {1}{x}\right )^5 x (c-a c x)^{7/2}}{9 a^5 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}} \]
-40/63*(a-1/x)^4*(-a*c*x+c)^(7/2)/a^5/(1-1/a/x)^(7/2)/(1+1/a/x)^(1/2)+1177 6/63*(-a*c*x+c)^(7/2)/a^5/(1-1/a/x)^(7/2)/x^4/(1+1/a/x)^(1/2)+5120/63*(-a* c*x+c)^(7/2)/a^4/(1-1/a/x)^(7/2)/x^3/(1+1/a/x)^(1/2)-512/63*(-a*c*x+c)^(7/ 2)/a^3/(1-1/a/x)^(7/2)/x^2/(1+1/a/x)^(1/2)+128/63*(a-1/x)^3*(-a*c*x+c)^(7/ 2)/a^5/(1-1/a/x)^(7/2)/x/(1+1/a/x)^(1/2)+2/9*(a-1/x)^5*x*(-a*c*x+c)^(7/2)/ a^5/(1-1/a/x)^(7/2)/(1+1/a/x)^(1/2)
Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.24 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2 c^3 \sqrt {c-a c x} \left (5797+2867 a x-638 a^2 x^2+214 a^3 x^3-55 a^4 x^4+7 a^5 x^5\right )}{63 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \]
(-2*c^3*Sqrt[c - a*c*x]*(5797 + 2867*a*x - 638*a^2*x^2 + 214*a^3*x^3 - 55* a^4*x^4 + 7*a^5*x^5))/(63*a^2*Sqrt[1 - 1/(a^2*x^2)]*x)
Time = 0.33 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.74, 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.
| \(\displaystyle \int (c-a c x)^{7/2} e^{-3 \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6727 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2} \int \frac {\left (a-\frac {1}{x}\right )^5}{a^5 \left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{11/2}}d\frac {1}{x}}{\left (1-\frac {1}{a x}\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2} \int \frac {\left (a-\frac {1}{x}\right )^5}{\left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{11/2}}d\frac {1}{x}}{a^5 \left (1-\frac {1}{a x}\right )^{7/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2} \left (-\frac {20}{9} \int \frac {\left (a-\frac {1}{x}\right )^4}{\left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{9/2}}d\frac {1}{x}-\frac {2 \left (a-\frac {1}{x}\right )^5}{9 \left (\frac {1}{x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^5 \left (1-\frac {1}{a x}\right )^{7/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2} \left (-\frac {20}{9} \left (-\frac {16}{7} \int \frac {\left (a-\frac {1}{x}\right )^3}{\left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}-\frac {2 \left (a-\frac {1}{x}\right )^4}{7 \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^5}{9 \left (\frac {1}{x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^5 \left (1-\frac {1}{a x}\right )^{7/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2} \left (-\frac {20}{9} \left (-\frac {16}{7} \left (-\frac {12}{5} \int \frac {\left (a-\frac {1}{x}\right )^2}{\left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{5/2}}d\frac {1}{x}-\frac {2 \left (a-\frac {1}{x}\right )^3}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^4}{7 \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^5}{9 \left (\frac {1}{x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^5 \left (1-\frac {1}{a x}\right )^{7/2}}\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2} \left (-\frac {20}{9} \left (-\frac {16}{7} \left (-\frac {12}{5} \left (\frac {2}{3} \int -\frac {10 a-\frac {3}{x}}{2 \left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}-\frac {2 a^2}{3 \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^3}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^4}{7 \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^5}{9 \left (\frac {1}{x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^5 \left (1-\frac {1}{a x}\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2} \left (-\frac {20}{9} \left (-\frac {16}{7} \left (-\frac {12}{5} \left (-\frac {1}{3} \int \frac {10 a-\frac {3}{x}}{\left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}-\frac {2 a^2}{3 \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^3}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^4}{7 \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^5}{9 \left (\frac {1}{x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^5 \left (1-\frac {1}{a x}\right )^{7/2}}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2} \left (-\frac {20}{9} \left (-\frac {16}{7} \left (-\frac {12}{5} \left (\frac {1}{3} \left (23 \int \frac {1}{\left (1+\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{x}}}d\frac {1}{x}+\frac {20 a}{\sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 a^2}{3 \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^3}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^4}{7 \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^5}{9 \left (\frac {1}{x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^5 \left (1-\frac {1}{a x}\right )^{7/2}}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle -\frac {\left (\frac {1}{x}\right )^{7/2} \left (-\frac {20}{9} \left (-\frac {16}{7} \left (-\frac {12}{5} \left (\frac {1}{3} \left (\frac {20 a}{\sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}}+\frac {46 \sqrt {\frac {1}{x}}}{\sqrt {\frac {1}{a x}+1}}\right )-\frac {2 a^2}{3 \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^3}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^4}{7 \left (\frac {1}{x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 \left (a-\frac {1}{x}\right )^5}{9 \left (\frac {1}{x}\right )^{9/2} \sqrt {\frac {1}{a x}+1}}\right ) (c-a c x)^{7/2}}{a^5 \left (1-\frac {1}{a x}\right )^{7/2}}\) |
-((((-20*((-16*((-12*(((20*a)/(Sqrt[1 + 1/(a*x)]*Sqrt[x^(-1)]) + (46*Sqrt[ x^(-1)])/Sqrt[1 + 1/(a*x)])/3 - (2*a^2)/(3*Sqrt[1 + 1/(a*x)]*(x^(-1))^(3/2 ))))/5 - (2*(a - x^(-1))^3)/(5*Sqrt[1 + 1/(a*x)]*(x^(-1))^(5/2))))/7 - (2* (a - x^(-1))^4)/(7*Sqrt[1 + 1/(a*x)]*(x^(-1))^(7/2))))/9 - (2*(a - x^(-1)) ^5)/(9*Sqrt[1 + 1/(a*x)]*(x^(-1))^(9/2)))*(x^(-1))^(7/2)*(c - a*c*x)^(7/2) )/(a^5*(1 - 1/(a*x))^(7/2)))
3.3.71.3.1 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 = 0.45 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.26
| method | result | size |
| gosper | \(\frac {2 \left (a x +1\right ) \left (7 a^{5} x^{5}-55 a^{4} x^{4}+214 a^{3} x^{3}-638 a^{2} x^{2}+2867 a x +5797\right ) \left (-a c x +c \right )^{\frac {7}{2}} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{63 a \left (a x -1\right )^{5}}\) | \(80\) |
| default | \(-\frac {2 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, c^{3} \left (7 a^{5} x^{5}-55 a^{4} x^{4}+214 a^{3} x^{3}-638 a^{2} x^{2}+2867 a x +5797\right )}{63 \left (a x -1\right )^{2} a}\) | \(84\) |
| risch | \(\frac {2 \left (7 a^{4} x^{4}-62 a^{3} x^{3}+276 a^{2} x^{2}-914 a x +3781\right ) \left (a x +1\right ) c^{4} \sqrt {\frac {a x -1}{a x +1}}}{63 a \sqrt {-c \left (a x -1\right )}}+\frac {64 c^{4} \sqrt {\frac {a x -1}{a x +1}}}{a \sqrt {-c \left (a x -1\right )}}\) | \(103\) |
2/63*(a*x+1)*(7*a^5*x^5-55*a^4*x^4+214*a^3*x^3-638*a^2*x^2+2867*a*x+5797)* (-a*c*x+c)^(7/2)*((a*x-1)/(a*x+1))^(3/2)/a/(a*x-1)^5
Time = 0.25 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.30 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2 \, {\left (7 \, a^{5} c^{3} x^{5} - 55 \, a^{4} c^{3} x^{4} + 214 \, a^{3} c^{3} x^{3} - 638 \, a^{2} c^{3} x^{2} + 2867 \, a c^{3} x + 5797 \, c^{3}\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{63 \, {\left (a^{2} x - a\right )}} \]
-2/63*(7*a^5*c^3*x^5 - 55*a^4*c^3*x^4 + 214*a^3*c^3*x^3 - 638*a^2*c^3*x^2 + 2867*a*c^3*x + 5797*c^3)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1))/(a^2 *x - a)
Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=\text {Timed out} \]
Time = 0.22 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.44 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2 \, {\left (7 \, a^{6} \sqrt {-c} c^{3} x^{6} - 48 \, a^{5} \sqrt {-c} c^{3} x^{5} + 159 \, a^{4} \sqrt {-c} c^{3} x^{4} - 424 \, a^{3} \sqrt {-c} c^{3} x^{3} + 2229 \, a^{2} \sqrt {-c} c^{3} x^{2} + 8664 \, a \sqrt {-c} c^{3} x + 5797 \, \sqrt {-c} c^{3}\right )} {\left (a x - 1\right )}^{2}}{63 \, {\left (a^{3} x^{2} - 2 \, a^{2} x + a\right )} {\left (a x + 1\right )}^{\frac {3}{2}}} \]
-2/63*(7*a^6*sqrt(-c)*c^3*x^6 - 48*a^5*sqrt(-c)*c^3*x^5 + 159*a^4*sqrt(-c) *c^3*x^4 - 424*a^3*sqrt(-c)*c^3*x^3 + 2229*a^2*sqrt(-c)*c^3*x^2 + 8664*a*s qrt(-c)*c^3*x + 5797*sqrt(-c)*c^3)*(a*x - 1)^2/((a^3*x^2 - 2*a^2*x + a)*(a *x + 1)^(3/2))
Exception generated. \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=\text {Exception raised: TypeError} \]
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 = 4.85 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.33 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2\,c^3\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (7\,a^4\,x^4-48\,a^3\,x^3+166\,a^2\,x^2-472\,a\,x+2395\right )}{63\,a}-\frac {16384\,c^3\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{63\,a\,\left (a\,x-1\right )} \]