Integrand size = 18, antiderivative size = 197 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {8 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{7/2}}{21 a \left (1-\frac {1}{a x}\right )^{7/2}}-\frac {568 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{7/2}}{315 a^3 \left (1-\frac {1}{a x}\right )^{7/2} x^2}+\frac {48 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{7/2}}{35 a^2 \left (1-\frac {1}{a x}\right )^{7/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{7/2}}{9 a^3 \left (1-\frac {1}{a x}\right )^{7/2}} \]
-8/21*(1+1/a/x)^(3/2)*(-a*c*x+c)^(7/2)/a/(1-1/a/x)^(7/2)-568/315*(1+1/a/x) ^(3/2)*(-a*c*x+c)^(7/2)/a^3/(1-1/a/x)^(7/2)/x^2+48/35*(1+1/a/x)^(3/2)*(-a* c*x+c)^(7/2)/a^2/(1-1/a/x)^(7/2)/x+2/9*(a-1/x)^3*(1+1/a/x)^(3/2)*x*(-a*c*x +c)^(7/2)/a^3/(1-1/a/x)^(7/2)
Time = 0.06 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.40 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2 c^3 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x} \left (-319+2 a x+156 a^2 x^2-130 a^3 x^3+35 a^4 x^4\right )}{315 a \sqrt {1-\frac {1}{a x}}} \]
(-2*c^3*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x]*(-319 + 2*a*x + 156*a^2*x^2 - 13 0*a^3*x^3 + 35*a^4*x^4))/(315*a*Sqrt[1 - 1/(a*x)])
Time = 0.30 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.80, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {6727, 27, 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.
\(\displaystyle \int (c-a c x)^{7/2} e^{\coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6727 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2} \int \frac {\left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}}}{a^3 \left (\frac {1}{x}\right )^{11/2}}d\frac {1}{x}}{\left (1-\frac {1}{a x}\right )^{7/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2} \int \frac {\left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}}}{\left (\frac {1}{x}\right )^{11/2}}d\frac {1}{x}}{a^3 \left (1-\frac {1}{a x}\right )^{7/2}}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2} \left (-\frac {4}{3} \int \frac {\left (a-\frac {1}{x}\right )^2 \sqrt {1+\frac {1}{a x}}}{\left (\frac {1}{x}\right )^{9/2}}d\frac {1}{x}-\frac {2 \left (\frac {1}{a x}+1\right )^{3/2} \left (a-\frac {1}{x}\right )^3}{9 \left (\frac {1}{x}\right )^{9/2}}\right )}{a^3 \left (1-\frac {1}{a x}\right )^{7/2}}\) |
\(\Big \downarrow \) 100 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2} \left (-\frac {4}{3} \left (\frac {2}{7} \int -\frac {\left (18 a-\frac {7}{x}\right ) \sqrt {1+\frac {1}{a x}}}{2 \left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{3/2}}{7 \left (\frac {1}{x}\right )^{7/2}}\right )-\frac {2 \left (\frac {1}{a x}+1\right )^{3/2} \left (a-\frac {1}{x}\right )^3}{9 \left (\frac {1}{x}\right )^{9/2}}\right )}{a^3 \left (1-\frac {1}{a x}\right )^{7/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2} \left (-\frac {4}{3} \left (-\frac {1}{7} \int \frac {\left (18 a-\frac {7}{x}\right ) \sqrt {1+\frac {1}{a x}}}{\left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{3/2}}{7 \left (\frac {1}{x}\right )^{7/2}}\right )-\frac {2 \left (\frac {1}{a x}+1\right )^{3/2} \left (a-\frac {1}{x}\right )^3}{9 \left (\frac {1}{x}\right )^{9/2}}\right )}{a^3 \left (1-\frac {1}{a x}\right )^{7/2}}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2} \left (-\frac {4}{3} \left (\frac {1}{7} \left (\frac {71}{5} \int \frac {\sqrt {1+\frac {1}{a x}}}{\left (\frac {1}{x}\right )^{5/2}}d\frac {1}{x}+\frac {36 a \left (\frac {1}{a x}+1\right )^{3/2}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{3/2}}{7 \left (\frac {1}{x}\right )^{7/2}}\right )-\frac {2 \left (\frac {1}{a x}+1\right )^{3/2} \left (a-\frac {1}{x}\right )^3}{9 \left (\frac {1}{x}\right )^{9/2}}\right )}{a^3 \left (1-\frac {1}{a x}\right )^{7/2}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{7/2} \left (-\frac {4}{3} \left (\frac {1}{7} \left (\frac {36 a \left (\frac {1}{a x}+1\right )^{3/2}}{5 \left (\frac {1}{x}\right )^{5/2}}-\frac {142 \left (\frac {1}{a x}+1\right )^{3/2}}{15 \left (\frac {1}{x}\right )^{3/2}}\right )-\frac {2 a^2 \left (\frac {1}{a x}+1\right )^{3/2}}{7 \left (\frac {1}{x}\right )^{7/2}}\right )-\frac {2 \left (\frac {1}{a x}+1\right )^{3/2} \left (a-\frac {1}{x}\right )^3}{9 \left (\frac {1}{x}\right )^{9/2}}\right ) (c-a c x)^{7/2}}{a^3 \left (1-\frac {1}{a x}\right )^{7/2}}\) |
-((((-4*(((36*a*(1 + 1/(a*x))^(3/2))/(5*(x^(-1))^(5/2)) - (142*(1 + 1/(a*x ))^(3/2))/(15*(x^(-1))^(3/2)))/7 - (2*a^2*(1 + 1/(a*x))^(3/2))/(7*(x^(-1)) ^(7/2))))/3 - (2*(a - x^(-1))^3*(1 + 1/(a*x))^(3/2))/(9*(x^(-1))^(9/2)))*( x^(-1))^(7/2)*(c - a*c*x)^(7/2))/(a^3*(1 - 1/(a*x))^(7/2)))
3.3.27.3.1 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 = 61, normalized size of antiderivative = 0.31
method | result | size |
default | \(-\frac {2 \sqrt {-c \left (a x -1\right )}\, c^{3} \left (a x +1\right ) \left (35 a^{3} x^{3}-165 a^{2} x^{2}+321 a x -319\right )}{315 \sqrt {\frac {a x -1}{a x +1}}\, a}\) | \(61\) |
gosper | \(\frac {2 \left (a x +1\right ) \left (35 a^{3} x^{3}-165 a^{2} x^{2}+321 a x -319\right ) \left (-a c x +c \right )^{\frac {7}{2}}}{315 a \left (a x -1\right )^{3} \sqrt {\frac {a x -1}{a x +1}}}\) | \(64\) |
risch | \(\frac {2 c^{4} \left (a x -1\right ) \left (35 a^{4} x^{4}-130 a^{3} x^{3}+156 a^{2} x^{2}+2 a x -319\right )}{315 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}\, a}\) | \(69\) |
-2/315/((a*x-1)/(a*x+1))^(1/2)*(-c*(a*x-1))^(1/2)*c^3*(a*x+1)*(35*a^3*x^3- 165*a^2*x^2+321*a*x-319)/a
Time = 0.24 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.48 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2 \, {\left (35 \, a^{5} c^{3} x^{5} - 95 \, a^{4} c^{3} x^{4} + 26 \, a^{3} c^{3} x^{3} + 158 \, a^{2} c^{3} x^{2} - 317 \, a c^{3} x - 319 \, c^{3}\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{315 \, {\left (a^{2} x - a\right )}} \]
-2/315*(35*a^5*c^3*x^5 - 95*a^4*c^3*x^4 + 26*a^3*c^3*x^3 + 158*a^2*c^3*x^2 - 317*a*c^3*x - 319*c^3)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1))/(a^2* x - a)
Timed out. \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=\text {Timed out} \]
Time = 0.22 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.42 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2 \, {\left (35 \, a^{4} \sqrt {-c} c^{3} x^{4} - 130 \, a^{3} \sqrt {-c} c^{3} x^{3} + 156 \, a^{2} \sqrt {-c} c^{3} x^{2} + 2 \, a \sqrt {-c} c^{3} x - 319 \, \sqrt {-c} c^{3}\right )} \sqrt {a x + 1}}{315 \, a} \]
-2/315*(35*a^4*sqrt(-c)*c^3*x^4 - 130*a^3*sqrt(-c)*c^3*x^3 + 156*a^2*sqrt( -c)*c^3*x^2 + 2*a*sqrt(-c)*c^3*x - 319*sqrt(-c)*c^3)*sqrt(a*x + 1)/a
Exception generated. \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=\text {Exception raised: TypeError} \]
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 = 4.45 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.52 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=\frac {2\,c^3\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (-35\,a^4\,x^4+60\,a^3\,x^3+34\,a^2\,x^2-124\,a\,x+193\right )}{315\,a}+\frac {1024\,c^3\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{315\,a\,\left (a\,x-1\right )} \]