Integrand size = 21, antiderivative size = 182 \[ \int e^{-3 \coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=-\frac {158 \sqrt {c-a c x}}{15 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {316 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{15 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {32 x \sqrt {c-a c x}}{15 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {2 x^2 \sqrt {c-a c x}}{5 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}} \]
-158/15*(-a*c*x+c)^(1/2)/a^2/(1-1/a/x)^(1/2)/(1+1/a/x)^(1/2)-32/15*x*(-a*c *x+c)^(1/2)/a/(1-1/a/x)^(1/2)/(1+1/a/x)^(1/2)+2/5*x^2*(-a*c*x+c)^(1/2)/(1- 1/a/x)^(1/2)/(1+1/a/x)^(1/2)+316/15*(1+1/a/x)^(1/2)*(-a*c*x+c)^(1/2)/a^2/( 1-1/a/x)^(1/2)
Time = 0.04 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.31 \[ \int e^{-3 \coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {2 \sqrt {c-a c x} \left (158+79 a x-16 a^2 x^2+3 a^3 x^3\right )}{15 a^3 \sqrt {1-\frac {1}{a^2 x^2}} x} \]
Time = 0.31 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.79, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6730, 27, 100, 27, 87, 55, 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.
\(\displaystyle \int x \sqrt {c-a c x} e^{-3 \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6730 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \int \frac {\left (a-\frac {1}{x}\right )^2}{a^2 \left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}}{\sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \int \frac {\left (a-\frac {1}{x}\right )^2}{\left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 100 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {2}{5} \int -\frac {16 a-\frac {5}{x}}{2 \left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{5/2}}d\frac {1}{x}-\frac {2 a^2}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (-\frac {1}{5} \int \frac {16 a-\frac {5}{x}}{\left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{5/2}}d\frac {1}{x}-\frac {2 a^2}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {1}{5} \left (\frac {79}{3} \int \frac {1}{\left (1+\frac {1}{a x}\right )^{3/2} \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}+\frac {32 a}{3 \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 a^2}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {1}{5} \left (\frac {79}{3} \left (2 \int \frac {1}{\sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}+\frac {2}{\sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}}\right )+\frac {32 a}{3 \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}\right )-\frac {2 a^2}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \left (\frac {1}{5} \left (\frac {32 a}{3 \left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}+\frac {79}{3} \left (\frac {2}{\sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}}-\frac {4 \sqrt {\frac {1}{a x}+1}}{\sqrt {\frac {1}{x}}}\right )\right )-\frac {2 a^2}{5 \left (\frac {1}{x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}\right ) \sqrt {c-a c x}}{a^2 \sqrt {1-\frac {1}{a x}}}\) |
-(((((79*(2/(Sqrt[1 + 1/(a*x)]*Sqrt[x^(-1)]) - (4*Sqrt[1 + 1/(a*x)])/Sqrt[ x^(-1)]))/3 + (32*a)/(3*Sqrt[1 + 1/(a*x)]*(x^(-1))^(3/2)))/5 - (2*a^2)/(5* Sqrt[1 + 1/(a*x)]*(x^(-1))^(5/2)))*Sqrt[x^(-1)]*Sqrt[c - a*c*x])/(a^2*Sqrt [1 - 1/(a*x)]))
3.4.52.3.1 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_))^(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] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 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[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(p _), x_Symbol] :> Simp[(-(e*x)^m)*(1/x)^(m + p)*((c + d*x)^p/(1 + c/(d*x))^p ) Subst[Int[((1 + c*(x/d))^p*((1 + x/a)^(n/2)/x^(m + p + 2)))/(1 - x/a)^( n/2), x], x, 1/x], x] /; FreeQ[{a, c, d, e, m, n, p}, x] && EqQ[a^2*c^2 - d ^2, 0] && !IntegerQ[p]
Time = 0.47 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.35
method | result | size |
gosper | \(\frac {2 \left (a x +1\right ) \left (3 a^{3} x^{3}-16 a^{2} x^{2}+79 a x +158\right ) \sqrt {-a c x +c}\, \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{15 a^{2} \left (a x -1\right )^{2}}\) | \(64\) |
default | \(\frac {2 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (3 a^{3} x^{3}-16 a^{2} x^{2}+79 a x +158\right )}{15 \left (a x -1\right )^{2} a^{2}}\) | \(65\) |
risch | \(-\frac {2 \left (3 a^{2} x^{2}-19 a x +98\right ) \left (a x +1\right ) c \sqrt {\frac {a x -1}{a x +1}}}{15 a^{2} \sqrt {-c \left (a x -1\right )}}-\frac {8 c \sqrt {\frac {a x -1}{a x +1}}}{a^{2} \sqrt {-c \left (a x -1\right )}}\) | \(83\) |
2/15*(a*x+1)*(3*a^3*x^3-16*a^2*x^2+79*a*x+158)*(-a*c*x+c)^(1/2)*((a*x-1)/( a*x+1))^(3/2)/a^2/(a*x-1)^2
Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.34 \[ \int e^{-3 \coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {2 \, {\left (3 \, a^{3} x^{3} - 16 \, a^{2} x^{2} + 79 \, a x + 158\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{15 \, {\left (a^{3} x - a^{2}\right )}} \]
2/15*(3*a^3*x^3 - 16*a^2*x^2 + 79*a*x + 158)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1))/(a^3*x - a^2)
Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\text {Timed out} \]
Time = 0.24 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.50 \[ \int e^{-3 \coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {2 \, {\left (3 \, a^{4} \sqrt {-c} x^{4} - 13 \, a^{3} \sqrt {-c} x^{3} + 63 \, a^{2} \sqrt {-c} x^{2} + 237 \, a \sqrt {-c} x + 158 \, \sqrt {-c}\right )} {\left (a x - 1\right )}^{2}}{15 \, {\left (a^{4} x^{2} - 2 \, a^{3} x + a^{2}\right )} {\left (a x + 1\right )}^{\frac {3}{2}}} \]
2/15*(3*a^4*sqrt(-c)*x^4 - 13*a^3*sqrt(-c)*x^3 + 63*a^2*sqrt(-c)*x^2 + 237 *a*sqrt(-c)*x + 158*sqrt(-c))*(a*x - 1)^2/((a^4*x^2 - 2*a^3*x + a^2)*(a*x + 1)^(3/2))
Exception generated. \[ \int e^{-3 \coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\text {Exception raised: TypeError} \]
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 = 4.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.32 \[ \int e^{-3 \coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {2\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (3\,a^3\,x^3-16\,a^2\,x^2+79\,a\,x+158\right )}{15\,a^2\,\left (a\,x-1\right )} \]