Integrand size = 21, antiderivative size = 104 \[ \int e^{-\coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=-\frac {8 c \sqrt {1-\frac {1}{a^2 x^2}} x}{5 a \sqrt {c-a c x}}-\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x \sqrt {c-a c x}}{5 a}-\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x (c-a c x)^{3/2}}{5 a c} \] Output:
-8/5*c*(1-1/a^2/x^2)^(1/2)*x/a/(-a*c*x+c)^(1/2)-2/5*(1-1/a^2/x^2)^(1/2)*x* (-a*c*x+c)^(1/2)/a-2/5*(1-1/a^2/x^2)^(1/2)*x*(-a*c*x+c)^(3/2)/a/c
Time = 0.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.56 \[ \int e^{-\coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {2 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x} \left (6-3 a x+a^2 x^2\right )}{5 a^2 \sqrt {1-\frac {1}{a x}}} \] Input:
Integrate[(x*Sqrt[c - a*c*x])/E^ArcCoth[a*x],x]
Output:
(2*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x]*(6 - 3*a*x + a^2*x^2))/(5*a^2*Sqrt[1 - 1/(a*x)])
Time = 0.48 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6730, 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.
\(\displaystyle \int x \sqrt {c-a c x} e^{-\coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6730 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \int \frac {a-\frac {1}{x}}{a \sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}}{\sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \int \frac {a-\frac {1}{x}}{\sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}}{a \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (-\frac {9}{5} \int \frac {1}{\sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{5/2}}d\frac {1}{x}-\frac {2 a \sqrt {\frac {1}{a x}+1}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )}{a \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (-\frac {9}{5} \left (-\frac {2 \int \frac {1}{\sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}}{3 a}-\frac {2 \sqrt {\frac {1}{a x}+1}}{3 \left (\frac {1}{x}\right )^{3/2}}\right )-\frac {2 a \sqrt {\frac {1}{a x}+1}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )}{a \sqrt {1-\frac {1}{a x}}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {\sqrt {\frac {1}{x}} \left (-\frac {2 a \sqrt {\frac {1}{a x}+1}}{5 \left (\frac {1}{x}\right )^{5/2}}-\frac {9}{5} \left (\frac {4 \sqrt {\frac {1}{a x}+1}}{3 a \sqrt {\frac {1}{x}}}-\frac {2 \sqrt {\frac {1}{a x}+1}}{3 \left (\frac {1}{x}\right )^{3/2}}\right )\right ) \sqrt {c-a c x}}{a \sqrt {1-\frac {1}{a x}}}\) |
Input:
Int[(x*Sqrt[c - a*c*x])/E^ArcCoth[a*x],x]
Output:
-((((-9*((-2*Sqrt[1 + 1/(a*x)])/(3*(x^(-1))^(3/2)) + (4*Sqrt[1 + 1/(a*x)]) /(3*a*Sqrt[x^(-1)])))/5 - (2*a*Sqrt[1 + 1/(a*x)])/(5*(x^(-1))^(5/2)))*Sqrt [x^(-1)]*Sqrt[c - a*c*x])/(a*Sqrt[1 - 1/(a*x)]))
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[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 = 0.15 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.48
method | result | size |
risch | \(-\frac {2 c \sqrt {\frac {a x -1}{a x +1}}\, \left (a^{2} x^{2}-3 a x +6\right ) \left (a x +1\right )}{5 \sqrt {-c \left (a x -1\right )}\, a^{2}}\) | \(50\) |
gosper | \(\frac {2 \left (a x +1\right ) \left (a^{2} x^{2}-3 a x +6\right ) \sqrt {-a c x +c}\, \sqrt {\frac {a x -1}{a x +1}}}{5 a^{2} \left (a x -1\right )}\) | \(55\) |
orering | \(\frac {2 \left (a x +1\right ) \left (a^{2} x^{2}-3 a x +6\right ) \sqrt {-a c x +c}\, \sqrt {\frac {a x -1}{a x +1}}}{5 a^{2} \left (a x -1\right )}\) | \(55\) |
default | \(\frac {2 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (a^{2} x^{2}-3 a x +6\right )}{5 \left (a x -1\right ) a^{2}}\) | \(56\) |
Input:
int(x*(-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/5*c*((a*x-1)/(a*x+1))^(1/2)/(-c*(a*x-1))^(1/2)*(a^2*x^2-3*a*x+6)/a^2*(a *x+1)
Time = 0.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.58 \[ \int e^{-\coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {2 \, {\left (a^{3} x^{3} - 2 \, a^{2} x^{2} + 3 \, a x + 6\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{5 \, {\left (a^{3} x - a^{2}\right )}} \] Input:
integrate(x*(-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(1/2),x, algorithm="fricas" )
Output:
2/5*(a^3*x^3 - 2*a^2*x^2 + 3*a*x + 6)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1))/(a^3*x - a^2)
\[ \int e^{-\coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\int x \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {- c \left (a x - 1\right )}\, dx \] Input:
integrate(x*(-a*c*x+c)**(1/2)*((a*x-1)/(a*x+1))**(1/2),x)
Output:
Integral(x*sqrt((a*x - 1)/(a*x + 1))*sqrt(-c*(a*x - 1)), x)
Time = 0.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.66 \[ \int e^{-\coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {2 \, {\left (a^{3} \sqrt {-c} x^{3} - 2 \, a^{2} \sqrt {-c} x^{2} + 3 \, a \sqrt {-c} x + 6 \, \sqrt {-c}\right )} {\left (a x - 1\right )}}{5 \, {\left (a^{3} x - a^{2}\right )} \sqrt {a x + 1}} \] Input:
integrate(x*(-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(1/2),x, algorithm="maxima" )
Output:
2/5*(a^3*sqrt(-c)*x^3 - 2*a^2*sqrt(-c)*x^2 + 3*a*sqrt(-c)*x + 6*sqrt(-c))* (a*x - 1)/((a^3*x - a^2)*sqrt(a*x + 1))
Time = 0.15 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.66 \[ \int e^{-\coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=-\frac {4 \, \sqrt {-a c x - c} {\left | c \right |}}{a^{2} c} - \frac {2 \, {\left ({\left (a c x + c\right )}^{2} \sqrt {-a c x - c} {\left | c \right |} + 5 \, {\left (-a c x - c\right )}^{\frac {3}{2}} c {\left | c \right |}\right )}}{5 \, a^{2} c^{3}} \] Input:
integrate(x*(-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(1/2),x, algorithm="giac")
Output:
-4*sqrt(-a*c*x - c)*abs(c)/(a^2*c) - 2/5*((a*c*x + c)^2*sqrt(-a*c*x - c)*a bs(c) + 5*(-a*c*x - c)^(3/2)*c*abs(c))/(a^2*c^3)
Time = 13.68 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.55 \[ \int e^{-\coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {2\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (a^3\,x^3-2\,a^2\,x^2+3\,a\,x+6\right )}{5\,a^2\,\left (a\,x-1\right )} \] Input:
int(x*(c - a*c*x)^(1/2)*((a*x - 1)/(a*x + 1))^(1/2),x)
Output:
(2*(c - a*c*x)^(1/2)*((a*x - 1)/(a*x + 1))^(1/2)*(3*a*x - 2*a^2*x^2 + a^3* x^3 + 6))/(5*a^2*(a*x - 1))
Time = 0.15 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.27 \[ \int e^{-\coth ^{-1}(a x)} x \sqrt {c-a c x} \, dx=\frac {2 \sqrt {c}\, \sqrt {a x +1}\, i \left (-a^{2} x^{2}+3 a x -6\right )}{5 a^{2}} \] Input:
int(x*(-a*c*x+c)^(1/2)*((a*x-1)/(a*x+1))^(1/2),x)
Output:
(2*sqrt(c)*sqrt(a*x + 1)*i*( - a**2*x**2 + 3*a*x - 6))/(5*a**2)