Integrand size = 25, antiderivative size = 99 \[ \int e^{-2 \coth ^{-1}(a x)} x \sqrt {c-a^2 c x^2} \, dx=\frac {5 \sqrt {c-a^2 c x^2}}{3 a^2}-\frac {x \sqrt {c-a^2 c x^2}}{a}+\frac {1}{3} x^2 \sqrt {c-a^2 c x^2}+\frac {\sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a^2} \] Output:
5/3*(-a^2*c*x^2+c)^(1/2)/a^2-x*(-a^2*c*x^2+c)^(1/2)/a+1/3*x^2*(-a^2*c*x^2+ c)^(1/2)+c^(1/2)*arctan(a*c^(1/2)*x/(-a^2*c*x^2+c)^(1/2))/a^2
Time = 0.14 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.80 \[ \int e^{-2 \coth ^{-1}(a x)} x \sqrt {c-a^2 c x^2} \, dx=\frac {\left (5-3 a x+a^2 x^2\right ) \sqrt {c-a^2 c x^2}-3 \sqrt {c} \arctan \left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (-1+a^2 x^2\right )}\right )}{3 a^2} \] Input:
Integrate[(x*Sqrt[c - a^2*c*x^2])/E^(2*ArcCoth[a*x]),x]
Output:
((5 - 3*a*x + a^2*x^2)*Sqrt[c - a^2*c*x^2] - 3*Sqrt[c]*ArcTan[(a*x*Sqrt[c - a^2*c*x^2])/(Sqrt[c]*(-1 + a^2*x^2))])/(3*a^2)
Time = 0.83 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.22, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6717, 6702, 541, 25, 27, 533, 27, 455, 224, 216}
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.
\(\displaystyle \int x \sqrt {c-a^2 c x^2} e^{-2 \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6717 |
\(\displaystyle -\int e^{-2 \text {arctanh}(a x)} x \sqrt {c-a^2 c x^2}dx\) |
\(\Big \downarrow \) 6702 |
\(\displaystyle -c \int \frac {x (1-a x)^2}{\sqrt {c-a^2 c x^2}}dx\) |
\(\Big \downarrow \) 541 |
\(\displaystyle -c \left (-\frac {\int -\frac {a^2 c x (5-6 a x)}{\sqrt {c-a^2 c x^2}}dx}{3 a^2 c}-\frac {x^2 \sqrt {c-a^2 c x^2}}{3 c}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -c \left (\frac {\int \frac {a^2 c x (5-6 a x)}{\sqrt {c-a^2 c x^2}}dx}{3 a^2 c}-\frac {x^2 \sqrt {c-a^2 c x^2}}{3 c}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -c \left (\frac {1}{3} \int \frac {x (5-6 a x)}{\sqrt {c-a^2 c x^2}}dx-\frac {x^2 \sqrt {c-a^2 c x^2}}{3 c}\right )\) |
\(\Big \downarrow \) 533 |
\(\displaystyle -c \left (\frac {1}{3} \left (\frac {\int -\frac {2 a c (3-5 a x)}{\sqrt {c-a^2 c x^2}}dx}{2 a^2 c}+\frac {3 x \sqrt {c-a^2 c x^2}}{a c}\right )-\frac {x^2 \sqrt {c-a^2 c x^2}}{3 c}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -c \left (\frac {1}{3} \left (\frac {3 x \sqrt {c-a^2 c x^2}}{a c}-\frac {\int \frac {3-5 a x}{\sqrt {c-a^2 c x^2}}dx}{a}\right )-\frac {x^2 \sqrt {c-a^2 c x^2}}{3 c}\right )\) |
\(\Big \downarrow \) 455 |
\(\displaystyle -c \left (\frac {1}{3} \left (\frac {3 x \sqrt {c-a^2 c x^2}}{a c}-\frac {3 \int \frac {1}{\sqrt {c-a^2 c x^2}}dx+\frac {5 \sqrt {c-a^2 c x^2}}{a c}}{a}\right )-\frac {x^2 \sqrt {c-a^2 c x^2}}{3 c}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -c \left (\frac {1}{3} \left (\frac {3 x \sqrt {c-a^2 c x^2}}{a c}-\frac {3 \int \frac {1}{\frac {a^2 c x^2}{c-a^2 c x^2}+1}d\frac {x}{\sqrt {c-a^2 c x^2}}+\frac {5 \sqrt {c-a^2 c x^2}}{a c}}{a}\right )-\frac {x^2 \sqrt {c-a^2 c x^2}}{3 c}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -c \left (\frac {1}{3} \left (\frac {3 x \sqrt {c-a^2 c x^2}}{a c}-\frac {\frac {3 \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a \sqrt {c}}+\frac {5 \sqrt {c-a^2 c x^2}}{a c}}{a}\right )-\frac {x^2 \sqrt {c-a^2 c x^2}}{3 c}\right )\) |
Input:
Int[(x*Sqrt[c - a^2*c*x^2])/E^(2*ArcCoth[a*x]),x]
Output:
-(c*(-1/3*(x^2*Sqrt[c - a^2*c*x^2])/c + ((3*x*Sqrt[c - a^2*c*x^2])/(a*c) - ((5*Sqrt[c - a^2*c*x^2])/(a*c) + (3*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x ^2]])/(a*Sqrt[c]))/a)/3))
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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*x^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[1/(b*(m + 2* p + 2)) Int[x^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[m, 0] && GtQ[p, -1] && Integer Q[2*p]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[d^n*x^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*(m + n + 2*p + 1))), x ] + Simp[1/(b*(m + n + 2*p + 1)) Int[x^m*(a + b*x^2)^p*ExpandToSum[b*(m + n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1)*x^n - a*d^n*(m + n - 1) *x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, m, p}, x] && IGtQ[n, 1] && IGt Q[m, -2] && GtQ[p, -1] && IntegerQ[2*p]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[1/c^(n/2) Int[x^m*((c + d*x^2)^(p + n/2)/(1 - a*x)^n), x] , x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ[c, 0]) && ILtQ[n/2, 0]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2) Int[ u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
Time = 0.19 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.80
method | result | size |
risch | \(-\frac {\left (a^{2} x^{2}-3 a x +5\right ) \left (a^{2} x^{2}-1\right ) c}{3 a^{2} \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\frac {\arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right ) c}{a \sqrt {a^{2} c}}\) | \(79\) |
default | \(-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3 a^{2} c}-\frac {2 \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{a}+\frac {2 \sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 \left (x +\frac {1}{a}\right ) a c}+\frac {2 a c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 \left (x +\frac {1}{a}\right ) a c}}\right )}{\sqrt {a^{2} c}}}{a^{2}}\) | \(154\) |
Input:
int(x*(-a^2*c*x^2+c)^(1/2)*(a*x-1)/(a*x+1),x,method=_RETURNVERBOSE)
Output:
-1/3*(a^2*x^2-3*a*x+5)*(a^2*x^2-1)/a^2/(-c*(a^2*x^2-1))^(1/2)*c+1/a/(a^2*c )^(1/2)*arctan((a^2*c)^(1/2)*x/(-a^2*c*x^2+c)^(1/2))*c
Time = 0.12 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.52 \[ \int e^{-2 \coth ^{-1}(a x)} x \sqrt {c-a^2 c x^2} \, dx=\left [\frac {2 \, \sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} - 3 \, a x + 5\right )} + 3 \, \sqrt {-c} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right )}{6 \, a^{2}}, \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} - 3 \, a x + 5\right )} - 3 \, \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right )}{3 \, a^{2}}\right ] \] Input:
integrate(x*(-a^2*c*x^2+c)^(1/2)*(a*x-1)/(a*x+1),x, algorithm="fricas")
Output:
[1/6*(2*sqrt(-a^2*c*x^2 + c)*(a^2*x^2 - 3*a*x + 5) + 3*sqrt(-c)*log(2*a^2* c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c))/a^2, 1/3*(sqrt(-a^2*c*x^ 2 + c)*(a^2*x^2 - 3*a*x + 5) - 3*sqrt(c)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqr t(c)*x/(a^2*c*x^2 - c)))/a^2]
\[ \int e^{-2 \coth ^{-1}(a x)} x \sqrt {c-a^2 c x^2} \, dx=\int \frac {x \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x - 1\right )}{a x + 1}\, dx \] Input:
integrate(x*(-a**2*c*x**2+c)**(1/2)*(a*x-1)/(a*x+1),x)
Output:
Integral(x*sqrt(-c*(a*x - 1)*(a*x + 1))*(a*x - 1)/(a*x + 1), x)
Time = 0.11 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.71 \[ \int e^{-2 \coth ^{-1}(a x)} x \sqrt {c-a^2 c x^2} \, dx=-\frac {\sqrt {-a^{2} c x^{2} + c} x}{a} + \frac {\sqrt {c} \arcsin \left (a x\right )}{a^{2}} + \frac {2 \, \sqrt {-a^{2} c x^{2} + c}}{a^{2}} - \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{3 \, a^{2} c} \] Input:
integrate(x*(-a^2*c*x^2+c)^(1/2)*(a*x-1)/(a*x+1),x, algorithm="maxima")
Output:
-sqrt(-a^2*c*x^2 + c)*x/a + sqrt(c)*arcsin(a*x)/a^2 + 2*sqrt(-a^2*c*x^2 + c)/a^2 - 1/3*(-a^2*c*x^2 + c)^(3/2)/(a^2*c)
Time = 0.16 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.74 \[ \int e^{-2 \coth ^{-1}(a x)} x \sqrt {c-a^2 c x^2} \, dx=\frac {1}{3} \, \sqrt {-a^{2} c x^{2} + c} {\left ({\left (x - \frac {3}{a}\right )} x + \frac {5}{a^{2}}\right )} - \frac {c \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{a \sqrt {-c} {\left | a \right |}} \] Input:
integrate(x*(-a^2*c*x^2+c)^(1/2)*(a*x-1)/(a*x+1),x, algorithm="giac")
Output:
1/3*sqrt(-a^2*c*x^2 + c)*((x - 3/a)*x + 5/a^2) - c*log(abs(-sqrt(-a^2*c)*x + sqrt(-a^2*c*x^2 + c)))/(a*sqrt(-c)*abs(a))
Timed out. \[ \int e^{-2 \coth ^{-1}(a x)} x \sqrt {c-a^2 c x^2} \, dx=\int \frac {x\,\sqrt {c-a^2\,c\,x^2}\,\left (a\,x-1\right )}{a\,x+1} \,d x \] Input:
int((x*(c - a^2*c*x^2)^(1/2)*(a*x - 1))/(a*x + 1),x)
Output:
int((x*(c - a^2*c*x^2)^(1/2)*(a*x - 1))/(a*x + 1), x)
Time = 0.16 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.62 \[ \int e^{-2 \coth ^{-1}(a x)} x \sqrt {c-a^2 c x^2} \, dx=\frac {\sqrt {c}\, \left (3 \mathit {asin} \left (a x \right )+\sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-3 \sqrt {-a^{2} x^{2}+1}\, a x +5 \sqrt {-a^{2} x^{2}+1}-5\right )}{3 a^{2}} \] Input:
int(x*(-a^2*c*x^2+c)^(1/2)*(a*x-1)/(a*x+1),x)
Output:
(sqrt(c)*(3*asin(a*x) + sqrt( - a**2*x**2 + 1)*a**2*x**2 - 3*sqrt( - a**2* x**2 + 1)*a*x + 5*sqrt( - a**2*x**2 + 1) - 5))/(3*a**2)