Integrand size = 25, antiderivative size = 84 \[ \int e^{2 \text {arctanh}(a x)} x \sqrt {c-a^2 c x^2} \, dx=-\frac {1}{3} x^2 \sqrt {c-a^2 c x^2}-\frac {(5+3 a x) \sqrt {c-a^2 c x^2}}{3 a^2}+\frac {\sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a^2} \]
arctan(a*x*c^(1/2)/(-a^2*c*x^2+c)^(1/2))*c^(1/2)/a^2-1/3*x^2*(-a^2*c*x^2+c )^(1/2)-1/3*(3*a*x+5)*(-a^2*c*x^2+c)^(1/2)/a^2
Time = 0.10 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.94 \[ \int e^{2 \text {arctanh}(a x)} x \sqrt {c-a^2 c x^2} \, dx=-\frac {\left (5+3 a x+a^2 x^2\right ) \sqrt {c-a^2 c x^2}+3 \sqrt {c} \arctan \left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (-1+a^2 x^2\right )}\right )}{3 a^2} \]
-1/3*((5 + 3*a*x + a^2*x^2)*Sqrt[c - a^2*c*x^2] + 3*Sqrt[c]*ArcTan[(a*x*Sq rt[c - a^2*c*x^2])/(Sqrt[c]*(-1 + a^2*x^2))])/a^2
Time = 0.37 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.42, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {6701, 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.
\(\displaystyle \int x e^{2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2} \, dx\) |
\(\Big \downarrow \) 6701 |
\(\displaystyle c \int \frac {x (a x+1)^2}{\sqrt {c-a^2 c x^2}}dx\) |
\(\Big \downarrow \) 541 |
\(\displaystyle c \left (-\frac {\int -\frac {a^2 c x (6 a x+5)}{\sqrt {c-a^2 c x^2}}dx}{3 a^2 c}-\frac {x^2 \sqrt {c-a^2 c x^2}}{3 c}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle c \left (\frac {\int \frac {a^2 c x (6 a x+5)}{\sqrt {c-a^2 c x^2}}dx}{3 a^2 c}-\frac {x^2 \sqrt {c-a^2 c x^2}}{3 c}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle c \left (\frac {1}{3} \int \frac {x (6 a x+5)}{\sqrt {c-a^2 c x^2}}dx-\frac {x^2 \sqrt {c-a^2 c x^2}}{3 c}\right )\) |
\(\Big \downarrow \) 533 |
\(\displaystyle c \left (\frac {1}{3} \left (\frac {\int \frac {2 a c (5 a x+3)}{\sqrt {c-a^2 c x^2}}dx}{2 a^2 c}-\frac {3 x \sqrt {c-a^2 c x^2}}{a c}\right )-\frac {x^2 \sqrt {c-a^2 c x^2}}{3 c}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle c \left (\frac {1}{3} \left (\frac {\int \frac {5 a x+3}{\sqrt {c-a^2 c x^2}}dx}{a}-\frac {3 x \sqrt {c-a^2 c x^2}}{a c}\right )-\frac {x^2 \sqrt {c-a^2 c x^2}}{3 c}\right )\) |
\(\Big \downarrow \) 455 |
\(\displaystyle c \left (\frac {1}{3} \left (\frac {3 \int \frac {1}{\sqrt {c-a^2 c x^2}}dx-\frac {5 \sqrt {c-a^2 c x^2}}{a c}}{a}-\frac {3 x \sqrt {c-a^2 c x^2}}{a c}\right )-\frac {x^2 \sqrt {c-a^2 c x^2}}{3 c}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle c \left (\frac {1}{3} \left (\frac {3 \int \frac {1}{\frac {a^2 c x^2}{c-a^2 c x^2}+1}d\frac {x}{\sqrt {c-a^2 c x^2}}-\frac {5 \sqrt {c-a^2 c x^2}}{a c}}{a}-\frac {3 x \sqrt {c-a^2 c x^2}}{a c}\right )-\frac {x^2 \sqrt {c-a^2 c x^2}}{3 c}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle c \left (\frac {1}{3} \left (\frac {\frac {3 \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a \sqrt {c}}-\frac {5 \sqrt {c-a^2 c x^2}}{a c}}{a}-\frac {3 x \sqrt {c-a^2 c x^2}}{a c}\right )-\frac {x^2 \sqrt {c-a^2 c x^2}}{3 c}\right )\) |
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)
3.11.79.3.1 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_)^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[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]) && IGtQ[n/2, 0]
Time = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.94
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 | \(\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3 a^{2} c}-\frac {2 \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{a}-\frac {2 \left (\sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}-\frac {a c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}\right )}{\sqrt {a^{2} c}}\right )}{a^{2}}\) | \(163\) |
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 = 0.27 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.79 \[ \int e^{2 \text {arctanh}(a x)} x \sqrt {c-a^2 c x^2} \, dx=\left [-\frac {2 \, \sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} + 3 \, a x + 5\right )} - 3 \, \sqrt {-c} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right )}{6 \, a^{2}}, -\frac {\sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} + 3 \, a x + 5\right )} + 3 \, \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right )}{3 \, a^{2}}\right ] \]
[-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*s qrt(c)*x/(a^2*c*x^2 - c)))/a^2]
\[ \int e^{2 \text {arctanh}(a x)} x \sqrt {c-a^2 c x^2} \, dx=- \int \frac {x \sqrt {- a^{2} c x^{2} + c}}{a x - 1}\, dx - \int \frac {a x^{2} \sqrt {- a^{2} c x^{2} + c}}{a x - 1}\, dx \]
-Integral(x*sqrt(-a**2*c*x**2 + c)/(a*x - 1), x) - Integral(a*x**2*sqrt(-a **2*c*x**2 + c)/(a*x - 1), x)
Time = 0.33 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.88 \[ \int e^{2 \text {arctanh}(a x)} x \sqrt {c-a^2 c x^2} \, dx=-\frac {1}{3} \, a {\left (\frac {3 \, \sqrt {-a^{2} c x^{2} + c} x}{a^{2}} - \frac {3 \, \sqrt {c} \arcsin \left (a x\right )}{a^{3}} + \frac {6 \, \sqrt {-a^{2} c x^{2} + c}}{a^{3}} - \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{a^{3} c}\right )} \]
-1/3*a*(3*sqrt(-a^2*c*x^2 + c)*x/a^2 - 3*sqrt(c)*arcsin(a*x)/a^3 + 6*sqrt( -a^2*c*x^2 + c)/a^3 - (-a^2*c*x^2 + c)^(3/2)/(a^3*c))
Time = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.87 \[ \int e^{2 \text {arctanh}(a x)} x \sqrt {c-a^2 c x^2} \, dx=-\frac {1}{3} \, \sqrt {-a^{2} c x^{2} + c} {\left ({\left (x + \frac {3}{a}\right )} x + \frac {5}{a^{2}}\right )} - \frac {c \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{a \sqrt {-c} {\left | a \right |}} \]
-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. \[ \int e^{2 \text {arctanh}(a x)} x \sqrt {c-a^2 c x^2} \, dx=\int -\frac {x\,\sqrt {c-a^2\,c\,x^2}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \]