Integrand size = 20, antiderivative size = 76 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=-\frac {4 \sqrt {c-a c x}}{a}-\frac {2 (c-a c x)^{3/2}}{3 a c}+\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a} \]
-2/3*(-a*c*x+c)^(3/2)/a/c+4*arctanh(1/2*(-a*c*x+c)^(1/2)*2^(1/2)/c^(1/2))* 2^(1/2)*c^(1/2)/a-4*(-a*c*x+c)^(1/2)/a
Time = 0.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.80 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 (-7+a x) \sqrt {c-a c x}+12 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{3 a} \]
(2*(-7 + a*x)*Sqrt[c - a*c*x] + 12*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x] /(Sqrt[2]*Sqrt[c])])/(3*a)
Time = 0.32 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {6717, 6680, 35, 60, 60, 73, 219}
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.
\(\displaystyle \int \sqrt {c-a c x} e^{-2 \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6717 |
\(\displaystyle -\int e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}dx\) |
\(\Big \downarrow \) 6680 |
\(\displaystyle -\int \frac {(1-a x) \sqrt {c-a c x}}{a x+1}dx\) |
\(\Big \downarrow \) 35 |
\(\displaystyle -\frac {\int \frac {(c-a c x)^{3/2}}{a x+1}dx}{c}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle -\frac {2 c \int \frac {\sqrt {c-a c x}}{a x+1}dx+\frac {2 (c-a c x)^{3/2}}{3 a}}{c}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle -\frac {2 c \left (2 c \int \frac {1}{(a x+1) \sqrt {c-a c x}}dx+\frac {2 \sqrt {c-a c x}}{a}\right )+\frac {2 (c-a c x)^{3/2}}{3 a}}{c}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {2 c \left (\frac {2 \sqrt {c-a c x}}{a}-\frac {4 \int \frac {1}{2-\frac {c-a c x}{c}}d\sqrt {c-a c x}}{a}\right )+\frac {2 (c-a c x)^{3/2}}{3 a}}{c}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {2 c \left (\frac {2 \sqrt {c-a c x}}{a}-\frac {2 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a}\right )+\frac {2 (c-a c x)^{3/2}}{3 a}}{c}\) |
-(((2*(c - a*c*x)^(3/2))/(3*a) + 2*c*((2*Sqrt[c - a*c*x])/a - (2*Sqrt[2]*S qrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/a))/c)
3.3.64.3.1 Defintions of rubi rules used
Int[(u_.)*((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.), x_Symbol] :> Simp[(b/d)^m Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n} , x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && !(IntegerQ[n] && SimplerQ[a + b*x, c + d*x])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol ] :> Int[u*(c + d*x)^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c , d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && !(IntegerQ[p] || GtQ[c, 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 = 0.53 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.75
method | result | size |
risch | \(-\frac {2 \left (a x -7\right ) \left (a x -1\right ) c}{3 a \sqrt {-c \left (a x -1\right )}}+\frac {4 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, \sqrt {c}}{a}\) | \(57\) |
derivativedivides | \(-\frac {2 \left (\frac {\left (-a c x +c \right )^{\frac {3}{2}}}{3}+2 c \sqrt {-a c x +c}-2 c^{\frac {3}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{c a}\) | \(59\) |
default | \(\frac {-\frac {2 \left (-a c x +c \right )^{\frac {3}{2}}}{3}-4 c \sqrt {-a c x +c}+4 c^{\frac {3}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{a c}\) | \(59\) |
pseudoelliptic | \(\frac {4 \sqrt {c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )+\frac {2 a x \sqrt {-c \left (a x -1\right )}}{3}-\frac {14 \sqrt {-c \left (a x -1\right )}}{3}}{a}\) | \(59\) |
-2/3*(a*x-7)*(a*x-1)/a/(-c*(a*x-1))^(1/2)*c+4*arctanh(1/2*(-a*c*x+c)^(1/2) *2^(1/2)/c^(1/2))*2^(1/2)*c^(1/2)/a
Time = 0.25 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.57 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\left [\frac {2 \, {\left (3 \, \sqrt {2} \sqrt {c} \log \left (\frac {a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + \sqrt {-a c x + c} {\left (a x - 7\right )}\right )}}{3 \, a}, -\frac {2 \, {\left (6 \, \sqrt {2} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) - \sqrt {-a c x + c} {\left (a x - 7\right )}\right )}}{3 \, a}\right ] \]
[2/3*(3*sqrt(2)*sqrt(c)*log((a*c*x - 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a*x + 1)) + sqrt(-a*c*x + c)*(a*x - 7))/a, -2/3*(6*sqrt(2)*sqrt(-c)* arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(-c)/c) - sqrt(-a*c*x + c)*(a*x - 7))/a]
Time = 2.58 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.30 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\begin {cases} - \frac {2 \cdot \left (\frac {2 \sqrt {2} c^{2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{\sqrt {- c}} + 2 c \sqrt {- a c x + c} + \frac {\left (- a c x + c\right )^{\frac {3}{2}}}{3}\right )}{a c} & \text {for}\: a c \neq 0 \\\sqrt {c} \left (\begin {cases} - x & \text {for}\: a = 0 \\\frac {a x - 2 \log {\left (a x + 1 \right )} + 1}{a} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
Piecewise((-2*(2*sqrt(2)*c**2*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/ sqrt(-c) + 2*c*sqrt(-a*c*x + c) + (-a*c*x + c)**(3/2)/3)/(a*c), Ne(a*c, 0) ), (sqrt(c)*Piecewise((-x, Eq(a, 0)), ((a*x - 2*log(a*x + 1) + 1)/a, True) ), True))
Time = 0.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.04 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=-\frac {2 \, {\left (3 \, \sqrt {2} c^{\frac {3}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right ) + {\left (-a c x + c\right )}^{\frac {3}{2}} + 6 \, \sqrt {-a c x + c} c\right )}}{3 \, a c} \]
-2/3*(3*sqrt(2)*c^(3/2)*log(-(sqrt(2)*sqrt(c) - sqrt(-a*c*x + c))/(sqrt(2) *sqrt(c) + sqrt(-a*c*x + c))) + (-a*c*x + c)^(3/2) + 6*sqrt(-a*c*x + c)*c) /(a*c)
Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.01 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=-\frac {4 \, \sqrt {2} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{a \sqrt {-c}} - \frac {2 \, {\left ({\left (-a c x + c\right )}^{\frac {3}{2}} a^{2} c^{2} + 6 \, \sqrt {-a c x + c} a^{2} c^{3}\right )}}{3 \, a^{3} c^{3}} \]
-4*sqrt(2)*c*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(-c))/(a*sqrt(-c)) - 2/3*((-a*c*x + c)^(3/2)*a^2*c^2 + 6*sqrt(-a*c*x + c)*a^2*c^3)/(a^3*c^3)
Time = 4.52 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.80 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {4\,\sqrt {2}\,\sqrt {c}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}}{2\,\sqrt {c}}\right )}{a}-\frac {2\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a\,c}-\frac {4\,\sqrt {c-a\,c\,x}}{a} \]