Integrand size = 20, antiderivative size = 95 \[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=-\frac {8 c \sqrt {c-a c x}}{a}-\frac {4 (c-a c x)^{3/2}}{3 a}-\frac {2 (c-a c x)^{5/2}}{5 a c}+\frac {8 \sqrt {2} c^{3/2} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a} \]
-4/3*(-a*c*x+c)^(3/2)/a-2/5*(-a*c*x+c)^(5/2)/a/c+8*c^(3/2)*arctanh(1/2*(-a *c*x+c)^(1/2)*2^(1/2)/c^(1/2))*2^(1/2)/a-8*c*(-a*c*x+c)^(1/2)/a
Time = 0.05 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.75 \[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=\frac {-2 c \sqrt {c-a c x} \left (73-16 a x+3 a^2 x^2\right )+120 \sqrt {2} c^{3/2} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{15 a} \]
(-2*c*Sqrt[c - a*c*x]*(73 - 16*a*x + 3*a^2*x^2) + 120*Sqrt[2]*c^(3/2)*ArcT anh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/(15*a)
Time = 0.34 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6717, 6680, 35, 60, 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.
\(\displaystyle \int (c-a c x)^{3/2} e^{-2 \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6717 |
\(\displaystyle -\int e^{-2 \text {arctanh}(a x)} (c-a c x)^{3/2}dx\) |
\(\Big \downarrow \) 6680 |
\(\displaystyle -\int \frac {(1-a x) (c-a c x)^{3/2}}{a x+1}dx\) |
\(\Big \downarrow \) 35 |
\(\displaystyle -\frac {\int \frac {(c-a c x)^{5/2}}{a x+1}dx}{c}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle -\frac {2 c \int \frac {(c-a c x)^{3/2}}{a x+1}dx+\frac {2 (c-a c x)^{5/2}}{5 a}}{c}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle -\frac {2 c \left (2 c \int \frac {\sqrt {c-a c x}}{a x+1}dx+\frac {2 (c-a c x)^{3/2}}{3 a}\right )+\frac {2 (c-a c x)^{5/2}}{5 a}}{c}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle -\frac {2 c \left (2 c \left (2 c \int \frac {1}{(a x+1) \sqrt {c-a c x}}dx+\frac {2 \sqrt {c-a c x}}{a}\right )+\frac {2 (c-a c x)^{3/2}}{3 a}\right )+\frac {2 (c-a c x)^{5/2}}{5 a}}{c}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {2 c \left (2 c \left (\frac {2 \sqrt {c-a c x}}{a}-\frac {4 \int \frac {1}{2-\frac {c-a c x}{c}}d\sqrt {c-a c x}}{a}\right )+\frac {2 (c-a c x)^{3/2}}{3 a}\right )+\frac {2 (c-a c x)^{5/2}}{5 a}}{c}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {2 c \left (2 c \left (\frac {2 \sqrt {c-a c x}}{a}-\frac {2 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a}\right )+\frac {2 (c-a c x)^{3/2}}{3 a}\right )+\frac {2 (c-a c x)^{5/2}}{5 a}}{c}\) |
-(((2*(c - a*c*x)^(5/2))/(5*a) + 2*c*((2*(c - a*c*x)^(3/2))/(3*a) + 2*c*(( 2*Sqrt[c - a*c*x])/a - (2*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2] *Sqrt[c])])/a)))/c)
3.3.63.3.1 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 = 0.56 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.64
method | result | size |
pseudoelliptic | \(-\frac {2 \left (-20 \sqrt {c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )+\frac {\left (3 a^{2} x^{2}-16 a x +73\right ) \sqrt {-c \left (a x -1\right )}}{3}\right ) c}{5 a}\) | \(61\) |
risch | \(\frac {2 \left (3 a^{2} x^{2}-16 a x +73\right ) \left (a x -1\right ) c^{2}}{15 a \sqrt {-c \left (a x -1\right )}}+\frac {8 c^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}}{a}\) | \(68\) |
derivativedivides | \(-\frac {2 \left (\frac {\left (-a c x +c \right )^{\frac {5}{2}}}{5}+\frac {2 c \left (-a c x +c \right )^{\frac {3}{2}}}{3}+4 c^{2} \sqrt {-a c x +c}-4 c^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{c a}\) | \(73\) |
default | \(\frac {-\frac {2 \left (-a c x +c \right )^{\frac {5}{2}}}{5}-\frac {4 c \left (-a c x +c \right )^{\frac {3}{2}}}{3}-8 c^{2} \sqrt {-a c x +c}+8 c^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{a c}\) | \(73\) |
-2/5*(-20*c^(1/2)*2^(1/2)*arctanh(1/2*(-c*(a*x-1))^(1/2)*2^(1/2)/c^(1/2))+ 1/3*(3*a^2*x^2-16*a*x+73)*(-c*(a*x-1))^(1/2))*c/a
Time = 0.27 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.54 \[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=\left [\frac {2 \, {\left (30 \, \sqrt {2} c^{\frac {3}{2}} \log \left (\frac {a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) - {\left (3 \, a^{2} c x^{2} - 16 \, a c x + 73 \, c\right )} \sqrt {-a c x + c}\right )}}{15 \, a}, -\frac {2 \, {\left (60 \, \sqrt {2} \sqrt {-c} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) + {\left (3 \, a^{2} c x^{2} - 16 \, a c x + 73 \, c\right )} \sqrt {-a c x + c}\right )}}{15 \, a}\right ] \]
[2/15*(30*sqrt(2)*c^(3/2)*log((a*c*x - 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a*x + 1)) - (3*a^2*c*x^2 - 16*a*c*x + 73*c)*sqrt(-a*c*x + c))/a, - 2/15*(60*sqrt(2)*sqrt(-c)*c*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(-c)/c ) + (3*a^2*c*x^2 - 16*a*c*x + 73*c)*sqrt(-a*c*x + c))/a]
Time = 2.61 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.22 \[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=\begin {cases} - \frac {2 \cdot \left (\frac {4 \sqrt {2} c^{3} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{\sqrt {- c}} + 4 c^{2} \sqrt {- a c x + c} + \frac {2 c \left (- a c x + c\right )^{\frac {3}{2}}}{3} + \frac {\left (- a c x + c\right )^{\frac {5}{2}}}{5}\right )}{a c} & \text {for}\: a c \neq 0 \\c^{\frac {3}{2}} \left (\begin {cases} - x & \text {for}\: a = 0 \\\frac {a x - 2 \log {\left (a x + 1 \right )} + 1}{a} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
Piecewise((-2*(4*sqrt(2)*c**3*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/ sqrt(-c) + 4*c**2*sqrt(-a*c*x + c) + 2*c*(-a*c*x + c)**(3/2)/3 + (-a*c*x + c)**(5/2)/5)/(a*c), Ne(a*c, 0)), (c**(3/2)*Piecewise((-x, Eq(a, 0)), ((a* x - 2*log(a*x + 1) + 1)/a, True)), True))
Time = 0.30 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00 \[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=-\frac {2 \, {\left (30 \, \sqrt {2} c^{\frac {5}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right ) + 3 \, {\left (-a c x + c\right )}^{\frac {5}{2}} + 10 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c + 60 \, \sqrt {-a c x + c} c^{2}\right )}}{15 \, a c} \]
-2/15*(30*sqrt(2)*c^(5/2)*log(-(sqrt(2)*sqrt(c) - sqrt(-a*c*x + c))/(sqrt( 2)*sqrt(c) + sqrt(-a*c*x + c))) + 3*(-a*c*x + c)^(5/2) + 10*(-a*c*x + c)^( 3/2)*c + 60*sqrt(-a*c*x + c)*c^2)/(a*c)
Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.13 \[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=-\frac {8 \, \sqrt {2} c^{2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{a \sqrt {-c}} - \frac {2 \, {\left (3 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} a^{4} c^{4} + 10 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{4} c^{5} + 60 \, \sqrt {-a c x + c} a^{4} c^{6}\right )}}{15 \, a^{5} c^{5}} \]
-8*sqrt(2)*c^2*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(-c))/(a*sqrt(-c)) - 2/15*(3*(a*c*x - c)^2*sqrt(-a*c*x + c)*a^4*c^4 + 10*(-a*c*x + c)^(3/2)*a ^4*c^5 + 60*sqrt(-a*c*x + c)*a^4*c^6)/(a^5*c^5)
Time = 4.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.82 \[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx=-\frac {4\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a}-\frac {8\,c\,\sqrt {c-a\,c\,x}}{a}-\frac {2\,{\left (c-a\,c\,x\right )}^{5/2}}{5\,a\,c}-\frac {\sqrt {2}\,c^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,8{}\mathrm {i}}{a} \]