Integrand size = 23, antiderivative size = 139 \[ \int e^{-2 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx=\frac {4 \sqrt {c-a c x}}{a^4}+\frac {2 (c-a c x)^{3/2}}{3 a^4 c}+\frac {2 (c-a c x)^{5/2}}{5 a^4 c^2}-\frac {2 (c-a c x)^{7/2}}{7 a^4 c^3}+\frac {2 (c-a c x)^{9/2}}{9 a^4 c^4}-\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a^4} \]
2/3*(-a*c*x+c)^(3/2)/a^4/c+2/5*(-a*c*x+c)^(5/2)/a^4/c^2-2/7*(-a*c*x+c)^(7/ 2)/a^4/c^3+2/9*(-a*c*x+c)^(9/2)/a^4/c^4-4*arctanh(1/2*(-a*c*x+c)^(1/2)*2^( 1/2)/c^(1/2))*2^(1/2)*c^(1/2)/a^4+4*(-a*c*x+c)^(1/2)/a^4
Time = 0.14 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.61 \[ \int e^{-2 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx=\frac {2 \left (\sqrt {c-a c x} \left (788-236 a x+138 a^2 x^2-95 a^3 x^3+35 a^4 x^4\right )-630 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )\right )}{315 a^4} \]
(2*(Sqrt[c - a*c*x]*(788 - 236*a*x + 138*a^2*x^2 - 95*a^3*x^3 + 35*a^4*x^4 ) - 630*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]))/(315* a^4)
Time = 0.51 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6717, 6680, 35, 99, 2009}
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.
\(\displaystyle \int x^3 \sqrt {c-a c x} e^{-2 \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6717 |
\(\displaystyle -\int e^{-2 \text {arctanh}(a x)} x^3 \sqrt {c-a c x}dx\) |
\(\Big \downarrow \) 6680 |
\(\displaystyle -\int \frac {x^3 (1-a x) \sqrt {c-a c x}}{a x+1}dx\) |
\(\Big \downarrow \) 35 |
\(\displaystyle -\frac {\int \frac {x^3 (c-a c x)^{3/2}}{a x+1}dx}{c}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle -\frac {\int \left (\frac {(c-a c x)^{7/2}}{a^3 c^2}-\frac {(c-a c x)^{5/2}}{a^3 c}-\frac {(c-a c x)^{3/2}}{a^3 (a x+1)}+\frac {(c-a c x)^{3/2}}{a^3}\right )dx}{c}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\frac {4 \sqrt {2} c^{3/2} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a^4}-\frac {2 (c-a c x)^{9/2}}{9 a^4 c^3}+\frac {2 (c-a c x)^{7/2}}{7 a^4 c^2}-\frac {2 (c-a c x)^{5/2}}{5 a^4 c}-\frac {2 (c-a c x)^{3/2}}{3 a^4}-\frac {4 c \sqrt {c-a c x}}{a^4}}{c}\) |
-(((-4*c*Sqrt[c - a*c*x])/a^4 - (2*(c - a*c*x)^(3/2))/(3*a^4) - (2*(c - a* c*x)^(5/2))/(5*a^4*c) + (2*(c - a*c*x)^(7/2))/(7*a^4*c^2) - (2*(c - a*c*x) ^(9/2))/(9*a^4*c^3) + (4*Sqrt[2]*c^(3/2)*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]* Sqrt[c])])/a^4)/c)
3.4.41.3.1 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_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
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 = 0.60 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.54
method | result | size |
pseudoelliptic | \(\frac {\frac {2 \left (35 a^{4} x^{4}-95 a^{3} x^{3}+138 a^{2} x^{2}-236 a x +788\right ) \sqrt {-c \left (a x -1\right )}}{315}-4 \sqrt {c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{a^{4}}\) | \(75\) |
risch | \(-\frac {2 \left (35 a^{4} x^{4}-95 a^{3} x^{3}+138 a^{2} x^{2}-236 a x +788\right ) \left (a x -1\right ) c}{315 a^{4} \sqrt {-c \left (a x -1\right )}}-\frac {4 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, \sqrt {c}}{a^{4}}\) | \(82\) |
derivativedivides | \(\frac {\frac {2 \left (-a c x +c \right )^{\frac {9}{2}}}{9}-\frac {2 c \left (-a c x +c \right )^{\frac {7}{2}}}{7}+\frac {2 c^{2} \left (-a c x +c \right )^{\frac {5}{2}}}{5}+\frac {2 c^{3} \left (-a c x +c \right )^{\frac {3}{2}}}{3}+4 c^{4} \sqrt {-a c x +c}-4 c^{\frac {9}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{a^{4} c^{4}}\) | \(101\) |
default | \(\frac {\frac {2 \left (-a c x +c \right )^{\frac {9}{2}}}{9}-\frac {2 c \left (-a c x +c \right )^{\frac {7}{2}}}{7}+\frac {2 c^{2} \left (-a c x +c \right )^{\frac {5}{2}}}{5}+\frac {2 c^{3} \left (-a c x +c \right )^{\frac {3}{2}}}{3}+4 c^{4} \sqrt {-a c x +c}-4 c^{\frac {9}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{a^{4} c^{4}}\) | \(101\) |
2/315*((35*a^4*x^4-95*a^3*x^3+138*a^2*x^2-236*a*x+788)*(-c*(a*x-1))^(1/2)- 630*c^(1/2)*2^(1/2)*arctanh(1/2*(-c*(a*x-1))^(1/2)*2^(1/2)/c^(1/2)))/a^4
Time = 0.25 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.21 \[ \int e^{-2 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx=\left [\frac {2 \, {\left (315 \, \sqrt {2} \sqrt {c} \log \left (\frac {a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + {\left (35 \, a^{4} x^{4} - 95 \, a^{3} x^{3} + 138 \, a^{2} x^{2} - 236 \, a x + 788\right )} \sqrt {-a c x + c}\right )}}{315 \, a^{4}}, \frac {2 \, {\left (630 \, \sqrt {2} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) + {\left (35 \, a^{4} x^{4} - 95 \, a^{3} x^{3} + 138 \, a^{2} x^{2} - 236 \, a x + 788\right )} \sqrt {-a c x + c}\right )}}{315 \, a^{4}}\right ] \]
[2/315*(315*sqrt(2)*sqrt(c)*log((a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c ) - 3*c)/(a*x + 1)) + (35*a^4*x^4 - 95*a^3*x^3 + 138*a^2*x^2 - 236*a*x + 7 88)*sqrt(-a*c*x + c))/a^4, 2/315*(630*sqrt(2)*sqrt(-c)*arctan(1/2*sqrt(2)* sqrt(-a*c*x + c)*sqrt(-c)/c) + (35*a^4*x^4 - 95*a^3*x^3 + 138*a^2*x^2 - 23 6*a*x + 788)*sqrt(-a*c*x + c))/a^4]
Time = 4.14 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.26 \[ \int e^{-2 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx=\begin {cases} \frac {2 \cdot \left (\frac {2 \sqrt {2} c^{5} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{\sqrt {- c}} + 2 c^{4} \sqrt {- a c x + c} + \frac {c^{3} \left (- a c x + c\right )^{\frac {3}{2}}}{3} + \frac {c^{2} \left (- a c x + c\right )^{\frac {5}{2}}}{5} - \frac {c \left (- a c x + c\right )^{\frac {7}{2}}}{7} + \frac {\left (- a c x + c\right )^{\frac {9}{2}}}{9}\right )}{a^{4} c^{4}} & \text {for}\: a c \neq 0 \\\sqrt {c} \left (\frac {x^{4}}{4} - \frac {2 x^{3}}{3 a} + \frac {x^{2}}{a^{2}} - \frac {2 x}{a^{3}} + \frac {2 \left (\begin {cases} x & \text {for}\: a = 0 \\\frac {\log {\left (a x + 1 \right )}}{a} & \text {otherwise} \end {cases}\right )}{a^{3}}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(2*sqrt(2)*c**5*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/s qrt(-c) + 2*c**4*sqrt(-a*c*x + c) + c**3*(-a*c*x + c)**(3/2)/3 + c**2*(-a* c*x + c)**(5/2)/5 - c*(-a*c*x + c)**(7/2)/7 + (-a*c*x + c)**(9/2)/9)/(a**4 *c**4), Ne(a*c, 0)), (sqrt(c)*(x**4/4 - 2*x**3/(3*a) + x**2/a**2 - 2*x/a** 3 + 2*Piecewise((x, Eq(a, 0)), (log(a*x + 1)/a, True))/a**3), True))
Time = 0.28 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.88 \[ \int e^{-2 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx=\frac {2 \, {\left (315 \, \sqrt {2} c^{\frac {9}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right ) + 35 \, {\left (-a c x + c\right )}^{\frac {9}{2}} - 45 \, {\left (-a c x + c\right )}^{\frac {7}{2}} c + 63 \, {\left (-a c x + c\right )}^{\frac {5}{2}} c^{2} + 105 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{3} + 630 \, \sqrt {-a c x + c} c^{4}\right )}}{315 \, a^{4} c^{4}} \]
2/315*(315*sqrt(2)*c^(9/2)*log(-(sqrt(2)*sqrt(c) - sqrt(-a*c*x + c))/(sqrt (2)*sqrt(c) + sqrt(-a*c*x + c))) + 35*(-a*c*x + c)^(9/2) - 45*(-a*c*x + c) ^(7/2)*c + 63*(-a*c*x + c)^(5/2)*c^2 + 105*(-a*c*x + c)^(3/2)*c^3 + 630*sq rt(-a*c*x + c)*c^4)/(a^4*c^4)
Time = 0.27 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.14 \[ \int e^{-2 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx=\frac {4 \, \sqrt {2} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{a^{4} \sqrt {-c}} + \frac {2 \, {\left (35 \, {\left (a c x - c\right )}^{4} \sqrt {-a c x + c} a^{32} c^{32} + 45 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} a^{32} c^{33} + 63 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} a^{32} c^{34} + 105 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{32} c^{35} + 630 \, \sqrt {-a c x + c} a^{32} c^{36}\right )}}{315 \, a^{36} c^{36}} \]
4*sqrt(2)*c*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(-c))/(a^4*sqrt(-c)) + 2/315*(35*(a*c*x - c)^4*sqrt(-a*c*x + c)*a^32*c^32 + 45*(a*c*x - c)^3*sqr t(-a*c*x + c)*a^32*c^33 + 63*(a*c*x - c)^2*sqrt(-a*c*x + c)*a^32*c^34 + 10 5*(-a*c*x + c)^(3/2)*a^32*c^35 + 630*sqrt(-a*c*x + c)*a^32*c^36)/(a^36*c^3 6)
Time = 0.10 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.82 \[ \int e^{-2 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx=\frac {4\,\sqrt {c-a\,c\,x}}{a^4}+\frac {2\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a^4\,c}+\frac {2\,{\left (c-a\,c\,x\right )}^{5/2}}{5\,a^4\,c^2}-\frac {2\,{\left (c-a\,c\,x\right )}^{7/2}}{7\,a^4\,c^3}+\frac {2\,{\left (c-a\,c\,x\right )}^{9/2}}{9\,a^4\,c^4}+\frac {\sqrt {2}\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,4{}\mathrm {i}}{a^4} \]