Integrand size = 20, antiderivative size = 116 \[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=-\frac {16 c^2 \sqrt {c-a c x}}{a}-\frac {8 c (c-a c x)^{3/2}}{3 a}-\frac {4 (c-a c x)^{5/2}}{5 a}-\frac {2 (c-a c x)^{7/2}}{7 a c}+\frac {16 \sqrt {2} c^{5/2} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a} \]
-8/3*c*(-a*c*x+c)^(3/2)/a-4/5*(-a*c*x+c)^(5/2)/a-2/7*(-a*c*x+c)^(7/2)/a/c+ 16*c^(5/2)*arctanh(1/2*(-a*c*x+c)^(1/2)*2^(1/2)/c^(1/2))*2^(1/2)/a-16*c^2* (-a*c*x+c)^(1/2)/a
Time = 0.07 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.69 \[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\frac {2 c^2 \left (\sqrt {c-a c x} \left (-1037+269 a x-87 a^2 x^2+15 a^3 x^3\right )+840 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )\right )}{105 a} \]
(2*c^2*(Sqrt[c - a*c*x]*(-1037 + 269*a*x - 87*a^2*x^2 + 15*a^3*x^3) + 840* Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]))/(105*a)
Time = 0.35 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6717, 6680, 35, 60, 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.
\(\displaystyle \int (c-a c x)^{5/2} e^{-2 \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6717 |
\(\displaystyle -\int e^{-2 \text {arctanh}(a x)} (c-a c x)^{5/2}dx\) |
\(\Big \downarrow \) 6680 |
\(\displaystyle -\int \frac {(1-a x) (c-a c x)^{5/2}}{a x+1}dx\) |
\(\Big \downarrow \) 35 |
\(\displaystyle -\frac {\int \frac {(c-a c x)^{7/2}}{a x+1}dx}{c}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle -\frac {2 c \int \frac {(c-a c x)^{5/2}}{a x+1}dx+\frac {2 (c-a c x)^{7/2}}{7 a}}{c}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle -\frac {2 c \left (2 c \int \frac {(c-a c x)^{3/2}}{a x+1}dx+\frac {2 (c-a c x)^{5/2}}{5 a}\right )+\frac {2 (c-a c x)^{7/2}}{7 a}}{c}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle -\frac {2 c \left (2 c \left (2 c \int \frac {\sqrt {c-a c x}}{a x+1}dx+\frac {2 (c-a c x)^{3/2}}{3 a}\right )+\frac {2 (c-a c x)^{5/2}}{5 a}\right )+\frac {2 (c-a c x)^{7/2}}{7 a}}{c}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle -\frac {2 c \left (2 c \left (2 c \left (2 c \int \frac {1}{(a x+1) \sqrt {c-a c x}}dx+\frac {2 \sqrt {c-a c x}}{a}\right )+\frac {2 (c-a c x)^{3/2}}{3 a}\right )+\frac {2 (c-a c x)^{5/2}}{5 a}\right )+\frac {2 (c-a c x)^{7/2}}{7 a}}{c}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {2 c \left (2 c \left (2 c \left (\frac {2 \sqrt {c-a c x}}{a}-\frac {4 \int \frac {1}{2-\frac {c-a c x}{c}}d\sqrt {c-a c x}}{a}\right )+\frac {2 (c-a c x)^{3/2}}{3 a}\right )+\frac {2 (c-a c x)^{5/2}}{5 a}\right )+\frac {2 (c-a c x)^{7/2}}{7 a}}{c}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {2 c \left (2 c \left (2 c \left (\frac {2 \sqrt {c-a c x}}{a}-\frac {2 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a}\right )+\frac {2 (c-a c x)^{3/2}}{3 a}\right )+\frac {2 (c-a c x)^{5/2}}{5 a}\right )+\frac {2 (c-a c x)^{7/2}}{7 a}}{c}\) |
-(((2*(c - a*c*x)^(7/2))/(7*a) + 2*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)
3.3.62.3.1 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 = 0.57 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.61
method | result | size |
pseudoelliptic | \(\frac {2 \left (56 \sqrt {c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )+\frac {\left (15 a^{3} x^{3}-87 a^{2} x^{2}+269 a x -1037\right ) \sqrt {-c \left (a x -1\right )}}{15}\right ) c^{2}}{7 a}\) | \(71\) |
risch | \(-\frac {2 \left (15 a^{3} x^{3}-87 a^{2} x^{2}+269 a x -1037\right ) \left (a x -1\right ) c^{3}}{105 a \sqrt {-c \left (a x -1\right )}}+\frac {16 c^{\frac {5}{2}} \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}}{a}\) | \(76\) |
derivativedivides | \(-\frac {2 \left (\frac {\left (-a c x +c \right )^{\frac {7}{2}}}{7}+\frac {2 c \left (-a c x +c \right )^{\frac {5}{2}}}{5}+\frac {4 c^{2} \left (-a c x +c \right )^{\frac {3}{2}}}{3}+8 c^{3} \sqrt {-a c x +c}-8 c^{\frac {7}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{c a}\) | \(87\) |
default | \(\frac {-\frac {2 \left (-a c x +c \right )^{\frac {7}{2}}}{7}-\frac {4 c \left (-a c x +c \right )^{\frac {5}{2}}}{5}-\frac {8 c^{2} \left (-a c x +c \right )^{\frac {3}{2}}}{3}-16 c^{3} \sqrt {-a c x +c}+16 c^{\frac {7}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{a c}\) | \(87\) |
2/7*(56*c^(1/2)*2^(1/2)*arctanh(1/2*(-c*(a*x-1))^(1/2)*2^(1/2)/c^(1/2))+1/ 15*(15*a^3*x^3-87*a^2*x^2+269*a*x-1037)*(-c*(a*x-1))^(1/2))*c^2/a
Time = 0.27 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.57 \[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\left [\frac {2 \, {\left (420 \, \sqrt {2} c^{\frac {5}{2}} \log \left (\frac {a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + {\left (15 \, a^{3} c^{2} x^{3} - 87 \, a^{2} c^{2} x^{2} + 269 \, a c^{2} x - 1037 \, c^{2}\right )} \sqrt {-a c x + c}\right )}}{105 \, a}, -\frac {2 \, {\left (840 \, \sqrt {2} \sqrt {-c} c^{2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) - {\left (15 \, a^{3} c^{2} x^{3} - 87 \, a^{2} c^{2} x^{2} + 269 \, a c^{2} x - 1037 \, c^{2}\right )} \sqrt {-a c x + c}\right )}}{105 \, a}\right ] \]
[2/105*(420*sqrt(2)*c^(5/2)*log((a*c*x - 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c ) - 3*c)/(a*x + 1)) + (15*a^3*c^2*x^3 - 87*a^2*c^2*x^2 + 269*a*c^2*x - 103 7*c^2)*sqrt(-a*c*x + c))/a, -2/105*(840*sqrt(2)*sqrt(-c)*c^2*arctan(1/2*sq rt(2)*sqrt(-a*c*x + c)*sqrt(-c)/c) - (15*a^3*c^2*x^3 - 87*a^2*c^2*x^2 + 26 9*a*c^2*x - 1037*c^2)*sqrt(-a*c*x + c))/a]
Time = 2.75 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.15 \[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=\begin {cases} - \frac {2 \cdot \left (\frac {8 \sqrt {2} c^{4} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{\sqrt {- c}} + 8 c^{3} \sqrt {- a c x + c} + \frac {4 c^{2} \left (- a c x + c\right )^{\frac {3}{2}}}{3} + \frac {2 c \left (- a c x + c\right )^{\frac {5}{2}}}{5} + \frac {\left (- a c x + c\right )^{\frac {7}{2}}}{7}\right )}{a c} & \text {for}\: a c \neq 0 \\c^{\frac {5}{2}} \left (\begin {cases} - x & \text {for}\: a = 0 \\\frac {a x - 2 \log {\left (a x + 1 \right )} + 1}{a} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
Piecewise((-2*(8*sqrt(2)*c**4*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/ sqrt(-c) + 8*c**3*sqrt(-a*c*x + c) + 4*c**2*(-a*c*x + c)**(3/2)/3 + 2*c*(- a*c*x + c)**(5/2)/5 + (-a*c*x + c)**(7/2)/7)/(a*c), Ne(a*c, 0)), (c**(5/2) *Piecewise((-x, Eq(a, 0)), ((a*x - 2*log(a*x + 1) + 1)/a, True)), True))
Time = 0.30 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.94 \[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=-\frac {2 \, {\left (420 \, \sqrt {2} c^{\frac {7}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right ) + 15 \, {\left (-a c x + c\right )}^{\frac {7}{2}} + 42 \, {\left (-a c x + c\right )}^{\frac {5}{2}} c + 140 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{2} + 840 \, \sqrt {-a c x + c} c^{3}\right )}}{105 \, a c} \]
-2/105*(420*sqrt(2)*c^(7/2)*log(-(sqrt(2)*sqrt(c) - sqrt(-a*c*x + c))/(sqr t(2)*sqrt(c) + sqrt(-a*c*x + c))) + 15*(-a*c*x + c)^(7/2) + 42*(-a*c*x + c )^(5/2)*c + 140*(-a*c*x + c)^(3/2)*c^2 + 840*sqrt(-a*c*x + c)*c^3)/(a*c)
Time = 0.27 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.16 \[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=-\frac {16 \, \sqrt {2} c^{3} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{a \sqrt {-c}} + \frac {2 \, {\left (15 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} a^{6} c^{6} - 42 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} a^{6} c^{7} - 140 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{6} c^{8} - 840 \, \sqrt {-a c x + c} a^{6} c^{9}\right )}}{105 \, a^{7} c^{7}} \]
-16*sqrt(2)*c^3*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(-c))/(a*sqrt(-c)) + 2/105*(15*(a*c*x - c)^3*sqrt(-a*c*x + c)*a^6*c^6 - 42*(a*c*x - c)^2*sqr t(-a*c*x + c)*a^6*c^7 - 140*(-a*c*x + c)^(3/2)*a^6*c^8 - 840*sqrt(-a*c*x + c)*a^6*c^9)/(a^7*c^7)
Time = 0.07 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.82 \[ \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx=-\frac {4\,{\left (c-a\,c\,x\right )}^{5/2}}{5\,a}-\frac {8\,c\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a}-\frac {16\,c^2\,\sqrt {c-a\,c\,x}}{a}-\frac {2\,{\left (c-a\,c\,x\right )}^{7/2}}{7\,a\,c}-\frac {\sqrt {2}\,c^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,16{}\mathrm {i}}{a} \]