Integrand size = 24, antiderivative size = 130 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=-\frac {7}{16} c^2 x \sqrt {c-a^2 c x^2}-\frac {7}{24} c x \left (c-a^2 c x^2\right )^{3/2}+\frac {7 \left (c-a^2 c x^2\right )^{5/2}}{30 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{5/2}}{6 a}-\frac {7 c^{5/2} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{16 a} \]
-7/24*c*x*(-a^2*c*x^2+c)^(3/2)+7/30*(-a^2*c*x^2+c)^(5/2)/a+1/6*(a*x+1)*(-a ^2*c*x^2+c)^(5/2)/a-7/16*c^(5/2)*arctan(a*x*c^(1/2)/(-a^2*c*x^2+c)^(1/2))/ a-7/16*c^2*x*(-a^2*c*x^2+c)^(1/2)
Time = 0.15 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.04 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=\frac {c^2 \sqrt {c-a^2 c x^2} \left (\sqrt {1+a x} \left (96-231 a x-57 a^2 x^2+182 a^3 x^3+106 a^4 x^4-56 a^5 x^5-40 a^6 x^6\right )+210 \sqrt {1-a x} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{240 a \sqrt {1-a x} \sqrt {1-a^2 x^2}} \]
(c^2*Sqrt[c - a^2*c*x^2]*(Sqrt[1 + a*x]*(96 - 231*a*x - 57*a^2*x^2 + 182*a ^3*x^3 + 106*a^4*x^4 - 56*a^5*x^5 - 40*a^6*x^6) + 210*Sqrt[1 - a*x]*ArcSin [Sqrt[1 - a*x]/Sqrt[2]]))/(240*a*Sqrt[1 - a*x]*Sqrt[1 - a^2*x^2])
Time = 0.39 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6717, 6691, 469, 455, 211, 211, 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.
| \(\displaystyle \int \left (c-a^2 c x^2\right )^{5/2} e^{2 \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6717 |
| \(\displaystyle -\int e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2}dx\) |
| \(\Big \downarrow \) 6691 |
| \(\displaystyle -c \int (a x+1)^2 \left (c-a^2 c x^2\right )^{3/2}dx\) |
| \(\Big \downarrow \) 469 |
| \(\displaystyle -c \left (\frac {7}{6} \int (a x+1) \left (c-a^2 c x^2\right )^{3/2}dx-\frac {(a x+1) \left (c-a^2 c x^2\right )^{5/2}}{6 a c}\right )\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle -c \left (\frac {7}{6} \left (\int \left (c-a^2 c x^2\right )^{3/2}dx-\frac {\left (c-a^2 c x^2\right )^{5/2}}{5 a c}\right )-\frac {(a x+1) \left (c-a^2 c x^2\right )^{5/2}}{6 a c}\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle -c \left (\frac {7}{6} \left (\frac {3}{4} c \int \sqrt {c-a^2 c x^2}dx-\frac {\left (c-a^2 c x^2\right )^{5/2}}{5 a c}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2}\right )-\frac {(a x+1) \left (c-a^2 c x^2\right )^{5/2}}{6 a c}\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle -c \left (\frac {7}{6} \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {1}{\sqrt {c-a^2 c x^2}}dx+\frac {1}{2} x \sqrt {c-a^2 c x^2}\right )-\frac {\left (c-a^2 c x^2\right )^{5/2}}{5 a c}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2}\right )-\frac {(a x+1) \left (c-a^2 c x^2\right )^{5/2}}{6 a c}\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle -c \left (\frac {7}{6} \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {1}{\frac {a^2 c x^2}{c-a^2 c x^2}+1}d\frac {x}{\sqrt {c-a^2 c x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2}\right )-\frac {\left (c-a^2 c x^2\right )^{5/2}}{5 a c}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2}\right )-\frac {(a x+1) \left (c-a^2 c x^2\right )^{5/2}}{6 a c}\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -c \left (\frac {7}{6} \left (\frac {3}{4} c \left (\frac {\sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{2 a}+\frac {1}{2} x \sqrt {c-a^2 c x^2}\right )-\frac {\left (c-a^2 c x^2\right )^{5/2}}{5 a c}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2}\right )-\frac {(a x+1) \left (c-a^2 c x^2\right )^{5/2}}{6 a c}\right )\) |
-(c*(-1/6*((1 + a*x)*(c - a^2*c*x^2)^(5/2))/(a*c) + (7*((x*(c - a^2*c*x^2) ^(3/2))/4 - (c - a^2*c*x^2)^(5/2)/(5*a*c) + (3*c*((x*Sqrt[c - a^2*c*x^2])/ 2 + (Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(2*a)))/4))/6))
3.7.25.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
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[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[2*c* ((n + p)/(n + 2*p + 1)) Int[(c + d*x)^(n - 1)*(a + b*x^2)^p, x], x] /; Fr eeQ[{a, b, c, d, p}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[n, 0] && NeQ[n + 2* p + 1, 0] && IntegerQ[2*p]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[c^(n/2) Int[(c + d*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c , d, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ[c, 0]) && IGtQ[n/ 2, 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 = 0.59 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.82
| method | result | size |
| risch | \(-\frac {\left (40 a^{5} x^{5}+96 a^{4} x^{4}-10 a^{3} x^{3}-192 a^{2} x^{2}-135 a x +96\right ) \left (a^{2} x^{2}-1\right ) c^{3}}{240 a \sqrt {-c \left (a^{2} x^{2}-1\right )}}-\frac {7 \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right ) c^{3}}{16 \sqrt {a^{2} c}}\) | \(106\) |
| default | \(\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )}{6}+\frac {\frac {2 \left (-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {5}{2}}}{5}-2 a c \left (-\frac {\left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right ) \left (-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}{8 a^{2} c}+\frac {3 c \left (-\frac {\left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right ) \sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )}{a}\) | \(296\) |
-1/240*(40*a^5*x^5+96*a^4*x^4-10*a^3*x^3-192*a^2*x^2-135*a*x+96)*(a^2*x^2- 1)/a/(-c*(a^2*x^2-1))^(1/2)*c^3-7/16/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/ (-a^2*c*x^2+c)^(1/2))*c^3
Time = 0.28 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.85 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=\left [\frac {105 \, \sqrt {-c} c^{2} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) + 2 \, {\left (40 \, a^{5} c^{2} x^{5} + 96 \, a^{4} c^{2} x^{4} - 10 \, a^{3} c^{2} x^{3} - 192 \, a^{2} c^{2} x^{2} - 135 \, a c^{2} x + 96 \, c^{2}\right )} \sqrt {-a^{2} c x^{2} + c}}{480 \, a}, \frac {105 \, c^{\frac {5}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) + {\left (40 \, a^{5} c^{2} x^{5} + 96 \, a^{4} c^{2} x^{4} - 10 \, a^{3} c^{2} x^{3} - 192 \, a^{2} c^{2} x^{2} - 135 \, a c^{2} x + 96 \, c^{2}\right )} \sqrt {-a^{2} c x^{2} + c}}{240 \, a}\right ] \]
[1/480*(105*sqrt(-c)*c^2*log(2*a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(- c)*x - c) + 2*(40*a^5*c^2*x^5 + 96*a^4*c^2*x^4 - 10*a^3*c^2*x^3 - 192*a^2* c^2*x^2 - 135*a*c^2*x + 96*c^2)*sqrt(-a^2*c*x^2 + c))/a, 1/240*(105*c^(5/2 )*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + (40*a^5*c^2*x ^5 + 96*a^4*c^2*x^4 - 10*a^3*c^2*x^3 - 192*a^2*c^2*x^2 - 135*a*c^2*x + 96* c^2)*sqrt(-a^2*c*x^2 + c))/a]
Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (117) = 234\).
Time = 2.53 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.42 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=\begin {cases} \frac {- 2 c^{2} \left (\begin {cases} \left (\frac {a^{2} x^{2}}{3} - \frac {1}{3}\right ) \sqrt {- a^{2} c x^{2} + c} & \text {for}\: c \neq 0 \\\frac {a^{2} \sqrt {c} x^{2}}{2} & \text {otherwise} \end {cases}\right ) + 2 c^{2} \left (\begin {cases} \sqrt {- a^{2} c x^{2} + c} \left (\frac {a^{4} x^{4}}{5} - \frac {a^{2} x^{2}}{15} - \frac {2}{15}\right ) & \text {for}\: c \neq 0 \\\frac {a^{4} \sqrt {c} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + c^{2} \left (\begin {cases} \frac {c \left (\begin {cases} \frac {\log {\left (- 2 a c x + 2 \sqrt {- c} \sqrt {- a^{2} c x^{2} + c} \right )}}{\sqrt {- c}} & \text {for}\: c \neq 0 \\\frac {a x \log {\left (a x \right )}}{\sqrt {- a^{2} c x^{2}}} & \text {otherwise} \end {cases}\right )}{16} + \sqrt {- a^{2} c x^{2} + c} \left (\frac {a^{5} x^{5}}{6} - \frac {a^{3} x^{3}}{24} - \frac {a x}{16}\right ) & \text {for}\: c \neq 0 \\\frac {a^{5} \sqrt {c} x^{5}}{5} & \text {otherwise} \end {cases}\right ) - c^{2} \left (\begin {cases} \frac {a x \sqrt {- a^{2} c x^{2} + c}}{2} + \frac {c \left (\begin {cases} \frac {\log {\left (- 2 a c x + 2 \sqrt {- c} \sqrt {- a^{2} c x^{2} + c} \right )}}{\sqrt {- c}} & \text {for}\: c \neq 0 \\\frac {a x \log {\left (a x \right )}}{\sqrt {- a^{2} c x^{2}}} & \text {otherwise} \end {cases}\right )}{2} & \text {for}\: c \neq 0 \\a \sqrt {c} x & \text {otherwise} \end {cases}\right )}{a} & \text {for}\: a \neq 0 \\- c^{\frac {5}{2}} x & \text {otherwise} \end {cases} \]
Piecewise(((-2*c**2*Piecewise(((a**2*x**2/3 - 1/3)*sqrt(-a**2*c*x**2 + c), Ne(c, 0)), (a**2*sqrt(c)*x**2/2, True)) + 2*c**2*Piecewise((sqrt(-a**2*c* x**2 + c)*(a**4*x**4/5 - a**2*x**2/15 - 2/15), Ne(c, 0)), (a**4*sqrt(c)*x* *4/4, True)) + c**2*Piecewise((c*Piecewise((log(-2*a*c*x + 2*sqrt(-c)*sqrt (-a**2*c*x**2 + c))/sqrt(-c), Ne(c, 0)), (a*x*log(a*x)/sqrt(-a**2*c*x**2), True))/16 + sqrt(-a**2*c*x**2 + c)*(a**5*x**5/6 - a**3*x**3/24 - a*x/16), Ne(c, 0)), (a**5*sqrt(c)*x**5/5, True)) - c**2*Piecewise((a*x*sqrt(-a**2* c*x**2 + c)/2 + c*Piecewise((log(-2*a*c*x + 2*sqrt(-c)*sqrt(-a**2*c*x**2 + c))/sqrt(-c), Ne(c, 0)), (a*x*log(a*x)/sqrt(-a**2*c*x**2), True))/2, Ne(c , 0)), (a*sqrt(c)*x, True)))/a, Ne(a, 0)), (-c**(5/2)*x, True))
Time = 0.28 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.18 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=\frac {1}{6} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x - \frac {7}{24} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c x - \frac {3}{4} \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c^{2} x + \frac {5}{16} \, \sqrt {-a^{2} c x^{2} + c} c^{2} x - \frac {3 \, c^{4} \arcsin \left (a x - 2\right )}{4 \, a \left (-c\right )^{\frac {3}{2}}} + \frac {5 \, c^{\frac {5}{2}} \arcsin \left (a x\right )}{16 \, a} + \frac {2 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}{5 \, a} + \frac {3 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c^{2}}{2 \, a} \]
1/6*(-a^2*c*x^2 + c)^(5/2)*x - 7/24*(-a^2*c*x^2 + c)^(3/2)*c*x - 3/4*sqrt( a^2*c*x^2 - 4*a*c*x + 3*c)*c^2*x + 5/16*sqrt(-a^2*c*x^2 + c)*c^2*x - 3/4*c ^4*arcsin(a*x - 2)/(a*(-c)^(3/2)) + 5/16*c^(5/2)*arcsin(a*x)/a + 2/5*(-a^2 *c*x^2 + c)^(5/2)/a + 3/2*sqrt(a^2*c*x^2 - 4*a*c*x + 3*c)*c^2/a
Time = 0.28 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.89 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=\frac {7 \, c^{3} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{16 \, \sqrt {-c} {\left | a \right |}} - \frac {1}{240} \, \sqrt {-a^{2} c x^{2} + c} {\left ({\left (135 \, c^{2} + 2 \, {\left (96 \, a c^{2} + {\left (5 \, a^{2} c^{2} - 4 \, {\left (5 \, a^{4} c^{2} x + 12 \, a^{3} c^{2}\right )} x\right )} x\right )} x\right )} x - \frac {96 \, c^{2}}{a}\right )} \]
7/16*c^3*log(abs(-sqrt(-a^2*c)*x + sqrt(-a^2*c*x^2 + c)))/(sqrt(-c)*abs(a) ) - 1/240*sqrt(-a^2*c*x^2 + c)*((135*c^2 + 2*(96*a*c^2 + (5*a^2*c^2 - 4*(5 *a^4*c^2*x + 12*a^3*c^2)*x)*x)*x)*x - 96*c^2/a)
Timed out. \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^{5/2}\,\left (a\,x+1\right )}{a\,x-1} \,d x \]