Integrand size = 22, antiderivative size = 83 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {3}{8} c^2 x \sqrt {1-a^2 x^2}+\frac {1}{4} c^2 x \left (1-a^2 x^2\right )^{3/2}+\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac {3 c^2 \arcsin (a x)}{8 a} \]
1/4*c^2*x*(-a^2*x^2+1)^(3/2)+1/5*c^2*(-a^2*x^2+1)^(5/2)/a+3/8*c^2*arcsin(a *x)/a+3/8*c^2*x*(-a^2*x^2+1)^(1/2)
Time = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.90 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {c^2 \left (\sqrt {1-a^2 x^2} \left (8+25 a x-16 a^2 x^2-10 a^3 x^3+8 a^4 x^4\right )-30 \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{40 a} \]
(c^2*(Sqrt[1 - a^2*x^2]*(8 + 25*a*x - 16*a^2*x^2 - 10*a^3*x^3 + 8*a^4*x^4) - 30*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(40*a)
Time = 0.25 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6689, 455, 211, 211, 223}
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.
\(\displaystyle \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx\) |
\(\Big \downarrow \) 6689 |
\(\displaystyle c^2 \int (1-a x) \left (1-a^2 x^2\right )^{3/2}dx\) |
\(\Big \downarrow \) 455 |
\(\displaystyle c^2 \left (\int \left (1-a^2 x^2\right )^{3/2}dx+\frac {\left (1-a^2 x^2\right )^{5/2}}{5 a}\right )\) |
\(\Big \downarrow \) 211 |
\(\displaystyle c^2 \left (\frac {3}{4} \int \sqrt {1-a^2 x^2}dx+\frac {\left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2}\right )\) |
\(\Big \downarrow \) 211 |
\(\displaystyle c^2 \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-a^2 x^2}}dx+\frac {1}{2} x \sqrt {1-a^2 x^2}\right )+\frac {\left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2}\right )\) |
\(\Big \downarrow \) 223 |
\(\displaystyle c^2 \left (\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2}+\frac {\arcsin (a x)}{2 a}\right )+\frac {\left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2}\right )\) |
c^2*((x*(1 - a^2*x^2)^(3/2))/4 + (1 - a^2*x^2)^(5/2)/(5*a) + (3*((x*Sqrt[1 - a^2*x^2])/2 + ArcSin[a*x]/(2*a)))/4)
3.12.94.3.1 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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
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[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[c^p Int[(1 - a^2*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] && ILtQ[(n - 1)/2, 0] && !In tegerQ[p - n/2]
Time = 0.36 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.10
method | result | size |
risch | \(-\frac {\left (8 a^{4} x^{4}-10 a^{3} x^{3}-16 a^{2} x^{2}+25 a x +8\right ) \left (a^{2} x^{2}-1\right ) c^{2}}{40 a \sqrt {-a^{2} x^{2}+1}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{2}}{8 \sqrt {a^{2}}}\) | \(91\) |
default | \(c^{2} \left (\frac {\sqrt {-a^{2} x^{2}+1}\, x}{2}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}+a^{3} \left (-\frac {x^{2} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{5 a^{2}}-\frac {2 \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{15 a^{4}}\right )-a^{2} \left (-\frac {x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4 a^{2}}+\frac {\frac {\sqrt {-a^{2} x^{2}+1}\, x}{2}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}}{4 a^{2}}\right )+\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 a}\right )\) | \(179\) |
-1/40*(8*a^4*x^4-10*a^3*x^3-16*a^2*x^2+25*a*x+8)*(a^2*x^2-1)/a/(-a^2*x^2+1 )^(1/2)*c^2+3/8/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))*c^2
Time = 0.26 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.12 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=-\frac {30 \, c^{2} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (8 \, a^{4} c^{2} x^{4} - 10 \, a^{3} c^{2} x^{3} - 16 \, a^{2} c^{2} x^{2} + 25 \, a c^{2} x + 8 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{40 \, a} \]
-1/40*(30*c^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (8*a^4*c^2*x^4 - 10 *a^3*c^2*x^3 - 16*a^2*c^2*x^2 + 25*a*c^2*x + 8*c^2)*sqrt(-a^2*x^2 + 1))/a
Time = 0.89 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.70 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=a^{3} c^{2} \left (\begin {cases} \sqrt {- a^{2} x^{2} + 1} \left (\frac {x^{4}}{5} - \frac {x^{2}}{15 a^{2}} - \frac {2}{15 a^{4}}\right ) & \text {for}\: a^{2} \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) - a^{2} c^{2} \left (\begin {cases} \left (\frac {x^{3}}{4} - \frac {x}{8 a^{2}}\right ) \sqrt {- a^{2} x^{2} + 1} + \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{8 a^{2} \sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{3}}{3} & \text {otherwise} \end {cases}\right ) - a c^{2} \left (\begin {cases} \left (\frac {x^{2}}{3} - \frac {1}{3 a^{2}}\right ) \sqrt {- a^{2} x^{2} + 1} & \text {for}\: a^{2} \neq 0 \\\frac {x^{2}}{2} & \text {otherwise} \end {cases}\right ) + c^{2} \left (\begin {cases} \frac {x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{2 \sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\x & \text {otherwise} \end {cases}\right ) \]
a**3*c**2*Piecewise((sqrt(-a**2*x**2 + 1)*(x**4/5 - x**2/(15*a**2) - 2/(15 *a**4)), Ne(a**2, 0)), (x**4/4, True)) - a**2*c**2*Piecewise(((x**3/4 - x/ (8*a**2))*sqrt(-a**2*x**2 + 1) + log(-2*a**2*x + 2*sqrt(-a**2)*sqrt(-a**2* x**2 + 1))/(8*a**2*sqrt(-a**2)), Ne(a**2, 0)), (x**3/3, True)) - a*c**2*Pi ecewise(((x**2/3 - 1/(3*a**2))*sqrt(-a**2*x**2 + 1), Ne(a**2, 0)), (x**2/2 , True)) + c**2*Piecewise((x*sqrt(-a**2*x**2 + 1)/2 + log(-2*a**2*x + 2*sq rt(-a**2)*sqrt(-a**2*x**2 + 1))/(2*sqrt(-a**2)), Ne(a**2, 0)), (x, True))
Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.08 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=-\frac {1}{5} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a c^{2} x^{2} + \frac {1}{4} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{2} x + \frac {3}{8} \, \sqrt {-a^{2} x^{2} + 1} c^{2} x + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{2}}{5 \, a} + \frac {3 \, c^{2} \arcsin \left (a x\right )}{8 \, a} \]
-1/5*(-a^2*x^2 + 1)^(3/2)*a*c^2*x^2 + 1/4*(-a^2*x^2 + 1)^(3/2)*c^2*x + 3/8 *sqrt(-a^2*x^2 + 1)*c^2*x + 1/5*(-a^2*x^2 + 1)^(3/2)*c^2/a + 3/8*c^2*arcsi n(a*x)/a
Time = 0.28 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.94 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {3 \, c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{8 \, {\left | a \right |}} + \frac {1}{40} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (25 \, c^{2} - 2 \, {\left (8 \, a c^{2} - {\left (4 \, a^{3} c^{2} x - 5 \, a^{2} c^{2}\right )} x\right )} x\right )} x + \frac {8 \, c^{2}}{a}\right )} \]
3/8*c^2*arcsin(a*x)*sgn(a)/abs(a) + 1/40*sqrt(-a^2*x^2 + 1)*((25*c^2 - 2*( 8*a*c^2 - (4*a^3*c^2*x - 5*a^2*c^2)*x)*x)*x + 8*c^2/a)
Time = 0.04 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.54 \[ \int e^{-\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {5\,c^2\,x\,\sqrt {1-a^2\,x^2}}{8}+\frac {3\,c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,\sqrt {-a^2}}+\frac {c^2\,\sqrt {1-a^2\,x^2}}{5\,a}-\frac {2\,a\,c^2\,x^2\,\sqrt {1-a^2\,x^2}}{5}-\frac {a^2\,c^2\,x^3\,\sqrt {1-a^2\,x^2}}{4}+\frac {a^3\,c^2\,x^4\,\sqrt {1-a^2\,x^2}}{5} \]