Integrand size = 24, antiderivative size = 108 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\frac {5}{8} c x \sqrt {c-a^2 c x^2}+\frac {5 \left (c-a^2 c x^2\right )^{3/2}}{12 a}+\frac {(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}+\frac {5 c^{3/2} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a} \]
5/12*(-a^2*c*x^2+c)^(3/2)/a+1/4*(-a*x+1)*(-a^2*c*x^2+c)^(3/2)/a+5/8*c^(3/2 )*arctan(a*x*c^(1/2)/(-a^2*c*x^2+c)^(1/2))/a+5/8*c*x*(-a^2*c*x^2+c)^(1/2)
Time = 0.07 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.08 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=-\frac {c \sqrt {c-a^2 c x^2} \left (\sqrt {1+a x} \left (-16+7 a x+25 a^2 x^2-22 a^3 x^3+6 a^4 x^4\right )+30 \sqrt {1-a x} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{24 a \sqrt {1-a x} \sqrt {1-a^2 x^2}} \]
-1/24*(c*Sqrt[c - a^2*c*x^2]*(Sqrt[1 + a*x]*(-16 + 7*a*x + 25*a^2*x^2 - 22 *a^3*x^3 + 6*a^4*x^4) + 30*Sqrt[1 - a*x]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/( a*Sqrt[1 - a*x]*Sqrt[1 - a^2*x^2])
Time = 0.30 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6692, 469, 455, 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.
\(\displaystyle \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 6692 |
\(\displaystyle c \int (1-a x)^2 \sqrt {c-a^2 c x^2}dx\) |
\(\Big \downarrow \) 469 |
\(\displaystyle c \left (\frac {5}{4} \int (1-a x) \sqrt {c-a^2 c x^2}dx+\frac {(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a c}\right )\) |
\(\Big \downarrow \) 455 |
\(\displaystyle c \left (\frac {5}{4} \left (\int \sqrt {c-a^2 c x^2}dx+\frac {\left (c-a^2 c x^2\right )^{3/2}}{3 a c}\right )+\frac {(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a c}\right )\) |
\(\Big \downarrow \) 211 |
\(\displaystyle c \left (\frac {5}{4} \left (\frac {1}{2} c \int \frac {1}{\sqrt {c-a^2 c x^2}}dx+\frac {\left (c-a^2 c x^2\right )^{3/2}}{3 a c}+\frac {1}{2} x \sqrt {c-a^2 c x^2}\right )+\frac {(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a c}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle c \left (\frac {5}{4} \left (\frac {1}{2} c \int \frac {1}{\frac {a^2 c x^2}{c-a^2 c x^2}+1}d\frac {x}{\sqrt {c-a^2 c x^2}}+\frac {\left (c-a^2 c x^2\right )^{3/2}}{3 a c}+\frac {1}{2} x \sqrt {c-a^2 c x^2}\right )+\frac {(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a c}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle c \left (\frac {5}{4} \left (\frac {\sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{2 a}+\frac {\left (c-a^2 c x^2\right )^{3/2}}{3 a c}+\frac {1}{2} x \sqrt {c-a^2 c x^2}\right )+\frac {(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a c}\right )\) |
c*(((1 - a*x)*(c - a^2*c*x^2)^(3/2))/(4*a*c) + (5*((x*Sqrt[c - a^2*c*x^2]) /2 + (c - a^2*c*x^2)^(3/2)/(3*a*c) + (Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(2*a)))/4)
3.13.45.3.1 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[1/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]) && ILtQ[ n/2, 0]
Time = 0.36 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83
method | result | size |
risch | \(-\frac {\left (6 a^{3} x^{3}-16 a^{2} x^{2}+9 a x +16\right ) \left (a^{2} x^{2}-1\right ) c^{2}}{24 a \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\frac {5 \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right ) c^{2}}{8 \sqrt {a^{2} c}}\) | \(90\) |
default | \(-\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}-\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}+\frac {\frac {2 \left (-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}{3}+2 a c \left (-\frac {\left (-2 a^{2} c \left (x +\frac {1}{a}\right )+2 a c \right ) \sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c}}\right )}{2 \sqrt {a^{2} c}}\right )}{a}\) | \(202\) |
-1/24*(6*a^3*x^3-16*a^2*x^2+9*a*x+16)*(a^2*x^2-1)/a/(-c*(a^2*x^2-1))^(1/2) *c^2+5/8/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-a^2*c*x^2+c)^(1/2))*c^2
Time = 0.30 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.67 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=\left [\frac {15 \, \sqrt {-c} c \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) + 2 \, {\left (6 \, a^{3} c x^{3} - 16 \, a^{2} c x^{2} + 9 \, a c x + 16 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{48 \, a}, -\frac {15 \, c^{\frac {3}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) - {\left (6 \, a^{3} c x^{3} - 16 \, a^{2} c x^{2} + 9 \, a c x + 16 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{24 \, a}\right ] \]
[1/48*(15*sqrt(-c)*c*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) + 2*(6*a^3*c*x^3 - 16*a^2*c*x^2 + 9*a*c*x + 16*c)*sqrt(-a^2*c*x^2 + c))/a, -1/24*(15*c^(3/2)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^ 2 - c)) - (6*a^3*c*x^3 - 16*a^2*c*x^2 + 9*a*c*x + 16*c)*sqrt(-a^2*c*x^2 + c))/a]
Time = 1.97 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.36 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=a^{2} c \left (\begin {cases} \left (\frac {x^{3}}{4} - \frac {x}{8 a^{2}}\right ) \sqrt {- a^{2} c x^{2} + c} + \frac {c \left (\begin {cases} \frac {\log {\left (- 2 a^{2} c x + 2 \sqrt {- a^{2} c} \sqrt {- a^{2} c x^{2} + c} \right )}}{\sqrt {- a^{2} c}} & \text {for}\: c \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- a^{2} c x^{2}}} & \text {otherwise} \end {cases}\right )}{8 a^{2}} & \text {for}\: a^{2} c \neq 0 \\\frac {\sqrt {c} x^{3}}{3} & \text {otherwise} \end {cases}\right ) - 2 a c \left (\begin {cases} \left (\frac {x^{2}}{3} - \frac {1}{3 a^{2}}\right ) \sqrt {- a^{2} c x^{2} + c} & \text {for}\: a^{2} c \neq 0 \\\frac {\sqrt {c} x^{2}}{2} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} \frac {c \left (\begin {cases} \frac {\log {\left (- 2 a^{2} c x + 2 \sqrt {- a^{2} c} \sqrt {- a^{2} c x^{2} + c} \right )}}{\sqrt {- a^{2} c}} & \text {for}\: c \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- a^{2} c x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {- a^{2} c x^{2} + c}}{2} & \text {for}\: a^{2} c \neq 0 \\\sqrt {c} x & \text {otherwise} \end {cases}\right ) \]
a**2*c*Piecewise(((x**3/4 - x/(8*a**2))*sqrt(-a**2*c*x**2 + c) + c*Piecewi se((log(-2*a**2*c*x + 2*sqrt(-a**2*c)*sqrt(-a**2*c*x**2 + c))/sqrt(-a**2*c ), Ne(c, 0)), (x*log(x)/sqrt(-a**2*c*x**2), True))/(8*a**2), Ne(a**2*c, 0) ), (sqrt(c)*x**3/3, True)) - 2*a*c*Piecewise(((x**2/3 - 1/(3*a**2))*sqrt(- a**2*c*x**2 + c), Ne(a**2*c, 0)), (sqrt(c)*x**2/2, True)) + c*Piecewise((c *Piecewise((log(-2*a**2*c*x + 2*sqrt(-a**2*c)*sqrt(-a**2*c*x**2 + c))/sqrt (-a**2*c), Ne(c, 0)), (x*log(x)/sqrt(-a**2*c*x**2), True))/2 + x*sqrt(-a** 2*c*x**2 + c)/2, Ne(a**2*c, 0)), (sqrt(c)*x, True))
Time = 0.28 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.20 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=-\frac {1}{4} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x + \sqrt {a^{2} c x^{2} + 4 \, a c x + 3 \, c} c x - \frac {3}{8} \, \sqrt {-a^{2} c x^{2} + c} c x - \frac {c^{3} \arcsin \left (a x + 2\right )}{a \left (-c\right )^{\frac {3}{2}}} - \frac {3 \, c^{\frac {3}{2}} \arcsin \left (a x\right )}{8 \, a} + \frac {2 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{3 \, a} + \frac {2 \, \sqrt {a^{2} c x^{2} + 4 \, a c x + 3 \, c} c}{a} \]
-1/4*(-a^2*c*x^2 + c)^(3/2)*x + sqrt(a^2*c*x^2 + 4*a*c*x + 3*c)*c*x - 3/8* sqrt(-a^2*c*x^2 + c)*c*x - c^3*arcsin(a*x + 2)/(a*(-c)^(3/2)) - 3/8*c^(3/2 )*arcsin(a*x)/a + 2/3*(-a^2*c*x^2 + c)^(3/2)/a + 2*sqrt(a^2*c*x^2 + 4*a*c* x + 3*c)*c/a
Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (87) = 174\).
Time = 0.34 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.07 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=-\frac {{\left (240 \, a^{5} c^{\frac {3}{2}} \arctan \left (\frac {\sqrt {-c + \frac {2 \, c}{a x + 1}}}{\sqrt {c}}\right ) \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) - \frac {{\left (15 \, a^{5} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{3} c^{2} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) + 73 \, a^{5} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{2} c^{3} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) + 15 \, a^{5} c^{5} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) + 55 \, a^{5} c^{4} {\left (-c + \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right )\right )} {\left (a x + 1\right )}^{4}}{c^{4}}\right )} {\left | a \right |}}{192 \, a^{7}} \]
-1/192*(240*a^5*c^(3/2)*arctan(sqrt(-c + 2*c/(a*x + 1))/sqrt(c))*sgn(1/(a* x + 1))*sgn(a) - (15*a^5*(c - 2*c/(a*x + 1))^3*c^2*sqrt(-c + 2*c/(a*x + 1) )*sgn(1/(a*x + 1))*sgn(a) + 73*a^5*(c - 2*c/(a*x + 1))^2*c^3*sqrt(-c + 2*c /(a*x + 1))*sgn(1/(a*x + 1))*sgn(a) + 15*a^5*c^5*sqrt(-c + 2*c/(a*x + 1))* sgn(1/(a*x + 1))*sgn(a) + 55*a^5*c^4*(-c + 2*c/(a*x + 1))^(3/2)*sgn(1/(a*x + 1))*sgn(a))*(a*x + 1)^4/c^4)*abs(a)/a^7
Timed out. \[ \int e^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx=-\int \frac {{\left (c-a^2\,c\,x^2\right )}^{3/2}\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \]