Integrand size = 20, antiderivative size = 82 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right ) \, dx=\frac {5}{2} c x \sqrt {1-a^2 x^2}+\frac {5 c \left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac {2 c \left (1-a^2 x^2\right )^{5/2}}{a (1+a x)^2}+\frac {5 c \arcsin (a x)}{2 a} \] Output:
5/2*c*x*(-a^2*x^2+1)^(1/2)+5/3*c*(-a^2*x^2+1)^(3/2)/a+2*c*(-a^2*x^2+1)^(5/ 2)/a/(a*x+1)^2+5/2*c*arcsin(a*x)/a
Time = 0.12 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.85 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right ) \, dx=\frac {c \left (\frac {\sqrt {1+a x} \left (22-31 a x+11 a^2 x^2-2 a^3 x^3\right )}{\sqrt {1-a x}}-30 \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{6 a} \] Input:
Integrate[(c - a^2*c*x^2)/E^(3*ArcTanh[a*x]),x]
Output:
(c*((Sqrt[1 + a*x]*(22 - 31*a*x + 11*a^2*x^2 - 2*a^3*x^3))/Sqrt[1 - a*x] - 30*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(6*a)
Time = 0.45 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.16, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6689, 469, 469, 455, 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.
\(\displaystyle \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right ) \, dx\) |
\(\Big \downarrow \) 6689 |
\(\displaystyle c \int \frac {(1-a x)^3}{\sqrt {1-a^2 x^2}}dx\) |
\(\Big \downarrow \) 469 |
\(\displaystyle c \left (\frac {5}{3} \int \frac {(1-a x)^2}{\sqrt {1-a^2 x^2}}dx+\frac {\sqrt {1-a^2 x^2} (1-a x)^2}{3 a}\right )\) |
\(\Big \downarrow \) 469 |
\(\displaystyle c \left (\frac {5}{3} \left (\frac {3}{2} \int \frac {1-a x}{\sqrt {1-a^2 x^2}}dx+\frac {\sqrt {1-a^2 x^2} (1-a x)}{2 a}\right )+\frac {\sqrt {1-a^2 x^2} (1-a x)^2}{3 a}\right )\) |
\(\Big \downarrow \) 455 |
\(\displaystyle c \left (\frac {5}{3} \left (\frac {3}{2} \left (\int \frac {1}{\sqrt {1-a^2 x^2}}dx+\frac {\sqrt {1-a^2 x^2}}{a}\right )+\frac {\sqrt {1-a^2 x^2} (1-a x)}{2 a}\right )+\frac {\sqrt {1-a^2 x^2} (1-a x)^2}{3 a}\right )\) |
\(\Big \downarrow \) 223 |
\(\displaystyle c \left (\frac {5}{3} \left (\frac {3}{2} \left (\frac {\sqrt {1-a^2 x^2}}{a}+\frac {\arcsin (a x)}{a}\right )+\frac {\sqrt {1-a^2 x^2} (1-a x)}{2 a}\right )+\frac {\sqrt {1-a^2 x^2} (1-a x)^2}{3 a}\right )\) |
Input:
Int[(c - a^2*c*x^2)/E^(3*ArcTanh[a*x]),x]
Output:
c*(((1 - a*x)^2*Sqrt[1 - a^2*x^2])/(3*a) + (5*(((1 - a*x)*Sqrt[1 - a^2*x^2 ])/(2*a) + (3*(Sqrt[1 - a^2*x^2]/a + ArcSin[a*x]/a))/2))/3)
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[((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^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 = 0.17 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.87
method | result | size |
risch | \(-\frac {\left (2 a^{2} x^{2}-9 a x +22\right ) \left (a^{2} x^{2}-1\right ) c}{6 a \sqrt {-a^{2} x^{2}+1}}+\frac {5 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c}{2 \sqrt {a^{2}}}\) | \(71\) |
default | \(-c \left (\frac {\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )}{a}-\frac {2 \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+3 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )}{a^{2}}\right )\) | \(270\) |
Input:
int((-a^2*c*x^2+c)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/6*(2*a^2*x^2-9*a*x+22)*(a^2*x^2-1)/a/(-a^2*x^2+1)^(1/2)*c+5/2/(a^2)^(1/ 2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))*c
Time = 0.13 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.77 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right ) \, dx=-\frac {30 \, c \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (2 \, a^{2} c x^{2} - 9 \, a c x + 22 \, c\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, a} \] Input:
integrate((-a^2*c*x^2+c)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="fricas ")
Output:
-1/6*(30*c*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (2*a^2*c*x^2 - 9*a*c*x + 22*c)*sqrt(-a^2*x^2 + 1))/a
\[ \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right ) \, dx=c \left (\int \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int \left (- \frac {2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\right )\, dx + \int \frac {a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx\right ) \] Input:
integrate((-a**2*c*x**2+c)/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)
Output:
c*(Integral(sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3*a*x + 1), x) + Integral(-2*a**2*x**2*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a**2*x**2 + 3 *a*x + 1), x) + Integral(a**4*x**4*sqrt(-a**2*x**2 + 1)/(a**3*x**3 + 3*a** 2*x**2 + 3*a*x + 1), x))
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.49 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right ) \, dx=-\frac {1}{2} \, \sqrt {a^{2} x^{2} + 4 \, a x + 3} c x + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c}{a^{2} x + a} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c}{3 \, a} + \frac {i \, c \arcsin \left (a x + 2\right )}{2 \, a} + \frac {3 \, c \arcsin \left (a x\right )}{a} - \frac {\sqrt {a^{2} x^{2} + 4 \, a x + 3} c}{a} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} c}{a} \] Input:
integrate((-a^2*c*x^2+c)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="maxima ")
Output:
-1/2*sqrt(a^2*x^2 + 4*a*x + 3)*c*x + (-a^2*x^2 + 1)^(3/2)*c/(a^2*x + a) - 1/3*(-a^2*x^2 + 1)^(3/2)*c/a + 1/2*I*c*arcsin(a*x + 2)/a + 3*c*arcsin(a*x) /a - sqrt(a^2*x^2 + 4*a*x + 3)*c/a + 3*sqrt(-a^2*x^2 + 1)*c/a
Time = 0.16 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.56 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right ) \, dx=\frac {5 \, c \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{2 \, {\left | a \right |}} + \frac {1}{6} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, a c x - 9 \, c\right )} x + \frac {22 \, c}{a}\right )} \] Input:
integrate((-a^2*c*x^2+c)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="giac")
Output:
5/2*c*arcsin(a*x)*sgn(a)/abs(a) + 1/6*sqrt(-a^2*x^2 + 1)*((2*a*c*x - 9*c)* x + 22*c/a)
Time = 0.04 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.90 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right ) \, dx=\frac {11\,c\,\sqrt {1-a^2\,x^2}}{3\,a}-\frac {3\,c\,x\,\sqrt {1-a^2\,x^2}}{2}+\frac {5\,c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,\sqrt {-a^2}}+\frac {a\,c\,x^2\,\sqrt {1-a^2\,x^2}}{3} \] Input:
int(((c - a^2*c*x^2)*(1 - a^2*x^2)^(3/2))/(a*x + 1)^3,x)
Output:
(11*c*(1 - a^2*x^2)^(1/2))/(3*a) - (3*c*x*(1 - a^2*x^2)^(1/2))/2 + (5*c*as inh(x*(-a^2)^(1/2)))/(2*(-a^2)^(1/2)) + (a*c*x^2*(1 - a^2*x^2)^(1/2))/3
Time = 0.15 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.74 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right ) \, dx=\frac {c \left (15 \mathit {asin} \left (a x \right )+2 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-9 \sqrt {-a^{2} x^{2}+1}\, a x +22 \sqrt {-a^{2} x^{2}+1}-22\right )}{6 a} \] Input:
int((-a^2*c*x^2+c)/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x)
Output:
(c*(15*asin(a*x) + 2*sqrt( - a**2*x**2 + 1)*a**2*x**2 - 9*sqrt( - a**2*x** 2 + 1)*a*x + 22*sqrt( - a**2*x**2 + 1) - 22))/(6*a)