Integrand size = 20, antiderivative size = 78 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right ) \, dx=-\frac {11 c \sqrt {1-a^2 x^2}}{3 a}-\frac {3}{2} c x \sqrt {1-a^2 x^2}-\frac {1}{3} a c x^2 \sqrt {1-a^2 x^2}+\frac {5 c \arcsin (a x)}{2 a} \] Output:
-11/3*c*(-a^2*x^2+1)^(1/2)/a-3/2*c*x*(-a^2*x^2+1)^(1/2)-1/3*a*c*x^2*(-a^2* x^2+1)^(1/2)+5/2*c*arcsin(a*x)/a
Time = 0.05 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.73 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right ) \, dx=-\frac {c \left (\sqrt {1-a^2 x^2} \left (22+9 a x+2 a^2 x^2\right )+30 \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{6 a} \] Input:
Integrate[E^(3*ArcTanh[a*x])*(c - a^2*c*x^2),x]
Output:
-1/6*(c*(Sqrt[1 - a^2*x^2]*(22 + 9*a*x + 2*a^2*x^2) + 30*ArcSin[Sqrt[1 - a *x]/Sqrt[2]]))/a
Time = 0.44 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.21, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6688, 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.
\(\displaystyle \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right ) \, dx\) |
\(\Big \downarrow \) 6688 |
\(\displaystyle c \int \frac {(a x+1)^3}{\sqrt {1-a^2 x^2}}dx\) |
\(\Big \downarrow \) 469 |
\(\displaystyle c \left (\frac {5}{3} \int \frac {(a x+1)^2}{\sqrt {1-a^2 x^2}}dx-\frac {(a x+1)^2 \sqrt {1-a^2 x^2}}{3 a}\right )\) |
\(\Big \downarrow \) 469 |
\(\displaystyle c \left (\frac {5}{3} \left (\frac {3}{2} \int \frac {a x+1}{\sqrt {1-a^2 x^2}}dx-\frac {(a x+1) \sqrt {1-a^2 x^2}}{2 a}\right )-\frac {(a x+1)^2 \sqrt {1-a^2 x^2}}{3 a}\right )\) |
\(\Big \downarrow \) 455 |
\(\displaystyle c \left (\frac {5}{3} \left (\frac {3}{2} \left (\int \frac {1}{\sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{a}\right )-\frac {(a x+1) \sqrt {1-a^2 x^2}}{2 a}\right )-\frac {(a x+1)^2 \sqrt {1-a^2 x^2}}{3 a}\right )\) |
\(\Big \downarrow \) 223 |
\(\displaystyle c \left (\frac {5}{3} \left (\frac {3}{2} \left (\frac {\arcsin (a x)}{a}-\frac {\sqrt {1-a^2 x^2}}{a}\right )-\frac {(a x+1) \sqrt {1-a^2 x^2}}{2 a}\right )-\frac {(a x+1)^2 \sqrt {1-a^2 x^2}}{3 a}\right )\) |
Input:
Int[E^(3*ArcTanh[a*x])*(c - a^2*c*x^2),x]
Output:
c*(-1/3*((1 + a*x)^2*Sqrt[1 - a^2*x^2])/a + (5*(-1/2*((1 + a*x)*Sqrt[1 - a ^2*x^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] && IGtQ[(n + 1)/2, 0] && !I ntegerQ[p - n/2]
Time = 0.20 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.91
method | result | size |
risch | \(\frac {\left (2 a^{2} x^{2}+9 a x +22\right ) \left (a^{2} x^{2}-1\right ) c}{6 a \sqrt {-a^{2} x^{2}+1}}+\frac {5 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c}{2 \sqrt {a^{2}}}\) | \(71\) |
meijerg | \(-\frac {2 c \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {c x}{\sqrt {-a^{2} x^{2}+1}}+\frac {c \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-2 a^{4} x^{4}-8 a^{2} x^{2}+16\right )}{6 \sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}-\frac {2 c \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{4 \sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}-\frac {3 c \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (-5 a^{2} x^{2}+15\right )}{10 a^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{2 a^{5}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {3 c \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}\) | \(269\) |
default | \(-c \left (a^{5} \left (-\frac {x^{4}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {4 x^{2}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {8}{3 a^{4} \sqrt {-a^{2} x^{2}+1}}}{a^{2}}\right )-\frac {x}{\sqrt {-a^{2} x^{2}+1}}-\frac {3}{a \sqrt {-a^{2} x^{2}+1}}-2 a^{2} \left (\frac {x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}\right )+2 a^{3} \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )+3 a^{4} \left (-\frac {x^{3}}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )\right )\) | \(280\) |
Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c),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 = 0.11 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.79 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right ) \, dx=-\frac {30 \, c \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (2 \, a^{2} c x^{2} + 9 \, a c x + 22 \, c\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, a} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c),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
Time = 5.35 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.44 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right ) \, dx=\begin {cases} \frac {5 c \log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{2 \sqrt {- a^{2}}} + \sqrt {- a^{2} x^{2} + 1} \left (- \frac {a c x^{2}}{3} - \frac {3 c x}{2} - \frac {11 c}{3 a}\right ) & \text {for}\: a^{2} \neq 0 \\\frac {a^{3} c x^{4}}{4} + a^{2} c x^{3} + \frac {3 a c x^{2}}{2} + c x & \text {otherwise} \end {cases} \] Input:
integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c),x)
Output:
Piecewise((5*c*log(-2*a**2*x + 2*sqrt(-a**2)*sqrt(-a**2*x**2 + 1))/(2*sqrt (-a**2)) + sqrt(-a**2*x**2 + 1)*(-a*c*x**2/3 - 3*c*x/2 - 11*c/(3*a)), Ne(a **2, 0)), (a**3*c*x**4/4 + a**2*c*x**3 + 3*a*c*x**2/2 + c*x, True))
Time = 0.11 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.36 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right ) \, dx=\frac {a^{3} c x^{4}}{3 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {3 \, a^{2} c x^{3}}{2 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {10 \, a c x^{2}}{3 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {3 \, c x}{2 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {5 \, c \arcsin \left (a x\right )}{2 \, a} - \frac {11 \, c}{3 \, \sqrt {-a^{2} x^{2} + 1} a} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c),x, algorithm="maxima ")
Output:
1/3*a^3*c*x^4/sqrt(-a^2*x^2 + 1) + 3/2*a^2*c*x^3/sqrt(-a^2*x^2 + 1) + 10/3 *a*c*x^2/sqrt(-a^2*x^2 + 1) - 3/2*c*x/sqrt(-a^2*x^2 + 1) + 5/2*c*arcsin(a* x)/a - 11/3*c/(sqrt(-a^2*x^2 + 1)*a)
Time = 0.15 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.59 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right ) \, dx=\frac {5 \, c \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{2 \, {\left | a \right |}} - \frac {1}{6} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, a c x + 9 \, c\right )} x + \frac {22 \, c}{a}\right )} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c),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 = 0.95 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right ) \, dx=\frac {5\,c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,\sqrt {-a^2}}-\frac {3\,c\,x\,\sqrt {1-a^2\,x^2}}{2}-\frac {11\,c\,\sqrt {1-a^2\,x^2}}{3\,a}-\frac {a\,c\,x^2\,\sqrt {1-a^2\,x^2}}{3} \] Input:
int(((c - a^2*c*x^2)*(a*x + 1)^3)/(1 - a^2*x^2)^(3/2),x)
Output:
(5*c*asinh(x*(-a^2)^(1/2)))/(2*(-a^2)^(1/2)) - (3*c*x*(1 - a^2*x^2)^(1/2)) /2 - (11*c*(1 - a^2*x^2)^(1/2))/(3*a) - (a*c*x^2*(1 - a^2*x^2)^(1/2))/3
Time = 0.16 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.78 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right ) \, dx=\frac {c \left (15 \mathit {asin} \left (a x \right )-2 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-9 \sqrt {-a^{2} x^{2}+1}\, a x -22 \sqrt {-a^{2} x^{2}+1}+22\right )}{6 a} \] Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c),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)