Integrand size = 20, antiderivative size = 83 \[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {3}{8} c^2 x \sqrt {1-a^2 x^2}+\frac {1}{4} c^2 x \left (1-a^2 x^2\right )^{3/2}-\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac {3 c^2 \arcsin (a x)}{8 a} \] Output:
3/8*c^2*x*(-a^2*x^2+1)^(1/2)+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
Time = 0.11 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.90 \[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=-\frac {c^2 \left (\sqrt {1-a^2 x^2} \left (8-25 a x-16 a^2 x^2+10 a^3 x^3+8 a^4 x^4\right )+30 \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{40 a} \] Input:
Integrate[E^ArcTanh[a*x]*(c - a^2*c*x^2)^2,x]
Output:
-1/40*(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]]))/a
Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6688, 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.
\(\displaystyle \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx\) |
\(\Big \downarrow \) 6688 |
\(\displaystyle c^2 \int (a x+1) \left (1-a^2 x^2\right )^{3/2}dx\) |
\(\Big \downarrow \) 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 |
\(\displaystyle c^2 \left (\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-a^2 x^2}+\frac {\arcsin (a x)}{2 a}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{5 a}+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2}\right )\) |
Input:
Int[E^ArcTanh[a*x]*(c - a^2*c*x^2)^2,x]
Output:
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)
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] && IGtQ[(n + 1)/2, 0] && !I ntegerQ[p - n/2]
Time = 0.26 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.10
method | result | size |
risch | \(\frac {\left (8 a^{4} x^{4}+10 a^{3} x^{3}-16 a^{2} x^{2}-25 a x +8\right ) \left (a^{2} x^{2}-1\right ) c^{2}}{40 a \sqrt {-a^{2} x^{2}+1}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{2}}{8 \sqrt {a^{2}}}\) | \(91\) |
meijerg | \(-\frac {c^{2} \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (6 a^{4} x^{4}+8 a^{2} x^{2}+16\right ) \sqrt {-a^{2} x^{2}+1}}{15}\right )}{2 a \sqrt {\pi }}-\frac {c^{2} \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{6}\right )}{a \sqrt {\pi }}-\frac {c^{2} \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{2 a \sqrt {\pi }}+\frac {c^{2} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (10 a^{2} x^{2}+15\right ) \sqrt {-a^{2} x^{2}+1}}{20 a^{4}}+\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{4 a^{5}}\right )}{2 \sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {c^{2} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}} \sqrt {-a^{2} x^{2}+1}}{a^{2}}+\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {c^{2} \arcsin \left (a x \right )}{a}\) | \(277\) |
default | \(c^{2} \left (\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-\frac {\sqrt {-a^{2} x^{2}+1}}{a}+a^{4} \left (-\frac {x^{3} \sqrt {-a^{2} x^{2}+1}}{4 a^{2}}+\frac {-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )+a^{5} \left (-\frac {x^{4} \sqrt {-a^{2} x^{2}+1}}{5 a^{2}}+\frac {-\frac {4 x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15 a^{4}}}{a^{2}}\right )-2 a^{3} \left (-\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}\right )-2 a^{2} \left (-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )\right )\) | \(293\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^2,x,method=_RETURNVERBOSE)
Output:
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 = 0.11 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.11 \[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=-\frac {30 \, c^{2} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (8 \, a^{4} c^{2} x^{4} + 10 \, a^{3} c^{2} x^{3} - 16 \, a^{2} c^{2} x^{2} - 25 \, a c^{2} x + 8 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{40 \, a} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^2,x, algorithm="fricas ")
Output:
-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
Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (70) = 140\).
Time = 0.92 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.22 \[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\begin {cases} \frac {3 c^{2} \log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{8 \sqrt {- a^{2}}} + \sqrt {- a^{2} x^{2} + 1} \left (- \frac {a^{3} c^{2} x^{4}}{5} - \frac {a^{2} c^{2} x^{3}}{4} + \frac {2 a c^{2} x^{2}}{5} + \frac {5 c^{2} x}{8} - \frac {c^{2}}{5 a}\right ) & \text {for}\: a^{2} \neq 0 \\\begin {cases} c^{2} x & \text {for}\: a = 0 \\\frac {\frac {a^{6} c^{2} x^{6}}{6} + \frac {a^{5} c^{2} x^{5}}{5} - \frac {a^{4} c^{2} x^{4}}{2} - \frac {2 a^{3} c^{2} x^{3}}{3} + \frac {a^{2} c^{2} x^{2}}{2} + a c^{2} x}{a} & \text {otherwise} \end {cases} & \text {otherwise} \end {cases} \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a**2*c*x**2+c)**2,x)
Output:
Piecewise((3*c**2*log(-2*a**2*x + 2*sqrt(-a**2)*sqrt(-a**2*x**2 + 1))/(8*s qrt(-a**2)) + sqrt(-a**2*x**2 + 1)*(-a**3*c**2*x**4/5 - a**2*c**2*x**3/4 + 2*a*c**2*x**2/5 + 5*c**2*x/8 - c**2/(5*a)), Ne(a**2, 0)), (Piecewise((c** 2*x, Eq(a, 0)), ((a**6*c**2*x**6/6 + a**5*c**2*x**5/5 - a**4*c**2*x**4/2 - 2*a**3*c**2*x**3/3 + a**2*c**2*x**2/2 + a*c**2*x)/a, True)), True))
Time = 0.11 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.42 \[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=-\frac {1}{5} \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{2} x^{4} - \frac {1}{4} \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{2} x^{3} + \frac {2}{5} \, \sqrt {-a^{2} x^{2} + 1} a c^{2} x^{2} + \frac {5}{8} \, \sqrt {-a^{2} x^{2} + 1} c^{2} x + \frac {3 \, c^{2} \arcsin \left (a x\right )}{8 \, a} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{5 \, a} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^2,x, algorithm="maxima ")
Output:
-1/5*sqrt(-a^2*x^2 + 1)*a^3*c^2*x^4 - 1/4*sqrt(-a^2*x^2 + 1)*a^2*c^2*x^3 + 2/5*sqrt(-a^2*x^2 + 1)*a*c^2*x^2 + 5/8*sqrt(-a^2*x^2 + 1)*c^2*x + 3/8*c^2 *arcsin(a*x)/a - 1/5*sqrt(-a^2*x^2 + 1)*c^2/a
Time = 0.17 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.94 \[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {3 \, c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{8 \, {\left | a \right |}} + \frac {1}{40} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (25 \, c^{2} + 2 \, {\left (8 \, a c^{2} - {\left (4 \, a^{3} c^{2} x + 5 \, a^{2} c^{2}\right )} x\right )} x\right )} x - \frac {8 \, c^{2}}{a}\right )} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^2,x, algorithm="giac")
Output:
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 = 82, normalized size of antiderivative = 0.99 \[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {3\,c^2\,x\,\sqrt {1-a^2\,x^2}}{8}+\frac {c^2\,x\,{\left (1-a^2\,x^2\right )}^{3/2}}{4}-\frac {c^2\,{\left (1-a^2\,x^2\right )}^{5/2}}{5\,a}-\frac {3\,c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}{8\,a^2} \] Input:
int(((c - a^2*c*x^2)^2*(a*x + 1))/(1 - a^2*x^2)^(1/2),x)
Output:
(3*c^2*x*(1 - a^2*x^2)^(1/2))/8 + (c^2*x*(1 - a^2*x^2)^(3/2))/4 - (c^2*(1 - a^2*x^2)^(5/2))/(5*a) - (3*c^2*asinh(x*(-a^2)^(1/2))*(-a^2)^(1/2))/(8*a^ 2)
Time = 0.16 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.22 \[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {c^{2} \left (15 \mathit {asin} \left (a x \right )-8 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-10 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+16 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+25 \sqrt {-a^{2} x^{2}+1}\, a x -8 \sqrt {-a^{2} x^{2}+1}+8\right )}{40 a} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^2,x)
Output:
(c**2*(15*asin(a*x) - 8*sqrt( - a**2*x**2 + 1)*a**4*x**4 - 10*sqrt( - a**2 *x**2 + 1)*a**3*x**3 + 16*sqrt( - a**2*x**2 + 1)*a**2*x**2 + 25*sqrt( - a* *2*x**2 + 1)*a*x - 8*sqrt( - a**2*x**2 + 1) + 8))/(40*a)