Integrand size = 22, antiderivative size = 114 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {7}{8} c^2 x \sqrt {1-a^2 x^2}+\frac {7}{12} c^2 x \left (1-a^2 x^2\right )^{3/2}+\frac {7 c^2 \left (1-a^2 x^2\right )^{5/2}}{15 a}+\frac {2 c^2 \left (1-a^2 x^2\right )^{7/2}}{3 a (1+a x)^2}+\frac {7 c^2 \arcsin (a x)}{8 a} \] Output:
7/8*c^2*x*(-a^2*x^2+1)^(1/2)+7/12*c^2*x*(-a^2*x^2+1)^(3/2)+7/15*c^2*(-a^2* x^2+1)^(5/2)/a+2/3*c^2*(-a^2*x^2+1)^(7/2)/a/(a*x+1)^2+7/8*c^2*arcsin(a*x)/ a
Time = 0.13 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.66 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=-\frac {c^2 \left (\sqrt {1-a^2 x^2} \left (-136-15 a x+112 a^2 x^2-90 a^3 x^3+24 a^4 x^4\right )+210 \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{120 a} \] Input:
Integrate[(c - a^2*c*x^2)^2/E^(3*ArcTanh[a*x]),x]
Output:
-1/120*(c^2*(Sqrt[1 - a^2*x^2]*(-136 - 15*a*x + 112*a^2*x^2 - 90*a^3*x^3 + 24*a^4*x^4) + 210*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/a
Time = 0.48 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6689, 469, 469, 455, 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.
\(\displaystyle \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx\) |
\(\Big \downarrow \) 6689 |
\(\displaystyle c^2 \int (1-a x)^3 \sqrt {1-a^2 x^2}dx\) |
\(\Big \downarrow \) 469 |
\(\displaystyle c^2 \left (\frac {7}{5} \int (1-a x)^2 \sqrt {1-a^2 x^2}dx+\frac {\left (1-a^2 x^2\right )^{3/2} (1-a x)^2}{5 a}\right )\) |
\(\Big \downarrow \) 469 |
\(\displaystyle c^2 \left (\frac {7}{5} \left (\frac {5}{4} \int (1-a x) \sqrt {1-a^2 x^2}dx+\frac {(1-a x) \left (1-a^2 x^2\right )^{3/2}}{4 a}\right )+\frac {\left (1-a^2 x^2\right )^{3/2} (1-a x)^2}{5 a}\right )\) |
\(\Big \downarrow \) 455 |
\(\displaystyle c^2 \left (\frac {7}{5} \left (\frac {5}{4} \left (\int \sqrt {1-a^2 x^2}dx+\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a}\right )+\frac {(1-a x) \left (1-a^2 x^2\right )^{3/2}}{4 a}\right )+\frac {\left (1-a^2 x^2\right )^{3/2} (1-a x)^2}{5 a}\right )\) |
\(\Big \downarrow \) 211 |
\(\displaystyle c^2 \left (\frac {7}{5} \left (\frac {5}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-a^2 x^2}}dx+\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac {1}{2} x \sqrt {1-a^2 x^2}\right )+\frac {(1-a x) \left (1-a^2 x^2\right )^{3/2}}{4 a}\right )+\frac {\left (1-a^2 x^2\right )^{3/2} (1-a x)^2}{5 a}\right )\) |
\(\Big \downarrow \) 223 |
\(\displaystyle c^2 \left (\frac {7}{5} \left (\frac {5}{4} \left (\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac {1}{2} x \sqrt {1-a^2 x^2}+\frac {\arcsin (a x)}{2 a}\right )+\frac {(1-a x) \left (1-a^2 x^2\right )^{3/2}}{4 a}\right )+\frac {\left (1-a^2 x^2\right )^{3/2} (1-a x)^2}{5 a}\right )\) |
Input:
Int[(c - a^2*c*x^2)^2/E^(3*ArcTanh[a*x]),x]
Output:
c^2*(((1 - a*x)^2*(1 - a^2*x^2)^(3/2))/(5*a) + (7*(((1 - a*x)*(1 - a^2*x^2 )^(3/2))/(4*a) + (5*((x*Sqrt[1 - a^2*x^2])/2 + (1 - a^2*x^2)^(3/2)/(3*a) + ArcSin[a*x]/(2*a)))/4))/5)
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[((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 = 0.20 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.80
method | result | size |
risch | \(\frac {\left (24 a^{4} x^{4}-90 a^{3} x^{3}+112 a^{2} x^{2}-15 a x -136\right ) \left (a^{2} x^{2}-1\right ) c^{2}}{120 a \sqrt {-a^{2} x^{2}+1}}+\frac {7 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{2}}{8 \sqrt {a^{2}}}\) | \(91\) |
default | \(c^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{5 a}-\frac {3 x \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4}-\frac {9 x \sqrt {-a^{2} x^{2}+1}}{8}-\frac {9 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}}+\frac {\frac {4 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+4 a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )}{a}\right )\) | \(194\) |
Input:
int((-a^2*c*x^2+c)^2/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/120*(24*a^4*x^4-90*a^3*x^3+112*a^2*x^2-15*a*x-136)*(a^2*x^2-1)/a/(-a^2*x ^2+1)^(1/2)*c^2+7/8/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))*c ^2
Time = 0.10 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.81 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=-\frac {210 \, c^{2} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (24 \, a^{4} c^{2} x^{4} - 90 \, a^{3} c^{2} x^{3} + 112 \, a^{2} c^{2} x^{2} - 15 \, a c^{2} x - 136 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{120 \, a} \] Input:
integrate((-a^2*c*x^2+c)^2/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="fric as")
Output:
-1/120*(210*c^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (24*a^4*c^2*x^4 - 90*a^3*c^2*x^3 + 112*a^2*c^2*x^2 - 15*a*c^2*x - 136*c^2)*sqrt(-a^2*x^2 + 1))/a
Time = 1.85 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.00 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=- a^{3} c^{2} \left (\begin {cases} \sqrt {- a^{2} x^{2} + 1} \left (\frac {x^{4}}{5} - \frac {x^{2}}{15 a^{2}} - \frac {2}{15 a^{4}}\right ) & \text {for}\: a^{2} \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) + 3 a^{2} c^{2} \left (\begin {cases} \left (\frac {x^{3}}{4} - \frac {x}{8 a^{2}}\right ) \sqrt {- a^{2} x^{2} + 1} + \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{8 a^{2} \sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{3}}{3} & \text {otherwise} \end {cases}\right ) - 3 a c^{2} \left (\begin {cases} \left (\frac {x^{2}}{3} - \frac {1}{3 a^{2}}\right ) \sqrt {- a^{2} x^{2} + 1} & \text {for}\: a^{2} \neq 0 \\\frac {x^{2}}{2} & \text {otherwise} \end {cases}\right ) + c^{2} \left (\begin {cases} \frac {x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{2 \sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\x & \text {otherwise} \end {cases}\right ) \] Input:
integrate((-a**2*c*x**2+c)**2/(a*x+1)**3*(-a**2*x**2+1)**(3/2),x)
Output:
-a**3*c**2*Piecewise((sqrt(-a**2*x**2 + 1)*(x**4/5 - x**2/(15*a**2) - 2/(1 5*a**4)), Ne(a**2, 0)), (x**4/4, True)) + 3*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)) - 3*a*c* *2*Piecewise(((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*sqrt(-a**2)*sqrt(-a**2*x**2 + 1))/(2*sqrt(-a**2)), Ne(a**2, 0)), (x, Tr ue))
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.29 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=-\frac {3}{4} \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{2} x - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} c^{2}}{5 \, a} + 2 \, \sqrt {a^{2} x^{2} + 4 \, a x + 3} c^{2} x - \frac {9}{8} \, \sqrt {-a^{2} x^{2} + 1} c^{2} x + \frac {4 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{2}}{3 \, a} - \frac {2 i \, c^{2} \arcsin \left (a x + 2\right )}{a} - \frac {9 \, c^{2} \arcsin \left (a x\right )}{8 \, a} + \frac {4 \, \sqrt {a^{2} x^{2} + 4 \, a x + 3} c^{2}}{a} \] Input:
integrate((-a^2*c*x^2+c)^2/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="maxi ma")
Output:
-3/4*(-a^2*x^2 + 1)^(3/2)*c^2*x - 1/5*(-a^2*x^2 + 1)^(5/2)*c^2/a + 2*sqrt( a^2*x^2 + 4*a*x + 3)*c^2*x - 9/8*sqrt(-a^2*x^2 + 1)*c^2*x + 4/3*(-a^2*x^2 + 1)^(3/2)*c^2/a - 2*I*c^2*arcsin(a*x + 2)/a - 9/8*c^2*arcsin(a*x)/a + 4*s qrt(a^2*x^2 + 4*a*x + 3)*c^2/a
Time = 0.14 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.68 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {7 \, c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{8 \, {\left | a \right |}} + \frac {1}{120} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (15 \, c^{2} - 2 \, {\left (56 \, a c^{2} + 3 \, {\left (4 \, a^{3} c^{2} x - 15 \, a^{2} c^{2}\right )} x\right )} x\right )} x + \frac {136 \, c^{2}}{a}\right )} \] Input:
integrate((-a^2*c*x^2+c)^2/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x, algorithm="giac ")
Output:
7/8*c^2*arcsin(a*x)*sgn(a)/abs(a) + 1/120*sqrt(-a^2*x^2 + 1)*((15*c^2 - 2* (56*a*c^2 + 3*(4*a^3*c^2*x - 15*a^2*c^2)*x)*x)*x + 136*c^2/a)
Time = 0.04 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.12 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {c^2\,x\,\sqrt {1-a^2\,x^2}}{8}+\frac {7\,c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,\sqrt {-a^2}}+\frac {17\,c^2\,\sqrt {1-a^2\,x^2}}{15\,a}-\frac {14\,a\,c^2\,x^2\,\sqrt {1-a^2\,x^2}}{15}+\frac {3\,a^2\,c^2\,x^3\,\sqrt {1-a^2\,x^2}}{4}-\frac {a^3\,c^2\,x^4\,\sqrt {1-a^2\,x^2}}{5} \] Input:
int(((c - a^2*c*x^2)^2*(1 - a^2*x^2)^(3/2))/(a*x + 1)^3,x)
Output:
(c^2*x*(1 - a^2*x^2)^(1/2))/8 + (7*c^2*asinh(x*(-a^2)^(1/2)))/(8*(-a^2)^(1 /2)) + (17*c^2*(1 - a^2*x^2)^(1/2))/(15*a) - (14*a*c^2*x^2*(1 - a^2*x^2)^( 1/2))/15 + (3*a^2*c^2*x^3*(1 - a^2*x^2)^(1/2))/4 - (a^3*c^2*x^4*(1 - a^2*x ^2)^(1/2))/5
Time = 0.15 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.89 \[ \int e^{-3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {c^{2} \left (105 \mathit {asin} \left (a x \right )-24 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+90 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-112 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+15 \sqrt {-a^{2} x^{2}+1}\, a x +136 \sqrt {-a^{2} x^{2}+1}-136\right )}{120 a} \] Input:
int((-a^2*c*x^2+c)^2/(a*x+1)^3*(-a^2*x^2+1)^(3/2),x)
Output:
(c**2*(105*asin(a*x) - 24*sqrt( - a**2*x**2 + 1)*a**4*x**4 + 90*sqrt( - a* *2*x**2 + 1)*a**3*x**3 - 112*sqrt( - a**2*x**2 + 1)*a**2*x**2 + 15*sqrt( - a**2*x**2 + 1)*a*x + 136*sqrt( - a**2*x**2 + 1) - 136))/(120*a)