Integrand size = 25, antiderivative size = 93 \[ \int \frac {e^{3 \text {arctanh}(a x)} x^2}{c-a^2 c x^2} \, dx=\frac {(1+a x)^3}{3 a^3 c \left (1-a^2 x^2\right )^{3/2}}-\frac {4 (1+a x)}{a^3 c \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a^3 c}+\frac {3 \arcsin (a x)}{a^3 c} \] Output:
1/3*(a*x+1)^3/a^3/c/(-a^2*x^2+1)^(3/2)-4*(a*x+1)/a^3/c/(-a^2*x^2+1)^(1/2)- (-a^2*x^2+1)^(1/2)/a^3/c+3*arcsin(a*x)/a^3/c
Time = 0.10 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.84 \[ \int \frac {e^{3 \text {arctanh}(a x)} x^2}{c-a^2 c x^2} \, dx=\frac {14-5 a x-16 a^2 x^2+3 a^3 x^3+9 (-1+a x) \sqrt {1-a^2 x^2} \arcsin (a x)}{3 a^3 c (-1+a x) \sqrt {1-a^2 x^2}} \] Input:
Integrate[(E^(3*ArcTanh[a*x])*x^2)/(c - a^2*c*x^2),x]
Output:
(14 - 5*a*x - 16*a^2*x^2 + 3*a^3*x^3 + 9*(-1 + a*x)*Sqrt[1 - a^2*x^2]*ArcS in[a*x])/(3*a^3*c*(-1 + a*x)*Sqrt[1 - a^2*x^2])
Time = 0.56 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6698, 529, 27, 462, 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.
\(\displaystyle \int \frac {x^2 e^{3 \text {arctanh}(a x)}}{c-a^2 c x^2} \, dx\) |
\(\Big \downarrow \) 6698 |
\(\displaystyle \frac {\int \frac {x^2 (a x+1)^3}{\left (1-a^2 x^2\right )^{5/2}}dx}{c}\) |
\(\Big \downarrow \) 529 |
\(\displaystyle \frac {\frac {(a x+1)^3}{3 a^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{3} \int \frac {3 (a x+1)^3}{a^2 \left (1-a^2 x^2\right )^{3/2}}dx}{c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {(a x+1)^3}{3 a^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {\int \frac {(a x+1)^3}{\left (1-a^2 x^2\right )^{3/2}}dx}{a^2}}{c}\) |
\(\Big \downarrow \) 462 |
\(\displaystyle \frac {\frac {(a x+1)^3}{3 a^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {\frac {4 (a x+1)}{a \sqrt {1-a^2 x^2}}-\int \frac {a x+3}{\sqrt {1-a^2 x^2}}dx}{a^2}}{c}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle \frac {\frac {(a x+1)^3}{3 a^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {-3 \int \frac {1}{\sqrt {1-a^2 x^2}}dx+\frac {4 (a x+1)}{a \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a}}{a^2}}{c}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {\frac {(a x+1)^3}{3 a^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {\frac {4 (a x+1)}{a \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a}-\frac {3 \arcsin (a x)}{a}}{a^2}}{c}\) |
Input:
Int[(E^(3*ArcTanh[a*x])*x^2)/(c - a^2*c*x^2),x]
Output:
((1 + a*x)^3/(3*a^3*(1 - a^2*x^2)^(3/2)) - ((4*(1 + a*x))/(a*Sqrt[1 - a^2* x^2]) + Sqrt[1 - a^2*x^2]/a - (3*ArcSin[a*x])/a)/a^2)/c
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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)^(3/2), x_Symbol] :> Simp [(-2^(n - 1))*d*c^(n - 2)*((c + d*x)/(b*Sqrt[a + b*x^2])), x] + Simp[d^2/b Int[(1/Sqrt[a + b*x^2])*ExpandToSum[(2^(n - 1)*c^(n - 1) - (c + d*x)^(n - 1))/(c - d*x), x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && IGtQ[n, 2]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m, a*d + b*c*x, x], R = PolynomialRem ainder[x^m, a*d + b*c*x, x]}, Simp[(-c)*R*(c + d*x)^n*((a + b*x^2)^(p + 1)/ (2*a*d*(p + 1))), x] + Simp[c/(2*a*(p + 1)) Int[(c + d*x)^(n - 1)*(a + b* x^2)^(p + 1)*ExpandToSum[2*a*d*(p + 1)*Qx + R*(n + 2*p + 2), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 1] && LtQ[p, -1] && EqQ[b* c^2 + a*d^2, 0]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[c^p Int[x^m*(1 - a^2*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0]) && IGtQ[(n + 1)/2, 0] && !IntegerQ[p - n/2]
Time = 0.23 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.58
method | result | size |
risch | \(\frac {a^{2} x^{2}-1}{a^{3} \sqrt {-a^{2} x^{2}+1}\, c}+\frac {\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}+\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a^{5} \left (x -\frac {1}{a}\right )^{2}}+\frac {13 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a^{4} \left (x -\frac {1}{a}\right )}}{c}\) | \(147\) |
default | \(-\frac {a \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )+\frac {7 x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}+\frac {4}{a^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {4}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {4 \left (-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a \right )}{3 a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}}{a^{3}}}{c}\) | \(207\) |
Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^2/(-a^2*c*x^2+c),x,method=_RETURNVERBOS E)
Output:
1/a^3*(a^2*x^2-1)/(-a^2*x^2+1)^(1/2)/c+(3/a^2/(a^2)^(1/2)*arctan((a^2)^(1/ 2)*x/(-a^2*x^2+1)^(1/2))+2/3/a^5/(x-1/a)^2*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1 /2)+13/3/a^4/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2))/c
Time = 0.09 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.11 \[ \int \frac {e^{3 \text {arctanh}(a x)} x^2}{c-a^2 c x^2} \, dx=-\frac {14 \, a^{2} x^{2} - 28 \, a x + 18 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (3 \, a^{2} x^{2} - 19 \, a x + 14\right )} \sqrt {-a^{2} x^{2} + 1} + 14}{3 \, {\left (a^{5} c x^{2} - 2 \, a^{4} c x + a^{3} c\right )}} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^2/(-a^2*c*x^2+c),x, algorithm="fr icas")
Output:
-1/3*(14*a^2*x^2 - 28*a*x + 18*(a^2*x^2 - 2*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (3*a^2*x^2 - 19*a*x + 14)*sqrt(-a^2*x^2 + 1) + 14)/(a^ 5*c*x^2 - 2*a^4*c*x + a^3*c)
\[ \int \frac {e^{3 \text {arctanh}(a x)} x^2}{c-a^2 c x^2} \, dx=\frac {\int \frac {x^{2}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a x^{3}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} x^{4}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{3} x^{5}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c} \] Input:
integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x**2/(-a**2*c*x**2+c),x)
Output:
(Integral(x**2/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x* *2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(3*a*x**3/(a**4*x**4*sqrt(-a **2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(3*a**2*x**4/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sq rt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a**3*x**5/(a**4* x**4*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2* x**2 + 1)), x))/c
Leaf count of result is larger than twice the leaf count of optimal. 364 vs. \(2 (85) = 170\).
Time = 0.21 (sec) , antiderivative size = 364, normalized size of antiderivative = 3.91 \[ \int \frac {e^{3 \text {arctanh}(a x)} x^2}{c-a^2 c x^2} \, dx=\frac {1}{6} \, {\left (\frac {a^{3} c^{3}}{\sqrt {-a^{2} x^{2} + 1} a^{8} c^{4} x + \sqrt {-a^{2} x^{2} + 1} a^{7} c^{4}} - \frac {a^{3} c^{3}}{\sqrt {-a^{2} x^{2} + 1} a^{8} c^{4} x - \sqrt {-a^{2} x^{2} + 1} a^{7} c^{4}} + \frac {3 \, a c}{\sqrt {-a^{2} x^{2} + 1} a^{6} c^{2} x + \sqrt {-a^{2} x^{2} + 1} a^{5} c^{2}} - \frac {3 \, a c}{\sqrt {-a^{2} x^{2} + 1} a^{6} c^{2} x - \sqrt {-a^{2} x^{2} + 1} a^{5} c^{2}} - \frac {4 \, c}{\sqrt {-a^{2} x^{2} + 1} a^{5} c^{2} x + \sqrt {-a^{2} x^{2} + 1} a^{4} c^{2}} - \frac {4 \, c}{\sqrt {-a^{2} x^{2} + 1} a^{5} c^{2} x - \sqrt {-a^{2} x^{2} + 1} a^{4} c^{2}} + \frac {6 \, x^{2}}{\sqrt {-a^{2} x^{2} + 1} a^{2} c} - \frac {26 \, x}{\sqrt {-a^{2} x^{2} + 1} a^{3} c} + \frac {18 \, \arcsin \left (a x\right )}{a^{4} c} - \frac {36}{\sqrt {-a^{2} x^{2} + 1} a^{4} c}\right )} a \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^2/(-a^2*c*x^2+c),x, algorithm="ma xima")
Output:
1/6*(a^3*c^3/(sqrt(-a^2*x^2 + 1)*a^8*c^4*x + sqrt(-a^2*x^2 + 1)*a^7*c^4) - a^3*c^3/(sqrt(-a^2*x^2 + 1)*a^8*c^4*x - sqrt(-a^2*x^2 + 1)*a^7*c^4) + 3*a *c/(sqrt(-a^2*x^2 + 1)*a^6*c^2*x + sqrt(-a^2*x^2 + 1)*a^5*c^2) - 3*a*c/(sq rt(-a^2*x^2 + 1)*a^6*c^2*x - sqrt(-a^2*x^2 + 1)*a^5*c^2) - 4*c/(sqrt(-a^2* x^2 + 1)*a^5*c^2*x + sqrt(-a^2*x^2 + 1)*a^4*c^2) - 4*c/(sqrt(-a^2*x^2 + 1) *a^5*c^2*x - sqrt(-a^2*x^2 + 1)*a^4*c^2) + 6*x^2/(sqrt(-a^2*x^2 + 1)*a^2*c ) - 26*x/(sqrt(-a^2*x^2 + 1)*a^3*c) + 18*arcsin(a*x)/(a^4*c) - 36/(sqrt(-a ^2*x^2 + 1)*a^4*c))*a
Exception generated. \[ \int \frac {e^{3 \text {arctanh}(a x)} x^2}{c-a^2 c x^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^2/(-a^2*c*x^2+c),x, algorithm="gi ac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Time = 23.91 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.43 \[ \int \frac {e^{3 \text {arctanh}(a x)} x^2}{c-a^2 c x^2} \, dx=\frac {2\,\sqrt {1-a^2\,x^2}}{3\,\left (c\,a^5\,x^2-2\,c\,a^4\,x+c\,a^3\right )}+\frac {13\,\sqrt {1-a^2\,x^2}}{3\,\left (a\,c\,\sqrt {-a^2}-a^2\,c\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a^3\,c}+\frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{a^2\,c\,\sqrt {-a^2}} \] Input:
int((x^2*(a*x + 1)^3)/((c - a^2*c*x^2)*(1 - a^2*x^2)^(3/2)),x)
Output:
(2*(1 - a^2*x^2)^(1/2))/(3*(a^3*c + a^5*c*x^2 - 2*a^4*c*x)) + (13*(1 - a^2 *x^2)^(1/2))/(3*(a*c*(-a^2)^(1/2) - a^2*c*x*(-a^2)^(1/2))*(-a^2)^(1/2)) - (1 - a^2*x^2)^(1/2)/(a^3*c) + (3*asinh(x*(-a^2)^(1/2)))/(a^2*c*(-a^2)^(1/2 ))
Time = 0.16 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.95 \[ \int \frac {e^{3 \text {arctanh}(a x)} x^2}{c-a^2 c x^2} \, dx=\frac {9 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a x -9 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )+9 \mathit {asin} \left (a x \right ) a^{2} x^{2}-18 \mathit {asin} \left (a x \right ) a x +9 \mathit {asin} \left (a x \right )-3 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+11 \sqrt {-a^{2} x^{2}+1}\, a x -6 \sqrt {-a^{2} x^{2}+1}+3 a^{3} x^{3}-24 a^{2} x^{2}+11 a x +6}{3 a^{3} c \left (\sqrt {-a^{2} x^{2}+1}\, a x -\sqrt {-a^{2} x^{2}+1}+a^{2} x^{2}-2 a x +1\right )} \] Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^2/(-a^2*c*x^2+c),x)
Output:
(9*sqrt( - a**2*x**2 + 1)*asin(a*x)*a*x - 9*sqrt( - a**2*x**2 + 1)*asin(a* x) + 9*asin(a*x)*a**2*x**2 - 18*asin(a*x)*a*x + 9*asin(a*x) - 3*sqrt( - a* *2*x**2 + 1)*a**2*x**2 + 11*sqrt( - a**2*x**2 + 1)*a*x - 6*sqrt( - a**2*x* *2 + 1) + 3*a**3*x**3 - 24*a**2*x**2 + 11*a*x + 6)/(3*a**3*c*(sqrt( - a**2 *x**2 + 1)*a*x - sqrt( - a**2*x**2 + 1) + a**2*x**2 - 2*a*x + 1))