Integrand size = 23, antiderivative size = 88 \[ \int \frac {e^{\text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {1+a x}{3 a^4 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{a^4 c^2 \sqrt {1-a^2 x^2}}-\frac {4 x}{3 a^3 c^2 \sqrt {1-a^2 x^2}}+\frac {\arcsin (a x)}{a^4 c^2} \] Output:
1/3*(a*x+1)/a^4/c^2/(-a^2*x^2+1)^(3/2)-1/a^4/c^2/(-a^2*x^2+1)^(1/2)-4/3*x/ a^3/c^2/(-a^2*x^2+1)^(1/2)+arcsin(a*x)/a^4/c^2
Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.78 \[ \int \frac {e^{\text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {2+a x-4 a^2 x^2+3 (-1+a x) \sqrt {1-a^2 x^2} \arcsin (a x)}{3 a^4 c^2 (-1+a x) \sqrt {1-a^2 x^2}} \] Input:
Integrate[(E^ArcTanh[a*x]*x^3)/(c - a^2*c*x^2)^2,x]
Output:
(2 + a*x - 4*a^2*x^2 + 3*(-1 + a*x)*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(3*a^4* c^2*(-1 + a*x)*Sqrt[1 - a^2*x^2])
Time = 0.36 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.80, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6698, 529, 2345, 27, 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.
\(\displaystyle \int \frac {x^3 e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 6698 |
\(\displaystyle \frac {\int \frac {x^3 (a x+1)}{\left (1-a^2 x^2\right )^{5/2}}dx}{c^2}\) |
\(\Big \downarrow \) 529 |
\(\displaystyle \frac {\frac {a x+1}{3 a^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{3} \int \frac {\frac {3 x^2}{a}+\frac {3 x}{a^2}+\frac {1}{a^3}}{\left (1-a^2 x^2\right )^{3/2}}dx}{c^2}\) |
\(\Big \downarrow \) 2345 |
\(\displaystyle \frac {\frac {1}{3} \left (\int \frac {3}{a^3 \sqrt {1-a^2 x^2}}dx-\frac {4 a x+3}{a^4 \sqrt {1-a^2 x^2}}\right )+\frac {a x+1}{3 a^4 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {1}{3} \left (\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}}dx}{a^3}-\frac {4 a x+3}{a^4 \sqrt {1-a^2 x^2}}\right )+\frac {a x+1}{3 a^4 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {\frac {1}{3} \left (\frac {3 \arcsin (a x)}{a^4}-\frac {4 a x+3}{a^4 \sqrt {1-a^2 x^2}}\right )+\frac {a x+1}{3 a^4 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\) |
Input:
Int[(E^ArcTanh[a*x]*x^3)/(c - a^2*c*x^2)^2,x]
Output:
((1 + a*x)/(3*a^4*(1 - a^2*x^2)^(3/2)) + (-((3 + 4*a*x)/(a^4*Sqrt[1 - a^2* x^2])) + (3*ArcSin[a*x])/a^4)/3)/c^2
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[(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[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(195\) vs. \(2(78)=156\).
Time = 0.24 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.23
method | result | size |
default | \(\frac {\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{3} \sqrt {a^{2}}}+\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{5} \left (x +\frac {1}{a}\right )}+\frac {\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{2 a^{5}}+\frac {5 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{4 a^{5} \left (x -\frac {1}{a}\right )}}{c^{2}}\) | \(196\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3/(-a^2*c*x^2+c)^2,x,method=_RETURNVERBOS E)
Output:
1/c^2*(1/a^3/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+1/4/a^5/ (x+1/a)*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)+1/2/a^5*(1/3/a/(x-1/a)^2*(-(x-1 /a)^2*a^2-2*a*(x-1/a))^(1/2)-1/3/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2 ))+5/4/a^5/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2))
Time = 0.10 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.56 \[ \int \frac {e^{\text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^2} \, dx=-\frac {2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} - 2 \, a x + 6 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (4 \, a^{2} x^{2} - a x - 2\right )} \sqrt {-a^{2} x^{2} + 1} + 2}{3 \, {\left (a^{7} c^{2} x^{3} - a^{6} c^{2} x^{2} - a^{5} c^{2} x + a^{4} c^{2}\right )}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3/(-a^2*c*x^2+c)^2,x, algorithm="fr icas")
Output:
-1/3*(2*a^3*x^3 - 2*a^2*x^2 - 2*a*x + 6*(a^3*x^3 - a^2*x^2 - a*x + 1)*arct an((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (4*a^2*x^2 - a*x - 2)*sqrt(-a^2*x^2 + 1) + 2)/(a^7*c^2*x^3 - a^6*c^2*x^2 - a^5*c^2*x + a^4*c^2)
\[ \int \frac {e^{\text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {\int \frac {x^{3}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{4}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{2}} \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**3/(-a**2*c*x**2+c)**2,x)
Output:
(Integral(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(a*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)), x) )/c**2
\[ \int \frac {e^{\text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^2} \, dx=\int { \frac {{\left (a x + 1\right )} x^{3}}{{\left (a^{2} c x^{2} - c\right )}^{2} \sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3/(-a^2*c*x^2+c)^2,x, algorithm="ma xima")
Output:
a*integrate(x^4/((a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2)*sqrt(a*x + 1)*sqrt(-a *x + 1)), x) + 1/3*(3*a^2*x^2 - 2)/((-a^2*x^2 + 1)^(3/2)*a^4*c^2)
Exception generated. \[ \int \frac {e^{\text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3/(-a^2*c*x^2+c)^2,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 = 14.48 (sec) , antiderivative size = 176, normalized size of antiderivative = 2.00 \[ \int \frac {e^{\text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {\sqrt {1-a^2\,x^2}}{6\,\left (a^6\,c^2\,x^2-2\,a^5\,c^2\,x+a^4\,c^2\right )}-\frac {\sqrt {1-a^2\,x^2}}{4\,\left (a^2\,c^2\,\sqrt {-a^2}+a^3\,c^2\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}+\frac {13\,\sqrt {1-a^2\,x^2}}{12\,\left (a^2\,c^2\,\sqrt {-a^2}-a^3\,c^2\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}+\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{a^3\,c^2\,\sqrt {-a^2}} \] Input:
int((x^3*(a*x + 1))/((c - a^2*c*x^2)^2*(1 - a^2*x^2)^(1/2)),x)
Output:
(1 - a^2*x^2)^(1/2)/(6*(a^4*c^2 - 2*a^5*c^2*x + a^6*c^2*x^2)) - (1 - a^2*x ^2)^(1/2)/(4*(a^2*c^2*(-a^2)^(1/2) + a^3*c^2*x*(-a^2)^(1/2))*(-a^2)^(1/2)) + (13*(1 - a^2*x^2)^(1/2))/(12*(a^2*c^2*(-a^2)^(1/2) - a^3*c^2*x*(-a^2)^( 1/2))*(-a^2)^(1/2)) + asinh(x*(-a^2)^(1/2))/(a^3*c^2*(-a^2)^(1/2))
Time = 0.15 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.19 \[ \int \frac {e^{\text {arctanh}(a x)} x^3}{\left (c-a^2 c x^2\right )^2} \, dx=\frac {3 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a x -3 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )-2 \sqrt {-a^{2} x^{2}+1}\, a x +2 \sqrt {-a^{2} x^{2}+1}-4 a^{2} x^{2}+a x +2}{3 \sqrt {-a^{2} x^{2}+1}\, a^{4} c^{2} \left (a x -1\right )} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3/(-a^2*c*x^2+c)^2,x)
Output:
(3*sqrt( - a**2*x**2 + 1)*asin(a*x)*a*x - 3*sqrt( - a**2*x**2 + 1)*asin(a* x) - 2*sqrt( - a**2*x**2 + 1)*a*x + 2*sqrt( - a**2*x**2 + 1) - 4*a**2*x**2 + a*x + 2)/(3*sqrt( - a**2*x**2 + 1)*a**4*c**2*(a*x - 1))