Integrand size = 19, antiderivative size = 107 \[ \int \frac {e^{\text {arctanh}(a x)} x^2}{(c-a c x)^3} \, dx=\frac {2 \sqrt {1-a^2 x^2}}{a^3 c^3 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a^3 c^3 (1-a x)^4}-\frac {3 \left (1-a^2 x^2\right )^{3/2}}{5 a^3 c^3 (1-a x)^3}-\frac {\arcsin (a x)}{a^3 c^3} \] Output:
2*(-a^2*x^2+1)^(1/2)/a^3/c^3/(-a*x+1)+1/5*(-a^2*x^2+1)^(3/2)/a^3/c^3/(-a*x +1)^4-3/5*(-a^2*x^2+1)^(3/2)/a^3/c^3/(-a*x+1)^3-arcsin(a*x)/a^3/c^3
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.72 \[ \int \frac {e^{\text {arctanh}(a x)} x^2}{(c-a c x)^3} \, dx=\frac {\sqrt {1+a x} \left (4+3 a x-a^2 x^2\right )+20 \sqrt {2} (-1+a x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},\frac {1}{2} (1-a x)\right )}{15 a^3 c^3 (1-a x)^{5/2}} \] Input:
Integrate[(E^ArcTanh[a*x]*x^2)/(c - a*c*x)^3,x]
Output:
(Sqrt[1 + a*x]*(4 + 3*a*x - a^2*x^2) + 20*Sqrt[2]*(-1 + a*x)*Hypergeometri c2F1[-3/2, -3/2, -1/2, (1 - a*x)/2])/(15*a^3*c^3*(1 - a*x)^(5/2))
Time = 0.37 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {6678, 27, 582, 2009}
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.
\(\displaystyle \int \frac {x^2 e^{\text {arctanh}(a x)}}{(c-a c x)^3} \, dx\) |
\(\Big \downarrow \) 6678 |
\(\displaystyle c \int \frac {x^2 \sqrt {1-a^2 x^2}}{c^4 (1-a x)^4}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {x^2 \sqrt {1-a^2 x^2}}{(1-a x)^4}dx}{c^3}\) |
\(\Big \downarrow \) 582 |
\(\displaystyle \frac {\int \left (\frac {\sqrt {1-a^2 x^2}}{a^2 (a x-1)^2}+\frac {2 \sqrt {1-a^2 x^2}}{a^2 (a x-1)^3}+\frac {\sqrt {1-a^2 x^2}}{a^2 (a x-1)^4}\right )dx}{c^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {\arcsin (a x)}{a^3}-\frac {3 \left (1-a^2 x^2\right )^{3/2}}{5 a^3 (1-a x)^3}+\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a^3 (1-a x)^4}+\frac {2 \sqrt {1-a^2 x^2}}{a^3 (1-a x)}}{c^3}\) |
Input:
Int[(E^ArcTanh[a*x]*x^2)/(c - a*c*x)^3,x]
Output:
((2*Sqrt[1 - a^2*x^2])/(a^3*(1 - a*x)) + (1 - a^2*x^2)^(3/2)/(5*a^3*(1 - a *x)^4) - (3*(1 - a^2*x^2)^(3/2))/(5*a^3*(1 - a*x)^3) - ArcSin[a*x]/a^3)/c^ 3
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Int[ExpandIntegrand[x^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a , b, c, d, p}, x] && EqQ[b*c^2 + a*d^2, 0] && IntegerQ[2*p] && IntegerQ[m] && ILtQ[n, 0]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.)*((e_.) + (f_.)* (x_))^(m_.), x_Symbol] :> Simp[c^n Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1 , 0]) && IntegerQ[2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(291\) vs. \(2(97)=194\).
Time = 0.24 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.73
method | result | size |
default | \(-\frac {\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}+\frac {\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {4 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{a^{5}}+\frac {\frac {5 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {5 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{a^{4}}+\frac {4 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{4} \left (x -\frac {1}{a}\right )}}{c^{3}}\) | \(292\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2/(-a*c*x+c)^3,x,method=_RETURNVERBOSE)
Output:
-1/c^3*(1/a^2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+2/a^5*( 1/5/a/(x-1/a)^3*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)-2/5*a*(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/a^4*(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))+4/a^4/(x-1/a)*(-(x-1/a)^2*a^2-2* a*(x-1/a))^(1/2))
Time = 0.09 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.29 \[ \int \frac {e^{\text {arctanh}(a x)} x^2}{(c-a c x)^3} \, dx=\frac {8 \, a^{3} x^{3} - 24 \, a^{2} x^{2} + 24 \, a x + 10 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (13 \, a^{2} x^{2} - 19 \, a x + 8\right )} \sqrt {-a^{2} x^{2} + 1} - 8}{5 \, {\left (a^{6} c^{3} x^{3} - 3 \, a^{5} c^{3} x^{2} + 3 \, a^{4} c^{3} x - a^{3} c^{3}\right )}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2/(-a*c*x+c)^3,x, algorithm="fricas ")
Output:
1/5*(8*a^3*x^3 - 24*a^2*x^2 + 24*a*x + 10*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1 )*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (13*a^2*x^2 - 19*a*x + 8)*sqrt( -a^2*x^2 + 1) - 8)/(a^6*c^3*x^3 - 3*a^5*c^3*x^2 + 3*a^4*c^3*x - a^3*c^3)
\[ \int \frac {e^{\text {arctanh}(a x)} x^2}{(c-a c x)^3} \, dx=- \frac {\int \frac {x^{2}}{a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{3}}{a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**2/(-a*c*x+c)**3,x)
Output:
-(Integral(x**2/(a**3*x**3*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x **2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integr al(a*x**3/(a**3*x**3*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c**3
Time = 0.11 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.35 \[ \int \frac {e^{\text {arctanh}(a x)} x^2}{(c-a c x)^3} \, dx=-\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{5 \, {\left (a^{6} c^{3} x^{3} - 3 \, a^{5} c^{3} x^{2} + 3 \, a^{4} c^{3} x - a^{3} c^{3}\right )}} - \frac {7 \, \sqrt {-a^{2} x^{2} + 1}}{5 \, {\left (a^{5} c^{3} x^{2} - 2 \, a^{4} c^{3} x + a^{3} c^{3}\right )}} - \frac {13 \, \sqrt {-a^{2} x^{2} + 1}}{5 \, {\left (a^{4} c^{3} x - a^{3} c^{3}\right )}} - \frac {\arcsin \left (a x\right )}{a^{3} c^{3}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2/(-a*c*x+c)^3,x, algorithm="maxima ")
Output:
-2/5*sqrt(-a^2*x^2 + 1)/(a^6*c^3*x^3 - 3*a^5*c^3*x^2 + 3*a^4*c^3*x - a^3*c ^3) - 7/5*sqrt(-a^2*x^2 + 1)/(a^5*c^3*x^2 - 2*a^4*c^3*x + a^3*c^3) - 13/5* sqrt(-a^2*x^2 + 1)/(a^4*c^3*x - a^3*c^3) - arcsin(a*x)/(a^3*c^3)
Exception generated. \[ \int \frac {e^{\text {arctanh}(a x)} x^2}{(c-a c x)^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2/(-a*c*x+c)^3,x, algorithm="giac")
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 = 0.03 (sec) , antiderivative size = 259, normalized size of antiderivative = 2.42 \[ \int \frac {e^{\text {arctanh}(a x)} x^2}{(c-a c x)^3} \, dx=\frac {4\,a^2\,\sqrt {1-a^2\,x^2}}{15\,\left (a^7\,c^3\,x^2-2\,a^6\,c^3\,x+a^5\,c^3\right )}-\frac {13\,\sqrt {1-a^2\,x^2}}{5\,\left (a\,c^3\,\sqrt {-a^2}-a^2\,c^3\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {5\,\sqrt {1-a^2\,x^2}}{3\,\left (a^5\,c^3\,x^2-2\,a^4\,c^3\,x+a^3\,c^3\right )}-\frac {2\,\sqrt {1-a^2\,x^2}}{5\,\sqrt {-a^2}\,\left (a\,c^3\,\sqrt {-a^2}+3\,a^3\,c^3\,x^2\,\sqrt {-a^2}-a^4\,c^3\,x^3\,\sqrt {-a^2}-3\,a^2\,c^3\,x\,\sqrt {-a^2}\right )}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{a^2\,c^3\,\sqrt {-a^2}} \] Input:
int((x^2*(a*x + 1))/((1 - a^2*x^2)^(1/2)*(c - a*c*x)^3),x)
Output:
(4*a^2*(1 - a^2*x^2)^(1/2))/(15*(a^5*c^3 - 2*a^6*c^3*x + a^7*c^3*x^2)) - ( 13*(1 - a^2*x^2)^(1/2))/(5*(a*c^3*(-a^2)^(1/2) - a^2*c^3*x*(-a^2)^(1/2))*( -a^2)^(1/2)) - (5*(1 - a^2*x^2)^(1/2))/(3*(a^3*c^3 - 2*a^4*c^3*x + a^5*c^3 *x^2)) - (2*(1 - a^2*x^2)^(1/2))/(5*(-a^2)^(1/2)*(a*c^3*(-a^2)^(1/2) + 3*a ^3*c^3*x^2*(-a^2)^(1/2) - a^4*c^3*x^3*(-a^2)^(1/2) - 3*a^2*c^3*x*(-a^2)^(1 /2))) - asinh(x*(-a^2)^(1/2))/(a^2*c^3*(-a^2)^(1/2))
Time = 0.15 (sec) , antiderivative size = 241, normalized size of antiderivative = 2.25 \[ \int \frac {e^{\text {arctanh}(a x)} x^2}{(c-a c x)^3} \, dx=\frac {-5 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a^{2} x^{2}+10 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a x -5 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )-5 \mathit {asin} \left (a x \right ) a^{3} x^{3}+15 \mathit {asin} \left (a x \right ) a^{2} x^{2}-15 \mathit {asin} \left (a x \right ) a x +5 \mathit {asin} \left (a x \right )-7 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+7 \sqrt {-a^{2} x^{2}+1}\, a x -2 \sqrt {-a^{2} x^{2}+1}+19 a^{3} x^{3}-24 a^{2} x^{2}+7 a x +2}{5 a^{3} c^{3} \left (\sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-2 \sqrt {-a^{2} x^{2}+1}\, a x +\sqrt {-a^{2} x^{2}+1}+a^{3} x^{3}-3 a^{2} x^{2}+3 a x -1\right )} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^2/(-a*c*x+c)^3,x)
Output:
( - 5*sqrt( - a**2*x**2 + 1)*asin(a*x)*a**2*x**2 + 10*sqrt( - a**2*x**2 + 1)*asin(a*x)*a*x - 5*sqrt( - a**2*x**2 + 1)*asin(a*x) - 5*asin(a*x)*a**3*x **3 + 15*asin(a*x)*a**2*x**2 - 15*asin(a*x)*a*x + 5*asin(a*x) - 7*sqrt( - a**2*x**2 + 1)*a**2*x**2 + 7*sqrt( - a**2*x**2 + 1)*a*x - 2*sqrt( - a**2*x **2 + 1) + 19*a**3*x**3 - 24*a**2*x**2 + 7*a*x + 2)/(5*a**3*c**3*(sqrt( - a**2*x**2 + 1)*a**2*x**2 - 2*sqrt( - a**2*x**2 + 1)*a*x + sqrt( - a**2*x** 2 + 1) + a**3*x**3 - 3*a**2*x**2 + 3*a*x - 1))