Integrand size = 19, antiderivative size = 138 \[ \int \frac {e^{\text {arctanh}(a x)} x^3}{(c-a c x)^4} \, dx=-\frac {2 \sqrt {1-a^2 x^2}}{a^4 c^4 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^4 c^4 (1-a x)^5}-\frac {19 \left (1-a^2 x^2\right )^{3/2}}{35 a^4 c^4 (1-a x)^4}+\frac {86 \left (1-a^2 x^2\right )^{3/2}}{105 a^4 c^4 (1-a x)^3}+\frac {\arcsin (a x)}{a^4 c^4} \] Output:
-2*(-a^2*x^2+1)^(1/2)/a^4/c^4/(-a*x+1)+1/7*(-a^2*x^2+1)^(3/2)/a^4/c^4/(-a* x+1)^5-19/35*(-a^2*x^2+1)^(3/2)/a^4/c^4/(-a*x+1)^4+86/105*(-a^2*x^2+1)^(3/ 2)/a^4/c^4/(-a*x+1)^3+arcsin(a*x)/a^4/c^4
Time = 0.25 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.68 \[ \int \frac {e^{\text {arctanh}(a x)} x^3}{(c-a c x)^4} \, dx=\frac {\sqrt {1+a x} \left (\sqrt {1-a^2 x^2} \left (-166+559 a x-659 a^2 x^2+296 a^3 x^3\right )+105 (-1+a x)^4 \arcsin (a x)\right )}{105 a^4 c^4 (1-a x)^{7/2} \sqrt {1-a^2 x^2}} \] Input:
Integrate[(E^ArcTanh[a*x]*x^3)/(c - a*c*x)^4,x]
Output:
(Sqrt[1 + a*x]*(Sqrt[1 - a^2*x^2]*(-166 + 559*a*x - 659*a^2*x^2 + 296*a^3* x^3) + 105*(-1 + a*x)^4*ArcSin[a*x]))/(105*a^4*c^4*(1 - a*x)^(7/2)*Sqrt[1 - a^2*x^2])
Time = 0.67 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.92, 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.
| \(\displaystyle \int \frac {x^3 e^{\text {arctanh}(a x)}}{(c-a c x)^4} \, dx\) |
| \(\Big \downarrow \) 6678 |
| \(\displaystyle c \int \frac {x^3 \sqrt {1-a^2 x^2}}{c^5 (1-a x)^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {x^3 \sqrt {1-a^2 x^2}}{(1-a x)^5}dx}{c^4}\) |
| \(\Big \downarrow \) 582 |
| \(\displaystyle \frac {\int \left (-\frac {\sqrt {1-a^2 x^2}}{a^3 (a x-1)^2}-\frac {3 \sqrt {1-a^2 x^2}}{a^3 (a x-1)^3}-\frac {3 \sqrt {1-a^2 x^2}}{a^3 (a x-1)^4}-\frac {\sqrt {1-a^2 x^2}}{a^3 (a x-1)^5}\right )dx}{c^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\arcsin (a x)}{a^4}+\frac {86 \left (1-a^2 x^2\right )^{3/2}}{105 a^4 (1-a x)^3}-\frac {19 \left (1-a^2 x^2\right )^{3/2}}{35 a^4 (1-a x)^4}+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^4 (1-a x)^5}-\frac {2 \sqrt {1-a^2 x^2}}{a^4 (1-a x)}}{c^4}\) |
Input:
Int[(E^ArcTanh[a*x]*x^3)/(c - a*c*x)^4,x]
Output:
((-2*Sqrt[1 - a^2*x^2])/(a^4*(1 - a*x)) + (1 - a^2*x^2)^(3/2)/(7*a^4*(1 - a*x)^5) - (19*(1 - a^2*x^2)^(3/2))/(35*a^4*(1 - a*x)^4) + (86*(1 - a^2*x^2 )^(3/2))/(105*a^4*(1 - a*x)^3) + ArcSin[a*x]/a^4)/c^4
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. \(465\) vs. \(2(124)=248\).
Time = 0.16 (sec) , antiderivative size = 466, normalized size of antiderivative = 3.38
| method | result | size |
| default | \(\frac {\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{3} \sqrt {a^{2}}}+\frac {\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{7 a \left (x -\frac {1}{a}\right )^{4}}-\frac {6 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )}{7}}{a^{7}}+\frac {\frac {7 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {14 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{a^{6}}+\frac {\frac {3 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{a \left (x -\frac {1}{a}\right )^{2}}-\frac {3 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{x -\frac {1}{a}}}{a^{5}}+\frac {5 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{5} \left (x -\frac {1}{a}\right )}}{c^{4}}\) | \(466\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3/(-a*c*x+c)^4,x,method=_RETURNVERBOSE)
Output:
1/c^4*(1/a^3/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+2/a^7*(1 /7/a/(x-1/a)^4*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)-3/7*a*(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))))+7/a^6* (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)))+9/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/a^5/(x-1/a)*(-(x-1/a)^2*a^2-2 *a*(x-1/a))^(1/2))
Time = 0.09 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.25 \[ \int \frac {e^{\text {arctanh}(a x)} x^3}{(c-a c x)^4} \, dx=-\frac {166 \, a^{4} x^{4} - 664 \, a^{3} x^{3} + 996 \, a^{2} x^{2} - 664 \, a x + 210 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (296 \, a^{3} x^{3} - 659 \, a^{2} x^{2} + 559 \, a x - 166\right )} \sqrt {-a^{2} x^{2} + 1} + 166}{105 \, {\left (a^{8} c^{4} x^{4} - 4 \, a^{7} c^{4} x^{3} + 6 \, a^{6} c^{4} x^{2} - 4 \, a^{5} c^{4} x + a^{4} c^{4}\right )}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3/(-a*c*x+c)^4,x, algorithm="fricas ")
Output:
-1/105*(166*a^4*x^4 - 664*a^3*x^3 + 996*a^2*x^2 - 664*a*x + 210*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (296*a^3*x^3 - 659*a^2*x^2 + 559*a*x - 166)*sqrt(-a^2*x^2 + 1) + 166)/( a^8*c^4*x^4 - 4*a^7*c^4*x^3 + 6*a^6*c^4*x^2 - 4*a^5*c^4*x + a^4*c^4)
\[ \int \frac {e^{\text {arctanh}(a x)} x^3}{(c-a c x)^4} \, dx=\frac {\int \frac {x^{3}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{4}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{4}} \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**3/(-a*c*x+c)**4,x)
Output:
(Integral(x**3/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**3*x**3*sqrt(-a**2*x* *2 + 1) + 6*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*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 ) - 4*a**3*x**3*sqrt(-a**2*x**2 + 1) + 6*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**4
Time = 0.27 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.51 \[ \int \frac {e^{\text {arctanh}(a x)} x^3}{(c-a c x)^4} \, dx=\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{7 \, {\left (a^{8} c^{4} x^{4} - 4 \, a^{7} c^{4} x^{3} + 6 \, a^{6} c^{4} x^{2} - 4 \, a^{5} c^{4} x + a^{4} c^{4}\right )}} + \frac {43 \, \sqrt {-a^{2} x^{2} + 1}}{35 \, {\left (a^{7} c^{4} x^{3} - 3 \, a^{6} c^{4} x^{2} + 3 \, a^{5} c^{4} x - a^{4} c^{4}\right )}} + \frac {229 \, \sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (a^{6} c^{4} x^{2} - 2 \, a^{5} c^{4} x + a^{4} c^{4}\right )}} + \frac {296 \, \sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (a^{5} c^{4} x - a^{4} c^{4}\right )}} + \frac {\arcsin \left (a x\right )}{a^{4} c^{4}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3/(-a*c*x+c)^4,x, algorithm="maxima ")
Output:
2/7*sqrt(-a^2*x^2 + 1)/(a^8*c^4*x^4 - 4*a^7*c^4*x^3 + 6*a^6*c^4*x^2 - 4*a^ 5*c^4*x + a^4*c^4) + 43/35*sqrt(-a^2*x^2 + 1)/(a^7*c^4*x^3 - 3*a^6*c^4*x^2 + 3*a^5*c^4*x - a^4*c^4) + 229/105*sqrt(-a^2*x^2 + 1)/(a^6*c^4*x^2 - 2*a^ 5*c^4*x + a^4*c^4) + 296/105*sqrt(-a^2*x^2 + 1)/(a^5*c^4*x - a^4*c^4) + ar csin(a*x)/(a^4*c^4)
Time = 0.16 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.59 \[ \int \frac {e^{\text {arctanh}(a x)} x^3}{(c-a c x)^4} \, dx=\frac {\arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{a^{3} c^{4} {\left | a \right |}} + \frac {2 \, {\left (\frac {1057 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - \frac {2751 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac {3640 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac {2170 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} + \frac {735 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5}}{a^{10} x^{5}} - \frac {105 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6}}{a^{12} x^{6}} - 166\right )}}{105 \, a^{3} c^{4} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{7} {\left | a \right |}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3/(-a*c*x+c)^4,x, algorithm="giac")
Output:
arcsin(a*x)*sgn(a)/(a^3*c^4*abs(a)) + 2/105*(1057*(sqrt(-a^2*x^2 + 1)*abs( a) + a)/(a^2*x) - 2751*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/(a^4*x^2) + 3640* (sqrt(-a^2*x^2 + 1)*abs(a) + a)^3/(a^6*x^3) - 2170*(sqrt(-a^2*x^2 + 1)*abs (a) + a)^4/(a^8*x^4) + 735*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^5/(a^10*x^5) - 105*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^6/(a^12*x^6) - 166)/(a^3*c^4*((sqrt(-a ^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)^7*abs(a))
Time = 0.07 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.04 \[ \int \frac {e^{\text {arctanh}(a x)} x^3}{(c-a c x)^4} \, dx=\frac {2\,\sqrt {1-a^2\,x^2}}{7\,\left (a^8\,c^4\,x^4-4\,a^7\,c^4\,x^3+6\,a^6\,c^4\,x^2-4\,a^5\,c^4\,x+a^4\,c^4\right )}+\frac {229\,\sqrt {1-a^2\,x^2}}{105\,\left (a^6\,c^4\,x^2-2\,a^5\,c^4\,x+a^4\,c^4\right )}+\frac {43\,\sqrt {1-a^2\,x^2}}{35\,\sqrt {-a^2}\,\left (a^2\,c^4\,\sqrt {-a^2}+3\,a^4\,c^4\,x^2\,\sqrt {-a^2}-a^5\,c^4\,x^3\,\sqrt {-a^2}-3\,a^3\,c^4\,x\,\sqrt {-a^2}\right )}+\frac {296\,\sqrt {1-a^2\,x^2}}{105\,\left (a^2\,c^4\,\sqrt {-a^2}-a^3\,c^4\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}+\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{a^3\,c^4\,\sqrt {-a^2}} \] Input:
int((x^3*(a*x + 1))/((1 - a^2*x^2)^(1/2)*(c - a*c*x)^4),x)
Output:
(2*(1 - a^2*x^2)^(1/2))/(7*(a^4*c^4 - 4*a^5*c^4*x + 6*a^6*c^4*x^2 - 4*a^7* c^4*x^3 + a^8*c^4*x^4)) + (229*(1 - a^2*x^2)^(1/2))/(105*(a^4*c^4 - 2*a^5* c^4*x + a^6*c^4*x^2)) + (43*(1 - a^2*x^2)^(1/2))/(35*(-a^2)^(1/2)*(a^2*c^4 *(-a^2)^(1/2) + 3*a^4*c^4*x^2*(-a^2)^(1/2) - a^5*c^4*x^3*(-a^2)^(1/2) - 3* a^3*c^4*x*(-a^2)^(1/2))) + (296*(1 - a^2*x^2)^(1/2))/(105*(a^2*c^4*(-a^2)^ (1/2) - a^3*c^4*x*(-a^2)^(1/2))*(-a^2)^(1/2)) + asinh(x*(-a^2)^(1/2))/(a^3 *c^4*(-a^2)^(1/2))
Time = 0.16 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.80 \[ \int \frac {e^{\text {arctanh}(a x)} x^3}{(c-a c x)^4} \, dx=\frac {-105 \mathit {atan} \left (\frac {2 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-\sqrt {-a^{2} x^{2}+1}}{2 a^{3} x^{3}-2 a x}\right ) a^{4} x^{4}+420 \mathit {atan} \left (\frac {2 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-\sqrt {-a^{2} x^{2}+1}}{2 a^{3} x^{3}-2 a x}\right ) a^{3} x^{3}-630 \mathit {atan} \left (\frac {2 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-\sqrt {-a^{2} x^{2}+1}}{2 a^{3} x^{3}-2 a x}\right ) a^{2} x^{2}+420 \mathit {atan} \left (\frac {2 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-\sqrt {-a^{2} x^{2}+1}}{2 a^{3} x^{3}-2 a x}\right ) a x -105 \mathit {atan} \left (\frac {2 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-\sqrt {-a^{2} x^{2}+1}}{2 a^{3} x^{3}-2 a x}\right )+592 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-1318 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+1118 \sqrt {-a^{2} x^{2}+1}\, a x -332 \sqrt {-a^{2} x^{2}+1}}{210 a^{4} c^{4} \left (a^{4} x^{4}-4 a^{3} x^{3}+6 a^{2} x^{2}-4 a x +1\right )} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^3/(-a*c*x+c)^4,x)
Output:
( - 105*atan((2*sqrt( - a**2*x**2 + 1)*a**2*x**2 - sqrt( - a**2*x**2 + 1)) /(2*a**3*x**3 - 2*a*x))*a**4*x**4 + 420*atan((2*sqrt( - a**2*x**2 + 1)*a** 2*x**2 - sqrt( - a**2*x**2 + 1))/(2*a**3*x**3 - 2*a*x))*a**3*x**3 - 630*at an((2*sqrt( - a**2*x**2 + 1)*a**2*x**2 - sqrt( - a**2*x**2 + 1))/(2*a**3*x **3 - 2*a*x))*a**2*x**2 + 420*atan((2*sqrt( - a**2*x**2 + 1)*a**2*x**2 - s qrt( - a**2*x**2 + 1))/(2*a**3*x**3 - 2*a*x))*a*x - 105*atan((2*sqrt( - a* *2*x**2 + 1)*a**2*x**2 - sqrt( - a**2*x**2 + 1))/(2*a**3*x**3 - 2*a*x)) + 592*sqrt( - a**2*x**2 + 1)*a**3*x**3 - 1318*sqrt( - a**2*x**2 + 1)*a**2*x* *2 + 1118*sqrt( - a**2*x**2 + 1)*a*x - 332*sqrt( - a**2*x**2 + 1))/(210*a* *4*c**4*(a**4*x**4 - 4*a**3*x**3 + 6*a**2*x**2 - 4*a*x + 1))