Integrand size = 19, antiderivative size = 156 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{(c-a c x)^3} \, dx=\frac {4 \sqrt {1-a^2 x^2}}{a^5 c^3}+\frac {x \sqrt {1-a^2 x^2}}{2 a^4 c^3}+\frac {12 \sqrt {1-a^2 x^2}}{a^5 c^3 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a^5 c^3 (1-a x)^4}-\frac {19 \left (1-a^2 x^2\right )^{3/2}}{15 a^5 c^3 (1-a x)^3}-\frac {19 \arcsin (a x)}{2 a^5 c^3} \] Output:
4*(-a^2*x^2+1)^(1/2)/a^5/c^3+1/2*x*(-a^2*x^2+1)^(1/2)/a^4/c^3+12*(-a^2*x^2 +1)^(1/2)/a^5/c^3/(-a*x+1)+1/5*(-a^2*x^2+1)^(3/2)/a^5/c^3/(-a*x+1)^4-19/15 *(-a^2*x^2+1)^(3/2)/a^5/c^3/(-a*x+1)^3-19/2*arcsin(a*x)/a^5/c^3
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.10 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.78 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{(c-a c x)^3} \, dx=\frac {\sqrt {1+a x} \left (308-639 a x+433 a^2 x^2-75 a^3 x^3-15 a^4 x^4\right )+360 (1-a x)^{5/2} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )+140 \sqrt {2} (-1+a x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},\frac {1}{2} (1-a x)\right )}{30 a^5 c^3 (1-a x)^{5/2}} \] Input:
Integrate[(E^ArcTanh[a*x]*x^4)/(c - a*c*x)^3,x]
Output:
(Sqrt[1 + a*x]*(308 - 639*a*x + 433*a^2*x^2 - 75*a^3*x^3 - 15*a^4*x^4) + 3 60*(1 - a*x)^(5/2)*ArcSin[Sqrt[1 - a*x]/Sqrt[2]] + 140*Sqrt[2]*(-1 + a*x)* Hypergeometric2F1[-3/2, -3/2, -1/2, (1 - a*x)/2])/(30*a^5*c^3*(1 - a*x)^(5 /2))
Time = 0.65 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.96, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {6678, 27, 570, 529, 2166, 27, 2166, 27, 676, 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.
| \(\displaystyle \int \frac {x^4 e^{\text {arctanh}(a x)}}{(c-a c x)^3} \, dx\) |
| \(\Big \downarrow \) 6678 |
| \(\displaystyle c \int \frac {x^4 \sqrt {1-a^2 x^2}}{c^4 (1-a x)^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {x^4 \sqrt {1-a^2 x^2}}{(1-a x)^4}dx}{c^3}\) |
| \(\Big \downarrow \) 570 |
| \(\displaystyle \frac {\int \frac {x^4 (a x+1)^4}{\left (1-a^2 x^2\right )^{7/2}}dx}{c^3}\) |
| \(\Big \downarrow \) 529 |
| \(\displaystyle \frac {\frac {(a x+1)^4}{5 a^5 \left (1-a^2 x^2\right )^{5/2}}-\frac {1}{5} \int \frac {(a x+1)^3 \left (\frac {5 x^3}{a}+\frac {5 x^2}{a^2}+\frac {5 x}{a^3}+\frac {4}{a^4}\right )}{\left (1-a^2 x^2\right )^{5/2}}dx}{c^3}\) |
| \(\Big \downarrow \) 2166 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \int \frac {15 (a x+1)^2 \left (\frac {x^2}{a^2}+\frac {2 x}{a^3}+\frac {3}{a^4}\right )}{\left (1-a^2 x^2\right )^{3/2}}dx-\frac {19 (a x+1)^3}{3 a^5 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {(a x+1)^4}{5 a^5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{5} \left (5 \int \frac {(a x+1)^2 \left (\frac {x^2}{a^2}+\frac {2 x}{a^3}+\frac {3}{a^4}\right )}{\left (1-a^2 x^2\right )^{3/2}}dx-\frac {19 (a x+1)^3}{3 a^5 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {(a x+1)^4}{5 a^5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 2166 |
| \(\displaystyle \frac {\frac {1}{5} \left (5 \left (\frac {6 (a x+1)^2}{a^5 \sqrt {1-a^2 x^2}}-\int \frac {(a x+1) (a x+9)}{a^4 \sqrt {1-a^2 x^2}}dx\right )-\frac {19 (a x+1)^3}{3 a^5 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {(a x+1)^4}{5 a^5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{5} \left (5 \left (\frac {6 (a x+1)^2}{a^5 \sqrt {1-a^2 x^2}}-\frac {\int \frac {(a x+1) (a x+9)}{\sqrt {1-a^2 x^2}}dx}{a^4}\right )-\frac {19 (a x+1)^3}{3 a^5 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {(a x+1)^4}{5 a^5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 676 |
| \(\displaystyle \frac {\frac {1}{5} \left (5 \left (\frac {6 (a x+1)^2}{a^5 \sqrt {1-a^2 x^2}}-\frac {\frac {19}{2} \int \frac {1}{\sqrt {1-a^2 x^2}}dx-\frac {1}{2} x \sqrt {1-a^2 x^2}-\frac {10 \sqrt {1-a^2 x^2}}{a}}{a^4}\right )-\frac {19 (a x+1)^3}{3 a^5 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {(a x+1)^4}{5 a^5 \left (1-a^2 x^2\right )^{5/2}}}{c^3}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {\frac {(a x+1)^4}{5 a^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {1}{5} \left (5 \left (\frac {6 (a x+1)^2}{a^5 \sqrt {1-a^2 x^2}}-\frac {-\frac {1}{2} x \sqrt {1-a^2 x^2}-\frac {10 \sqrt {1-a^2 x^2}}{a}+\frac {19 \arcsin (a x)}{2 a}}{a^4}\right )-\frac {19 (a x+1)^3}{3 a^5 \left (1-a^2 x^2\right )^{3/2}}\right )}{c^3}\) |
Input:
Int[(E^ArcTanh[a*x]*x^4)/(c - a*c*x)^3,x]
Output:
((1 + a*x)^4/(5*a^5*(1 - a^2*x^2)^(5/2)) + ((-19*(1 + a*x)^3)/(3*a^5*(1 - a^2*x^2)^(3/2)) + 5*((6*(1 + a*x)^2)/(a^5*Sqrt[1 - a^2*x^2]) - ((-10*Sqrt[ 1 - a^2*x^2])/a - (x*Sqrt[1 - a^2*x^2])/2 + (19*ArcSin[a*x])/(2*a))/a^4))/ 5)/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[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[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c^(2*n)/a^n Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I LtQ[n, -1] && !(IGtQ[m, 0] && ILtQ[m + n, 0] && !GtQ[p, 1])
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + (Sim p[e*g*x*((a + c*x^2)^(p + 1)/(c*(2*p + 3))), x] - Simp[(a*e*g - c*d*f*(2*p + 3))/(c*(2*p + 3)) Int[(a + c*x^2)^p, x], x]) /; FreeQ[{a, c, d, e, f, g , p}, x] && !LeQ[p, -1]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{Qx = PolynomialQuotient[Pq, a*e + b*d*x, x], R = PolynomialRemainde r[Pq, a*e + b*d*x, x]}, Simp[(-d)*R*(d + e*x)^m*((a + b*x^2)^(p + 1)/(2*a*e *(p + 1))), x] + Simp[d/(2*a*(p + 1)) Int[(d + e*x)^(m - 1)*(a + b*x^2)^( p + 1)*ExpandToSum[2*a*e*(p + 1)*Qx + R*(m + 2*p + 2), x], x], x]] /; FreeQ [{a, b, d, e}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 0] && ILtQ[p + 1/2, 0] && GtQ[m, 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]
Time = 0.29 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.25
| method | result | size |
| risch | \(-\frac {\left (a x +8\right ) \left (a^{2} x^{2}-1\right )}{2 a^{5} \sqrt {-a^{2} x^{2}+1}\, c^{3}}-\frac {\frac {19 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{4} \sqrt {a^{2}}}+\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a^{8} \left (x -\frac {1}{a}\right )^{3}}+\frac {41 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{15 a^{7} \left (x -\frac {1}{a}\right )^{2}}+\frac {199 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{15 a^{6} \left (x -\frac {1}{a}\right )}}{c^{3}}\) | \(195\) |
| default | \(-\frac {\frac {-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}}{a^{2}}+\frac {9 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{4} \sqrt {a^{2}}}+\frac {\frac {3 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{a \left (x -\frac {1}{a}\right )^{2}}-\frac {3 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{x -\frac {1}{a}}}{a^{6}}+\frac {16 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{6} \left (x -\frac {1}{a}\right )}-\frac {4 \sqrt {-a^{2} x^{2}+1}}{a^{5}}+\frac {\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {4 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{a^{7}}}{c^{3}}\) | \(363\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a*c*x+c)^3,x,method=_RETURNVERBOSE)
Output:
-1/2*(a*x+8)*(a^2*x^2-1)/a^5/(-a^2*x^2+1)^(1/2)/c^3-(19/2/a^4/(a^2)^(1/2)* arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+2/5/a^8/(x-1/a)^3*(-(x-1/a)^2*a^2 -2*a*(x-1/a))^(1/2)+41/15/a^7/(x-1/a)^2*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2) +199/15/a^6/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2))/c^3
Time = 0.13 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.98 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{(c-a c x)^3} \, dx=\frac {448 \, a^{3} x^{3} - 1344 \, a^{2} x^{2} + 1344 \, a x + 570 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (15 \, a^{4} x^{4} + 75 \, a^{3} x^{3} - 713 \, a^{2} x^{2} + 1059 \, a x - 448\right )} \sqrt {-a^{2} x^{2} + 1} - 448}{30 \, {\left (a^{8} c^{3} x^{3} - 3 \, a^{7} c^{3} x^{2} + 3 \, a^{6} c^{3} x - a^{5} c^{3}\right )}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a*c*x+c)^3,x, algorithm="fricas ")
Output:
1/30*(448*a^3*x^3 - 1344*a^2*x^2 + 1344*a*x + 570*(a^3*x^3 - 3*a^2*x^2 + 3 *a*x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (15*a^4*x^4 + 75*a^3*x^ 3 - 713*a^2*x^2 + 1059*a*x - 448)*sqrt(-a^2*x^2 + 1) - 448)/(a^8*c^3*x^3 - 3*a^7*c^3*x^2 + 3*a^6*c^3*x - a^5*c^3)
\[ \int \frac {e^{\text {arctanh}(a x)} x^4}{(c-a c x)^3} \, dx=- \frac {\int \frac {x^{4}}{a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{5}}{a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**4/(-a*c*x+c)**3,x)
Output:
-(Integral(x**4/(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**5/(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 = 0.12 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.19 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{(c-a c x)^3} \, dx=-\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{5 \, {\left (a^{8} c^{3} x^{3} - 3 \, a^{7} c^{3} x^{2} + 3 \, a^{6} c^{3} x - a^{5} c^{3}\right )}} - \frac {41 \, \sqrt {-a^{2} x^{2} + 1}}{15 \, {\left (a^{7} c^{3} x^{2} - 2 \, a^{6} c^{3} x + a^{5} c^{3}\right )}} - \frac {199 \, \sqrt {-a^{2} x^{2} + 1}}{15 \, {\left (a^{6} c^{3} x - a^{5} c^{3}\right )}} + \frac {\sqrt {-a^{2} x^{2} + 1} x}{2 \, a^{4} c^{3}} - \frac {19 \, \arcsin \left (a x\right )}{2 \, a^{5} c^{3}} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1}}{a^{5} c^{3}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a*c*x+c)^3,x, algorithm="maxima ")
Output:
-2/5*sqrt(-a^2*x^2 + 1)/(a^8*c^3*x^3 - 3*a^7*c^3*x^2 + 3*a^6*c^3*x - a^5*c ^3) - 41/15*sqrt(-a^2*x^2 + 1)/(a^7*c^3*x^2 - 2*a^6*c^3*x + a^5*c^3) - 199 /15*sqrt(-a^2*x^2 + 1)/(a^6*c^3*x - a^5*c^3) + 1/2*sqrt(-a^2*x^2 + 1)*x/(a ^4*c^3) - 19/2*arcsin(a*x)/(a^5*c^3) + 4*sqrt(-a^2*x^2 + 1)/(a^5*c^3)
Exception generated. \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{(c-a c x)^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-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 = 14.02 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.94 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{(c-a c x)^3} \, dx=\frac {4\,a^4\,\sqrt {1-a^2\,x^2}}{15\,\left (a^{11}\,c^3\,x^2-2\,a^{10}\,c^3\,x+a^9\,c^3\right )}-\frac {2\,\sqrt {1-a^2\,x^2}}{5\,\sqrt {-a^2}\,\left (a^3\,c^3\,\sqrt {-a^2}+3\,a^5\,c^3\,x^2\,\sqrt {-a^2}-a^6\,c^3\,x^3\,\sqrt {-a^2}-3\,a^4\,c^3\,x\,\sqrt {-a^2}\right )}-\frac {3\,\sqrt {1-a^2\,x^2}}{a^7\,c^3\,x^2-2\,a^6\,c^3\,x+a^5\,c^3}-\frac {199\,\sqrt {1-a^2\,x^2}}{15\,\left (a^3\,c^3\,\sqrt {-a^2}-a^4\,c^3\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}+\frac {4\,\sqrt {1-a^2\,x^2}}{a^5\,c^3}+\frac {x\,\sqrt {1-a^2\,x^2}}{2\,a^4\,c^3}-\frac {19\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a^4\,c^3\,\sqrt {-a^2}} \] Input:
int((x^4*(a*x + 1))/((1 - a^2*x^2)^(1/2)*(c - a*c*x)^3),x)
Output:
(4*a^4*(1 - a^2*x^2)^(1/2))/(15*(a^9*c^3 - 2*a^10*c^3*x + a^11*c^3*x^2)) - (2*(1 - a^2*x^2)^(1/2))/(5*(-a^2)^(1/2)*(a^3*c^3*(-a^2)^(1/2) + 3*a^5*c^3 *x^2*(-a^2)^(1/2) - a^6*c^3*x^3*(-a^2)^(1/2) - 3*a^4*c^3*x*(-a^2)^(1/2))) - (3*(1 - a^2*x^2)^(1/2))/(a^5*c^3 - 2*a^6*c^3*x + a^7*c^3*x^2) - (199*(1 - a^2*x^2)^(1/2))/(15*(a^3*c^3*(-a^2)^(1/2) - a^4*c^3*x*(-a^2)^(1/2))*(-a^ 2)^(1/2)) + (4*(1 - a^2*x^2)^(1/2))/(a^5*c^3) + (x*(1 - a^2*x^2)^(1/2))/(2 *a^4*c^3) - (19*asinh(x*(-a^2)^(1/2)))/(2*a^4*c^3*(-a^2)^(1/2))
Time = 0.15 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.89 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{(c-a c x)^3} \, dx=\frac {-285 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a^{2} x^{2}+570 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a x -285 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )-285 \mathit {asin} \left (a x \right ) a^{3} x^{3}+855 \mathit {asin} \left (a x \right ) a^{2} x^{2}-855 \mathit {asin} \left (a x \right ) a x +285 \mathit {asin} \left (a x \right )+15 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+75 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-379 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+391 \sqrt {-a^{2} x^{2}+1}\, a x -114 \sqrt {-a^{2} x^{2}+1}-15 a^{5} x^{5}-90 a^{4} x^{4}+972 a^{3} x^{3}-1348 a^{2} x^{2}+391 a x +114}{30 a^{5} c^{3} \left (\sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-2 \sqrt {-a^{2} x^{2}+1}\, a x +\sqrt {-a^{2} x^{2}+1}+a^{3} x^{3}-3 a^{2} x^{2}+3 a x -1\right )} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a*c*x+c)^3,x)
Output:
( - 285*sqrt( - a**2*x**2 + 1)*asin(a*x)*a**2*x**2 + 570*sqrt( - a**2*x**2 + 1)*asin(a*x)*a*x - 285*sqrt( - a**2*x**2 + 1)*asin(a*x) - 285*asin(a*x) *a**3*x**3 + 855*asin(a*x)*a**2*x**2 - 855*asin(a*x)*a*x + 285*asin(a*x) + 15*sqrt( - a**2*x**2 + 1)*a**4*x**4 + 75*sqrt( - a**2*x**2 + 1)*a**3*x**3 - 379*sqrt( - a**2*x**2 + 1)*a**2*x**2 + 391*sqrt( - a**2*x**2 + 1)*a*x - 114*sqrt( - a**2*x**2 + 1) - 15*a**5*x**5 - 90*a**4*x**4 + 972*a**3*x**3 - 1348*a**2*x**2 + 391*a*x + 114)/(30*a**5*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))