\(\int \frac {e^{\text {arctanh}(a x)} x^4}{(c-a c x)^2} \, dx\) [372]

Optimal result

Integrand size = 19, antiderivative size = 158 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{(c-a c x)^2} \, dx=-\frac {34 \sqrt {1-a^2 x^2}}{3 a^5 c^2}-\frac {17 x \sqrt {1-a^2 x^2}}{6 a^4 c^2}+\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^5 c^2 (1-a x)^3}-\frac {2 \left (1-a^2 x^2\right )^{5/2}}{a^5 c^2 (1-a x)^3}-\frac {\left (1-a^2 x^2\right )^{5/2}}{3 a^5 c^2 (1-a x)^2}+\frac {17 \arcsin (a x)}{2 a^5 c^2} \] Output:

-34/3*(-a^2*x^2+1)^(1/2)/a^5/c^2-17/6*x*(-a^2*x^2+1)^(1/2)/a^4/c^2+1/3*(-a 
^2*x^2+1)^(3/2)/a^5/c^2/(-a*x+1)^3-2*(-a^2*x^2+1)^(5/2)/a^5/c^2/(-a*x+1)^3 
-1/3*(-a^2*x^2+1)^(5/2)/a^5/c^2/(-a*x+1)^2+17/2*arcsin(a*x)/a^5/c^2
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.51 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{(c-a c x)^2} \, dx=-\frac {\frac {\sqrt {1+a x} \left (80-109 a x+18 a^2 x^2+5 a^3 x^3+2 a^4 x^4\right )}{(1-a x)^{3/2}}+102 \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{6 a^5 c^2} \] Input:

Integrate[(E^ArcTanh[a*x]*x^4)/(c - a*c*x)^2,x]
 

Output:

-1/6*((Sqrt[1 + a*x]*(80 - 109*a*x + 18*a^2*x^2 + 5*a^3*x^3 + 2*a^4*x^4))/ 
(1 - a*x)^(3/2) + 102*ArcSin[Sqrt[1 - a*x]/Sqrt[2]])/(a^5*c^2)
 

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.98, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {6678, 27, 570, 529, 27, 2156, 2166, 27, 672, 469, 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.

\(\displaystyle \int \frac {x^4 e^{\text {arctanh}(a x)}}{(c-a c x)^2} \, dx\)

\(\Big \downarrow \) 6678

\(\displaystyle c \int \frac {x^4 \sqrt {1-a^2 x^2}}{c^3 (1-a x)^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {x^4 \sqrt {1-a^2 x^2}}{(1-a x)^3}dx}{c^2}\)

\(\Big \downarrow \) 570

\(\displaystyle \frac {\int \frac {x^4 (a x+1)^3}{\left (1-a^2 x^2\right )^{5/2}}dx}{c^2}\)

\(\Big \downarrow \) 529

\(\displaystyle \frac {\frac {(a x+1)^3}{3 a^5 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{3} \int \frac {3 (a x+1)^2 \left (\frac {x^3}{a}+\frac {x^2}{a^2}+\frac {x}{a^3}+\frac {1}{a^4}\right )}{\left (1-a^2 x^2\right )^{3/2}}dx}{c^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {(a x+1)^3}{3 a^5 \left (1-a^2 x^2\right )^{3/2}}-\int \frac {(a x+1)^2 \left (\frac {x^3}{a}+\frac {x^2}{a^2}+\frac {x}{a^3}+\frac {1}{a^4}\right )}{\left (1-a^2 x^2\right )^{3/2}}dx}{c^2}\)

\(\Big \downarrow \) 2156

\(\displaystyle \frac {\frac {(a x+1)^3}{3 a^5 \left (1-a^2 x^2\right )^{3/2}}-\int \frac {(a x+1)^3 \left (\frac {x^2}{a^2}+\frac {1}{a^4}\right )}{\left (1-a^2 x^2\right )^{3/2}}dx}{c^2}\)

\(\Big \downarrow \) 2166

\(\displaystyle \frac {\int \frac {(a x+1)^2 (a x+5)}{a^4 \sqrt {1-a^2 x^2}}dx-\frac {2 (a x+1)^3}{a^5 \sqrt {1-a^2 x^2}}+\frac {(a x+1)^3}{3 a^5 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {(a x+1)^2 (a x+5)}{\sqrt {1-a^2 x^2}}dx}{a^4}-\frac {2 (a x+1)^3}{a^5 \sqrt {1-a^2 x^2}}+\frac {(a x+1)^3}{3 a^5 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\)

\(\Big \downarrow \) 672

\(\displaystyle \frac {\frac {\frac {17}{3} \int \frac {(a x+1)^2}{\sqrt {1-a^2 x^2}}dx-\frac {(a x+1)^2 \sqrt {1-a^2 x^2}}{3 a}}{a^4}-\frac {2 (a x+1)^3}{a^5 \sqrt {1-a^2 x^2}}+\frac {(a x+1)^3}{3 a^5 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\)

\(\Big \downarrow \) 469

\(\displaystyle \frac {\frac {\frac {17}{3} \left (\frac {3}{2} \int \frac {a x+1}{\sqrt {1-a^2 x^2}}dx-\frac {(a x+1) \sqrt {1-a^2 x^2}}{2 a}\right )-\frac {(a x+1)^2 \sqrt {1-a^2 x^2}}{3 a}}{a^4}-\frac {2 (a x+1)^3}{a^5 \sqrt {1-a^2 x^2}}+\frac {(a x+1)^3}{3 a^5 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {\frac {\frac {17}{3} \left (\frac {3}{2} \left (\int \frac {1}{\sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{a}\right )-\frac {(a x+1) \sqrt {1-a^2 x^2}}{2 a}\right )-\frac {(a x+1)^2 \sqrt {1-a^2 x^2}}{3 a}}{a^4}-\frac {2 (a x+1)^3}{a^5 \sqrt {1-a^2 x^2}}+\frac {(a x+1)^3}{3 a^5 \left (1-a^2 x^2\right )^{3/2}}}{c^2}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {-\frac {2 (a x+1)^3}{a^5 \sqrt {1-a^2 x^2}}+\frac {(a x+1)^3}{3 a^5 \left (1-a^2 x^2\right )^{3/2}}+\frac {\frac {17}{3} \left (\frac {3}{2} \left (\frac {\arcsin (a x)}{a}-\frac {\sqrt {1-a^2 x^2}}{a}\right )-\frac {(a x+1) \sqrt {1-a^2 x^2}}{2 a}\right )-\frac {(a x+1)^2 \sqrt {1-a^2 x^2}}{3 a}}{a^4}}{c^2}\)

Input:

Int[(E^ArcTanh[a*x]*x^4)/(c - a*c*x)^2,x]
 

Output:

((1 + a*x)^3/(3*a^5*(1 - a^2*x^2)^(3/2)) - (2*(1 + a*x)^3)/(a^5*Sqrt[1 - a 
^2*x^2]) + (-1/3*((1 + a*x)^2*Sqrt[1 - a^2*x^2])/a + (17*(-1/2*((1 + a*x)* 
Sqrt[1 - a^2*x^2])/a + (3*(-(Sqrt[1 - a^2*x^2]/a) + ArcSin[a*x]/a))/2))/3) 
/a^4)/c^2
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 223

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]
 

rule 455

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]
 

rule 469

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[2*c* 
((n + p)/(n + 2*p + 1))   Int[(c + d*x)^(n - 1)*(a + b*x^2)^p, x], x] /; Fr 
eeQ[{a, b, c, d, p}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[n, 0] && NeQ[n + 2* 
p + 1, 0] && IntegerQ[2*p]
 

rule 529

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]
 

rule 570

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])
 

rule 672

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_ 
), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), 
 x] + Simp[(m*(d*g + e*f) + 2*e*f*(p + 1))/(e*(m + 2*p + 2))   Int[(d + e*x 
)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^ 
2 + a*e^2, 0] && NeQ[m + 2*p + 2, 0]
 

rule 2156

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] 
:> Int[(d + e*x)^(m + 1)*PolynomialQuotient[Pq, d + e*x, x]*(a + b*x^2)^p, 
x] /; FreeQ[{a, b, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[PolynomialRemaind 
er[Pq, d + e*x, x], 0]
 

rule 2166

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]
 

rule 6678

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]
 

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.03

method	result	size
risch	\(\frac {\left (2 a^{2} x^{2}+9 a x +34\right ) \left (a^{2} x^{2}-1\right )}{6 a^{5} \sqrt {-a^{2} x^{2}+1}\, c^{2}}+\frac {\frac {17 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{4} \sqrt {a^{2}}}+\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a^{7} \left (x -\frac {1}{a}\right )^{2}}+\frac {25 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a^{6} \left (x -\frac {1}{a}\right )}}{c^{2}}\)	\(162\)
default	\(\frac {\frac {-\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}}{a}+\frac {7 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{4} \sqrt {a^{2}}}+\frac {9 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{6} \left (x -\frac {1}{a}\right )}-\frac {5 \sqrt {-a^{2} x^{2}+1}}{a^{5}}+\frac {-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}}{a^{2}}+\frac {\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{a^{6}}}{c^{2}}\)	\(275\)

Input:

int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a*c*x+c)^2,x,method=_RETURNVERBOSE)
 

Output:

1/6*(2*a^2*x^2+9*a*x+34)*(a^2*x^2-1)/a^5/(-a^2*x^2+1)^(1/2)/c^2+(17/2/a^4/ 
(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+2/3/a^7/(x-1/a)^2*(-( 
x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)+25/3/a^6/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1/ 
a))^(1/2))/c^2
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.79 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{(c-a c x)^2} \, dx=-\frac {80 \, a^{2} x^{2} - 160 \, a x + 102 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (2 \, a^{4} x^{4} + 5 \, a^{3} x^{3} + 18 \, a^{2} x^{2} - 109 \, a x + 80\right )} \sqrt {-a^{2} x^{2} + 1} + 80}{6 \, {\left (a^{7} c^{2} x^{2} - 2 \, a^{6} c^{2} x + a^{5} c^{2}\right )}} \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a*c*x+c)^2,x, algorithm="fricas 
")
 

Output:

-1/6*(80*a^2*x^2 - 160*a*x + 102*(a^2*x^2 - 2*a*x + 1)*arctan((sqrt(-a^2*x 
^2 + 1) - 1)/(a*x)) + (2*a^4*x^4 + 5*a^3*x^3 + 18*a^2*x^2 - 109*a*x + 80)* 
sqrt(-a^2*x^2 + 1) + 80)/(a^7*c^2*x^2 - 2*a^6*c^2*x + a^5*c^2)
 

Sympy [F]

\[ \int \frac {e^{\text {arctanh}(a x)} x^4}{(c-a c x)^2} \, dx=\frac {\int \frac {x^{4}}{a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{5}}{a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{2}} \] Input:

integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**4/(-a*c*x+c)**2,x)
 

Output:

(Integral(x**4/(a**2*x**2*sqrt(-a**2*x**2 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1 
) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**5/(a**2*x**2*sqrt(-a**2*x**2 
 + 1) - 2*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**2
 

Maxima [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.97 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{(c-a c x)^2} \, dx=\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{3 \, {\left (a^{7} c^{2} x^{2} - 2 \, a^{6} c^{2} x + a^{5} c^{2}\right )}} + \frac {25 \, \sqrt {-a^{2} x^{2} + 1}}{3 \, {\left (a^{6} c^{2} x - a^{5} c^{2}\right )}} - \frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{3 \, a^{3} c^{2}} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x}{2 \, a^{4} c^{2}} + \frac {17 \, \arcsin \left (a x\right )}{2 \, a^{5} c^{2}} - \frac {17 \, \sqrt {-a^{2} x^{2} + 1}}{3 \, a^{5} c^{2}} \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a*c*x+c)^2,x, algorithm="maxima 
")
 

Output:

2/3*sqrt(-a^2*x^2 + 1)/(a^7*c^2*x^2 - 2*a^6*c^2*x + a^5*c^2) + 25/3*sqrt(- 
a^2*x^2 + 1)/(a^6*c^2*x - a^5*c^2) - 1/3*sqrt(-a^2*x^2 + 1)*x^2/(a^3*c^2) 
- 3/2*sqrt(-a^2*x^2 + 1)*x/(a^4*c^2) + 17/2*arcsin(a*x)/(a^5*c^2) - 17/3*s 
qrt(-a^2*x^2 + 1)/(a^5*c^2)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{(c-a c x)^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a*c*x+c)^2,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
 

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.20 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{(c-a c x)^2} \, dx=\frac {2\,\sqrt {1-a^2\,x^2}}{3\,\left (a^7\,c^2\,x^2-2\,a^6\,c^2\,x+a^5\,c^2\right )}+\frac {25\,\sqrt {1-a^2\,x^2}}{3\,\left (a^3\,c^2\,\sqrt {-a^2}-a^4\,c^2\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {17\,\sqrt {1-a^2\,x^2}}{3\,a^5\,c^2}-\frac {3\,x\,\sqrt {1-a^2\,x^2}}{2\,a^4\,c^2}+\frac {17\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a^4\,c^2\,\sqrt {-a^2}}-\frac {x^2\,\sqrt {1-a^2\,x^2}}{3\,a^3\,c^2} \] Input:

int((x^4*(a*x + 1))/((1 - a^2*x^2)^(1/2)*(c - a*c*x)^2),x)
 

Output:

(2*(1 - a^2*x^2)^(1/2))/(3*(a^5*c^2 - 2*a^6*c^2*x + a^7*c^2*x^2)) + (25*(1 
 - a^2*x^2)^(1/2))/(3*(a^3*c^2*(-a^2)^(1/2) - a^4*c^2*x*(-a^2)^(1/2))*(-a^ 
2)^(1/2)) - (17*(1 - a^2*x^2)^(1/2))/(3*a^5*c^2) - (3*x*(1 - a^2*x^2)^(1/2 
))/(2*a^4*c^2) + (17*asinh(x*(-a^2)^(1/2)))/(2*a^4*c^2*(-a^2)^(1/2)) - (x^ 
2*(1 - a^2*x^2)^(1/2))/(3*a^3*c^2)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.49 \[ \int \frac {e^{\text {arctanh}(a x)} x^4}{(c-a c x)^2} \, dx=\frac {51 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a x -51 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )+51 \mathit {asin} \left (a x \right ) a^{2} x^{2}-102 \mathit {asin} \left (a x \right ) a x +51 \mathit {asin} \left (a x \right )-2 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-5 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-18 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+63 \sqrt {-a^{2} x^{2}+1}\, a x -34 \sqrt {-a^{2} x^{2}+1}+2 a^{5} x^{5}+7 a^{4} x^{4}+23 a^{3} x^{3}-137 a^{2} x^{2}+63 a x +34}{6 a^{5} c^{2} \left (\sqrt {-a^{2} x^{2}+1}\, a x -\sqrt {-a^{2} x^{2}+1}+a^{2} x^{2}-2 a x +1\right )} \] Input:

int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^4/(-a*c*x+c)^2,x)
 

Output:

(51*sqrt( - a**2*x**2 + 1)*asin(a*x)*a*x - 51*sqrt( - a**2*x**2 + 1)*asin( 
a*x) + 51*asin(a*x)*a**2*x**2 - 102*asin(a*x)*a*x + 51*asin(a*x) - 2*sqrt( 
 - a**2*x**2 + 1)*a**4*x**4 - 5*sqrt( - a**2*x**2 + 1)*a**3*x**3 - 18*sqrt 
( - a**2*x**2 + 1)*a**2*x**2 + 63*sqrt( - a**2*x**2 + 1)*a*x - 34*sqrt( - 
a**2*x**2 + 1) + 2*a**5*x**5 + 7*a**4*x**4 + 23*a**3*x**3 - 137*a**2*x**2 
+ 63*a*x + 34)/(6*a**5*c**2*(sqrt( - a**2*x**2 + 1)*a*x - sqrt( - a**2*x** 
2 + 1) + a**2*x**2 - 2*a*x + 1))