\(\int \frac {e^{3 \text {arctanh}(a x)}}{(c-a c x)^4} \, dx\) [197]

Optimal result

Integrand size = 18, antiderivative size = 97 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{(c-a c x)^4} \, dx=\frac {\left (1-a^2 x^2\right )^{5/2}}{9 a c^4 (1-a x)^7}+\frac {2 \left (1-a^2 x^2\right )^{5/2}}{63 a c^4 (1-a x)^6}+\frac {2 \left (1-a^2 x^2\right )^{5/2}}{315 a c^4 (1-a x)^5} \] Output:

1/9*(-a^2*x^2+1)^(5/2)/a/c^4/(-a*x+1)^7+2/63*(-a^2*x^2+1)^(5/2)/a/c^4/(-a* 
x+1)^6+2/315*(-a^2*x^2+1)^(5/2)/a/c^4/(-a*x+1)^5
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.44 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{(c-a c x)^4} \, dx=\frac {(1+a x)^{5/2} \left (47-14 a x+2 a^2 x^2\right )}{315 a c^4 (1-a x)^{9/2}} \] Input:

Integrate[E^(3*ArcTanh[a*x])/(c - a*c*x)^4,x]
 

Output:

((1 + a*x)^(5/2)*(47 - 14*a*x + 2*a^2*x^2))/(315*a*c^4*(1 - a*x)^(9/2))
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6677, 27, 461, 461, 460}

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.

\(\displaystyle \int \frac {e^{3 \text {arctanh}(a x)}}{(c-a c x)^4} \, dx\)

\(\Big \downarrow \) 6677

\(\displaystyle c^3 \int \frac {\left (1-a^2 x^2\right )^{3/2}}{c^7 (1-a x)^7}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\left (1-a^2 x^2\right )^{3/2}}{(1-a x)^7}dx}{c^4}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {\frac {2}{9} \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(1-a x)^6}dx+\frac {\left (1-a^2 x^2\right )^{5/2}}{9 a (1-a x)^7}}{c^4}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {\frac {2}{9} \left (\frac {1}{7} \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(1-a x)^5}dx+\frac {\left (1-a^2 x^2\right )^{5/2}}{7 a (1-a x)^6}\right )+\frac {\left (1-a^2 x^2\right )^{5/2}}{9 a (1-a x)^7}}{c^4}\)

\(\Big \downarrow \) 460

\(\displaystyle \frac {\frac {\left (1-a^2 x^2\right )^{5/2}}{9 a (1-a x)^7}+\frac {2}{9} \left (\frac {\left (1-a^2 x^2\right )^{5/2}}{35 a (1-a x)^5}+\frac {\left (1-a^2 x^2\right )^{5/2}}{7 a (1-a x)^6}\right )}{c^4}\)

Input:

Int[E^(3*ArcTanh[a*x])/(c - a*c*x)^4,x]
 

Output:

((1 - a^2*x^2)^(5/2)/(9*a*(1 - a*x)^7) + (2*((1 - a^2*x^2)^(5/2)/(7*a*(1 - 
 a*x)^6) + (1 - a^2*x^2)^(5/2)/(35*a*(1 - a*x)^5)))/9)/c^4
 

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 460

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

rule 461

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-d)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*b*c*(n + p + 1))), x] + Simp[Simpl 
ify[n + 2*p + 2]/(2*c*(n + p + 1))   Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x 
], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && ILtQ[Simp 
lify[n + 2*p + 2], 0] && (LtQ[n, -1] || GtQ[n + p, 0])
 

rule 6677

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> S 
imp[c^n   Int[(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, 
 d, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[2*p]
 

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.51

method	result	size
gosper	\(-\frac {\left (2 a^{2} x^{2}-14 a x +47\right ) \left (a x +1\right )^{4}}{315 \left (a x -1\right )^{3} c^{4} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} a}\)	\(49\)
orering	\(-\frac {\left (2 a^{2} x^{2}-14 a x +47\right ) \left (a x -1\right ) \left (a x +1\right )^{4}}{315 a \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \left (-a c x +c \right )^{4}}\)	\(53\)
trager	\(-\frac {\left (2 a^{4} x^{4}-10 a^{3} x^{3}+21 a^{2} x^{2}+80 a x +47\right ) \sqrt {-a^{2} x^{2}+1}}{315 c^{4} \left (a x -1\right )^{5} a}\)	\(58\)
default	\(\frac {\frac {\frac {1}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a}{3 a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}}{a}+\frac {\frac {8}{9 a \left (x -\frac {1}{a}\right )^{4} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {40 a \left (\frac {1}{7 a \left (x -\frac {1}{a}\right )^{3} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {4 a \left (\frac {1}{5 a \left (x -\frac {1}{a}\right )^{2} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {3 a \left (\frac {1}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a}{3 a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{5}\right )}{7}\right )}{9}}{a^{4}}+\frac {\frac {12}{7 a \left (x -\frac {1}{a}\right )^{3} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {48 a \left (\frac {1}{5 a \left (x -\frac {1}{a}\right )^{2} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {3 a \left (\frac {1}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a}{3 a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{5}\right )}{7}}{a^{3}}+\frac {\frac {6}{5 a \left (x -\frac {1}{a}\right )^{2} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {18 a \left (\frac {1}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a}{3 a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{5}}{a^{2}}}{c^{4}}\)	\(655\)

Input:

int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^4,x,method=_RETURNVERBOSE)
 

Output:

-1/315*(2*a^2*x^2-14*a*x+47)*(a*x+1)^4/(a*x-1)^3/c^4/(-a^2*x^2+1)^(3/2)/a
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.49 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{(c-a c x)^4} \, dx=\frac {47 \, a^{5} x^{5} - 235 \, a^{4} x^{4} + 470 \, a^{3} x^{3} - 470 \, a^{2} x^{2} + 235 \, a x - {\left (2 \, a^{4} x^{4} - 10 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 80 \, a x + 47\right )} \sqrt {-a^{2} x^{2} + 1} - 47}{315 \, {\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right )}} \] Input:

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^4,x, algorithm="fricas")
 

Output:

1/315*(47*a^5*x^5 - 235*a^4*x^4 + 470*a^3*x^3 - 470*a^2*x^2 + 235*a*x - (2 
*a^4*x^4 - 10*a^3*x^3 + 21*a^2*x^2 + 80*a*x + 47)*sqrt(-a^2*x^2 + 1) - 47) 
/(a^6*c^4*x^5 - 5*a^5*c^4*x^4 + 10*a^4*c^4*x^3 - 10*a^3*c^4*x^2 + 5*a^2*c^ 
4*x - a*c^4)
 

Sympy [F]

\[ \int \frac {e^{3 \text {arctanh}(a x)}}{(c-a c x)^4} \, dx=\frac {\int \frac {3 a x}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 4 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 5 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} x^{2}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 4 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 5 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{3} x^{3}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 4 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 5 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 4 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 5 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{4}} \] Input:

integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(-a*c*x+c)**4,x)
 

Output:

(Integral(3*a*x/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 4*a**5*x**5*sqrt(-a**2* 
x**2 + 1) - 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 5*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( 
3*a**2*x**2/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 4*a**5*x**5*sqrt(-a**2*x**2 
 + 1) - 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 5*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**3 
*x**3/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 4*a**5*x**5*sqrt(-a**2*x**2 + 1) 
- 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 5*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(1/(-a**6*x 
**6*sqrt(-a**2*x**2 + 1) + 4*a**5*x**5*sqrt(-a**2*x**2 + 1) - 5*a**4*x**4* 
sqrt(-a**2*x**2 + 1) + 5*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
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 327 vs. \(2 (82) = 164\).

Time = 0.04 (sec) , antiderivative size = 327, normalized size of antiderivative = 3.37 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{(c-a c x)^4} \, dx=\frac {8}{9 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{5} c^{4} x^{4} - 4 \, \sqrt {-a^{2} x^{2} + 1} a^{4} c^{4} x^{3} + 6 \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{4} x^{2} - 4 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{4} x + \sqrt {-a^{2} x^{2} + 1} a c^{4}\right )}} + \frac {68}{63 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{4} c^{4} x^{3} - 3 \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{4} x^{2} + 3 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{4} x - \sqrt {-a^{2} x^{2} + 1} a c^{4}\right )}} + \frac {106}{315 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{3} c^{4} x^{2} - 2 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{4} x + \sqrt {-a^{2} x^{2} + 1} a c^{4}\right )}} - \frac {1}{315 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{2} c^{4} x - \sqrt {-a^{2} x^{2} + 1} a c^{4}\right )}} + \frac {2 \, x}{315 \, \sqrt {-a^{2} x^{2} + 1} c^{4}} \] Input:

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^4,x, algorithm="maxima")
 

Output:

8/9/(sqrt(-a^2*x^2 + 1)*a^5*c^4*x^4 - 4*sqrt(-a^2*x^2 + 1)*a^4*c^4*x^3 + 6 
*sqrt(-a^2*x^2 + 1)*a^3*c^4*x^2 - 4*sqrt(-a^2*x^2 + 1)*a^2*c^4*x + sqrt(-a 
^2*x^2 + 1)*a*c^4) + 68/63/(sqrt(-a^2*x^2 + 1)*a^4*c^4*x^3 - 3*sqrt(-a^2*x 
^2 + 1)*a^3*c^4*x^2 + 3*sqrt(-a^2*x^2 + 1)*a^2*c^4*x - sqrt(-a^2*x^2 + 1)* 
a*c^4) + 106/315/(sqrt(-a^2*x^2 + 1)*a^3*c^4*x^2 - 2*sqrt(-a^2*x^2 + 1)*a^ 
2*c^4*x + sqrt(-a^2*x^2 + 1)*a*c^4) - 1/315/(sqrt(-a^2*x^2 + 1)*a^2*c^4*x 
- sqrt(-a^2*x^2 + 1)*a*c^4) + 2/315*x/(sqrt(-a^2*x^2 + 1)*c^4)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (82) = 164\).

Time = 0.15 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.61 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{(c-a c x)^4} \, dx=-\frac {2 \, {\left (\frac {108 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - \frac {1062 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac {1638 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac {3402 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} + \frac {2520 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5}}{a^{10} x^{5}} - \frac {2310 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6}}{a^{12} x^{6}} + \frac {630 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{7}}{a^{14} x^{7}} - \frac {315 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{8}}{a^{16} x^{8}} - 47\right )}}{315 \, c^{4} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{9} {\left | a \right |}} \] Input:

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^4,x, algorithm="giac")
 

Output:

-2/315*(108*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1062*(sqrt(-a^2*x^2 
+ 1)*abs(a) + a)^2/(a^4*x^2) + 1638*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3/(a^6 
*x^3) - 3402*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4/(a^8*x^4) + 2520*(sqrt(-a^2 
*x^2 + 1)*abs(a) + a)^5/(a^10*x^5) - 2310*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^ 
6/(a^12*x^6) + 630*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^7/(a^14*x^7) - 315*(sqr 
t(-a^2*x^2 + 1)*abs(a) + a)^8/(a^16*x^8) - 47)/(c^4*((sqrt(-a^2*x^2 + 1)*a 
bs(a) + a)/(a^2*x) - 1)^9*abs(a))
 

Mupad [B] (verification not implemented)

Time = 22.54 (sec) , antiderivative size = 492, normalized size of antiderivative = 5.07 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{(c-a c x)^4} \, dx=\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {12\,a^4}{35\,c^4\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^3}-\frac {8\,a^4}{35\,c^4\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}+\frac {4\,a^5}{7\,c^4\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^4\,\sqrt {-a^2}}+\frac {8\,a^7}{35\,c^4\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^2\,{\left (-a^2\right )}^{3/2}}\right )}{a^4\,\sqrt {-a^2}}+\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {32\,a^5}{315\,c^4\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}-\frac {16\,a^5}{105\,c^4\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^3}+\frac {4\,a^5}{9\,c^4\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^5}-\frac {16\,a^2\,{\left (-a^2\right )}^{3/2}}{63\,c^4\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^4}+\frac {32\,a^6}{315\,c^4\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^2\,\sqrt {-a^2}}\right )}{a^5\,\sqrt {-a^2}}+\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {2\,a^3}{15\,c^4\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}-\frac {a^3}{5\,c^4\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^3}+\frac {2\,a^4}{15\,c^4\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^2\,\sqrt {-a^2}}\right )}{a^3\,\sqrt {-a^2}} \] Input:

int((a*x + 1)^3/((1 - a^2*x^2)^(3/2)*(c - a*c*x)^4),x)
 

Output:

((1 - a^2*x^2)^(1/2)*((12*a^4)/(35*c^4*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)^3 
) - (8*a^4)/(35*c^4*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)) + (4*a^5)/(7*c^4*(x 
*(-a^2)^(1/2) - (-a^2)^(1/2)/a)^4*(-a^2)^(1/2)) + (8*a^7)/(35*c^4*(x*(-a^2 
)^(1/2) - (-a^2)^(1/2)/a)^2*(-a^2)^(3/2))))/(a^4*(-a^2)^(1/2)) + ((1 - a^2 
*x^2)^(1/2)*((32*a^5)/(315*c^4*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)) - (16*a^ 
5)/(105*c^4*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)^3) + (4*a^5)/(9*c^4*(x*(-a^2 
)^(1/2) - (-a^2)^(1/2)/a)^5) - (16*a^2*(-a^2)^(3/2))/(63*c^4*(x*(-a^2)^(1/ 
2) - (-a^2)^(1/2)/a)^4) + (32*a^6)/(315*c^4*(x*(-a^2)^(1/2) - (-a^2)^(1/2) 
/a)^2*(-a^2)^(1/2))))/(a^5*(-a^2)^(1/2)) + ((1 - a^2*x^2)^(1/2)*((2*a^3)/( 
15*c^4*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)) - a^3/(5*c^4*(x*(-a^2)^(1/2) - ( 
-a^2)^(1/2)/a)^3) + (2*a^4)/(15*c^4*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)^2*(- 
a^2)^(1/2))))/(a^3*(-a^2)^(1/2))
 

Reduce [F]

\[ \int \frac {e^{3 \text {arctanh}(a x)}}{(c-a c x)^4} \, dx=\int \frac {\left (a x +1\right )^{3}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \left (-a c x +c \right )^{4}}d x \] Input:

int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^4,x)
 

Output:

int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^4,x)