\(\int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^6} \, dx\) [225]

Optimal result

Integrand size = 18, antiderivative size = 143 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^6} \, dx=\frac {1}{a c^6 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {a^2}{7 c^6 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )^3}+\frac {24 a}{35 c^6 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )^2}-\frac {46}{35 c^6 \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )}+\frac {13}{35 a^2 c^6 \sqrt {1-\frac {1}{a^2 x^2}} x} \] Output:

1/a/c^6/(1-1/a^2/x^2)^(1/2)-1/7*a^2/c^6/(1-1/a^2/x^2)^(1/2)/(a-1/x)^3+24/3 
5*a/c^6/(1-1/a^2/x^2)^(1/2)/(a-1/x)^2-46/35/c^6/(1-1/a^2/x^2)^(1/2)/(a-1/x 
)+13/35/a^2/c^6/(1-1/a^2/x^2)^(1/2)/x
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.46 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^6} \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (-13+4 a x+20 a^2 x^2-24 a^3 x^3+8 a^4 x^4\right )}{35 c^6 (-1+a x)^4 (1+a x)} \] Input:

Integrate[1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^6),x]
 

Output:

(Sqrt[1 - 1/(a^2*x^2)]*x*(-13 + 4*a*x + 20*a^2*x^2 - 24*a^3*x^3 + 8*a^4*x^ 
4))/(35*c^6*(-1 + a*x)^4*(1 + a*x))
 

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.90, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6724, 25, 27, 570, 529, 2166, 2166, 27, 453}

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.

\(\displaystyle \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^6} \, dx\)

\(\Big \downarrow \) 6724

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {1}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (a-\frac {1}{x}\right )^3 x^4}d\frac {1}{x}}{a^3 c^6}\)

\(\Big \downarrow \) 570

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

\(\Big \downarrow \) 529

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

\(\Big \downarrow \) 2166

\(\displaystyle -\frac {\frac {a^5 \left (a+\frac {1}{x}\right )^3}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {1}{7} a \left (\frac {24 a^5 \left (a+\frac {1}{x}\right )^2}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {1}{5} a \int \frac {\left (a+\frac {1}{x}\right ) \left (33 a^4+\frac {70 a^3}{x}+\frac {35 a^2}{x^2}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2}}d\frac {1}{x}\right )}{a^9 c^6}\)

\(\Big \downarrow \) 2166

\(\displaystyle -\frac {\frac {a^5 \left (a+\frac {1}{x}\right )^3}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {1}{7} a \left (\frac {24 a^5 \left (a+\frac {1}{x}\right )^2}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {1}{5} a \left (\frac {46 a^5 \left (a+\frac {1}{x}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {1}{3} a \int \frac {3 a^3 \left (13 a+\frac {35}{x}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}\right )\right )}{a^9 c^6}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {a^5 \left (a+\frac {1}{x}\right )^3}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {1}{7} a \left (\frac {24 a^5 \left (a+\frac {1}{x}\right )^2}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {1}{5} a \left (\frac {46 a^5 \left (a+\frac {1}{x}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-a^4 \int \frac {13 a+\frac {35}{x}}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}\right )\right )}{a^9 c^6}\)

\(\Big \downarrow \) 453

\(\displaystyle -\frac {\frac {a^5 \left (a+\frac {1}{x}\right )^3}{7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {1}{7} a \left (\frac {24 a^5 \left (a+\frac {1}{x}\right )^2}{5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {1}{5} a \left (\frac {46 a^5 \left (a+\frac {1}{x}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {a^5 \left (35 a+\frac {13}{x}\right )}{\sqrt {1-\frac {1}{a^2 x^2}}}\right )\right )}{a^9 c^6}\)

Input:

Int[1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^6),x]
 

Output:

-((-1/7*(a*(-1/5*(a*((46*a^5*(a + x^(-1)))/(1 - 1/(a^2*x^2))^(3/2) - (a^5* 
(35*a + 13/x))/Sqrt[1 - 1/(a^2*x^2)])) + (24*a^5*(a + x^(-1))^2)/(5*(1 - 1 
/(a^2*x^2))^(5/2)))) + (a^5*(a + x^(-1))^3)/(7*(1 - 1/(a^2*x^2))^(7/2)))/( 
a^9*c^6))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 453

Int[((c_) + (d_.)*(x_))/((a_) + (b_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[-(a* 
d - b*c*x)/(a*b*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b, c, d}, x]
 

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 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 6724

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

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.44

method	result	size
trager	\(\frac {\left (8 a^{4} x^{4}-24 a^{3} x^{3}+20 a^{2} x^{2}+4 a x -13\right ) \sqrt {-\frac {-a x +1}{a x +1}}}{35 a \,c^{6} \left (a x -1\right )^{4}}\)	\(63\)
gosper	\(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (8 a^{4} x^{4}-24 a^{3} x^{3}+20 a^{2} x^{2}+4 a x -13\right ) \left (a x +1\right )}{35 \left (a x -1\right )^{5} c^{6} a}\)	\(66\)
default	\(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (8 a^{5} x^{5}-16 a^{4} x^{4}-4 a^{3} x^{3}+24 a^{2} x^{2}-9 a x -13\right )}{35 \left (a x -1\right )^{5} c^{6} a}\)	\(69\)
orering	\(\frac {\left (8 a^{4} x^{4}-24 a^{3} x^{3}+20 a^{2} x^{2}+4 a x -13\right ) \left (a x -1\right ) \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{35 a \left (-a c x +c \right )^{6}}\)	\(70\)

Input:

int(((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c)^6,x,method=_RETURNVERBOSE)
 

Output:

1/35/a/c^6*(8*a^4*x^4-24*a^3*x^3+20*a^2*x^2+4*a*x-13)/(a*x-1)^4*(-(-a*x+1) 
/(a*x+1))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.67 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^6} \, dx=\frac {{\left (8 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 20 \, a^{2} x^{2} + 4 \, a x - 13\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{35 \, {\left (a^{5} c^{6} x^{4} - 4 \, a^{4} c^{6} x^{3} + 6 \, a^{3} c^{6} x^{2} - 4 \, a^{2} c^{6} x + a c^{6}\right )}} \] Input:

integrate(((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c)^6,x, algorithm="fricas")
 

Output:

1/35*(8*a^4*x^4 - 24*a^3*x^3 + 20*a^2*x^2 + 4*a*x - 13)*sqrt((a*x - 1)/(a* 
x + 1))/(a^5*c^6*x^4 - 4*a^4*c^6*x^3 + 6*a^3*c^6*x^2 - 4*a^2*c^6*x + a*c^6 
)
 

Sympy [F]

\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^6} \, dx=\frac {\int \left (- \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{7} x^{7} - 5 a^{6} x^{6} + 9 a^{5} x^{5} - 5 a^{4} x^{4} - 5 a^{3} x^{3} + 9 a^{2} x^{2} - 5 a x + 1}\right )\, dx + \int \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{7} x^{7} - 5 a^{6} x^{6} + 9 a^{5} x^{5} - 5 a^{4} x^{4} - 5 a^{3} x^{3} + 9 a^{2} x^{2} - 5 a x + 1}\, dx}{c^{6}} \] Input:

integrate(((a*x-1)/(a*x+1))**(3/2)/(-a*c*x+c)**6,x)
 

Output:

(Integral(-sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**7*x**7 - 5*a**6*x**6 + 9* 
a**5*x**5 - 5*a**4*x**4 - 5*a**3*x**3 + 9*a**2*x**2 - 5*a*x + 1), x) + Int 
egral(a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a**7*x**7 - 5*a**6*x**6 + 9*a 
**5*x**5 - 5*a**4*x**4 - 5*a**3*x**3 + 9*a**2*x**2 - 5*a*x + 1), x))/c**6
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.68 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^6} \, dx=\frac {1}{560} \, a {\left (\frac {35 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{6}} + \frac {\frac {28 \, {\left (a x - 1\right )}}{a x + 1} - \frac {70 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {140 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 5}{a^{2} c^{6} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}}}\right )} \] Input:

integrate(((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c)^6,x, algorithm="maxima")
 

Output:

1/560*a*(35*sqrt((a*x - 1)/(a*x + 1))/(a^2*c^6) + (28*(a*x - 1)/(a*x + 1) 
- 70*(a*x - 1)^2/(a*x + 1)^2 + 140*(a*x - 1)^3/(a*x + 1)^3 - 5)/(a^2*c^6*( 
(a*x - 1)/(a*x + 1))^(7/2)))
 

Giac [F]

\[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^6} \, dx=\int { \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{{\left (a c x - c\right )}^{6}} \,d x } \] Input:

integrate(((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c)^6,x, algorithm="giac")
 

Output:

integrate(((a*x - 1)/(a*x + 1))^(3/2)/(a*c*x - c)^6, x)
 

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.42 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^6} \, dx=\frac {8\,a^4\,x^4-24\,a^3\,x^3+20\,a^2\,x^2+4\,a\,x-13}{35\,a\,c^6\,{\left (a\,x+1\right )}^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}} \] Input:

int(((a*x - 1)/(a*x + 1))^(3/2)/(c - a*c*x)^6,x)
 

Output:

(4*a*x + 20*a^2*x^2 - 24*a^3*x^3 + 8*a^4*x^4 - 13)/(35*a*c^6*(a*x + 1)^4*( 
(a*x - 1)/(a*x + 1))^(7/2))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.02 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^6} \, dx=\frac {-8 \sqrt {a x -1}\, a^{4} x^{4}+16 \sqrt {a x -1}\, a^{3} x^{3}-16 \sqrt {a x -1}\, a x +8 \sqrt {a x -1}+8 \sqrt {a x +1}\, a^{4} x^{4}-24 \sqrt {a x +1}\, a^{3} x^{3}+20 \sqrt {a x +1}\, a^{2} x^{2}+4 \sqrt {a x +1}\, a x -13 \sqrt {a x +1}}{35 \sqrt {a x -1}\, a \,c^{6} \left (a^{4} x^{4}-2 a^{3} x^{3}+2 a x -1\right )} \] Input:

int(((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c)^6,x)
 

Output:

( - 8*sqrt(a*x - 1)*a**4*x**4 + 16*sqrt(a*x - 1)*a**3*x**3 - 16*sqrt(a*x - 
 1)*a*x + 8*sqrt(a*x - 1) + 8*sqrt(a*x + 1)*a**4*x**4 - 24*sqrt(a*x + 1)*a 
**3*x**3 + 20*sqrt(a*x + 1)*a**2*x**2 + 4*sqrt(a*x + 1)*a*x - 13*sqrt(a*x 
+ 1))/(35*sqrt(a*x - 1)*a*c**6*(a**4*x**4 - 2*a**3*x**3 + 2*a*x - 1))