\(\int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx\) [258]

Optimal result

Integrand size = 20, antiderivative size = 194 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=-\frac {a^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x^2}{4 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{7/2}}+\frac {3 a^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x^3}{16 \left (a-\frac {1}{x}\right ) (c-a c x)^{7/2}}-\frac {3 \left (1-\frac {1}{a x}\right )^{7/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}\right )}{16 \sqrt {2} a \left (\frac {1}{a x}\right )^{7/2} (c-a c x)^{7/2}} \] Output:

-1/4*a^3*(1-1/a/x)^(7/2)*(1+1/a/x)^(1/2)*x^2/(a-1/x)^2/(-a*c*x+c)^(7/2)+3/ 
16*a^3*(1-1/a/x)^(7/2)*(1+1/a/x)^(1/2)*x^3/(a-1/x)/(-a*c*x+c)^(7/2)-3/32*( 
1-1/a/x)^(7/2)*arctanh(2^(1/2)*(1/a/x)^(1/2)/(1+1/a/x)^(1/2))*2^(1/2)/a/(1 
/a/x)^(7/2)/(-a*c*x+c)^(7/2)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.64 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\frac {\sqrt {1-\frac {1}{a x}} x \left (2 \sqrt {a} \sqrt {1+\frac {1}{a x}} (7-3 a x)+3 \sqrt {2} \sqrt {\frac {1}{x}} (-1+a x)^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )\right )}{32 \sqrt {a} c^3 (-1+a x)^2 \sqrt {c-a c x}} \] Input:

Integrate[1/(E^ArcCoth[a*x]*(c - a*c*x)^(7/2)),x]
 

Output:

(Sqrt[1 - 1/(a*x)]*x*(2*Sqrt[a]*Sqrt[1 + 1/(a*x)]*(7 - 3*a*x) + 3*Sqrt[2]* 
Sqrt[x^(-1)]*(-1 + a*x)^2*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 
 1/(a*x)])]))/(32*Sqrt[a]*c^3*(-1 + a*x)^2*Sqrt[c - a*c*x])
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.80, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6727, 27, 105, 105, 104, 219}

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.

\(\displaystyle \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx\)

\(\Big \downarrow \) 6727

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 105

\(\displaystyle -\frac {a^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {\left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}{4 \left (a-\frac {1}{x}\right )^2}-\frac {3}{8} \int \frac {\sqrt {\frac {1}{x}}}{\left (a-\frac {1}{x}\right )^2 \sqrt {1+\frac {1}{a x}}}d\frac {1}{x}\right )}{\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}}\)

\(\Big \downarrow \) 105

\(\displaystyle -\frac {a^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {\left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}{4 \left (a-\frac {1}{x}\right )^2}-\frac {3}{8} \left (\frac {\sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}}{2 \left (a-\frac {1}{x}\right )}-\frac {1}{4} \int \frac {1}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}\right )\right )}{\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}}\)

\(\Big \downarrow \) 104

\(\displaystyle -\frac {a^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {\left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}{4 \left (a-\frac {1}{x}\right )^2}-\frac {3}{8} \left (\frac {\sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}}{2 \left (a-\frac {1}{x}\right )}-\frac {1}{2} \int \frac {1}{a-\frac {2}{x^2}}d\frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{a x}}}\right )\right )}{\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {a^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {\left (\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}{4 \left (a-\frac {1}{x}\right )^2}-\frac {3}{8} \left (\frac {\sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1}}{2 \left (a-\frac {1}{x}\right )}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{2 \sqrt {2} \sqrt {a}}\right )\right )}{\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}}\)

Input:

Int[1/(E^ArcCoth[a*x]*(c - a*c*x)^(7/2)),x]
 

Output:

-((a^3*(1 - 1/(a*x))^(7/2)*((Sqrt[1 + 1/(a*x)]*(x^(-1))^(3/2))/(4*(a - x^( 
-1))^2) - (3*((Sqrt[1 + 1/(a*x)]*Sqrt[x^(-1)])/(2*(a - x^(-1))) - ArcTanh[ 
(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]/(2*Sqrt[2]*Sqrt[a])))/ 
8))/((x^(-1))^(7/2)*(c - a*c*x)^(7/2)))
 

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 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f)))   Int[(a 
+ b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, 
e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] 
 ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 6727

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

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.89

method	result	size
default	\(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, \left (-3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a^{2} c \,x^{2}+6 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a c x +6 a x \sqrt {-c \left (a x +1\right )}\, \sqrt {c}-3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c -14 \sqrt {-c \left (a x +1\right )}\, \sqrt {c}\right )}{32 c^{\frac {9}{2}} \left (a x -1\right )^{3} \sqrt {-c \left (a x +1\right )}\, a}\)	\(172\)

Input:

int(((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

1/32*((a*x-1)/(a*x+1))^(1/2)*(a*x+1)*(-c*(a*x-1))^(1/2)*(-3*2^(1/2)*arctan 
(1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*a^2*c*x^2+6*2^(1/2)*arctan(1/2*(- 
c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*a*c*x+6*a*x*(-c*(a*x+1))^(1/2)*c^(1/2)-3 
*2^(1/2)*arctan(1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*c-14*(-c*(a*x+1))^ 
(1/2)*c^(1/2))/c^(9/2)/(a*x-1)^3/(-c*(a*x+1))^(1/2)/a
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.79 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\left [-\frac {3 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 4 \, {\left (3 \, a^{2} x^{2} - 4 \, a x - 7\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{64 \, {\left (a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}}, -\frac {3 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, {\left (a c x - c\right )}}\right ) - 2 \, {\left (3 \, a^{2} x^{2} - 4 \, a x - 7\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{32 \, {\left (a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}}\right ] \] Input:

integrate(((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^(7/2),x, algorithm="fricas")
 

Output:

[-1/64*(3*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(-c)*log(-(a^2*c*x 
^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a*c*x + c)*(a*x + 1)*sqrt(-c)*sqrt((a*x - 1 
)/(a*x + 1)) - 3*c)/(a^2*x^2 - 2*a*x + 1)) - 4*(3*a^2*x^2 - 4*a*x - 7)*sqr 
t(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a^4*c^4*x^3 - 3*a^3*c^4*x^2 + 3* 
a^2*c^4*x - a*c^4), -1/32*(3*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqr 
t(c)*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*(a*x + 1)*sqrt(c)*sqrt((a*x - 1)/ 
(a*x + 1))/(a*c*x - c)) - 2*(3*a^2*x^2 - 4*a*x - 7)*sqrt(-a*c*x + c)*sqrt( 
(a*x - 1)/(a*x + 1)))/(a^4*c^4*x^3 - 3*a^3*c^4*x^2 + 3*a^2*c^4*x - a*c^4)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\text {Timed out} \] Input:

integrate(((a*x-1)/(a*x+1))**(1/2)/(-a*c*x+c)**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^(7/2),x, algorithm="maxima")
 

Output:

integrate(sqrt((a*x - 1)/(a*x + 1))/(-a*c*x + c)^(7/2), x)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.45 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\frac {{\left (\frac {3 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x - c}}{2 \, \sqrt {c}}\right )}{c^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, {\left (-a c x - c\right )}^{\frac {3}{2}} + 10 \, \sqrt {-a c x - c} c\right )}}{{\left (a c x - c\right )}^{2} c^{2}}\right )} {\left | c \right |} \mathrm {sgn}\left (a x + 1\right )}{32 \, a c^{2}} \] Input:

integrate(((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^(7/2),x, algorithm="giac")
 

Output:

1/32*(3*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-a*c*x - c)/sqrt(c))/c^(5/2) + 2*( 
3*(-a*c*x - c)^(3/2) + 10*sqrt(-a*c*x - c)*c)/((a*c*x - c)^2*c^2))*abs(c)* 
sgn(a*x + 1)/(a*c^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\int \frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{{\left (c-a\,c\,x\right )}^{7/2}} \,d x \] Input:

int(((a*x - 1)/(a*x + 1))^(1/2)/(c - a*c*x)^(7/2),x)
 

Output:

int(((a*x - 1)/(a*x + 1))^(1/2)/(c - a*c*x)^(7/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.60 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx=\frac {\sqrt {c}\, i \left (-6 \sqrt {a x +1}\, a x +14 \sqrt {a x +1}-3 \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )\right ) a^{2} x^{2}+6 \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )\right ) a x -3 \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )\right )\right )}{32 a \,c^{4} \left (a^{2} x^{2}-2 a x +1\right )} \] Input:

int(((a*x-1)/(a*x+1))^(1/2)/(-a*c*x+c)^(7/2),x)
 

Output:

(sqrt(c)*i*( - 6*sqrt(a*x + 1)*a*x + 14*sqrt(a*x + 1) - 3*sqrt(2)*log(tan( 
asin(sqrt( - a*x + 1)/sqrt(2))/2))*a**2*x**2 + 6*sqrt(2)*log(tan(asin(sqrt 
( - a*x + 1)/sqrt(2))/2))*a*x - 3*sqrt(2)*log(tan(asin(sqrt( - a*x + 1)/sq 
rt(2))/2))))/(32*a*c**4*(a**2*x**2 - 2*a*x + 1))