\(\int e^{3 \coth ^{-1}(a x)} (c-\frac {c}{a x})^{7/2} \, dx\) [471]

Optimal result

Integrand size = 24, antiderivative size = 195 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\frac {23 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{15 a \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{a \sqrt {c-\frac {c}{a x}}}+\frac {7 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{5 a \sqrt {c-\frac {c}{a x}}}+c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \sqrt {c-\frac {c}{a x}} x-\frac {c^{7/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a} \] Output:

23/15*c^5*(1-1/a^2/x^2)^(3/2)/a/(c-c/a/x)^(3/2)+c^4*(1-1/a^2/x^2)^(1/2)/a/ 
(c-c/a/x)^(1/2)+7/5*c^4*(1-1/a^2/x^2)^(3/2)/a/(c-c/a/x)^(1/2)+c^3*(1-1/a^2 
/x^2)^(3/2)*(c-c/a/x)^(1/2)*x-c^(7/2)*arctanh(c^(1/2)*(1-1/a^2/x^2)^(1/2)/ 
(c-c/a/x)^(1/2))/a
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.52 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\frac {c^3 \sqrt {c-\frac {c}{a x}} \left (\sqrt {1+\frac {1}{a x}} \left (-6+8 a x+44 a^2 x^2+15 a^3 x^3\right )-15 a^2 x^2 \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )\right )}{15 a^3 \sqrt {1-\frac {1}{a x}} x^2} \] Input:

Integrate[E^(3*ArcCoth[a*x])*(c - c/(a*x))^(7/2),x]
 

Output:

(c^3*Sqrt[c - c/(a*x)]*(Sqrt[1 + 1/(a*x)]*(-6 + 8*a*x + 44*a^2*x^2 + 15*a^ 
3*x^3) - 15*a^2*x^2*ArcTanh[Sqrt[1 + 1/(a*x)]]))/(15*a^3*Sqrt[1 - 1/(a*x)] 
*x^2)
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.68, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6731, 585, 27, 100, 27, 90, 60, 60, 73, 221}

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.

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

\(\Big \downarrow \) 6731

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

\(\Big \downarrow \) 585

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 100

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

Int[E^(3*ArcCoth[a*x])*(c - c/(a*x))^(7/2),x]
 

Output:

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

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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 90

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

rule 100

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d 
*e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1))   Int[(c + d*x)^ 
(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( 
p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n 
 + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))
 

rule 221

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

rule 585

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

rule 6731

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

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.83

method	result	size
default	\(\frac {\left (a x -1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{3} \left (30 a^{\frac {7}{2}} x^{3} \sqrt {x \left (a x +1\right )}+88 a^{\frac {5}{2}} x^{2} \sqrt {x \left (a x +1\right )}-15 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{3} x^{3}+16 a^{\frac {3}{2}} x \sqrt {x \left (a x +1\right )}-12 \sqrt {x \left (a x +1\right )}\, \sqrt {a}\right )}{30 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) x^{2} a^{\frac {7}{2}} \sqrt {x \left (a x +1\right )}}\)	\(161\)
risch	\(\frac {\left (15 a^{4} x^{4}+59 a^{3} x^{3}+52 a^{2} x^{2}+2 a x -6\right ) c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}}{15 x^{2} a^{3} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}-\frac {\ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right ) c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{2 \sqrt {a^{2} c}\, \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\)	\(176\)

Input:

int(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

1/30/((a*x-1)/(a*x+1))^(3/2)*(a*x-1)/(a*x+1)*(c*(a*x-1)/a/x)^(1/2)*c^3*(30 
*a^(7/2)*x^3*(x*(a*x+1))^(1/2)+88*a^(5/2)*x^2*(x*(a*x+1))^(1/2)-15*ln(1/2* 
(2*(x*(a*x+1))^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2))*a^3*x^3+16*a^(3/2)*x*(x*(a* 
x+1))^(1/2)-12*(x*(a*x+1))^(1/2)*a^(1/2))/x^2/a^(7/2)/(x*(a*x+1))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 415, normalized size of antiderivative = 2.13 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\left [\frac {15 \, {\left (a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2}\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (15 \, a^{4} c^{3} x^{4} + 59 \, a^{3} c^{3} x^{3} + 52 \, a^{2} c^{3} x^{2} + 2 \, a c^{3} x - 6 \, c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{60 \, {\left (a^{4} x^{3} - a^{3} x^{2}\right )}}, \frac {15 \, {\left (a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (15 \, a^{4} c^{3} x^{4} + 59 \, a^{3} c^{3} x^{3} + 52 \, a^{2} c^{3} x^{2} + 2 \, a c^{3} x - 6 \, c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{30 \, {\left (a^{4} x^{3} - a^{3} x^{2}\right )}}\right ] \] Input:

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(7/2),x, algorithm="fricas")
 

Output:

[1/60*(15*(a^3*c^3*x^3 - a^2*c^3*x^2)*sqrt(c)*log(-(8*a^3*c*x^3 - 7*a*c*x 
- 4*(2*a^3*x^3 + 3*a^2*x^2 + a*x)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt(( 
a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(15*a^4*c^3*x^4 + 59*a^3*c^3*x^3 + 5 
2*a^2*c^3*x^2 + 2*a*c^3*x - 6*c^3)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - 
 c)/(a*x)))/(a^4*x^3 - a^3*x^2), 1/30*(15*(a^3*c^3*x^3 - a^2*c^3*x^2)*sqrt 
(-c)*arctan(2*(a^2*x^2 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c 
*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) + 2*(15*a^4*c^3*x^4 + 59*a^3*c^3 
*x^3 + 52*a^2*c^3*x^2 + 2*a*c^3*x - 6*c^3)*sqrt((a*x - 1)/(a*x + 1))*sqrt( 
(a*c*x - c)/(a*x)))/(a^4*x^3 - a^3*x^2)]
 

Sympy [F(-1)]

Timed out. \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\text {Timed out} \] Input:

integrate(1/((a*x-1)/(a*x+1))**(3/2)*(c-c/a/x)**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(7/2),x, algorithm="maxima")
 

Output:

integrate((c - c/(a*x))^(7/2)/((a*x - 1)/(a*x + 1))^(3/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(7/2),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 [F(-1)]

Timed out. \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\int \frac {{\left (c-\frac {c}{a\,x}\right )}^{7/2}}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \] Input:

int((c - c/(a*x))^(7/2)/((a*x - 1)/(a*x + 1))^(3/2),x)
                                                                                    
                                                                                    
 

Output:

int((c - c/(a*x))^(7/2)/((a*x - 1)/(a*x + 1))^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.54 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\frac {\sqrt {c}\, c^{3} \left (60 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{3} x^{3}+176 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{2} x^{2}+32 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a x -24 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}-60 \,\mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}\right ) a^{3} x^{3}-53 a^{3} x^{3}\right )}{60 a^{4} x^{3}} \] Input:

int(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(7/2),x)
 

Output:

(sqrt(c)*c**3*(60*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a**3*x**3 + 176*sqrt(x)*sq 
rt(a)*sqrt(a*x + 1)*a**2*x**2 + 32*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a*x - 24* 
sqrt(x)*sqrt(a)*sqrt(a*x + 1) - 60*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a))*a* 
*3*x**3 - 53*a**3*x**3))/(60*a**4*x**3)