\(\int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx\) [498]

Optimal result

Integrand size = 24, antiderivative size = 115 \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\frac {10 \sqrt {c-\frac {c}{a x}}}{a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\left (c-\frac {c}{a x}\right )^{3/2} x}{c \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {7 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a} \] Output:

10*(c-c/a/x)^(1/2)/a/(1-1/a^2/x^2)^(1/2)+(c-c/a/x)^(3/2)*x/c/(1-1/a^2/x^2) 
^(1/2)-7*c^(1/2)*arctanh(c^(1/2)*(1-1/a^2/x^2)^(1/2)/(c-c/a/x)^(1/2))/a
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.58 \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\frac {\sqrt {c-\frac {c}{a x}} \left (9+a x-7 \sqrt {1+\frac {1}{a x}} \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )\right )}{a \sqrt {1-\frac {1}{a^2 x^2}}} \] Input:

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

Output:

(Sqrt[c - c/(a*x)]*(9 + a*x - 7*Sqrt[1 + 1/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x) 
]]))/(a*Sqrt[1 - 1/(a^2*x^2)])
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.80, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6731, 585, 27, 100, 27, 87, 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.

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

\(\Big \downarrow \) 6731

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

\(\Big \downarrow \) 585

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 100

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 87

\(\displaystyle -\frac {\sqrt {c-\frac {c}{a x}} \left (\frac {1}{2} \left (-7 a \int \frac {x}{\sqrt {1+\frac {1}{a x}}}d\frac {1}{x}-\frac {18 a}{\sqrt {\frac {1}{a x}+1}}\right )-\frac {a^2 x}{\sqrt {\frac {1}{a x}+1}}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {\sqrt {c-\frac {c}{a x}} \left (\frac {1}{2} \left (-14 a^2 \int \frac {1}{\frac {a}{x^2}-a}d\sqrt {1+\frac {1}{a x}}-\frac {18 a}{\sqrt {\frac {1}{a x}+1}}\right )-\frac {a^2 x}{\sqrt {\frac {1}{a x}+1}}\right )}{a^2 \sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\left (\frac {1}{2} \left (14 a \text {arctanh}\left (\sqrt {\frac {1}{a x}+1}\right )-\frac {18 a}{\sqrt {\frac {1}{a x}+1}}\right )-\frac {a^2 x}{\sqrt {\frac {1}{a x}+1}}\right ) \sqrt {c-\frac {c}{a x}}}{a^2 \sqrt {1-\frac {1}{a x}}}\)

Input:

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

Output:

-((Sqrt[c - c/(a*x)]*(-((a^2*x)/Sqrt[1 + 1/(a*x)]) + ((-18*a)/Sqrt[1 + 1/( 
a*x)] + 14*a*ArcTanh[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 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 87

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

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 = 0.11 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.27

method	result	size
default	\(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (2 a^{\frac {3}{2}} x \sqrt {x \left (a x +1\right )}-7 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a x +18 \sqrt {x \left (a x +1\right )}\, \sqrt {a}-7 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right )\right )}{2 \left (a x -1\right )^{2} \sqrt {a}\, \sqrt {x \left (a x +1\right )}}\)	\(146\)
risch	\(\frac {\left (a x +1\right ) x \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}}{a x -1}+\frac {\left (-\frac {7 \ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{2 \sqrt {a^{2} c}}+\frac {8 \sqrt {\left (x +\frac {1}{a}\right )^{2} a^{2} c -\left (x +\frac {1}{a}\right ) a c}}{a^{2} c \left (x +\frac {1}{a}\right )}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{a x -1}\)	\(180\)

Input:

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

Output:

1/2*((a*x-1)/(a*x+1))^(3/2)/(a*x-1)^2*(a*x+1)*(c*(a*x-1)/a/x)^(1/2)*x*(2*a 
^(3/2)*x*(x*(a*x+1))^(1/2)-7*ln(1/2*(2*(x*(a*x+1))^(1/2)*a^(1/2)+2*a*x+1)/ 
a^(1/2))*a*x+18*(x*(a*x+1))^(1/2)*a^(1/2)-7*ln(1/2*(2*(x*(a*x+1))^(1/2)*a^ 
(1/2)+2*a*x+1)/a^(1/2)))/a^(1/2)/(x*(a*x+1))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.60 \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\left [\frac {7 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (a^{2} x^{2} + 9 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{2} x - a\right )}}, \frac {7 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (a^{2} x^{2} + 9 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} x - a\right )}}\right ] \] Input:

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

Output:

[1/4*(7*(a*x - 1)*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*(a^2*x^2 + 9*a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x 
- c)/(a*x)))/(a^2*x - a), 1/2*(7*(a*x - 1)*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*(a^2*x^2 + 9*a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x 
 - c)/(a*x)))/(a^2*x - a)]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.51 \[ \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\frac {\sqrt {c}\, \left (-28 \sqrt {a x +1}\, \mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}\right )+33 \sqrt {a x +1}+4 \sqrt {x}\, \sqrt {a}\, a x +36 \sqrt {x}\, \sqrt {a}\right )}{4 \sqrt {a x +1}\, a} \] Input:

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

Output:

(sqrt(c)*( - 28*sqrt(a*x + 1)*log(sqrt(a*x + 1) + sqrt(x)*sqrt(a)) + 33*sq 
rt(a*x + 1) + 4*sqrt(x)*sqrt(a)*a*x + 36*sqrt(x)*sqrt(a)))/(4*sqrt(a*x + 1 
)*a)