\(\int e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx\) [336]

Optimal result

Integrand size = 20, antiderivative size = 164 \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {4 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{a \sqrt {1-\frac {1}{a x}}}+\frac {2 \left (1+\frac {1}{a x}\right )^{3/2} x \sqrt {c-a c x}}{3 \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} \sqrt {\frac {1}{a x}} \sqrt {c-a c x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}\right )}{a \sqrt {1-\frac {1}{a x}}} \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.64 \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 \sqrt {c-a c x} \left (\sqrt {a} \sqrt {1+\frac {1}{a x}} (7+a x)-6 \sqrt {2} \sqrt {\frac {1}{x}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )\right )}{3 a^{3/2} \sqrt {1-\frac {1}{a x}}} \] Input:

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

Output:

(2*Sqrt[c - a*c*x]*(Sqrt[a]*Sqrt[1 + 1/(a*x)]*(7 + a*x) - 6*Sqrt[2]*Sqrt[x 
^(-1)]*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(3*a^ 
(3/2)*Sqrt[1 - 1/(a*x)])
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.84, 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.

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

\(\Big \downarrow \) 6727

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

-((a*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*((-2*(1 + 1/(a*x))^(3/2))/(3*a*(x^(-1))^ 
(3/2)) + (2*((-2*Sqrt[1 + 1/(a*x)])/(a*Sqrt[x^(-1)]) + (2*Sqrt[2]*ArcTanh[ 
(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/a^(3/2)))/a))/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 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 = 107, normalized size of antiderivative = 0.65

method	result	size
default	\(-\frac {2 \left (a x -1\right ) \sqrt {-c \left (a x -1\right )}\, \left (6 \sqrt {c}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-a x \sqrt {-c \left (a x +1\right )}-7 \sqrt {-c \left (a x +1\right )}\right )}{3 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x +1\right )}\, a}\)	\(107\)
risch	\(-\frac {2 \left (a x +7\right ) c \left (a x -1\right )}{3 a \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}}-\frac {4 \sqrt {2}\, \sqrt {c}\, \arctan \left (\frac {\sqrt {-a c x -c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {-c \left (a x +1\right )}\, \left (a x -1\right )}{a \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}}\)	\(121\)

Input:

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

Output:

-2/3/((a*x-1)/(a*x+1))^(3/2)*(a*x-1)/(a*x+1)*(-c*(a*x-1))^(1/2)*(6*c^(1/2) 
*2^(1/2)*arctan(1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))-a*x*(-c*(a*x+1))^( 
1/2)-7*(-c*(a*x+1))^(1/2))/(-c*(a*x+1))^(1/2)/a
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.56 \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\left [\frac {2 \, {\left (3 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + {\left (a^{2} x^{2} + 8 \, a x + 7\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{3 \, {\left (a^{2} x - a\right )}}, -\frac {2 \, {\left (6 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, {\left (a c x - c\right )}}\right ) - {\left (a^{2} x^{2} + 8 \, a x + 7\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{3 \, {\left (a^{2} x - a\right )}}\right ] \] Input:

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

Output:

[2/3*(3*sqrt(2)*(a*x - 1)*sqrt(-c)*log(-(a^2*c*x^2 + 2*a*c*x + 2*sqrt(2)*s 
qrt(-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)) + (a^2*x^2 + 8*a*x + 7)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/( 
a*x + 1)))/(a^2*x - a), -2/3*(6*sqrt(2)*(a*x - 1)*sqrt(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 
)) - (a^2*x^2 + 8*a*x + 7)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a^ 
2*x - a)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.32 \[ \int e^{3 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 \sqrt {c}\, i \left (-\sqrt {a x +1}\, a x -7 \sqrt {a x +1}-6 \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )\right )+8 \sqrt {2}\right )}{3 a} \] Input:

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

Output:

(2*sqrt(c)*i*( - sqrt(a*x + 1)*a*x - 7*sqrt(a*x + 1) - 6*sqrt(2)*log(tan(a 
sin(sqrt( - a*x + 1)/sqrt(2))/2)) + 8*sqrt(2)))/(3*a)