\(\int e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx\) [621]

Optimal result

Integrand size = 24, antiderivative size = 155 \[ \int e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=-\frac {\sqrt {c-\frac {c}{a x}} x \sqrt {1+a x}}{\sqrt {1-a x}}-\frac {5 \sqrt {c-\frac {c}{a x}} \sqrt {x} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a} \sqrt {1-a x}}+\frac {4 \sqrt {2} \sqrt {c-\frac {c}{a x}} \sqrt {x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{\sqrt {a} \sqrt {1-a x}} \] Output:

-(c-c/a/x)^(1/2)*x*(a*x+1)^(1/2)/(-a*x+1)^(1/2)-5*(c-c/a/x)^(1/2)*x^(1/2)* 
arcsinh(a^(1/2)*x^(1/2))/a^(1/2)/(-a*x+1)^(1/2)+4*2^(1/2)*(c-c/a/x)^(1/2)* 
x^(1/2)*arctanh(2^(1/2)*a^(1/2)*x^(1/2)/(a*x+1)^(1/2))/a^(1/2)/(-a*x+1)^(1 
/2)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.68 \[ \int e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\frac {\sqrt {c-\frac {c}{a x}} \sqrt {x} \left (-\sqrt {x} \sqrt {1+a x}-\frac {5 \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}}+\frac {4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{\sqrt {a}}\right )}{\sqrt {1-a x}} \] Input:

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

Output:

(Sqrt[c - c/(a*x)]*Sqrt[x]*(-(Sqrt[x]*Sqrt[1 + a*x]) - (5*ArcSinh[Sqrt[a]* 
Sqrt[x]])/Sqrt[a] + (4*Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + 
a*x]])/Sqrt[a]))/Sqrt[1 - a*x]
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.71, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6684, 6679, 113, 27, 175, 63, 104, 219, 222}

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.

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

\(\Big \downarrow \) 6684

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

\(\Big \downarrow \) 6679

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

\(\Big \downarrow \) 113

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 175

\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{2} \left (8 \int \frac {1}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx-5 \int \frac {1}{\sqrt {x} \sqrt {a x+1}}dx\right )-\sqrt {x} \sqrt {a x+1}\right )}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 63

\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{2} \left (8 \int \frac {1}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx-10 \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}\right )-\sqrt {x} \sqrt {a x+1}\right )}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{2} \left (16 \int \frac {1}{1-\frac {2 a x}{a x+1}}d\frac {\sqrt {x}}{\sqrt {a x+1}}-10 \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}\right )-\sqrt {x} \sqrt {a x+1}\right )}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{2} \left (\frac {8 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{\sqrt {a}}-10 \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}\right )-\sqrt {x} \sqrt {a x+1}\right )}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{2} \left (\frac {8 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{\sqrt {a}}-\frac {10 \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}}\right )-\sqrt {x} \sqrt {a x+1}\right )}{\sqrt {1-a x}}\)

Input:

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

Output:

(Sqrt[c - c/(a*x)]*Sqrt[x]*(-(Sqrt[x]*Sqrt[1 + a*x]) + ((-10*ArcSinh[Sqrt[ 
a]*Sqrt[x]])/Sqrt[a] + (8*Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 
 + a*x]])/Sqrt[a])/2))/Sqrt[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 63

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2/b   S 
ubst[Int[1/Sqrt[c + d*(x^2/b)], x], x, Sqrt[b*x]], x] /; FreeQ[{b, c, d}, x 
] && GtQ[c, 0]
 

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 113

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

rule 175

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ 
)))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b   Int[(c + d*x)^n*(e + f*x)^p, x] 
, x] + Simp[(b*g - a*h)/b   Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] 
 /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
 

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 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

rule 6679

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol 
] :> Simp[c^p   Int[u*(1 + d*(x/c))^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] 
, x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ[p] 
|| GtQ[c, 0])
 

rule 6684

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] 
 :> Simp[x^p*((c + d/x)^p/(1 + c*(x/d))^p)   Int[u*(1 + c*(x/d))^p*(E^(n*Ar 
cTanh[a*x])/x^p), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 
 0] &&  !IntegerQ[p]
 

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.06

method	result	size
default	\(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \sqrt {-a^{2} x^{2}+1}\, \left (2 \sqrt {-x \left (a x +1\right )}\, a^{\frac {3}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}-5 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right ) a \sqrt {2}\, \sqrt {-\frac {1}{a}}+8 \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-x \left (a x +1\right )}\, a -3 a x -1}{a x -1}\right ) \sqrt {a}\right ) \sqrt {2}}{4 \left (a x -1\right ) \sqrt {-x \left (a x +1\right )}\, a^{\frac {3}{2}} \sqrt {-\frac {1}{a}}}\)	\(165\)
risch	\(-\frac {\left (a x +1\right ) x \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{\sqrt {-\left (a x +1\right ) a c x}\, \sqrt {-a^{2} x^{2}+1}}-\frac {\left (\frac {5 \arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-a c x}}\right )}{2 \sqrt {a^{2} c}}-\frac {4 \ln \left (\frac {-4 c -3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {-2 c}\, \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -3 \left (x -\frac {1}{a}\right ) a c -2 c}}{x -\frac {1}{a}}\right )}{a \sqrt {-2 c}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{\sqrt {-a^{2} x^{2}+1}}\)	\(242\)

Input:

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

Output:

1/4*(c*(a*x-1)/a/x)^(1/2)*x*(-a^2*x^2+1)^(1/2)*(2*(-x*(a*x+1))^(1/2)*a^(3/ 
2)*2^(1/2)*(-1/a)^(1/2)-5*arctan(1/2/a^(1/2)*(2*a*x+1)/(-x*(a*x+1))^(1/2)) 
*a*2^(1/2)*(-1/a)^(1/2)+8*ln((2*2^(1/2)*(-1/a)^(1/2)*(-x*(a*x+1))^(1/2)*a- 
3*a*x-1)/(a*x-1))*a^(1/2))*2^(1/2)/(a*x-1)/(-x*(a*x+1))^(1/2)/a^(3/2)/(-1/ 
a)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.85 \[ \int e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\left [\frac {4 \, \sqrt {-a^{2} x^{2} + 1} a x \sqrt {\frac {a c x - c}{a x}} + 4 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt {2} {\left (3 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 5 \, {\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right )}{4 \, {\left (a^{2} x - a\right )}}, \frac {2 \, \sqrt {-a^{2} x^{2} + 1} a x \sqrt {\frac {a c x - c}{a x}} - 4 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) + 5 \, {\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right )}{2 \, {\left (a^{2} x - a\right )}}\right ] \] Input:

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

Output:

[1/4*(4*sqrt(-a^2*x^2 + 1)*a*x*sqrt((a*c*x - c)/(a*x)) + 4*sqrt(2)*(a*x - 
1)*sqrt(-c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x + 4*sqrt(2)*(3*a^2 
*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a^3* 
x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 5*(a*x - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7 
*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/ 
(a*x)) - c)/(a*x - 1)))/(a^2*x - a), 1/2*(2*sqrt(-a^2*x^2 + 1)*a*x*sqrt((a 
*c*x - c)/(a*x)) - 4*sqrt(2)*(a*x - 1)*sqrt(c)*arctan(2*sqrt(2)*sqrt(-a^2* 
x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) 
+ 5*(a*x - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x 
- c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)))/(a^2*x - a)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.43 \[ \int e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx=\frac {\sqrt {c}\, \left (-4 \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, \sqrt {2}\, i}{a x +1}\right )+5 \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i}{a x +1}\right )+\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i \right )}{a} \] Input:

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

Output:

(sqrt(c)*( - 4*sqrt(2)*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*sqrt(2)*i)/(a*x 
 + 1)) + 5*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*i)/(a*x + 1)) + sqrt(x)*sqr 
t(a)*sqrt(a*x + 1)*i))/a