\(\int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x} \, dx\) [425]

Optimal result

Integrand size = 23, antiderivative size = 117 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x} \, dx=-\frac {2 c \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}-2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right )+4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.56 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x} \, dx=-\frac {2 \sqrt {c-a c x} \left (\sqrt {1+a x}+\text {arctanh}\left (\sqrt {1+a x}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+a x}}{\sqrt {2}}\right )\right )}{\sqrt {1-a x}} \] Input:

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

Output:

(-2*Sqrt[c - a*c*x]*(Sqrt[1 + a*x] + ArcTanh[Sqrt[1 + a*x]] - 2*Sqrt[2]*Ar 
cTanh[Sqrt[1 + a*x]/Sqrt[2]]))/Sqrt[1 - a*x]
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.59, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {6680, 37, 95, 25, 27, 174, 73, 219, 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.

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

\(\Big \downarrow \) 6680

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

\(\Big \downarrow \) 37

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

\(\Big \downarrow \) 95

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

\(\displaystyle \frac {\sqrt {c-a c x} \left (\int \frac {1}{x \sqrt {a x+1}}dx+4 a \int \frac {1}{(1-a x) \sqrt {a x+1}}dx-2 \sqrt {a x+1}\right )}{\sqrt {1-a x}}\)

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 37

Int[(u_.)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> S 
imp[(a + b*x)^m/(c + d*x)^m   Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, 
b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] &&  !SimplerQ[a + b*x, c + d*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 95

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

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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 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 6680

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

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.84

method	result	size
default	\(-\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (2 \sqrt {c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-\sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (a x +1\right )}}{\sqrt {c}}\right )-\sqrt {c \left (a x +1\right )}\right )}{\left (a x -1\right ) \sqrt {c \left (a x +1\right )}}\)	\(98\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.68 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x} \, dx=\left [\frac {2 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{a x^{2} - x}\right ) + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{a x - 1}, \frac {2 \, {\left (2 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{2 \, {\left (a c x - c\right )}}\right ) - {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a c x - c}\right ) + \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}\right )}}{a x - 1}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.10 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x} \, dx=\sqrt {c}\, \left (-2 \sqrt {a x +1}-4 \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )\right )+2 \sqrt {2}-\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )-1\right )+\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )+1\right )+\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )-1\right )-\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )+1\right )\right ) \] Input:

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

Output:

sqrt(c)*( - 2*sqrt(a*x + 1) - 4*sqrt(2)*log(tan(asin(sqrt( - a*x + 1)/sqrt 
(2))/2)) + 2*sqrt(2) - log( - sqrt(2) + tan(asin(sqrt( - a*x + 1)/sqrt(2)) 
/2) - 1) + log( - sqrt(2) + tan(asin(sqrt( - a*x + 1)/sqrt(2))/2) + 1) + l 
og(sqrt(2) + tan(asin(sqrt( - a*x + 1)/sqrt(2))/2) - 1) - log(sqrt(2) + ta 
n(asin(sqrt( - a*x + 1)/sqrt(2))/2) + 1))