\(\int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx\) [411]

Optimal result

Integrand size = 23, antiderivative size = 115 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=-\frac {\sqrt {1+a x} \sqrt {c-a c x}}{x \sqrt {1-a x}}-\frac {5 a \sqrt {c-a c x} \text {arctanh}\left (\sqrt {1+a x}\right )}{\sqrt {1-a x}}+\frac {4 \sqrt {2} a \sqrt {c-a c x} \text {arctanh}\left (\frac {\sqrt {1+a x}}{\sqrt {2}}\right )}{\sqrt {1-a x}} \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.65 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=-\frac {\sqrt {c-a c x} \left (\sqrt {1+a x}+5 a x \text {arctanh}\left (\sqrt {1+a x}\right )-4 \sqrt {2} a x \text {arctanh}\left (\frac {\sqrt {1+a x}}{\sqrt {2}}\right )\right )}{x \sqrt {1-a x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.68, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6680, 37, 109, 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.

\(\displaystyle \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx\)

\(\Big \downarrow \) 6680

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

\(\Big \downarrow \) 37

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(Sqrt[c - a*c*x]*(-(Sqrt[1 + a*x]/x) + (a*(-10*ArcTanh[Sqrt[1 + a*x]] + 8* 
Sqrt[2]*ArcTanh[Sqrt[1 + a*x]/Sqrt[2]]))/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 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 109

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

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 = 0.22 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.91

method	result	size
default	\(\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (-4 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a c x +5 \,\operatorname {arctanh}\left (\frac {\sqrt {c \left (a x +1\right )}}{\sqrt {c}}\right ) a c x +\sqrt {c \left (a x +1\right )}\, \sqrt {c}\right )}{\left (a x -1\right ) \sqrt {c \left (a x +1\right )}\, x \sqrt {c}}\)	\(105\)
risch	\(\frac {\left (a x +1\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{x \sqrt {c \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}-\frac {a \left (\frac {8 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{\sqrt {c}}-\frac {10 \,\operatorname {arctanh}\left (\frac {\sqrt {a c x +c}}{\sqrt {c}}\right )}{\sqrt {c}}\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}\)	\(167\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 351, normalized size of antiderivative = 3.05 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\left [\frac {4 \, \sqrt {2} {\left (a^{2} x^{2} - a x\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 5 \, {\left (a^{2} x^{2} - a x\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{a x^{2} - x}\right ) + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{2 \, {\left (a x^{2} - x\right )}}, \frac {4 \, \sqrt {2} {\left (a^{2} x^{2} - a x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{2 \, {\left (a c x - c\right )}}\right ) - 5 \, {\left (a^{2} x^{2} - a x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a c x - c}\right ) + \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{a x^{2} - x}\right ] \] Input:

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

Output:

[1/2*(4*sqrt(2)*(a^2*x^2 - a*x)*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)) + 5*(a^2*x^2 - a*x)*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)*s 
qrt(-a*c*x + c))/(a*x^2 - x), (4*sqrt(2)*(a^2*x^2 - a*x)*sqrt(-c)*arctan(1 
/2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a*c*x - c)) - 5*( 
a^2*x^2 - a*x)*sqrt(-c)*arctan(sqrt(-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^2 - x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\text {Exception raised: TypeError} \] Input:

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.64 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^2} \, dx=\frac {\sqrt {c}\, \left (-2 \sqrt {a x +1}-8 \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )\right ) a x +2 \sqrt {2}\, a x -7 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )-1\right ) a x +7 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )+1\right ) a x +7 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )-1\right ) a x -7 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )+1\right ) a x -2 \,\mathrm {log}\left (\frac {2 \sqrt {a x +1}-2}{\sqrt {2}}\right ) a x +2 \,\mathrm {log}\left (\frac {2 \sqrt {a x +1}+2}{\sqrt {2}}\right ) a x \right )}{2 x} \] Input:

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

Output:

(sqrt(c)*( - 2*sqrt(a*x + 1) - 8*sqrt(2)*log(tan(asin(sqrt( - a*x + 1)/sqr 
t(2))/2))*a*x + 2*sqrt(2)*a*x - 7*log( - sqrt(2) + tan(asin(sqrt( - a*x + 
1)/sqrt(2))/2) - 1)*a*x + 7*log( - sqrt(2) + tan(asin(sqrt( - a*x + 1)/sqr 
t(2))/2) + 1)*a*x + 7*log(sqrt(2) + tan(asin(sqrt( - a*x + 1)/sqrt(2))/2) 
- 1)*a*x - 7*log(sqrt(2) + tan(asin(sqrt( - a*x + 1)/sqrt(2))/2) + 1)*a*x 
- 2*log((2*sqrt(a*x + 1) - 2)/sqrt(2))*a*x + 2*log((2*sqrt(a*x + 1) + 2)/s 
qrt(2))*a*x))/(2*x)