\(\int \frac {e^{-3 \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx\) [576]

Optimal result

Integrand size = 24, antiderivative size = 126 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\frac {4 x \sqrt {1-a x}}{\sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}-\frac {5 \sqrt {1-a x} \sqrt {1+a x}}{a \sqrt {c-\frac {c}{a x}}}+\frac {5 \sqrt {1-a x} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{a^{3/2} \sqrt {c-\frac {c}{a x}} \sqrt {x}} \] Output:

4*x*(-a*x+1)^(1/2)/(c-c/a/x)^(1/2)/(a*x+1)^(1/2)-5*(-a*x+1)^(1/2)*(a*x+1)^ 
(1/2)/a/(c-c/a/x)^(1/2)+5*(-a*x+1)^(1/2)*arcsinh(a^(1/2)*x^(1/2))/a^(3/2)/ 
(c-c/a/x)^(1/2)/x^(1/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.68 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\frac {\sqrt {1-a x} \left (\frac {4 x^{3/2}}{\sqrt {1+a x}}-\frac {5 \sqrt {x} \sqrt {1+a x}}{a}+\frac {5 \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{a^{3/2}}\right )}{\sqrt {c-\frac {c}{a x}} \sqrt {x}} \] Input:

Integrate[1/(E^(3*ArcTanh[a*x])*Sqrt[c - c/(a*x)]),x]
 

Output:

(Sqrt[1 - a*x]*((4*x^(3/2))/Sqrt[1 + a*x] - (5*Sqrt[x]*Sqrt[1 + a*x])/a + 
(5*ArcSinh[Sqrt[a]*Sqrt[x]])/a^(3/2)))/(Sqrt[c - c/(a*x)]*Sqrt[x])
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.70, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6684, 6678, 516, 87, 60, 63, 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.

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

\(\Big \downarrow \) 6684

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

\(\Big \downarrow \) 6678

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

\(\Big \downarrow \) 516

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

\(\Big \downarrow \) 87

\(\displaystyle \frac {\sqrt {1-a x} \left (\frac {4 x^{3/2}}{\sqrt {a x+1}}-5 \int \frac {\sqrt {x}}{\sqrt {a x+1}}dx\right )}{\sqrt {x} \sqrt {c-\frac {c}{a x}}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\sqrt {1-a x} \left (\frac {4 x^{3/2}}{\sqrt {a x+1}}-5 \left (\frac {\sqrt {x} \sqrt {a x+1}}{a}-\frac {\int \frac {1}{\sqrt {x} \sqrt {a x+1}}dx}{2 a}\right )\right )}{\sqrt {x} \sqrt {c-\frac {c}{a x}}}\)

\(\Big \downarrow \) 63

\(\displaystyle \frac {\sqrt {1-a x} \left (\frac {4 x^{3/2}}{\sqrt {a x+1}}-5 \left (\frac {\sqrt {x} \sqrt {a x+1}}{a}-\frac {\int \frac {1}{\sqrt {a x+1}}d\sqrt {x}}{a}\right )\right )}{\sqrt {x} \sqrt {c-\frac {c}{a x}}}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {\sqrt {1-a x} \left (\frac {4 x^{3/2}}{\sqrt {a x+1}}-5 \left (\frac {\sqrt {x} \sqrt {a x+1}}{a}-\frac {\text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{a^{3/2}}\right )\right )}{\sqrt {x} \sqrt {c-\frac {c}{a x}}}\)

Input:

Int[1/(E^(3*ArcTanh[a*x])*Sqrt[c - c/(a*x)]),x]
 

Output:

(Sqrt[1 - a*x]*((4*x^(3/2))/Sqrt[1 + a*x] - 5*((Sqrt[x]*Sqrt[1 + a*x])/a - 
 ArcSinh[Sqrt[a]*Sqrt[x]]/a^(3/2))))/(Sqrt[c - c/(a*x)]*Sqrt[x])
 

Defintions of rubi rules used

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, 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 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 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 516

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. 
), x_Symbol] :> Int[(e*x)^m*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; Free 
Q[{a, b, c, d, e, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || 
(GtQ[a, 0] && GtQ[c, 0] &&  !IntegerQ[n]))
 

rule 6678

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

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 = 143, normalized size of antiderivative = 1.13

method	result	size
default	\(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (2 a^{\frac {3}{2}} x \sqrt {-x \left (a x +1\right )}+5 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right ) a x +10 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}+5 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right )\right ) \sqrt {-a^{2} x^{2}+1}}{2 \sqrt {a}\, c \left (a x +1\right ) \sqrt {-x \left (a x +1\right )}\, \left (a x -1\right )}\)	\(143\)
risch	\(\frac {\left (a x +1\right ) \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}\, \left (a x -1\right )}{a \sqrt {-\left (a x +1\right ) a c x}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}}+\frac {\left (-\frac {5 \arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-a c x}}\right )}{2 a \sqrt {a^{2} c}}-\frac {4 \sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +\left (x +\frac {1}{a}\right ) a c}}{a^{3} c \left (x +\frac {1}{a}\right )}\right ) \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}\, \left (a x -1\right )}{\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \sqrt {-a^{2} x^{2}+1}}\)	\(222\)

Input:

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

Output:

-1/2*(c*(a*x-1)/a/x)^(1/2)*x/a^(1/2)/c*(2*a^(3/2)*x*(-x*(a*x+1))^(1/2)+5*a 
rctan(1/2/a^(1/2)*(2*a*x+1)/(-x*(a*x+1))^(1/2))*a*x+10*a^(1/2)*(-x*(a*x+1) 
)^(1/2)+5*arctan(1/2/a^(1/2)*(2*a*x+1)/(-x*(a*x+1))^(1/2)))*(-a^2*x^2+1)^( 
1/2)/(a*x+1)/(-x*(a*x+1))^(1/2)/(a*x-1)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.27 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\left [-\frac {5 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (a^{2} x^{2} + 5 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{3} c x^{2} - a c\right )}}, \frac {5 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (a^{2} x^{2} + 5 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{3} c x^{2} - a c\right )}}\right ] \] Input:

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

Output:

[-1/4*(5*(a^2*x^2 - 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) 
) + 4*(a^2*x^2 + 5*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*c 
*x^2 - a*c), 1/2*(5*(a^2*x^2 - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sq 
rt(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(a^2*x^2 + 
5*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*c*x^2 - a*c)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.72 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx=\frac {\sqrt {c}\, \left (5 \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i}{a x +1}\right ) a x +5 \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i}{a x +1}\right )-\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a i x -5 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i \right )}{a c \left (a x +1\right )} \] Input:

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

Output:

(sqrt(c)*(5*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*i)/(a*x + 1))*a*x + 5*atan 
((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*i)/(a*x + 1)) - sqrt(x)*sqrt(a)*sqrt(a*x + 
 1)*a*i*x - 5*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*i))/(a*c*(a*x + 1))