\(\int \frac {e^{-2 \text {arctanh}(a x)}}{(c-\frac {c}{a x})^{3/2}} \, dx\) [567]

Optimal result

Integrand size = 24, antiderivative size = 95 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=-\frac {\sqrt {c-\frac {c}{a x}} x}{c^2}+\frac {\text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{3/2}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a c^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=-\frac {\sqrt {c-\frac {c}{a x}} x}{c^2}+\frac {\text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{3/2}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a c^{3/2}} \] Input:

Integrate[1/(E^(2*ArcTanh[a*x])*(c - c/(a*x))^(3/2)),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.21, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6683, 1035, 281, 899, 114, 27, 94, 73, 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.

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

\(\Big \downarrow \) 6683

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

\(\Big \downarrow \) 1035

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

\(\Big \downarrow \) 281

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

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 94

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

Int[1/(E^(2*ArcTanh[a*x])*(c - c/(a*x))^(3/2)),x]
 

Output:

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

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 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 94

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

rule 114

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 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 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 281

Int[(u_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_ 
Symbol] :> Simp[(b/d)^p   Int[u*(c + d*x^n)^(p + q), x], x] /; FreeQ[{a, b, 
 c, d, n, p, q}, x] && EqQ[b*c - a*d, 0] && IntegerQ[p] &&  !(IntegerQ[q] & 
& SimplerQ[a + b*x^n, c + d*x^n])
 

rule 899

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol 
] :> -Subst[Int[(a + b/x^n)^p*((c + d/x^n)^q/x^2), x], x, 1/x] /; FreeQ[{a, 
 b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]
 

rule 1035

Int[((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_.))^(p_.)*((e_) 
 + (f_.)*(x_)^(n_.))^(r_.), x_Symbol] :> Int[x^(n*(p + r))*(b + a/x^n)^p*(c 
 + d/x^n)^q*(f + e/x^n)^r, x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && EqQ[ 
mn, -n] && IntegerQ[p] && IntegerQ[r]
 

rule 6683

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, p}, x] && EqQ[c^2 - a^2*d^2, 0] &&  !IntegerQ[p] && IntegerQ[n/2] &&  !G 
tQ[c, 0]
 

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.41

method	result	size
default	\(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (-2 \sqrt {x \left (a x -1\right )}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}+\ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}+\sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x -1\right )}\, a -3 a x +1}{a x +1}\right ) \sqrt {a}\right )}{2 a^{\frac {3}{2}} \sqrt {x \left (a x -1\right )}\, c^{2} \sqrt {\frac {1}{a}}}\)	\(134\)
risch	\(-\frac {a x -1}{a c \sqrt {\frac {c \left (a x -1\right )}{a x}}}-\frac {\left (-\frac {\ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right )}{2 a^{2} \sqrt {a^{2} c}}-\frac {\sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {\left (x +\frac {1}{a}\right )^{2} a^{2} c -3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right )}{2 a^{3} \sqrt {c}}\right ) a \sqrt {c \left (a x -1\right ) a x}}{c \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x}\)	\(183\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.44 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\left [-\frac {2 \, a x \sqrt {\frac {a c x - c}{a x}} - \sqrt {2} \sqrt {c} \log \left (\frac {\frac {2 \, \sqrt {2} a x \sqrt {\frac {a c x - c}{a x}}}{\sqrt {c}} - 3 \, a x + 1}{a x + 1}\right ) - \sqrt {c} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right )}{2 \, a c^{2}}, \frac {\sqrt {2} c \sqrt {-\frac {1}{c}} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{c}} \sqrt {\frac {a c x - c}{a x}}\right ) - a x \sqrt {\frac {a c x - c}{a x}} - \sqrt {-c} \arctan \left (\frac {a \sqrt {-c} x \sqrt {\frac {a c x - c}{a x}}}{a c x - c}\right )}{a c^{2}}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.17 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\frac {\sqrt {c}\, \left (-2 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}-\sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}\, i +i \right )-\sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}\, i -i \right )+\sqrt {2}\, \mathrm {log}\left (2 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}+2 \sqrt {2}+2 a x +2\right )+2 \,\mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}\right )\right )}{2 a \,c^{2}} \] Input:

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

Output:

(sqrt(c)*( - 2*sqrt(x)*sqrt(a)*sqrt(a*x - 1) - sqrt(2)*log(sqrt(a*x - 1) + 
 sqrt(x)*sqrt(a) - sqrt(2)*i + i) - sqrt(2)*log(sqrt(a*x - 1) + sqrt(x)*sq 
rt(a) + sqrt(2)*i - i) + sqrt(2)*log(2*sqrt(x)*sqrt(a)*sqrt(a*x - 1) + 2*s 
qrt(2) + 2*a*x + 2) + 2*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a))))/(2*a*c**2)