\(\int e^{-\text {arctanh}(a x)} (c-\frac {c}{a x})^{5/2} \, dx\) [554]

Optimal result

Integrand size = 24, antiderivative size = 173 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\frac {6 a \left (c-\frac {c}{a x}\right )^{5/2} x^2 \sqrt {1+a x}}{(1-a x)^{5/2}}-\frac {a^2 \left (c-\frac {c}{a x}\right )^{5/2} x^3 \sqrt {1+a x}}{3 (1-a x)^{5/2}}-\frac {2 \left (c-\frac {c}{a x}\right )^{5/2} x \sqrt {1+a x}}{3 \sqrt {1-a x}}+\frac {7 a^{3/2} \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{5/2}} \] Output:

6*a*(c-c/a/x)^(5/2)*x^2*(a*x+1)^(1/2)/(-a*x+1)^(5/2)-1/3*a^2*(c-c/a/x)^(5/ 
2)*x^3*(a*x+1)^(1/2)/(-a*x+1)^(5/2)-2/3*(c-c/a/x)^(5/2)*x*(a*x+1)^(1/2)/(- 
a*x+1)^(1/2)+7*a^(3/2)*(c-c/a/x)^(5/2)*x^(5/2)*arcsinh(a^(1/2)*x^(1/2))/(- 
a*x+1)^(5/2)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.50 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=-\frac {c^2 \sqrt {c-\frac {c}{a x}} \left (\sqrt {1+a x} \left (2-22 a x+3 a^2 x^2\right )-21 a^{3/2} x^{3/2} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )\right )}{3 a^2 x \sqrt {1-a x}} \] Input:

Integrate[(c - c/(a*x))^(5/2)/E^ArcTanh[a*x],x]
 

Output:

-1/3*(c^2*Sqrt[c - c/(a*x)]*(Sqrt[1 + a*x]*(2 - 22*a*x + 3*a^2*x^2) - 21*a 
^(3/2)*x^(3/2)*ArcSinh[Sqrt[a]*Sqrt[x]]))/(a^2*x*Sqrt[1 - a*x])
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.61, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6684, 6679, 109, 27, 160, 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.

\(\displaystyle \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx\)

\(\Big \downarrow \) 6684

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

\(\Big \downarrow \) 6679

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

\(\Big \downarrow \) 109

\(\displaystyle \frac {x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2} \left (-\frac {2}{3} \int \frac {a (1-a x) (9-a x)}{2 x^{3/2} \sqrt {a x+1}}dx-\frac {2 \sqrt {a x+1} (1-a x)^2}{3 x^{3/2}}\right )}{(1-a x)^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2} \left (-\frac {1}{3} a \int \frac {(1-a x) (9-a x)}{x^{3/2} \sqrt {a x+1}}dx-\frac {2 \sqrt {a x+1} (1-a x)^2}{3 x^{3/2}}\right )}{(1-a x)^{5/2}}\)

\(\Big \downarrow \) 160

\(\displaystyle \frac {x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2} \left (-\frac {1}{3} a \left (-\frac {21}{2} a \int \frac {1}{\sqrt {x} \sqrt {a x+1}}dx-\frac {\sqrt {a x+1} (18-a x)}{\sqrt {x}}\right )-\frac {2 \sqrt {a x+1} (1-a x)^2}{3 x^{3/2}}\right )}{(1-a x)^{5/2}}\)

\(\Big \downarrow \) 63

\(\displaystyle \frac {x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2} \left (-\frac {1}{3} a \left (-21 a \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}-\frac {\sqrt {a x+1} (18-a x)}{\sqrt {x}}\right )-\frac {2 \sqrt {a x+1} (1-a x)^2}{3 x^{3/2}}\right )}{(1-a x)^{5/2}}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {x^{5/2} \left (-\frac {1}{3} a \left (-21 \sqrt {a} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )-\frac {\sqrt {a x+1} (18-a x)}{\sqrt {x}}\right )-\frac {2 \sqrt {a x+1} (1-a x)^2}{3 x^{3/2}}\right ) \left (c-\frac {c}{a x}\right )^{5/2}}{(1-a x)^{5/2}}\)

Input:

Int[(c - c/(a*x))^(5/2)/E^ArcTanh[a*x],x]
 

Output:

((c - c/(a*x))^(5/2)*x^(5/2)*((-2*(1 - a*x)^2*Sqrt[1 + a*x])/(3*x^(3/2)) - 
 (a*(-(((18 - a*x)*Sqrt[1 + a*x])/Sqrt[x]) - 21*Sqrt[a]*ArcSinh[Sqrt[a]*Sq 
rt[x]]))/3))/(1 - a*x)^(5/2)
 

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 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 160

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) 
)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b^2*d*e*g - a^2*d*f*h*m - a*b*(d*(f*g 
+ e*h) - c*f*h*(m + 1)) + b*f*h*(b*c - a*d)*(m + 1)*x)*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d*(b*c - a*d)*(m + 1))), x] + Simp[(a*d*f*h*m + b*(d* 
(f*g + e*h) - c*f*h*(m + 2)))/(b^2*d)   Int[(a + b*x)^(m + 1)*(c + d*x)^n, 
x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[m + n + 2, 0] && 
NeQ[m, -1] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])
 

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 = 136, normalized size of antiderivative = 0.79

method	result	size
default	\(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{2} \sqrt {-a^{2} x^{2}+1}\, \left (6 a^{\frac {5}{2}} x^{2} \sqrt {-x \left (a x +1\right )}+21 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right ) a^{2} x^{2}-44 a^{\frac {3}{2}} x \sqrt {-x \left (a x +1\right )}+4 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}\right )}{6 x \,a^{\frac {5}{2}} \left (a x -1\right ) \sqrt {-x \left (a x +1\right )}}\)	\(136\)
risch	\(-\frac {\left (3 a^{3} x^{3}-19 a^{2} x^{2}-20 a x +2\right ) c^{2} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{3 x \sqrt {-\left (a x +1\right ) a c x}\, \sqrt {-a^{2} x^{2}+1}\, a^{2}}+\frac {7 \arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-a c x}}\right ) c^{2} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{2 \sqrt {a^{2} c}\, \sqrt {-a^{2} x^{2}+1}}\)	\(192\)

Input:

int((c-c/a/x)^(5/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/6*(c*(a*x-1)/a/x)^(1/2)/x*c^2/a^(5/2)*(-a^2*x^2+1)^(1/2)*(6*a^(5/2)*x^2* 
(-x*(a*x+1))^(1/2)+21*arctan(1/2/a^(1/2)*(2*a*x+1)/(-x*(a*x+1))^(1/2))*a^2 
*x^2-44*a^(3/2)*x*(-x*(a*x+1))^(1/2)+4*a^(1/2)*(-x*(a*x+1))^(1/2))/(a*x-1) 
/(-x*(a*x+1))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.91 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\left [\frac {21 \, {\left (a^{2} c^{2} x^{2} - a c^{2} x\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (3 \, a^{2} c^{2} x^{2} - 22 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{12 \, {\left (a^{3} x^{2} - a^{2} x\right )}}, -\frac {21 \, {\left (a^{2} c^{2} x^{2} - a c^{2} x\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (3 \, a^{2} c^{2} x^{2} - 22 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{6 \, {\left (a^{3} x^{2} - a^{2} x\right )}}\right ] \] Input:

integrate((c-c/a/x)^(5/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="fricas" 
)
 

Output:

[1/12*(21*(a^2*c^2*x^2 - a*c^2*x)*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*(3*a^2*c^2*x^2 - 22*a*c^2*x + 2*c^2)*sqrt(-a^2*x^2 + 1)*s 
qrt((a*c*x - c)/(a*x)))/(a^3*x^2 - a^2*x), -1/6*(21*(a^2*c^2*x^2 - a*c^2*x 
)*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)) - 2*(3*a^2*c^2*x^2 - 22*a*c^2*x + 2*c^2)*sqrt(- 
a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2 - a^2*x)]
 

Sympy [F]

\[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\int \frac {\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {5}{2}} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{a x + 1}\, dx \] Input:

integrate((c-c/a/x)**(5/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)
 

Output:

Integral((-c*(-1 + 1/(a*x)))**(5/2)*sqrt(-(a*x - 1)*(a*x + 1))/(a*x + 1), 
x)
 

Maxima [F]

\[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} {\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}}}{a x + 1} \,d x } \] Input:

integrate((c-c/a/x)^(5/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="maxima" 
)
 

Output:

integrate(sqrt(-a^2*x^2 + 1)*(c - c/(a*x))^(5/2)/(a*x + 1), x)
 

Giac [F(-2)]

Exception generated. \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((c-c/a/x)^(5/2)/(a*x+1)*(-a^2*x^2+1)^(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. \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\int \frac {{\left (c-\frac {c}{a\,x}\right )}^{5/2}\,\sqrt {1-a^2\,x^2}}{a\,x+1} \,d x \] Input:

int(((c - c/(a*x))^(5/2)*(1 - a^2*x^2)^(1/2))/(a*x + 1),x)
 

Output:

int(((c - c/(a*x))^(5/2)*(1 - a^2*x^2)^(1/2))/(a*x + 1), x)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.53 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\frac {\sqrt {c}\, c^{2} i \left (6 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{2} x^{2}-44 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a x +4 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}-42 \,\mathrm {log}\left (\sqrt {a x +1}\, i +\sqrt {x}\, \sqrt {a}\, i \right ) a^{2} x^{2}+21 a^{2} x^{2}\right )}{6 a^{3} x^{2}} \] Input:

int((c-c/a/x)^(5/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x)
 

Output:

(sqrt(c)*c**2*i*(6*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a**2*x**2 - 44*sqrt(x)*sq 
rt(a)*sqrt(a*x + 1)*a*x + 4*sqrt(x)*sqrt(a)*sqrt(a*x + 1) - 42*log(sqrt(a* 
x + 1)*i + sqrt(x)*sqrt(a)*i)*a**2*x**2 + 21*a**2*x**2))/(6*a**3*x**2)