\(\int \frac {e^{-2 \text {arctanh}(a x)}}{(c-\frac {c}{a x})^{5/2}} \, dx\) [568]

Optimal result

Integrand size = 24, antiderivative size = 119 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\frac {2}{a c^2 \sqrt {c-\frac {c}{a x}}}-\frac {x}{c^2 \sqrt {c-\frac {c}{a x}}}-\frac {\text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{5/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {2} a c^{5/2}} \] Output:

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.56 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\frac {-a x+\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {a-\frac {1}{x}}{2 a}\right )+\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1-\frac {1}{a x}\right )}{a c^2 \sqrt {c-\frac {c}{a x}}} \] Input:

Integrate[1/(E^(2*ArcTanh[a*x])*(c - c/(a*x))^(5/2)),x]
 

Output:

(-(a*x) + Hypergeometric2F1[-1/2, 1, 1/2, (a - x^(-1))/(2*a)] + Hypergeome 
tric2F1[-1/2, 1, 1/2, 1 - 1/(a*x)])/(a*c^2*Sqrt[c - c/(a*x)])
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.10, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6683, 1035, 281, 899, 114, 27, 169, 25, 27, 174, 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.

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

\(\Big \downarrow \) 6683

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

\(\Big \downarrow \) 1035

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

\(\Big \downarrow \) 281

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

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 169

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

Int[1/(E^(2*ArcTanh[a*x])*(c - c/(a*x))^(5/2)),x]
 

Output:

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

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 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 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 169

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(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] + S 
imp[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*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

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 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(228\) vs. \(2(100)=200\).

Time = 0.22 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.92

method	result	size
risch	\(-\frac {a x -1}{a \,c^{2} \sqrt {\frac {c \left (a x -1\right )}{a x}}}-\frac {\left (\frac {\ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right )}{2 a^{3} \sqrt {a^{2} c}}-\frac {\sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2} c +\left (x -\frac {1}{a}\right ) a c}}{a^{5} c \left (x -\frac {1}{a}\right )}-\frac {\sqrt {2}\, \ln \left (\frac {4 c -3 \left (x +\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {\left (x +\frac {1}{a}\right )^{2} a^{2} c -3 \left (x +\frac {1}{a}\right ) a c +2 c}}{x +\frac {1}{a}}\right )}{4 a^{4} \sqrt {c}}\right ) a^{2} \sqrt {c \left (a x -1\right ) a x}}{c^{2} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x}\)	\(229\)
default	\(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (-8 \sqrt {x \left (a x -1\right )}\, a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, x^{2}+4 \left (x \left (a x -1\right )\right )^{\frac {3}{2}} a^{\frac {5}{2}} \sqrt {\frac {1}{a}}-2 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{3} x^{2}+a^{\frac {5}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x -1\right )}\, a -3 a x +1}{a x +1}\right ) x^{2}+16 \sqrt {x \left (a x -1\right )}\, a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, x +4 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{2} x -2 a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x -1\right )}\, a -3 a x +1}{a x +1}\right ) x -8 \sqrt {x \left (a x -1\right )}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}-2 \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}+\sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x -1\right )}\, a -3 a x +1}{a x +1}\right ) \sqrt {a}\right )}{4 a^{\frac {3}{2}} \sqrt {x \left (a x -1\right )}\, c^{3} \left (a x -1\right )^{2} \sqrt {\frac {1}{a}}}\)	\(368\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.53 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\left [\frac {\sqrt {2} {\left (a x - 1\right )} \sqrt {c} \log \left (\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right ) + 2 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-2 \, a c x + 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) - 4 \, {\left (a^{2} x^{2} - 2 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{2} c^{3} x - a c^{3}\right )}}, \frac {\sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} a \sqrt {-c} x \sqrt {\frac {a c x - c}{a x}}}{a c x - c}\right ) + 2 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {a \sqrt {-c} x \sqrt {\frac {a c x - c}{a x}}}{a c x - c}\right ) - 2 \, {\left (a^{2} x^{2} - 2 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} c^{3} x - a c^{3}\right )}}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.29 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx=\frac {\sqrt {c}\, \left (-\sqrt {a x -1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}-\sqrt {2}\, i +i \right )-\sqrt {a x -1}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}+\sqrt {2}\, i -i \right )+\sqrt {a x -1}\, \sqrt {2}\, \mathrm {log}\left (2 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}+2 \sqrt {2}+2 a x +2\right )-4 \sqrt {a x -1}\, \mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}\right )+5 \sqrt {a x -1}-4 \sqrt {x}\, \sqrt {a}\, a x +8 \sqrt {x}\, \sqrt {a}\right )}{4 \sqrt {a x -1}\, a \,c^{3}} \] Input:

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

Output:

(sqrt(c)*( - sqrt(a*x - 1)*sqrt(2)*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a) - s 
qrt(2)*i + i) - sqrt(a*x - 1)*sqrt(2)*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a) 
+ sqrt(2)*i - i) + sqrt(a*x - 1)*sqrt(2)*log(2*sqrt(x)*sqrt(a)*sqrt(a*x - 
1) + 2*sqrt(2) + 2*a*x + 2) - 4*sqrt(a*x - 1)*log(sqrt(a*x - 1) + sqrt(x)* 
sqrt(a)) + 5*sqrt(a*x - 1) - 4*sqrt(x)*sqrt(a)*a*x + 8*sqrt(x)*sqrt(a)))/( 
4*sqrt(a*x - 1)*a*c**3)