\(\int e^{3 \coth ^{-1}(a x)} (c-\frac {c}{a x})^{5/2} \, dx\) [472]

Optimal result

Integrand size = 24, antiderivative size = 156 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=-\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 a \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{a \sqrt {c-\frac {c}{a x}}}+\frac {c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x}{\left (c-\frac {c}{a x}\right )^{5/2}}+\frac {c^{5/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.57 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\frac {c^2 \sqrt {c-\frac {c}{a x}} \left (\sqrt {1+\frac {1}{a x}} \left (2+2 a x+3 a^2 x^2\right )+3 a x \text {arctanh}\left (\sqrt {1+\frac {1}{a x}}\right )\right )}{3 a^2 \sqrt {1-\frac {1}{a x}} x} \] Input:

Integrate[E^(3*ArcCoth[a*x])*(c - c/(a*x))^(5/2),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6731, 580, 576, 576, 573, 219}

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.

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

\(\Big \downarrow \) 6731

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

\(\Big \downarrow \) 580

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

\(\Big \downarrow \) 576

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

\(\Big \downarrow \) 576

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

\(\Big \downarrow \) 573

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

\(\Big \downarrow \) 219

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

Input:

Int[E^(3*ArcCoth[a*x])*(c - c/(a*x))^(5/2),x]
 

Output:

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

Defintions of rubi rules used

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 573

Int[Sqrt[(c_) + (d_.)*(x_)]/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> 
Simp[-2*c   Subst[Int[1/(a - c*x^2), x], x, Sqrt[a + b*x^2]/Sqrt[c + d*x]], 
 x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0]
 

rule 576

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
x_Symbol] :> Simp[(-(e*x)^(m + 1))*(c + d*x)^n*((a + b*x^2)^p/(e*(n - m - 1 
))), x] - Simp[b*c*(n/(d^2*(n - m - 1)))   Int[(e*x)^m*(c + d*x)^(n + 1)*(a 
 + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[b*c^2 + a* 
d^2, 0] && EqQ[n + p, 0] && GtQ[p, 0] && NeQ[m - n + 1, 0] &&  !IGtQ[m, 0] 
&&  !(IntegerQ[m + p] && LtQ[m + p + 2, 0]) && RationalQ[m]
 

rule 580

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
x_Symbol] :> Simp[(-d^2)*(e*x)^(m + 1)*(c + d*x)^(n - 2)*((a + b*x^2)^(p + 
1)/(b*e*(m + 1))), x] + Simp[d*((2*m + p + 3)/(e*(m + 1)))   Int[(e*x)^(m + 
 1)*(c + d*x)^(n - 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p}, 
 x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + p - 1, 0] && LtQ[m, -1] && IntegerQ 
[p + 1/2]
 

rule 6731

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> S 
imp[-c^n   Subst[Int[(c + d*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^2), x], x, 1/ 
x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && IntegerQ[(n - 1)/2] 
&& IntegerQ[2*p]
 

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.92

method	result	size
default	\(\frac {\left (a x -1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{2} \left (6 a^{\frac {5}{2}} x^{2} \sqrt {x \left (a x +1\right )}+3 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{2} x^{2}+4 a^{\frac {3}{2}} x \sqrt {x \left (a x +1\right )}+4 \sqrt {x \left (a x +1\right )}\, \sqrt {a}\right )}{6 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) x \,a^{\frac {5}{2}} \sqrt {x \left (a x +1\right )}}\)	\(144\)
risch	\(\frac {\left (3 a^{3} x^{3}+5 a^{2} x^{2}+4 a x +2\right ) c^{2} \sqrt {\frac {c \left (a x -1\right )}{a x}}}{3 x \,a^{2} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}+\frac {\ln \left (\frac {\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right ) c^{2} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\left (a x +1\right ) a c x}}{2 \sqrt {a^{2} c}\, \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\)	\(168\)

Input:

int(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 381, normalized size of antiderivative = 2.44 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\left [\frac {3 \, {\left (a^{2} c^{2} x^{2} - a c^{2} x\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (3 \, a^{3} c^{2} x^{3} + 5 \, a^{2} c^{2} x^{2} + 4 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{12 \, {\left (a^{3} x^{2} - a^{2} x\right )}}, -\frac {3 \, {\left (a^{2} c^{2} x^{2} - a c^{2} x\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (3 \, a^{3} c^{2} x^{3} + 5 \, a^{2} c^{2} x^{2} + 4 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{6 \, {\left (a^{3} x^{2} - a^{2} x\right )}}\right ] \] Input:

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(5/2),x, algorithm="fricas")
 

Output:

[1/12*(3*(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^3*x^3 + 3*a^2*x^2 + a*x)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x 
 - c)/(a*x)) - c)/(a*x - 1)) + 4*(3*a^3*c^2*x^3 + 5*a^2*c^2*x^2 + 4*a*c^2* 
x + 2*c^2)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2 - a 
^2*x), -1/6*(3*(a^2*c^2*x^2 - a*c^2*x)*sqrt(-c)*arctan(2*(a^2*x^2 + a*x)*s 
qrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a 
*c*x - c)) - 2*(3*a^3*c^2*x^3 + 5*a^2*c^2*x^2 + 4*a*c^2*x + 2*c^2)*sqrt((a 
*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2 - a^2*x)]
 

Sympy [F(-1)]

Timed out. \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\text {Timed out} \] Input:

integrate(1/((a*x-1)/(a*x+1))**(3/2)*(c-c/a/x)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\int { \frac {{\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(5/2),x, algorithm="maxima")
 

Output:

integrate((c - c/(a*x))^(5/2)/((a*x - 1)/(a*x + 1))^(3/2), x)
 

Giac [F]

\[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\int { \frac {{\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(5/2),x, algorithm="giac")
 

Output:

integrate((c - c/(a*x))^(5/2)/((a*x - 1)/(a*x + 1))^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\int \frac {{\left (c-\frac {c}{a\,x}\right )}^{5/2}}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \] Input:

int((c - c/(a*x))^(5/2)/((a*x - 1)/(a*x + 1))^(3/2),x)
 

Output:

int((c - c/(a*x))^(5/2)/((a*x - 1)/(a*x + 1))^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.56 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx=\frac {\sqrt {c}\, c^{2} \left (6 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{2} x^{2}+4 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a x +4 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}+6 \,\mathrm {log}\left (\sqrt {a x +1}+\sqrt {x}\, \sqrt {a}\right ) a^{2} x^{2}+5 a^{2} x^{2}\right )}{6 a^{3} x^{2}} \] Input:

int(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a/x)^(5/2),x)
 

Output:

(sqrt(c)*c**2*(6*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a**2*x**2 + 4*sqrt(x)*sqrt( 
a)*sqrt(a*x + 1)*a*x + 4*sqrt(x)*sqrt(a)*sqrt(a*x + 1) + 6*log(sqrt(a*x + 
1) + sqrt(x)*sqrt(a))*a**2*x**2 + 5*a**2*x**2))/(6*a**3*x**2)