\(\int e^{2 \coth ^{-1}(a x)} (c-\frac {c}{a x})^{7/2} \, dx\) [462]

Optimal result

Integrand size = 24, antiderivative size = 118 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\frac {3 c^3 \sqrt {c-\frac {c}{a x}}}{a}+\frac {c^2 \left (c-\frac {c}{a x}\right )^{3/2}}{a}+\frac {3 c \left (c-\frac {c}{a x}\right )^{5/2}}{5 a}+\left (c-\frac {c}{a x}\right )^{7/2} x-\frac {3 c^{7/2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.70 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\frac {c^3 \sqrt {c-\frac {c}{a x}} \left (-2+4 a x+8 a^2 x^2+5 a^3 x^3\right )}{5 a^3 x^2}-\frac {3 c^{7/2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a} \] Input:

Integrate[E^(2*ArcCoth[a*x])*(c - c/(a*x))^(7/2),x]
 

Output:

(c^3*Sqrt[c - c/(a*x)]*(-2 + 4*a*x + 8*a^2*x^2 + 5*a^3*x^3))/(5*a^3*x^2) - 
 (3*c^(7/2)*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {6717, 6683, 1035, 281, 899, 87, 60, 60, 60, 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.

\(\displaystyle \int \left (c-\frac {c}{a x}\right )^{7/2} e^{2 \coth ^{-1}(a x)} \, dx\)

\(\Big \downarrow \) 6717

\(\displaystyle -\int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{7/2}dx\)

\(\Big \downarrow \) 6683

\(\displaystyle -\int \frac {\left (c-\frac {c}{a x}\right )^{7/2} (a x+1)}{1-a x}dx\)

\(\Big \downarrow \) 1035

\(\displaystyle -\int \frac {\left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{7/2}}{\frac {1}{x}-a}dx\)

\(\Big \downarrow \) 281

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

\(\Big \downarrow \) 899

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

\(\Big \downarrow \) 87

\(\displaystyle -\frac {c \left (-\frac {3}{2} \int \left (c-\frac {c}{a x}\right )^{5/2} xd\frac {1}{x}-\frac {a x \left (c-\frac {c}{a x}\right )^{7/2}}{c}\right )}{a}\)

\(\Big \downarrow \) 60

\(\displaystyle -\frac {c \left (-\frac {3}{2} \left (c \int \left (c-\frac {c}{a x}\right )^{3/2} xd\frac {1}{x}+\frac {2}{5} \left (c-\frac {c}{a x}\right )^{5/2}\right )-\frac {a x \left (c-\frac {c}{a x}\right )^{7/2}}{c}\right )}{a}\)

\(\Big \downarrow \) 60

\(\displaystyle -\frac {c \left (-\frac {3}{2} \left (c \left (c \int \sqrt {c-\frac {c}{a x}} xd\frac {1}{x}+\frac {2}{3} \left (c-\frac {c}{a x}\right )^{3/2}\right )+\frac {2}{5} \left (c-\frac {c}{a x}\right )^{5/2}\right )-\frac {a x \left (c-\frac {c}{a x}\right )^{7/2}}{c}\right )}{a}\)

\(\Big \downarrow \) 60

\(\displaystyle -\frac {c \left (-\frac {3}{2} \left (c \left (c \left (c \int \frac {x}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}+2 \sqrt {c-\frac {c}{a x}}\right )+\frac {2}{3} \left (c-\frac {c}{a x}\right )^{3/2}\right )+\frac {2}{5} \left (c-\frac {c}{a x}\right )^{5/2}\right )-\frac {a x \left (c-\frac {c}{a x}\right )^{7/2}}{c}\right )}{a}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {c \left (-\frac {3}{2} \left (c \left (c \left (2 \sqrt {c-\frac {c}{a x}}-2 a \int \frac {1}{a-\frac {a}{c x^2}}d\sqrt {c-\frac {c}{a x}}\right )+\frac {2}{3} \left (c-\frac {c}{a x}\right )^{3/2}\right )+\frac {2}{5} \left (c-\frac {c}{a x}\right )^{5/2}\right )-\frac {a x \left (c-\frac {c}{a x}\right )^{7/2}}{c}\right )}{a}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {c \left (-\frac {3}{2} \left (c \left (c \left (2 \sqrt {c-\frac {c}{a x}}-2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )\right )+\frac {2}{3} \left (c-\frac {c}{a x}\right )^{3/2}\right )+\frac {2}{5} \left (c-\frac {c}{a x}\right )^{5/2}\right )-\frac {a x \left (c-\frac {c}{a x}\right )^{7/2}}{c}\right )}{a}\)

Input:

Int[E^(2*ArcCoth[a*x])*(c - c/(a*x))^(7/2),x]
 

Output:

-((c*(-((a*(c - c/(a*x))^(7/2)*x)/c) - (3*((2*(c - c/(a*x))^(5/2))/5 + c*( 
(2*(c - c/(a*x))^(3/2))/3 + c*(2*Sqrt[c - c/(a*x)] - 2*Sqrt[c]*ArcTanh[Sqr 
t[c - c/(a*x)]/Sqrt[c]]))))/2))/a)
 

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 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 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 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]
 

rule 6717

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2)   Int[ 
u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
 

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.22

method	result	size
default	\(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{3} \left (-30 \sqrt {a \,x^{2}-x}\, a^{\frac {7}{2}} x^{4}+20 a^{\frac {5}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x^{2}+15 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{3} x^{4}+4 a^{\frac {3}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x -4 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\right )}{10 x^{3} \sqrt {x \left (a x -1\right )}\, a^{\frac {7}{2}}}\)	\(144\)
risch	\(\frac {\left (5 a^{4} x^{4}+3 a^{3} x^{3}-4 a^{2} x^{2}-6 a x +2\right ) c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}}{5 x^{2} a^{3} \left (a x -1\right )}-\frac {3 \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right ) c^{3} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {c \left (a x -1\right ) a x}}{2 \sqrt {a^{2} c}\, \left (a x -1\right )}\)	\(147\)

Input:

int(1/(a*x-1)*(a*x+1)*(c-c/a/x)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

-1/10*(c*(a*x-1)/a/x)^(1/2)/x^3*c^3*(-30*(a*x^2-x)^(1/2)*a^(7/2)*x^4+20*a^ 
(5/2)*(a*x^2-x)^(3/2)*x^2+15*ln(1/2*(2*(a*x^2-x)^(1/2)*a^(1/2)+2*a*x-1)/a^ 
(1/2))*a^3*x^4+4*a^(3/2)*(a*x^2-x)^(3/2)*x-4*(a*x^2-x)^(3/2)*a^(1/2))/(x*( 
a*x-1))^(1/2)/a^(7/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.87 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\left [\frac {15 \, a^{2} c^{\frac {7}{2}} x^{2} \log \left (-2 \, a c x + 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) + 2 \, {\left (5 \, a^{3} c^{3} x^{3} + 8 \, a^{2} c^{3} x^{2} + 4 \, a c^{3} x - 2 \, c^{3}\right )} \sqrt {\frac {a c x - c}{a x}}}{10 \, a^{3} x^{2}}, \frac {15 \, a^{2} \sqrt {-c} c^{3} x^{2} \arctan \left (\frac {a \sqrt {-c} x \sqrt {\frac {a c x - c}{a x}}}{a c x - c}\right ) + {\left (5 \, a^{3} c^{3} x^{3} + 8 \, a^{2} c^{3} x^{2} + 4 \, a c^{3} x - 2 \, c^{3}\right )} \sqrt {\frac {a c x - c}{a x}}}{5 \, a^{3} x^{2}}\right ] \] Input:

integrate(1/(a*x-1)*(a*x+1)*(c-c/a/x)^(7/2),x, algorithm="fricas")
 

Output:

[1/10*(15*a^2*c^(7/2)*x^2*log(-2*a*c*x + 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a 
*x)) + c) + 2*(5*a^3*c^3*x^3 + 8*a^2*c^3*x^2 + 4*a*c^3*x - 2*c^3)*sqrt((a* 
c*x - c)/(a*x)))/(a^3*x^2), 1/5*(15*a^2*sqrt(-c)*c^3*x^2*arctan(a*sqrt(-c) 
*x*sqrt((a*c*x - c)/(a*x))/(a*c*x - c)) + (5*a^3*c^3*x^3 + 8*a^2*c^3*x^2 + 
 4*a*c^3*x - 2*c^3)*sqrt((a*c*x - c)/(a*x)))/(a^3*x^2)]
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 6.91 (sec) , antiderivative size = 740, normalized size of antiderivative = 6.27 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx =\text {Too large to display} \] Input:

integrate(1/(a*x-1)*(a*x+1)*(c-c/a/x)**(7/2),x)
 

Output:

c**3*Piecewise((-sqrt(c)*acosh(sqrt(a)*sqrt(x))/a + sqrt(c)*sqrt(x)*sqrt(a 
*x - 1)/sqrt(a), Abs(a*x) > 1), (-I*sqrt(a)*sqrt(c)*x**(3/2)/sqrt(-a*x + 1 
) + I*sqrt(c)*asin(sqrt(a)*sqrt(x))/a + I*sqrt(c)*sqrt(x)/(sqrt(a)*sqrt(-a 
*x + 1)), True)) + c**3*Piecewise((2*c*atan(sqrt(c - c/(a*x))/sqrt(-c))/sq 
rt(-c) + 2*sqrt(c - c/(a*x)), Ne(c/a, 0)), (-sqrt(c)*log(x), True))/a - c* 
*3*Piecewise((0, Eq(c, 0)), (2*a*(c - c/(a*x))**(3/2)/(3*c), True))/a**2 + 
 c**3*Piecewise((-4*a**(11/2)*sqrt(c)*x**(7/2)/(15*a**(7/2)*x**(7/2) - 15* 
a**(5/2)*x**(5/2)) + 4*a**(9/2)*sqrt(c)*x**(5/2)/(15*a**(7/2)*x**(7/2) - 1 
5*a**(5/2)*x**(5/2)) + 4*a**5*sqrt(c)*x**3*sqrt(a*x - 1)/(15*a**(7/2)*x**( 
7/2) - 15*a**(5/2)*x**(5/2)) - 2*a**4*sqrt(c)*x**2*sqrt(a*x - 1)/(15*a**(7 
/2)*x**(7/2) - 15*a**(5/2)*x**(5/2)) - 8*a**3*sqrt(c)*x*sqrt(a*x - 1)/(15* 
a**(7/2)*x**(7/2) - 15*a**(5/2)*x**(5/2)) + 6*a**2*sqrt(c)*sqrt(a*x - 1)/( 
15*a**(7/2)*x**(7/2) - 15*a**(5/2)*x**(5/2)), Abs(a*x) > 1), (-4*a**(11/2) 
*sqrt(c)*x**(7/2)/(15*a**(7/2)*x**(7/2) - 15*a**(5/2)*x**(5/2)) + 4*a**(9/ 
2)*sqrt(c)*x**(5/2)/(15*a**(7/2)*x**(7/2) - 15*a**(5/2)*x**(5/2)) + 4*I*a* 
*5*sqrt(c)*x**3*sqrt(-a*x + 1)/(15*a**(7/2)*x**(7/2) - 15*a**(5/2)*x**(5/2 
)) - 2*I*a**4*sqrt(c)*x**2*sqrt(-a*x + 1)/(15*a**(7/2)*x**(7/2) - 15*a**(5 
/2)*x**(5/2)) - 8*I*a**3*sqrt(c)*x*sqrt(-a*x + 1)/(15*a**(7/2)*x**(7/2) - 
15*a**(5/2)*x**(5/2)) + 6*I*a**2*sqrt(c)*sqrt(-a*x + 1)/(15*a**(7/2)*x**(7 
/2) - 15*a**(5/2)*x**(5/2)), True))/a**3
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\int { \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {7}{2}}}{a x - 1} \,d x } \] Input:

integrate(1/(a*x-1)*(a*x+1)*(c-c/a/x)^(7/2),x, algorithm="maxima")
 

Output:

integrate((a*x + 1)*(c - c/(a*x))^(7/2)/(a*x - 1), x)
 

Giac [F(-2)]

Exception generated. \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/(a*x-1)*(a*x+1)*(c-c/a/x)^(7/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable 
to make series expansion Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\int \frac {{\left (c-\frac {c}{a\,x}\right )}^{7/2}\,\left (a\,x+1\right )}{a\,x-1} \,d x \] Input:

int(((c - c/(a*x))^(7/2)*(a*x + 1))/(a*x - 1),x)
 

Output:

int(((c - c/(a*x))^(7/2)*(a*x + 1))/(a*x - 1), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.89 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx=\frac {\sqrt {c}\, c^{3} \left (20 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}\, a^{3} x^{3}+32 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}\, a^{2} x^{2}+16 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}\, a x -8 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}-60 \,\mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}\right ) a^{3} x^{3}-41 a^{3} x^{3}\right )}{20 a^{4} x^{3}} \] Input:

int(1/(a*x-1)*(a*x+1)*(c-c/a/x)^(7/2),x)
 

Output:

(sqrt(c)*c**3*(20*sqrt(x)*sqrt(a)*sqrt(a*x - 1)*a**3*x**3 + 32*sqrt(x)*sqr 
t(a)*sqrt(a*x - 1)*a**2*x**2 + 16*sqrt(x)*sqrt(a)*sqrt(a*x - 1)*a*x - 8*sq 
rt(x)*sqrt(a)*sqrt(a*x - 1) - 60*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a))*a**3 
*x**3 - 41*a**3*x**3))/(20*a**4*x**3)