\(\int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx\) [513]

Optimal result

Integrand size = 25, antiderivative size = 80 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\frac {7 \sqrt {c-\frac {c}{a x}} x}{4 a}+\frac {1}{2} \sqrt {c-\frac {c}{a x}} x^2+\frac {7 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{4 a^2} \] Output:

7/4*(c-c/a/x)^(1/2)*x/a+1/2*(c-c/a/x)^(1/2)*x^2+7/4*c^(1/2)*arctanh((c-c/a 
/x)^(1/2)/c^(1/2))/a^2
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.96 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\frac {\sqrt {c-\frac {c}{a x}} \left (a \sqrt {1-\frac {1}{a x}} x (7+2 a x)+7 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}}\right )\right )}{4 a^2 \sqrt {1-\frac {1}{a x}}} \] Input:

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

Output:

(Sqrt[c - c/(a*x)]*(a*Sqrt[1 - 1/(a*x)]*x*(7 + 2*a*x) + 7*ArcTanh[Sqrt[1 - 
 1/(a*x)]]))/(4*a^2*Sqrt[1 - 1/(a*x)])
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.14, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {6717, 6683, 1070, 281, 948, 87, 52, 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.

\(\displaystyle \int x \sqrt {c-\frac {c}{a x}} e^{2 \coth ^{-1}(a x)} \, dx\)

\(\Big \downarrow \) 6717

\(\displaystyle -\int e^{2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} xdx\)

\(\Big \downarrow \) 6683

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

\(\Big \downarrow \) 1070

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

\(\Big \downarrow \) 281

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

\(\Big \downarrow \) 948

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

\(\Big \downarrow \) 87

\(\displaystyle -\frac {c \left (\frac {7}{4} \int \frac {x^2}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}-\frac {a x^2 \sqrt {c-\frac {c}{a x}}}{2 c}\right )}{a}\)

\(\Big \downarrow \) 52

\(\displaystyle -\frac {c \left (\frac {7}{4} \left (\frac {\int \frac {x}{\sqrt {c-\frac {c}{a x}}}d\frac {1}{x}}{2 a}-\frac {x \sqrt {c-\frac {c}{a x}}}{c}\right )-\frac {a x^2 \sqrt {c-\frac {c}{a x}}}{2 c}\right )}{a}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {c \left (\frac {7}{4} \left (-\frac {\int \frac {1}{a-\frac {a}{c x^2}}d\sqrt {c-\frac {c}{a x}}}{c}-\frac {x \sqrt {c-\frac {c}{a x}}}{c}\right )-\frac {a x^2 \sqrt {c-\frac {c}{a x}}}{2 c}\right )}{a}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {c \left (\frac {7}{4} \left (-\frac {\text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a \sqrt {c}}-\frac {x \sqrt {c-\frac {c}{a x}}}{c}\right )-\frac {a x^2 \sqrt {c-\frac {c}{a x}}}{2 c}\right )}{a}\)

Input:

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

Output:

-((c*(-1/2*(a*Sqrt[c - c/(a*x)]*x^2)/c + (7*(-((Sqrt[c - c/(a*x)]*x)/c) - 
ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]]/(a*Sqrt[c])))/4))/a)
 

Defintions of rubi rules used

rule 52

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

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 948

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

rule 1070

Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_.) + (b_.)*(x_)^(n_.))^ 
(p_.)*((e_) + (f_.)*(x_)^(n_.))^(r_.), x_Symbol] :> Int[x^(m + 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, m, 
 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 = 0.14 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.39

method	result	size
risch	\(\frac {\left (2 a x +7\right ) x \sqrt {\frac {c \left (a x -1\right )}{a x}}}{4 a}+\frac {7 \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {c \left (a x -1\right ) a x}}{8 a \sqrt {a^{2} c}\, \left (a x -1\right )}\)	\(111\)
default	\(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (4 \sqrt {a \,x^{2}-x}\, a^{\frac {5}{2}} x +16 \sqrt {x \left (a x -1\right )}\, a^{\frac {3}{2}}-2 \sqrt {a \,x^{2}-x}\, a^{\frac {3}{2}}+8 a \ln \left (\frac {2 \sqrt {x \left (a x -1\right )}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right )-\ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a \right )}{8 \sqrt {x \left (a x -1\right )}\, a^{\frac {5}{2}}}\)	\(139\)

Input:

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

Output:

1/4*(2*a*x+7)/a*x*(c*(a*x-1)/a/x)^(1/2)+7/8/a*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)/(a*x-1)*(c*(a*x-1)/a/x)^(1/ 
2)*(c*(a*x-1)*a*x)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.95 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\left [\frac {2 \, {\left (2 \, a^{2} x^{2} + 7 \, a x\right )} \sqrt {\frac {a c x - c}{a x}} + 7 \, \sqrt {c} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right )}{8 \, a^{2}}, \frac {{\left (2 \, a^{2} x^{2} + 7 \, a x\right )} \sqrt {\frac {a c x - c}{a x}} - 7 \, \sqrt {-c} \arctan \left (\frac {a \sqrt {-c} x \sqrt {\frac {a c x - c}{a x}}}{a c x - c}\right )}{4 \, a^{2}}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.40 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\frac {1}{4} \, \sqrt {a^{2} c x^{2} - a c x} {\left (\frac {2 \, x {\left | a \right |}}{a^{2} \mathrm {sgn}\left (x\right )} + \frac {7 \, {\left | a \right |}}{a^{3} \mathrm {sgn}\left (x\right )}\right )} + \frac {7 \, \sqrt {c} \log \left ({\left | a \right |} {\left | c \right |}\right ) \mathrm {sgn}\left (x\right )}{8 \, a^{2}} - \frac {7 \, \sqrt {c} \log \left ({\left | -2 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} \sqrt {c} {\left | a \right |} + a c \right |}\right )}{8 \, a^{2} \mathrm {sgn}\left (x\right )} \] Input:

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

Output:

1/4*sqrt(a^2*c*x^2 - a*c*x)*(2*x*abs(a)/(a^2*sgn(x)) + 7*abs(a)/(a^3*sgn(x 
))) + 7/8*sqrt(c)*log(abs(a)*abs(c))*sgn(x)/a^2 - 7/8*sqrt(c)*log(abs(-2*( 
sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))*sqrt(c)*abs(a) + a*c))/(a^2*sgn(x 
))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.61 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx=\frac {\sqrt {c}\, \left (2 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}\, a x +7 \sqrt {x}\, \sqrt {a}\, \sqrt {a x -1}+7 \,\mathrm {log}\left (\sqrt {a x -1}+\sqrt {x}\, \sqrt {a}\right )\right )}{4 a^{2}} \] Input:

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

Output:

(sqrt(c)*(2*sqrt(x)*sqrt(a)*sqrt(a*x - 1)*a*x + 7*sqrt(x)*sqrt(a)*sqrt(a*x 
 - 1) + 7*log(sqrt(a*x - 1) + sqrt(x)*sqrt(a))))/(4*a**2)