3.6.59 \(\int e^{\coth ^{-1}(a x)} (c-a^2 c x^2) \, dx\) [559]

3.6.59.1 Optimal result

Integrand size = 18, antiderivative size = 145 \[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx=\frac {1}{2} c \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x+\frac {1}{6} a c \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {1}{3} a^2 c \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3+\frac {c \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{2 a} \]

output

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

3.6.59.2 Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.42 \[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx=\frac {c \left (a \sqrt {1-\frac {1}{a^2 x^2}} x \left (2-3 a x-2 a^2 x^2\right )+3 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{6 a} \]

input

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

output

(c*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(2 - 3*a*x - 2*a^2*x^2) + 3*Log[(1 + Sqrt[1 
- 1/(a^2*x^2)])*x]))/(6*a)
 

3.6.59.3 Rubi [F]

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.

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

\(\Big \downarrow \) 6745

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2005

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

\(\Big \downarrow \) 6745

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2005

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

\(\Big \downarrow \) 6745

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2005

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

\(\Big \downarrow \) 6745

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2005

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

\(\Big \downarrow \) 6745

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2005

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

\(\Big \downarrow \) 6745

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2005

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

\(\Big \downarrow \) 6745

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2005

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

\(\Big \downarrow \) 6745

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2005

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

\(\Big \downarrow \) 6745

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2005

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

\(\Big \downarrow \) 6745

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2005

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

input

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

output

$Aborted
 

3.6.59.3.1 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 2005

Int[(Fx_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(m 
+ n*p)*(b + a/x^n)^p*Fx, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && Neg 
Q[n]
 

rule 6745

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symb 
ol] :> Simp[d^p   Int[u*x^(2*p)*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], 
 x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] &&  !IntegerQ[n/2] && In 
tegerQ[p]
 

3.6.59.4 Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.74

method	result	size
risch	\(-\frac {\left (2 a^{2} x^{2}+3 a x -2\right ) \left (a x -1\right ) c}{6 a \sqrt {\frac {a x -1}{a x +1}}}+\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right ) c \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{2 \sqrt {a^{2}}\, \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\)	\(108\)
default	\(-\frac {\left (a x -1\right ) c \left (3 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x +2 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-3 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a \right )}{6 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a}\)	\(119\)

input

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

output

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

3.6.59.5 Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.63 \[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx=\frac {3 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 3 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (2 \, a^{3} c x^{3} + 5 \, a^{2} c x^{2} + a c x - 2 \, c\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, a} \]

input

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

output

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

3.6.59.6 Sympy [F]

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

input

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

output

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

3.6.59.7 Maxima [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.18 \[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx=-\frac {1}{6} \, a {\left (\frac {2 \, {\left (3 \, c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 8 \, c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 3 \, c \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {3 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {3 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - a^{2}} - \frac {3 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {3 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} \]

input

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

output

-1/6*a*(2*(3*c*((a*x - 1)/(a*x + 1))^(5/2) - 8*c*((a*x - 1)/(a*x + 1))^(3/ 
2) - 3*c*sqrt((a*x - 1)/(a*x + 1)))/(3*(a*x - 1)*a^2/(a*x + 1) - 3*(a*x - 
1)^2*a^2/(a*x + 1)^2 + (a*x - 1)^3*a^2/(a*x + 1)^3 - a^2) - 3*c*log(sqrt(( 
a*x - 1)/(a*x + 1)) + 1)/a^2 + 3*c*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2)
 

3.6.59.8 Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.62 \[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx=-\frac {1}{6} \, \sqrt {a^{2} x^{2} - 1} {\left ({\left (\frac {2 \, a c x}{\mathrm {sgn}\left (a x + 1\right )} + \frac {3 \, c}{\mathrm {sgn}\left (a x + 1\right )}\right )} x - \frac {2 \, c}{a \mathrm {sgn}\left (a x + 1\right )}\right )} - \frac {c \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{2 \, {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} \]

input

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

output

-1/6*sqrt(a^2*x^2 - 1)*((2*a*c*x/sgn(a*x + 1) + 3*c/sgn(a*x + 1))*x - 2*c/ 
(a*sgn(a*x + 1))) - 1/2*c*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))/(abs(a)* 
sgn(a*x + 1))
 

3.6.59.9 Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.90 \[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx=\frac {c\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}-\frac {c\,\sqrt {\frac {a\,x-1}{a\,x+1}}+\frac {8\,c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3}-c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{a-\frac {3\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {3\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}} \]

input

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

output

(c*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/a - (c*((a*x - 1)/(a*x + 1))^(1/2) 
+ (8*c*((a*x - 1)/(a*x + 1))^(3/2))/3 - c*((a*x - 1)/(a*x + 1))^(5/2))/(a 
- (3*a*(a*x - 1))/(a*x + 1) + (3*a*(a*x - 1)^2)/(a*x + 1)^2 - (a*(a*x - 1) 
^3)/(a*x + 1)^3)