\(\int e^{-3 \coth ^{-1}(a x)} (c-\frac {c}{a^2 x^2}) \, dx\) [801]

Optimal result

Integrand size = 20, antiderivative size = 69 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=-\frac {c \sqrt {1-\frac {1}{a^2 x^2}}}{a}+c \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {3 c \csc ^{-1}(a x)}{a}-\frac {3 c \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \] Output:

-c*(1-1/a^2/x^2)^(1/2)/a+c*(1-1/a^2/x^2)^(1/2)*x-3*c*arccsc(a*x)/a-3*c*arc 
tanh((1-1/a^2/x^2)^(1/2))/a
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.83 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {c \left (\sqrt {1-\frac {1}{a^2 x^2}} (-1+a x)-3 \arcsin \left (\frac {1}{a x}\right )-3 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{a} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.17, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6748, 109, 27, 140, 39, 103, 221, 223}

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.

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

\(\Big \downarrow \) 6748

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 140

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

\(\Big \downarrow \) 39

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

\(\Big \downarrow \) 103

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 223

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

Input:

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

Output:

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

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 39

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[( 
a*c + b*d*x^2)^m, x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && ( 
IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
 

rule 103

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

rule 109

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

rule 140

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*d^(m + n)*f^p   Int[(a + b*x)^(m - 1)/(c + d*x)^m, x] 
, x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandToSum[(a + b*x 
)*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, 
b, c, d, e, f, m, n}, x] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 
0] || SumSimplerQ[m, -1] ||  !(GtQ[n, 0] || SumSimplerQ[n, -1]))
 

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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 6748

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(129\) vs. \(2(63)=126\).

Time = 0.10 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.88

method	result	size
risch	\(-\frac {\left (a x +1\right ) c \sqrt {\frac {a x -1}{a x +1}}}{x \,a^{2}}+\frac {\left (\sqrt {\left (a x -1\right ) \left (a x +1\right )}-3 \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-\frac {3 a \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}\right ) c \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a \left (a x -1\right )}\)	\(130\)
default	\(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right )^{2} c \left (-\sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{2} x^{2}+4 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a x +\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-3 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a x +\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x -3 a \sqrt {a^{2}}\, x \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-4 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x \right )}{\left (a x -1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x \sqrt {a^{2}}}\)	\(234\)

Input:

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

Output:

-(a*x+1)/x*c/a^2*((a*x-1)/(a*x+1))^(1/2)+1/a*(((a*x-1)*(a*x+1))^(1/2)-3*ar 
ctan(1/(a^2*x^2-1)^(1/2))-3*a*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))^(1/2)/(a*x-1)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.49 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {6 \, a c x \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - 3 \, a c x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 3 \, a c x \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (a^{2} c x^{2} - c\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} x} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

c*(Integral(sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x**3 + x**2), x) + Integr 
al(-a*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x**2 + x), x) + Integral(-a**2* 
sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1), x) + Integral(a**3*x*sqrt(a*x 
/(a*x + 1) - 1/(a*x + 1))/(a*x + 1), x))/a**2
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.71 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=-{\left (\frac {4 \, c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{\frac {{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - a^{2}} - \frac {6 \, c \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} + \frac {3 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {3 \, c \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} a \] Input:

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

Output:

-(4*c*((a*x - 1)/(a*x + 1))^(3/2)/((a*x - 1)^2*a^2/(a*x + 1)^2 - a^2) - 6* 
c*arctan(sqrt((a*x - 1)/(a*x + 1)))/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)*a
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.77 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {6 \, c \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right ) \mathrm {sgn}\left (a x + 1\right )}{a} + \frac {3 \, c \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{{\left | a \right |}} + \frac {\sqrt {a^{2} x^{2} - 1} c \mathrm {sgn}\left (a x + 1\right )}{a} - \frac {2 \, c \mathrm {sgn}\left (a x + 1\right )}{{\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )} {\left | a \right |}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.22 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {6\,c\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}-\frac {6\,c\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}+\frac {4\,c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{a-\frac {a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}} \] Input:

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

Output:

(6*c*atan(((a*x - 1)/(a*x + 1))^(1/2)))/a - (6*c*atanh(((a*x - 1)/(a*x + 1 
))^(1/2)))/a + (4*c*((a*x - 1)/(a*x + 1))^(3/2))/(a - (a*(a*x - 1)^2)/(a*x 
 + 1)^2)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.49 \[ \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {c \left (6 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}-1\right ) a x -6 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}+1\right ) a x +\sqrt {a x +1}\, \sqrt {a x -1}\, a x -\sqrt {a x +1}\, \sqrt {a x -1}-6 \,\mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a x -a x \right )}{a^{2} x} \] Input:

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

Output:

(c*(6*atan(sqrt(a*x - 1) + sqrt(a*x + 1) - 1)*a*x - 6*atan(sqrt(a*x - 1) + 
 sqrt(a*x + 1) + 1)*a*x + sqrt(a*x + 1)*sqrt(a*x - 1)*a*x - sqrt(a*x + 1)* 
sqrt(a*x - 1) - 6*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a*x - a*x)) 
/(a**2*x)