\(\int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx\) [865]

Optimal result

Integrand size = 24, antiderivative size = 88 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=\sqrt {c-\frac {c}{a^2 x^2}} x-\frac {\sqrt {c} \arctan \left (\frac {\sqrt {c}}{a \sqrt {c-\frac {c}{a^2 x^2}} x}\right )}{a}+\frac {2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {c}}\right )}{a} \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.91 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} x \left (\sqrt {-1+a^2 x^2}-\arctan \left (\frac {1}{\sqrt {-1+a^2 x^2}}\right )+2 \log \left (a x+\sqrt {-1+a^2 x^2}\right )\right )}{\sqrt {-1+a^2 x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.82, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6717, 6709, 541, 25, 27, 538, 223, 243, 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.

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

\(\Big \downarrow \) 6717

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

\(\Big \downarrow \) 6709

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

\(\Big \downarrow \) 541

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 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 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 538

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

rule 541

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Simp[d^n*x^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*(m + n + 2*p + 1))), x 
] + Simp[1/(b*(m + n + 2*p + 1))   Int[x^m*(a + b*x^2)^p*ExpandToSum[b*(m + 
 n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1)*x^n - a*d^n*(m + n - 1) 
*x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, m, p}, x] && IGtQ[n, 1] && IGt 
Q[m, -2] && GtQ[p, -1] && IntegerQ[2*p]
 

rule 6709

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbo 
l] :> Simp[x^(2*p)*((c + d/x^2)^p/(1 - a^2*x^2)^p)   Int[u*((1 + a*x)^n/(x^ 
(2*p)*(1 - a^2*x^2)^(n/2 - p))), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c 
+ a^2*d, 0] &&  !IntegerQ[p] && IntegerQ[n/2] &&  !GtQ[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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(195\) vs. \(2(74)=148\).

Time = 0.22 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.23

method	result	size
default	\(-\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \left (\sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a^{2} \sqrt {-\frac {c}{a^{2}}}-2 \sqrt {c}\, \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}+x c}{\sqrt {c}}\right ) a \sqrt {-\frac {c}{a^{2}}}+c \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right )\right )}{\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2} \sqrt {-\frac {c}{a^{2}}}}\)	\(196\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.86 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=\left [\frac {2 \, a x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 4 \, \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right )}{2 \, a}, \frac {a x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} + \sqrt {c} \arctan \left (\frac {a x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{\sqrt {c}}\right ) + \sqrt {c} \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right )}{a}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.58 \[ \int e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=\frac {\sqrt {c}\, \left (2 \mathit {atan} \left (\sqrt {a^{2} x^{2}-1}+a x \right )+\sqrt {a^{2} x^{2}-1}+2 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right )\right )}{a} \] Input:

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

Output:

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