\(\int \frac {e^{2 \text {arctanh}(a x)}}{x^2 \sqrt {c-a^2 c x^2}} \, dx\) [1135]

Optimal result

Integrand size = 27, antiderivative size = 77 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{x^2 \sqrt {c-a^2 c x^2}} \, dx=\frac {2 a (1+a x)}{\sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{c x}-\frac {2 a \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{\sqrt {c}} \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.01 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{x^2 \sqrt {c-a^2 c x^2}} \, dx=\frac {(1-3 a x) \sqrt {c-a^2 c x^2}}{c x (-1+a x)}+\frac {2 a \log (x)}{\sqrt {c}}-\frac {2 a \log \left (c+\sqrt {c} \sqrt {c-a^2 c x^2}\right )}{\sqrt {c}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6701, 528, 534, 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.

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

\(\Big \downarrow \) 6701

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

\(\Big \downarrow \) 528

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

\(\Big \downarrow \) 534

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

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 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 528

Int[((x_)^(m_)*((c_) + (d_.)*(x_))^(n_.))/((a_) + (b_.)*(x_)^2)^(3/2), x_Sy 
mbol] :> Simp[(-2^(n - 1))*c^(m + n - 2)*((c + d*x)/(b*d^(m - 1)*Sqrt[a + b 
*x^2])), x] + Simp[c^2/a   Int[(x^m/Sqrt[a + b*x^2])*ExpandToSum[((c + d*x) 
^(n - 1) - (2^(n - 1)*c^(m + n - 1))/(d^m*x^m))/(c - d*x), x], x], x] /; Fr 
eeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && EqQ[b*c^2 + a*d^2, 0]
 

rule 534

Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d   Int[ 
x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 
0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
 

rule 6701

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

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.29

method	result	size
default	\(-\frac {\sqrt {-a^{2} c \,x^{2}+c}}{c x}-\frac {2 a \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{\sqrt {c}}-\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}{c \left (x -\frac {1}{a}\right )}\)	\(99\)
risch	\(\frac {a^{2} x^{2}-1}{x \sqrt {-c \left (a^{2} x^{2}-1\right )}}-\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}{c \left (x -\frac {1}{a}\right )}-\frac {2 a \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{\sqrt {c}}\)	\(105\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.17 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{x^2 \sqrt {c-a^2 c x^2}} \, dx=\left [\frac {{\left (a^{2} x^{2} - a x\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) - \sqrt {-a^{2} c x^{2} + c} {\left (3 \, a x - 1\right )}}{a c x^{2} - c x}, \frac {2 \, {\left (a^{2} x^{2} - a x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{c}\right ) - \sqrt {-a^{2} c x^{2} + c} {\left (3 \, a x - 1\right )}}{a c x^{2} - c x}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

undef
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.12 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{x^2 \sqrt {c-a^2 c x^2}} \, dx=\frac {\sqrt {c}\, a \left (4 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{2}-4 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )+\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{3}-10 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{2}+1\right )}{2 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right ) c \left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )-1\right )} \] Input:

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

Output:

(sqrt(c)*a*(4*log(tan(asin(a*x)/2))*tan(asin(a*x)/2)**2 - 4*log(tan(asin(a 
*x)/2))*tan(asin(a*x)/2) + tan(asin(a*x)/2)**3 - 10*tan(asin(a*x)/2)**2 + 
1))/(2*tan(asin(a*x)/2)*c*(tan(asin(a*x)/2) - 1))