\(\int \frac {e^{2 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx\) [842]

Optimal result

Integrand size = 24, antiderivative size = 111 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=-\frac {2 (1-a x) (1+a x)}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}-\frac {(1+a x)^2}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}+\frac {2 \sqrt {1-a x} \sqrt {1+a x} \arcsin (a x)}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6302, 6294, 6264, 79, 52, 41, 222} \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\frac {2 \sqrt {1-a x} \sqrt {a x+1} \arcsin (a x)}{a^2 x \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {(a x+1)^2}{a^2 x \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {2 (1-a x) (a x+1)}{a^2 x \sqrt {c-\frac {c}{a^2 x^2}}} \]

[In]

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

[Out]

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

Rule 41

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 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(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] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 222

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 6264

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

Rule 6294

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

Rule 6302

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{2 \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx \\ & = -\frac {\left (\sqrt {1-a x} \sqrt {1+a x}\right ) \int \frac {e^{2 \text {arctanh}(a x)} x}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{\sqrt {c-\frac {c}{a^2 x^2}} x} \\ & = -\frac {\left (\sqrt {1-a x} \sqrt {1+a x}\right ) \int \frac {x \sqrt {1+a x}}{(1-a x)^{3/2}} \, dx}{\sqrt {c-\frac {c}{a^2 x^2}} x} \\ & = -\frac {(1+a x)^2}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}+\frac {\left (2 \sqrt {1-a x} \sqrt {1+a x}\right ) \int \frac {\sqrt {1+a x}}{\sqrt {1-a x}} \, dx}{a \sqrt {c-\frac {c}{a^2 x^2}} x} \\ & = -\frac {2 (1-a x) (1+a x)}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}-\frac {(1+a x)^2}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}+\frac {\left (2 \sqrt {1-a x} \sqrt {1+a x}\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{a \sqrt {c-\frac {c}{a^2 x^2}} x} \\ & = -\frac {2 (1-a x) (1+a x)}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}-\frac {(1+a x)^2}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}+\frac {\left (2 \sqrt {1-a x} \sqrt {1+a x}\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a \sqrt {c-\frac {c}{a^2 x^2}} x} \\ & = -\frac {2 (1-a x) (1+a x)}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}-\frac {(1+a x)^2}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x}+\frac {2 \sqrt {1-a x} \sqrt {1+a x} \arcsin (a x)}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.61 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\frac {-3-2 a x+a^2 x^2+2 \sqrt {-1+a^2 x^2} \log \left (a x+\sqrt {-1+a^2 x^2}\right )}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.44

method	result	size
risch	\(\frac {a^{2} x^{2}-1}{a^{2} x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}+\frac {\left (\frac {2 \ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{a \sqrt {a^{2} c}}-\frac {2 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+2 \left (x -\frac {1}{a}\right ) a c}}{a^{3} c \left (x -\frac {1}{a}\right )}\right ) \sqrt {c \left (a^{2} x^{2}-1\right )}}{\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x}\)	\(160\)
default	\(\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \left (\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \sqrt {c}\, a^{2} x +2 \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) a c x -2 a \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, \sqrt {c}-\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a \sqrt {c}-2 \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) c \right )}{\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \,c^{\frac {3}{2}} a \left (a x -1\right )}\)	\(177\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.95 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\left [\frac {{\left (a x - 1\right )} \sqrt {c} \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + {\left (a^{2} x^{2} - 3 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x - a c}, -\frac {2 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - {\left (a^{2} x^{2} - 3 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x - a c}\right ] \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:index.cc index_m i_lex_is_greater Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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