Integrand size = 24, antiderivative size = 80 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\frac {2 \sqrt {c-\frac {c}{a^2 x^2}}}{c \left (a+\frac {1}{x}\right )}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} x}{c}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {c}}\right )}{a \sqrt {c}} \] Output:
2*(c-c/a^2/x^2)^(1/2)/c/(a+1/x)+(c-c/a^2/x^2)^(1/2)*x/c-2*arctanh((c-c/a^2 /x^2)^(1/2)/c^(1/2))/a/c^(1/2)
Time = 0.18 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.85 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\frac {-3+2 a x+a^2 x^2-2 \sqrt {-1+a^2 x^2} \log \left (a x+\sqrt {-1+a^2 x^2}\right )}{a^2 \sqrt {c-\frac {c}{a^2 x^2}} x} \] Input:
Integrate[1/(E^(2*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)]),x]
Output:
(-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)
Time = 0.68 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.16, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6717, 6709, 563, 455, 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.
\(\displaystyle \int \frac {e^{-2 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx\) |
\(\Big \downarrow \) 6717 |
\(\displaystyle -\int \frac {e^{-2 \text {arctanh}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}}dx\) |
\(\Big \downarrow \) 6709 |
\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \int \frac {x \sqrt {1-a^2 x^2}}{(a x+1)^2}dx}{x \sqrt {c-\frac {c}{a^2 x^2}}}\) |
\(\Big \downarrow \) 563 |
\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \left (\frac {\int \frac {2-a x}{\sqrt {1-a^2 x^2}}dx}{a}+\frac {2 \sqrt {1-a^2 x^2}}{a^2 (a x+1)}\right )}{x \sqrt {c-\frac {c}{a^2 x^2}}}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \left (\frac {2 \int \frac {1}{\sqrt {1-a^2 x^2}}dx+\frac {\sqrt {1-a^2 x^2}}{a}}{a}+\frac {2 \sqrt {1-a^2 x^2}}{a^2 (a x+1)}\right )}{x \sqrt {c-\frac {c}{a^2 x^2}}}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle -\frac {\sqrt {1-a^2 x^2} \left (\frac {\frac {\sqrt {1-a^2 x^2}}{a}+\frac {2 \arcsin (a x)}{a}}{a}+\frac {2 \sqrt {1-a^2 x^2}}{a^2 (a x+1)}\right )}{x \sqrt {c-\frac {c}{a^2 x^2}}}\) |
Input:
Int[1/(E^(2*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)]),x]
Output:
-((Sqrt[1 - a^2*x^2]*((2*Sqrt[1 - a^2*x^2])/(a^2*(1 + a*x)) + (Sqrt[1 - a^ 2*x^2]/a + (2*ArcSin[a*x])/a)/a))/(Sqrt[c - c/(a^2*x^2)]*x))
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]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[(-(-c)^(m - n - 2))*d^(2*n - m + 3)*(Sqrt[a + b*x^2]/(2^(n + 1)* b^(n + 2)*(c + d*x))), x] - Simp[d^(2*n - m + 2)/b^(n + 1) Int[(1/Sqrt[a + b*x^2])*ExpandToSum[(2^(-n - 1)*(-c)^(m - n - 1) - d^m*x^m*(-c + d*x)^(-n - 1))/(c + d*x), x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2 , 0] && IGtQ[m, 0] && ILtQ[n, 0] && EqQ[n + p, -3/2]
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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(153\) vs. \(2(70)=140\).
Time = 0.24 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.92
method | result | size |
risch | \(\frac {a^{2} x^{2}-1}{a^{2} x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}+\frac {\left (-\frac {2 \ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{a \sqrt {a^{2} c}}+\frac {2 \sqrt {\left (x +\frac {1}{a}\right )^{2} a^{2} c -2 \left (x +\frac {1}{a}\right ) a c}}{a^{3} c \left (x +\frac {1}{a}\right )}\right ) \sqrt {c \left (a^{2} x^{2}-1\right )}}{\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x}\) | \(154\) |
default | \(-\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \left (-\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \sqrt {c}\, a^{2} x +2 \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) a c x -2 a \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, \sqrt {c}-\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a \sqrt {c}+2 \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) c \right )}{\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \,c^{\frac {3}{2}} a \left (a x +1\right )}\) | \(179\) |
Input:
int((a*x-1)/(a*x+1)/(c-c/a^2/x^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
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)*((x+1/a)^2*a^2*c -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
Time = 0.11 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.65 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\left [\frac {{\left (a x + 1\right )} \sqrt {c} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + {\left (a^{2} x^{2} + 3 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x + a c}, \frac {2 \, {\left (a x + 1\right )} \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + {\left (a^{2} x^{2} + 3 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x + a c}\right ] \] Input:
integrate((a*x-1)/(a*x+1)/(c-c/a^2/x^2)^(1/2),x, algorithm="fricas")
Output:
[((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)]
\[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\int \frac {a x - 1}{\sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x + 1\right )}\, dx \] Input:
integrate((a*x-1)/(a*x+1)/(c-c/a**2/x**2)**(1/2),x)
Output:
Integral((a*x - 1)/(sqrt(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))*(a*x + 1)), x)
\[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\int { \frac {a x - 1}{{\left (a x + 1\right )} \sqrt {c - \frac {c}{a^{2} x^{2}}}} \,d x } \] Input:
integrate((a*x-1)/(a*x+1)/(c-c/a^2/x^2)^(1/2),x, algorithm="maxima")
Output:
integrate((a*x - 1)/((a*x + 1)*sqrt(c - c/(a^2*x^2))), x)
Exception generated. \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((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
Timed out. \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\int \frac {a\,x-1}{\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a\,x+1\right )} \,d x \] Input:
int((a*x - 1)/((c - c/(a^2*x^2))^(1/2)*(a*x + 1)),x)
Output:
int((a*x - 1)/((c - c/(a^2*x^2))^(1/2)*(a*x + 1)), x)
Time = 0.15 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.06 \[ \int \frac {e^{-2 \coth ^{-1}(a x)}}{\sqrt {c-\frac {c}{a^2 x^2}}} \, dx=\frac {\sqrt {c}\, \left (2 \sqrt {a^{2} x^{2}-1}\, a x +6 \sqrt {a^{2} x^{2}-1}-4 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right ) a x -4 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right )+5 a x +5\right )}{2 a c \left (a x +1\right )} \] Input:
int((a*x-1)/(a*x+1)/(c-c/a^2/x^2)^(1/2),x)
Output:
(sqrt(c)*(2*sqrt(a**2*x**2 - 1)*a*x + 6*sqrt(a**2*x**2 - 1) - 4*log(sqrt(a **2*x**2 - 1) + a*x)*a*x - 4*log(sqrt(a**2*x**2 - 1) + a*x) + 5*a*x + 5))/ (2*a*c*(a*x + 1))