Integrand size = 24, antiderivative size = 88 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=-\sqrt {c-\frac {c}{a^2 x^2}} x+\frac {\sqrt {c} \arctan \left (\frac {\sqrt {c}}{a \sqrt {c-\frac {c}{a^2 x^2}} x}\right )}{a}+\frac {2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {c}}\right )}{a} \] 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
Time = 0.05 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.91 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} x \left (-\sqrt {-1+a^2 x^2}+\arctan \left (\frac {1}{\sqrt {-1+a^2 x^2}}\right )+2 \log \left (a x+\sqrt {-1+a^2 x^2}\right )\right )}{\sqrt {-1+a^2 x^2}} \] Input:
Integrate[Sqrt[c - c/(a^2*x^2)]/E^(2*ArcTanh[a*x]),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]
Time = 0.37 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.81, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6709, 570, 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.
\(\displaystyle \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx\) |
\(\Big \downarrow \) 6709 |
\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \int \frac {\left (1-a^2 x^2\right )^{3/2}}{x (a x+1)^2}dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 570 |
\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \int \frac {(1-a x)^2}{x \sqrt {1-a^2 x^2}}dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 541 |
\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (-\frac {\int -\frac {a^2 (1-2 a x)}{x \sqrt {1-a^2 x^2}}dx}{a^2}-\sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {\int \frac {a^2 (1-2 a x)}{x \sqrt {1-a^2 x^2}}dx}{a^2}-\sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\int \frac {1-2 a x}{x \sqrt {1-a^2 x^2}}dx-\sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 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[Sqrt[c - c/(a^2*x^2)]/E^(2*ArcTanh[a*x]),x]
Output:
(Sqrt[c - c/(a^2*x^2)]*x*(-Sqrt[1 - a^2*x^2] - 2*ArcSin[a*x] - ArcTanh[Sqr t[1 - a^2*x^2]]))/Sqrt[1 - a^2*x^2]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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[(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]
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]
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]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c^(2*n)/a^n Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I LtQ[n, -1] && !(IGtQ[m, 0] && ILtQ[m + n, 0] && !GtQ[p, 1])
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]
Leaf count of result is larger than twice the leaf count of optimal. \(194\) vs. \(2(74)=148\).
Time = 0.22 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.22
method | result | size |
default | \(\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \left (\sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a^{2} \sqrt {-\frac {c}{a^{2}}}+2 \sqrt {c}\, \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}+x c}{\sqrt {c}}\right ) a \sqrt {-\frac {c}{a^{2}}}+c \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right )\right )}{\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2} \sqrt {-\frac {c}{a^{2}}}}\) | \(195\) |
Input:
int((c-c/a^2/x^2)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),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)
Time = 0.11 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.90 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=\left [-\frac {2 \, a x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} + 4 \, \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} - 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right )}{2 \, a}, -\frac {a x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} + \sqrt {c} \arctan \left (\frac {a x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{\sqrt {c}}\right ) - \sqrt {c} \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right )}{a}\right ] \] Input:
integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),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*s qrt((a^2*c*x^2 - c)/(a^2*x^2)) - c))/a]
\[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=- \int \left (- \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x + 1}\right )\, dx - \int \frac {a x \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x + 1}\, dx \] Input:
integrate((c-c/a**2/x**2)**(1/2)/(a*x+1)**2*(-a**2*x**2+1),x)
Output:
-Integral(-sqrt(c - c/(a**2*x**2))/(a*x + 1), x) - Integral(a*x*sqrt(c - c /(a**2*x**2))/(a*x + 1), x)
\[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=\int { -\frac {{\left (a^{2} x^{2} - 1\right )} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{{\left (a x + 1\right )}^{2}} \,d x } \] Input:
integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="maxima" )
Output:
-integrate((a^2*x^2 - 1)*sqrt(c - c/(a^2*x^2))/(a*x + 1)^2, x)
Exception generated. \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c-c/a^2/x^2)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),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. \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=-\int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \] Input:
int(-((c - c/(a^2*x^2))^(1/2)*(a^2*x^2 - 1))/(a*x + 1)^2,x)
Output:
-int(((c - c/(a^2*x^2))^(1/2)*(a^2*x^2 - 1))/(a*x + 1)^2, x)
Time = 0.15 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.60 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx=\frac {\sqrt {c}\, \left (-2 \mathit {atan} \left (\sqrt {a^{2} x^{2}-1}+a x \right )-\sqrt {a^{2} x^{2}-1}+2 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right )\right )}{a} \] Input:
int((c-c/a^2/x^2)^(1/2)/(a*x+1)^2*(-a^2*x^2+1),x)
Output:
(sqrt(c)*( - 2*atan(sqrt(a**2*x**2 - 1) + a*x) - sqrt(a**2*x**2 - 1) + 2*l og(sqrt(a**2*x**2 - 1) + a*x)))/a