Integrand size = 24, antiderivative size = 123 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\frac {(1+a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}-\frac {2 (5-2 a x) (1-a x) (1+a x)^2}{3 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}+\frac {2 (1-a x)^{3/2} (1+a x)^{3/2} \arcsin (a x)}{a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3} \]
1/3*(a*x+1)^2/a^2/(c-c/a^2/x^2)^(3/2)/x-2/3*(-2*a*x+5)*(-a*x+1)*(a*x+1)^2/ a^4/(c-c/a^2/x^2)^(3/2)/x^3+2*(-a*x+1)^(3/2)*(a*x+1)^(3/2)*arcsin(a*x)/a^4 /(c-c/a^2/x^2)^(3/2)/x^3
Time = 0.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.77 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\frac {-10+4 a x+11 a^2 x^2-3 a^3 x^3-6 (-1+a x) \sqrt {-1+a^2 x^2} \log \left (a x+\sqrt {-1+a^2 x^2}\right )}{3 a^2 c \sqrt {c-\frac {c}{a^2 x^2}} x (-1+a x)} \]
(-10 + 4*a*x + 11*a^2*x^2 - 3*a^3*x^3 - 6*(-1 + a*x)*Sqrt[-1 + a^2*x^2]*Lo g[a*x + Sqrt[-1 + a^2*x^2]])/(3*a^2*c*Sqrt[c - c/(a^2*x^2)]*x*(-1 + a*x))
Time = 0.44 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6709, 529, 2166, 27, 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.
| \(\displaystyle \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6709 |
| \(\displaystyle \frac {\left (1-a^2 x^2\right )^{3/2} \int \frac {x^3 (a x+1)^2}{\left (1-a^2 x^2\right )^{5/2}}dx}{x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}\) |
| \(\Big \downarrow \) 529 |
| \(\displaystyle \frac {\left (1-a^2 x^2\right )^{3/2} \left (\frac {(a x+1)^2}{3 a^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{3} \int \frac {(a x+1) \left (\frac {3 x^2}{a}+\frac {3 x}{a^2}+\frac {2}{a^3}\right )}{\left (1-a^2 x^2\right )^{3/2}}dx\right )}{x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}\) |
| \(\Big \downarrow \) 2166 |
| \(\displaystyle \frac {\left (1-a^2 x^2\right )^{3/2} \left (\frac {1}{3} \left (\int \frac {3 (a x+2)}{a^3 \sqrt {1-a^2 x^2}}dx-\frac {8 (a x+1)}{a^4 \sqrt {1-a^2 x^2}}\right )+\frac {(a x+1)^2}{3 a^4 \left (1-a^2 x^2\right )^{3/2}}\right )}{x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (1-a^2 x^2\right )^{3/2} \left (\frac {1}{3} \left (\frac {3 \int \frac {a x+2}{\sqrt {1-a^2 x^2}}dx}{a^3}-\frac {8 (a x+1)}{a^4 \sqrt {1-a^2 x^2}}\right )+\frac {(a x+1)^2}{3 a^4 \left (1-a^2 x^2\right )^{3/2}}\right )}{x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {\left (1-a^2 x^2\right )^{3/2} \left (\frac {1}{3} \left (\frac {3 \left (2 \int \frac {1}{\sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a^3}-\frac {8 (a x+1)}{a^4 \sqrt {1-a^2 x^2}}\right )+\frac {(a x+1)^2}{3 a^4 \left (1-a^2 x^2\right )^{3/2}}\right )}{x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {\left (1-a^2 x^2\right )^{3/2} \left (\frac {(a x+1)^2}{3 a^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {1}{3} \left (\frac {3 \left (\frac {2 \arcsin (a x)}{a}-\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a^3}-\frac {8 (a x+1)}{a^4 \sqrt {1-a^2 x^2}}\right )\right )}{x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}\) |
((1 - a^2*x^2)^(3/2)*((1 + a*x)^2/(3*a^4*(1 - a^2*x^2)^(3/2)) + ((-8*(1 + a*x))/(a^4*Sqrt[1 - a^2*x^2]) + (3*(-(Sqrt[1 - a^2*x^2]/a) + (2*ArcSin[a*x ])/a))/a^3)/3))/((c - c/(a^2*x^2))^(3/2)*x^3)
3.8.1.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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] :> With[{Qx = PolynomialQuotient[x^m, a*d + b*c*x, x], R = PolynomialRem ainder[x^m, a*d + b*c*x, x]}, Simp[(-c)*R*(c + d*x)^n*((a + b*x^2)^(p + 1)/ (2*a*d*(p + 1))), x] + Simp[c/(2*a*(p + 1)) Int[(c + d*x)^(n - 1)*(a + b* x^2)^(p + 1)*ExpandToSum[2*a*d*(p + 1)*Qx + R*(n + 2*p + 2), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 1] && LtQ[p, -1] && EqQ[b* c^2 + a*d^2, 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{Qx = PolynomialQuotient[Pq, a*e + b*d*x, x], R = PolynomialRemainde r[Pq, a*e + b*d*x, x]}, Simp[(-d)*R*(d + e*x)^m*((a + b*x^2)^(p + 1)/(2*a*e *(p + 1))), x] + Simp[d/(2*a*(p + 1)) Int[(d + e*x)^(m - 1)*(a + b*x^2)^( p + 1)*ExpandToSum[2*a*e*(p + 1)*Qx + R*(m + 2*p + 2), x], x], x]] /; FreeQ [{a, b, d, e}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 0] && ILtQ[p + 1/2, 0] && GtQ[m, 0]
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]
Time = 0.16 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.76
| method | result | size |
| risch | \(-\frac {a^{2} x^{2}-1}{a^{2} c x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}-\frac {\left (\frac {2 \ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{a^{3} \sqrt {a^{2} c}}-\frac {\sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+2 \left (x -\frac {1}{a}\right ) a c}}{3 a^{6} c \left (x -\frac {1}{a}\right )^{2}}-\frac {8 \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+2 \left (x -\frac {1}{a}\right ) a c}}{3 a^{5} c \left (x -\frac {1}{a}\right )}\right ) a^{2} \sqrt {c \left (a^{2} x^{2}-1\right )}}{c x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}\) | \(216\) |
| default | \(-\frac {\left (3 c^{\frac {3}{2}} \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a^{3} x^{3}+4 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c^{\frac {3}{2}} a^{2} x^{2}-15 x^{2} a^{2} c^{\frac {3}{2}} \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}+6 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a^{2} c x -4 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c^{\frac {3}{2}} a x -6 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \ln \left (\sqrt {c}\, x +\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a c -2 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c^{\frac {3}{2}}+12 c^{\frac {3}{2}} \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\right ) \left (a x +1\right )}{3 \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, x^{3} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {3}{2}} a^{4} c^{\frac {3}{2}}}\) | \(326\) |
-1/a^2*(a^2*x^2-1)/c/x/(c*(a^2*x^2-1)/a^2/x^2)^(1/2)-(2/a^3*ln(a^2*c*x/(a^ 2*c)^(1/2)+(a^2*c*x^2-c)^(1/2))/(a^2*c)^(1/2)-1/3/a^6/c/(x-1/a)^2*(a^2*c*( x-1/a)^2+2*(x-1/a)*a*c)^(1/2)-8/3/a^5/c/(x-1/a)*(a^2*c*(x-1/a)^2+2*(x-1/a) *a*c)^(1/2))*a^2/c/x/(c*(a^2*x^2-1)/a^2/x^2)^(1/2)*(c*(a^2*x^2-1))^(1/2)
Time = 0.29 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.28 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) - {\left (3 \, a^{3} x^{3} - 14 \, a^{2} x^{2} + 10 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}, \frac {6 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - {\left (3 \, a^{3} x^{3} - 14 \, a^{2} x^{2} + 10 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}\right ] \]
[1/3*(3*(a^2*x^2 - 2*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) - (3*a^3*x^3 - 14*a^2*x^2 + 10*a*x)*s qrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2), 1/3*( 6*(a^2*x^2 - 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)) - (3*a^3*x^3 - 14*a^2*x^2 + 10*a*x)*sqrt(( a^2*c*x^2 - c)/(a^2*x^2)))/(a^3*c^2*x^2 - 2*a^2*c^2*x + a*c^2)]
\[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=- \int \frac {a x}{a c x \sqrt {c - \frac {c}{a^{2} x^{2}}} - c \sqrt {c - \frac {c}{a^{2} x^{2}}} - \frac {c \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x} + \frac {c \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{2} x^{2}}}\, dx - \int \frac {1}{a c x \sqrt {c - \frac {c}{a^{2} x^{2}}} - c \sqrt {c - \frac {c}{a^{2} x^{2}}} - \frac {c \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x} + \frac {c \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{2} x^{2}}}\, dx \]
-Integral(a*x/(a*c*x*sqrt(c - c/(a**2*x**2)) - c*sqrt(c - c/(a**2*x**2)) - c*sqrt(c - c/(a**2*x**2))/(a*x) + c*sqrt(c - c/(a**2*x**2))/(a**2*x**2)), x) - Integral(1/(a*c*x*sqrt(c - c/(a**2*x**2)) - c*sqrt(c - c/(a**2*x**2) ) - c*sqrt(c - c/(a**2*x**2))/(a*x) + c*sqrt(c - c/(a**2*x**2))/(a**2*x**2 )), x)
\[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\int { -\frac {{\left (a x + 1\right )}^{2}}{{\left (a^{2} x^{2} - 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {3}{2}}} \,d x } \]
Exception generated. \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {e^{2 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx=\int -\frac {{\left (a\,x+1\right )}^2}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{3/2}\,\left (a^2\,x^2-1\right )} \,d x \]