Integrand size = 22, antiderivative size = 63 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{c-\frac {c}{a x}} \, dx=-\frac {2 (1-a x)}{a c \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a c}-\frac {2 \arcsin (a x)}{a c} \] Output:
(2*a*x-2)/a/c/(-a^2*x^2+1)^(1/2)-(-a^2*x^2+1)^(1/2)/a/c-2*arcsin(a*x)/a/c
Time = 0.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.06 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {-3+2 a x+a^2 x^2+4 \sqrt {1-a^2 x^2} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{a c \sqrt {1-a^2 x^2}} \] Input:
Integrate[1/(E^(3*ArcTanh[a*x])*(c - c/(a*x))),x]
Output:
(-3 + 2*a*x + a^2*x^2 + 4*Sqrt[1 - a^2*x^2]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]) /(a*c*Sqrt[1 - a^2*x^2])
Time = 0.57 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6681, 6678, 527, 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.
\(\displaystyle \int \frac {e^{-3 \text {arctanh}(a x)}}{c-\frac {c}{a x}} \, dx\) |
\(\Big \downarrow \) 6681 |
\(\displaystyle -\frac {a \int \frac {e^{-3 \text {arctanh}(a x)} x}{1-a x}dx}{c}\) |
\(\Big \downarrow \) 6678 |
\(\displaystyle -\frac {a \int \frac {x (1-a x)^2}{\left (1-a^2 x^2\right )^{3/2}}dx}{c}\) |
\(\Big \downarrow \) 527 |
\(\displaystyle -\frac {a \left (\frac {\int \frac {2-a x}{\sqrt {1-a^2 x^2}}dx}{a}+\frac {2 (1-a x)}{a^2 \sqrt {1-a^2 x^2}}\right )}{c}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle -\frac {a \left (\frac {2 \int \frac {1}{\sqrt {1-a^2 x^2}}dx+\frac {\sqrt {1-a^2 x^2}}{a}}{a}+\frac {2 (1-a x)}{a^2 \sqrt {1-a^2 x^2}}\right )}{c}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle -\frac {a \left (\frac {\frac {\sqrt {1-a^2 x^2}}{a}+\frac {2 \arcsin (a x)}{a}}{a}+\frac {2 (1-a x)}{a^2 \sqrt {1-a^2 x^2}}\right )}{c}\) |
Input:
Int[1/(E^(3*ArcTanh[a*x])*(c - c/(a*x))),x]
Output:
-((a*((2*(1 - a*x))/(a^2*Sqrt[1 - a^2*x^2]) + (Sqrt[1 - a^2*x^2]/a + (2*Ar cSin[a*x])/a)/a))/c)
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)^(3/2), x_S ymbol] :> Simp[(-2^(n - 1))*c^(m + n - 2)*((c + d*x)/(b*d^(m - 1)*Sqrt[a + b*x^2])), x] + Simp[1/(b*d^(m - 2)) 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] && IGtQ[n, 0] && IGtQ[m, 0] && EqQ[b*c^2 + a*d^2, 0]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.)*((e_.) + (f_.)* (x_))^(m_.), x_Symbol] :> Simp[c^n Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1 , 0]) && IntegerQ[2*p]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_.), x_Symbol ] :> Simp[d^p Int[u*(1 + c*(x/d))^p*(E^(n*ArcTanh[a*x])/x^p), x], x] /; F reeQ[{a, c, d, n}, x] && EqQ[c^2 - a^2*d^2, 0] && IntegerQ[p]
Time = 0.20 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.62
method | result | size |
risch | \(\frac {a^{2} x^{2}-1}{a c \sqrt {-a^{2} x^{2}+1}}-\frac {\left (\frac {2 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {a^{2}}}+\frac {2 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{a^{3} \left (x +\frac {1}{a}\right )}\right ) a}{c}\) | \(102\) |
default | \(\frac {a \left (\frac {-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{3}}-2 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+3 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )}{2 a^{4}}-\frac {\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )}{8 a^{2}}+\frac {\frac {\left (-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}-a \left (-\frac {\left (-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a \right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )}{8 a^{2}}-\frac {\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+3 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )}{4 a^{3}}\right )}{c}\) | \(593\) |
Input:
int(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x),x,method=_RETURNVERBOSE)
Output:
1/a/c*(a^2*x^2-1)/(-a^2*x^2+1)^(1/2)-(2/a/(a^2)^(1/2)*arctan((a^2)^(1/2)*x /(-a^2*x^2+1)^(1/2))+2/a^3/(x+1/a)*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2))*a/c
Time = 0.07 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.06 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{c-\frac {c}{a x}} \, dx=-\frac {3 \, a x - 4 \, {\left (a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} {\left (a x + 3\right )} + 3}{a^{2} c x + a c} \] Input:
integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x),x, algorithm="fricas")
Output:
-(3*a*x - 4*(a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + sqrt(-a^2*x ^2 + 1)*(a*x + 3) + 3)/(a^2*c*x + a*c)
\[ \int \frac {e^{-3 \text {arctanh}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {a \left (\int \frac {x \sqrt {- a^{2} x^{2} + 1}}{a^{4} x^{4} + 2 a^{3} x^{3} - 2 a x - 1}\, dx + \int \left (- \frac {a^{2} x^{3} \sqrt {- a^{2} x^{2} + 1}}{a^{4} x^{4} + 2 a^{3} x^{3} - 2 a x - 1}\right )\, dx\right )}{c} \] Input:
integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(c-c/a/x),x)
Output:
a*(Integral(x*sqrt(-a**2*x**2 + 1)/(a**4*x**4 + 2*a**3*x**3 - 2*a*x - 1), x) + Integral(-a**2*x**3*sqrt(-a**2*x**2 + 1)/(a**4*x**4 + 2*a**3*x**3 - 2 *a*x - 1), x))/c
\[ \int \frac {e^{-3 \text {arctanh}(a x)}}{c-\frac {c}{a x}} \, dx=\int { \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} {\left (c - \frac {c}{a x}\right )}} \,d x } \] Input:
integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x),x, algorithm="maxima")
Output:
integrate((-a^2*x^2 + 1)^(3/2)/((a*x + 1)^3*(c - c/(a*x))), x)
Time = 0.13 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.16 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{c-\frac {c}{a x}} \, dx=-\frac {2 \, \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{c {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a c} + \frac {4}{c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} \] Input:
integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x),x, algorithm="giac")
Output:
-2*arcsin(a*x)*sgn(a)/(c*abs(a)) - sqrt(-a^2*x^2 + 1)/(a*c) + 4/(c*((sqrt( -a^2*x^2 + 1)*abs(a) + a)/(a^2*x) + 1)*abs(a))
Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.43 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {2\,\sqrt {1-a^2\,x^2}}{c\,\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a\,c}-\frac {2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c\,\sqrt {-a^2}} \] Input:
int((1 - a^2*x^2)^(3/2)/((c - c/(a*x))*(a*x + 1)^3),x)
Output:
(2*(1 - a^2*x^2)^(1/2))/(c*(x*(-a^2)^(1/2) + (-a^2)^(1/2)/a)*(-a^2)^(1/2)) - (1 - a^2*x^2)^(1/2)/(a*c) - (2*asinh(x*(-a^2)^(1/2)))/(c*(-a^2)^(1/2))
Time = 0.16 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.52 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {-2 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )+2 \mathit {asin} \left (a x \right ) a x +2 \mathit {asin} \left (a x \right )+\sqrt {-a^{2} x^{2}+1}\, a x +4 \sqrt {-a^{2} x^{2}+1}+a^{2} x^{2}+a x -4}{a c \left (\sqrt {-a^{2} x^{2}+1}-a x -1\right )} \] Input:
int(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x),x)
Output:
( - 2*sqrt( - a**2*x**2 + 1)*asin(a*x) + 2*asin(a*x)*a*x + 2*asin(a*x) + s qrt( - a**2*x**2 + 1)*a*x + 4*sqrt( - a**2*x**2 + 1) + a**2*x**2 + a*x - 4 )/(a*c*(sqrt( - a**2*x**2 + 1) - a*x - 1))