Integrand size = 20, antiderivative size = 65 \[ \int \frac {e^{\text {arctanh}(a x)}}{c-\frac {c}{a x}} \, dx=-\frac {\sqrt {1-a^2 x^2}}{a c}-\frac {2 \sqrt {1-a^2 x^2}}{a c (1-a x)}+\frac {2 \arcsin (a x)}{a c} \] Output:
-(-a^2*x^2+1)^(1/2)/a/c-2*(-a^2*x^2+1)^(1/2)/a/c/(-a*x+1)+2*arcsin(a*x)/a/ c
Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.80 \[ \int \frac {e^{\text {arctanh}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {\frac {(-3+a x) \sqrt {1+a x}}{\sqrt {1-a x}}-4 \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{a c} \] Input:
Integrate[E^ArcTanh[a*x]/(c - c/(a*x)),x]
Output:
(((-3 + a*x)*Sqrt[1 + a*x])/Sqrt[1 - a*x] - 4*ArcSin[Sqrt[1 - a*x]/Sqrt[2] ])/(a*c)
Time = 0.36 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6681, 6678, 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.
\(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{c-\frac {c}{a x}} \, dx\) |
\(\Big \downarrow \) 6681 |
\(\displaystyle -\frac {a \int \frac {e^{\text {arctanh}(a x)} x}{1-a x}dx}{c}\) |
\(\Big \downarrow \) 6678 |
\(\displaystyle -\frac {a \int \frac {x \sqrt {1-a^2 x^2}}{(1-a x)^2}dx}{c}\) |
\(\Big \downarrow \) 563 |
\(\displaystyle -\frac {a \left (\frac {2 \sqrt {1-a^2 x^2}}{a^2 (1-a x)}-\frac {\int \frac {a x+2}{\sqrt {1-a^2 x^2}}dx}{a}\right )}{c}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle -\frac {a \left (\frac {2 \sqrt {1-a^2 x^2}}{a^2 (1-a x)}-\frac {2 \int \frac {1}{\sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{a}}{a}\right )}{c}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle -\frac {a \left (\frac {2 \sqrt {1-a^2 x^2}}{a^2 (1-a x)}-\frac {\frac {2 \arcsin (a x)}{a}-\frac {\sqrt {1-a^2 x^2}}{a}}{a}\right )}{c}\) |
Input:
Int[E^ArcTanh[a*x]/(c - c/(a*x)),x]
Output:
-((a*((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))/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)^(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_.))*((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 = 95, normalized size of antiderivative = 1.46
method | result | size |
default | \(\frac {a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{a^{2}}+\frac {2 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {a^{2}}}+\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{3} \left (x -\frac {1}{a}\right )}\right )}{c}\) | \(95\) |
risch | \(\frac {a^{2} x^{2}-1}{a c \sqrt {-a^{2} x^{2}+1}}+\frac {\left (\frac {2 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {a^{2}}}+\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{3} \left (x -\frac {1}{a}\right )}\right ) a}{c}\) | \(107\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a/x),x,method=_RETURNVERBOSE)
Output:
a/c*(-(-a^2*x^2+1)^(1/2)/a^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)*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2))
Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.05 \[ \int \frac {e^{\text {arctanh}(a x)}}{c-\frac {c}{a x}} \, dx=-\frac {3 \, a x + 4 \, {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} {\left (a x - 3\right )} - 3}{a^{2} c x - a c} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(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)
\[ \int \frac {e^{\text {arctanh}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {a \left (\int \frac {x}{a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{2}}{a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx\right )}{c} \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(c-c/a/x),x)
Output:
a*(Integral(x/(a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Inte gral(a*x**2/(a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c
\[ \int \frac {e^{\text {arctanh}(a x)}}{c-\frac {c}{a x}} \, dx=\int { \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (c - \frac {c}{a x}\right )}} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a/x),x, algorithm="maxima")
Output:
integrate((a*x + 1)/(sqrt(-a^2*x^2 + 1)*(c - c/(a*x))), x)
Time = 0.13 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.12 \[ \int \frac {e^{\text {arctanh}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {2 \, \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{c {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a c} - \frac {4}{c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(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 = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.40 \[ \int \frac {e^{\text {arctanh}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a\,c}-\frac {2\,\sqrt {1-a^2\,x^2}}{c\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \] Input:
int((a*x + 1)/((c - c/(a*x))*(1 - a^2*x^2)^(1/2)),x)
Output:
(2*asinh(x*(-a^2)^(1/2)))/(c*(-a^2)^(1/2)) - (1 - a^2*x^2)^(1/2)/(a*c) - ( 2*(1 - a^2*x^2)^(1/2))/(c*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)*(-a^2)^(1/2))
Time = 0.14 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.49 \[ \int \frac {e^{\text {arctanh}(a x)}}{c-\frac {c}{a x}} \, dx=\frac {2 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )+2 \mathit {asin} \left (a x \right ) a x -2 \mathit {asin} \left (a x \right )-\sqrt {-a^{2} x^{2}+1}\, a x +4 \sqrt {-a^{2} x^{2}+1}+a^{2} x^{2}-a x -4}{a c \left (\sqrt {-a^{2} x^{2}+1}+a x -1\right )} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/(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) - sqrt ( - 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))