Integrand size = 18, antiderivative size = 41 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{c-a c x} \, dx=-\frac {2 (1-a x)}{a c \sqrt {1-a^2 x^2}}-\frac {\arcsin (a x)}{a c} \] Output:
(2*a*x-2)/a/c/(-a^2*x^2+1)^(1/2)-arcsin(a*x)/a/c
Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.44 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{c-a c x} \, dx=\frac {2 \left (-1+a x+\sqrt {1-a^2 x^2} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{a c \sqrt {1-a^2 x^2}} \] Input:
Integrate[1/(E^(3*ArcTanh[a*x])*(c - a*c*x)),x]
Output:
(2*(-1 + a*x + Sqrt[1 - a^2*x^2]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(a*c*Sqrt [1 - a^2*x^2])
Time = 0.38 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6677, 27, 457, 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.
\(\displaystyle \int \frac {e^{-3 \text {arctanh}(a x)}}{c-a c x} \, dx\) |
\(\Big \downarrow \) 6677 |
\(\displaystyle \frac {\int \frac {c^2 (1-a x)^2}{\left (1-a^2 x^2\right )^{3/2}}dx}{c^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {(1-a x)^2}{\left (1-a^2 x^2\right )^{3/2}}dx}{c}\) |
\(\Big \downarrow \) 457 |
\(\displaystyle \frac {-\int \frac {1}{\sqrt {1-a^2 x^2}}dx-\frac {2 (1-a x)}{a \sqrt {1-a^2 x^2}}}{c}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {-\frac {2 (1-a x)}{a \sqrt {1-a^2 x^2}}-\frac {\arcsin (a x)}{a}}{c}\) |
Input:
Int[1/(E^(3*ArcTanh[a*x])*(c - a*c*x)),x]
Output:
((-2*(1 - a*x))/(a*Sqrt[1 - a^2*x^2]) - ArcSin[a*x]/a)/c
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_))^2*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*( c + d*x)*((a + b*x^2)^(p + 1)/(b*(p + 1))), x] - Simp[d^2*((p + 2)/(b*(p + 1))) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[ b*c^2 + a*d^2, 0] && LtQ[p, -1]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> S imp[c^n Int[(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(592\) vs. \(2(38)=76\).
Time = 0.17 (sec) , antiderivative size = 593, normalized size of antiderivative = 14.46
method | result | size |
default | \(-\frac {\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}-\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^{3}}-\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^{2}}-\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}}{c}\) | \(593\) |
Input:
int(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c),x,method=_RETURNVERBOSE)
Output:
-1/c*(1/8/a*(1/3*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(3/2)-a*(-1/4*(-2*(x-1/a)*a^ 2-2*a)/a^2*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)+1/2/(a^2)^(1/2)*arctan((a^2) ^(1/2)*x/(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2))))-1/2/a^3*(-1/a/(x+1/a)^3*(-a ^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)-2*a*(1/a/(x+1/a)^2*(-a^2*(x+1/a)^2+2*a*(x+ 1/a))^(5/2)+3*a*(1/3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)+a*(-1/4*(-2*a^2*(x +1/a)+2*a)/a^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)+1/2/(a^2)^(1/2)*arctan(( a^2)^(1/2)*x/(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2))))))-1/4/a^2*(1/a/(x+1/a)^ 2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(5/2)+3*a*(1/3*(-a^2*(x+1/a)^2+2*a*(x+1/a)) ^(3/2)+a*(-1/4*(-2*a^2*(x+1/a)+2*a)/a^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2) +1/2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2))) ))-1/8/a*(1/3*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(3/2)+a*(-1/4*(-2*a^2*(x+1/a)+2 *a)/a^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)+1/2/(a^2)^(1/2)*arctan((a^2)^(1 /2)*x/(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)))))
Time = 0.11 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.46 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{c-a c x} \, dx=-\frac {2 \, {\left (a x - {\left (a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} + 1\right )}}{a^{2} c x + a c} \] Input:
integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c),x, algorithm="fricas")
Output:
-2*(a*x - (a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + sqrt(-a^2*x^2 + 1) + 1)/(a^2*c*x + a*c)
\[ \int \frac {e^{-3 \text {arctanh}(a x)}}{c-a c x} \, dx=- \frac {\int \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{4} x^{4} + 2 a^{3} x^{3} - 2 a x - 1}\, dx + \int \left (- \frac {a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{4} x^{4} + 2 a^{3} x^{3} - 2 a x - 1}\right )\, dx}{c} \] Input:
integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(-a*c*x+c),x)
Output:
-(Integral(sqrt(-a**2*x**2 + 1)/(a**4*x**4 + 2*a**3*x**3 - 2*a*x - 1), x) + Integral(-a**2*x**2*sqrt(-a**2*x**2 + 1)/(a**4*x**4 + 2*a**3*x**3 - 2*a* x - 1), x))/c
\[ \int \frac {e^{-3 \text {arctanh}(a x)}}{c-a c x} \, dx=\int { -\frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a c x - c\right )} {\left (a x + 1\right )}^{3}} \,d x } \] Input:
integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c),x, algorithm="maxima")
Output:
-integrate((-a^2*x^2 + 1)^(3/2)/((a*c*x - c)*(a*x + 1)^3), x)
Time = 0.15 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.29 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{c-a c x} \, dx=-\frac {\arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{c {\left | a \right |}} + \frac {4}{c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} \] Input:
integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c),x, algorithm="giac")
Output:
-arcsin(a*x)*sgn(a)/(c*abs(a)) + 4/(c*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^ 2*x) + 1)*abs(a))
Time = 22.69 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.71 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{c-a c x} \, dx=\frac {2\,\sqrt {1-a^2\,x^2}}{c\,\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c\,\sqrt {-a^2}} \] Input:
int((1 - a^2*x^2)^(3/2)/((c - a*c*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)) - asinh(x*(-a^2)^(1/2))/(c*(-a^2)^(1/2))
Time = 0.15 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.15 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{c-a c x} \, dx=\frac {-\mathit {asin} \left (a x \right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )-\mathit {asin} \left (a x \right )+4 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )}{a c \left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )+1\right )} \] Input:
int(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(-a*c*x+c),x)
Output:
( - asin(a*x)*tan(asin(a*x)/2) - asin(a*x) + 4*tan(asin(a*x)/2))/(a*c*(tan (asin(a*x)/2) + 1))