Integrand size = 16, antiderivative size = 97 \[ \int \frac {e^{\text {arctanh}(a x)}}{(c-a c x)^4} \, dx=\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a c^4 (1-a x)^5}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{35 a c^4 (1-a x)^4}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{105 a c^4 (1-a x)^3} \] Output:
1/7*(-a^2*x^2+1)^(3/2)/a/c^4/(-a*x+1)^5+2/35*(-a^2*x^2+1)^(3/2)/a/c^4/(-a* x+1)^4+2/105*(-a^2*x^2+1)^(3/2)/a/c^4/(-a*x+1)^3
Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.44 \[ \int \frac {e^{\text {arctanh}(a x)}}{(c-a c x)^4} \, dx=-\frac {(1+a x)^{3/2} \left (-23+10 a x-2 a^2 x^2\right )}{105 a c^4 (1-a x)^{7/2}} \] Input:
Integrate[E^ArcTanh[a*x]/(c - a*c*x)^4,x]
Output:
-1/105*((1 + a*x)^(3/2)*(-23 + 10*a*x - 2*a^2*x^2))/(a*c^4*(1 - a*x)^(7/2) )
Time = 0.28 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6677, 27, 461, 461, 460}
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.
\(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{(c-a c x)^4} \, dx\) |
\(\Big \downarrow \) 6677 |
\(\displaystyle c \int \frac {\sqrt {1-a^2 x^2}}{c^5 (1-a x)^5}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\sqrt {1-a^2 x^2}}{(1-a x)^5}dx}{c^4}\) |
\(\Big \downarrow \) 461 |
\(\displaystyle \frac {\frac {2}{7} \int \frac {\sqrt {1-a^2 x^2}}{(1-a x)^4}dx+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a (1-a x)^5}}{c^4}\) |
\(\Big \downarrow \) 461 |
\(\displaystyle \frac {\frac {2}{7} \left (\frac {1}{5} \int \frac {\sqrt {1-a^2 x^2}}{(1-a x)^3}dx+\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a (1-a x)^4}\right )+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a (1-a x)^5}}{c^4}\) |
\(\Big \downarrow \) 460 |
\(\displaystyle \frac {\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a (1-a x)^5}+\frac {2}{7} \left (\frac {\left (1-a^2 x^2\right )^{3/2}}{15 a (1-a x)^3}+\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a (1-a x)^4}\right )}{c^4}\) |
Input:
Int[E^ArcTanh[a*x]/(c - a*c*x)^4,x]
Output:
((1 - a^2*x^2)^(3/2)/(7*a*(1 - a*x)^5) + (2*((1 - a^2*x^2)^(3/2)/(5*a*(1 - a*x)^4) + (1 - a^2*x^2)^(3/2)/(15*a*(1 - a*x)^3)))/7)/c^4
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-d)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(b*c*n)), x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + 2*p + 2, 0]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-d)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*b*c*(n + p + 1))), x] + Simp[Simpl ify[n + 2*p + 2]/(2*c*(n + p + 1)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x ], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && ILtQ[Simp lify[n + 2*p + 2], 0] && (LtQ[n, -1] || GtQ[n + p, 0])
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]
Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.51
method | result | size |
gosper | \(-\frac {\left (2 a^{2} x^{2}-10 a x +23\right ) \left (a x +1\right )^{2}}{105 \left (a x -1\right )^{3} c^{4} \sqrt {-a^{2} x^{2}+1}\, a}\) | \(49\) |
trager | \(\frac {\left (2 a^{3} x^{3}-8 a^{2} x^{2}+13 a x +23\right ) \sqrt {-a^{2} x^{2}+1}}{105 c^{4} \left (a x -1\right )^{4} a}\) | \(50\) |
orering | \(-\frac {\left (2 a^{2} x^{2}-10 a x +23\right ) \left (a x -1\right ) \left (a x +1\right )^{2}}{105 a \sqrt {-a^{2} x^{2}+1}\, \left (-a c x +c \right )^{4}}\) | \(53\) |
default | \(\frac {\frac {\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{a^{3}}+\frac {\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{7 a \left (x -\frac {1}{a}\right )^{4}}-\frac {6 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )}{7}}{a^{4}}}{c^{4}}\) | \(310\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^4,x,method=_RETURNVERBOSE)
Output:
-1/105*(2*a^2*x^2-10*a*x+23)*(a*x+1)^2/(a*x-1)^3/c^4/(-a^2*x^2+1)^(1/2)/a
Time = 0.08 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.20 \[ \int \frac {e^{\text {arctanh}(a x)}}{(c-a c x)^4} \, dx=\frac {23 \, a^{4} x^{4} - 92 \, a^{3} x^{3} + 138 \, a^{2} x^{2} - 92 \, a x + {\left (2 \, a^{3} x^{3} - 8 \, a^{2} x^{2} + 13 \, a x + 23\right )} \sqrt {-a^{2} x^{2} + 1} + 23}{105 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^4,x, algorithm="fricas")
Output:
1/105*(23*a^4*x^4 - 92*a^3*x^3 + 138*a^2*x^2 - 92*a*x + (2*a^3*x^3 - 8*a^2 *x^2 + 13*a*x + 23)*sqrt(-a^2*x^2 + 1) + 23)/(a^5*c^4*x^4 - 4*a^4*c^4*x^3 + 6*a^3*c^4*x^2 - 4*a^2*c^4*x + a*c^4)
\[ \int \frac {e^{\text {arctanh}(a x)}}{(c-a c x)^4} \, dx=\frac {\int \frac {a x}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{4}} \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(-a*c*x+c)**4,x)
Output:
(Integral(a*x/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**3*x**3*sqrt(-a**2*x** 2 + 1) + 6*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + s qrt(-a**2*x**2 + 1)), x) + Integral(1/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 4* a**3*x**3*sqrt(-a**2*x**2 + 1) + 6*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x* sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**4
Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (82) = 164\).
Time = 0.11 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.95 \[ \int \frac {e^{\text {arctanh}(a x)}}{(c-a c x)^4} \, dx=\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{7 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{35 \, {\left (a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (a^{2} c^{4} x - a c^{4}\right )}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^4,x, algorithm="maxima")
Output:
2/7*sqrt(-a^2*x^2 + 1)/(a^5*c^4*x^4 - 4*a^4*c^4*x^3 + 6*a^3*c^4*x^2 - 4*a^ 2*c^4*x + a*c^4) + 1/35*sqrt(-a^2*x^2 + 1)/(a^4*c^4*x^3 - 3*a^3*c^4*x^2 + 3*a^2*c^4*x - a*c^4) - 2/105*sqrt(-a^2*x^2 + 1)/(a^3*c^4*x^2 - 2*a^2*c^4*x + a*c^4) + 2/105*sqrt(-a^2*x^2 + 1)/(a^2*c^4*x - a*c^4)
Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (82) = 164\).
Time = 0.13 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.05 \[ \int \frac {e^{\text {arctanh}(a x)}}{(c-a c x)^4} \, dx=-\frac {2 \, {\left (\frac {56 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - \frac {273 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac {350 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac {455 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} + \frac {210 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5}}{a^{10} x^{5}} - \frac {105 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6}}{a^{12} x^{6}} - 23\right )}}{105 \, c^{4} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{7} {\left | a \right |}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^4,x, algorithm="giac")
Output:
-2/105*(56*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 273*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/(a^4*x^2) + 350*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3/(a^6*x^ 3) - 455*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4/(a^8*x^4) + 210*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^5/(a^10*x^5) - 105*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^6/(a^1 2*x^6) - 23)/(c^4*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)^7*abs(a))
Time = 14.00 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.51 \[ \int \frac {e^{\text {arctanh}(a x)}}{(c-a c x)^4} \, dx=\frac {\sqrt {1-a^2\,x^2}\,\left (2\,a^3\,x^3-8\,a^2\,x^2+13\,a\,x+23\right )}{105\,a\,c^4\,{\left (a\,x-1\right )}^4} \] Input:
int((a*x + 1)/((1 - a^2*x^2)^(1/2)*(c - a*c*x)^4),x)
Output:
((1 - a^2*x^2)^(1/2)*(13*a*x - 8*a^2*x^2 + 2*a^3*x^3 + 23))/(105*a*c^4*(a* x - 1)^4)
Time = 0.16 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.74 \[ \int \frac {e^{\text {arctanh}(a x)}}{(c-a c x)^4} \, dx=\frac {\sqrt {-a^{2} x^{2}+1}\, \left (2 a^{3} x^{3}-8 a^{2} x^{2}+13 a x +23\right )}{105 a \,c^{4} \left (a^{4} x^{4}-4 a^{3} x^{3}+6 a^{2} x^{2}-4 a x +1\right )} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^4,x)
Output:
(sqrt( - a**2*x**2 + 1)*(2*a**3*x**3 - 8*a**2*x**2 + 13*a*x + 23))/(105*a* c**4*(a**4*x**4 - 4*a**3*x**3 + 6*a**2*x**2 - 4*a*x + 1))