Integrand size = 16, antiderivative size = 129 \[ \int \frac {e^{\text {arctanh}(a x)}}{(c-a c x)^5} \, dx=\frac {\left (1-a^2 x^2\right )^{3/2}}{9 a c^5 (1-a x)^6}+\frac {\left (1-a^2 x^2\right )^{3/2}}{21 a c^5 (1-a x)^5}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{105 a c^5 (1-a x)^4}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{315 a c^5 (1-a x)^3} \] Output:
1/9*(-a^2*x^2+1)^(3/2)/a/c^5/(-a*x+1)^6+1/21*(-a^2*x^2+1)^(3/2)/a/c^5/(-a* x+1)^5+2/105*(-a^2*x^2+1)^(3/2)/a/c^5/(-a*x+1)^4+2/315*(-a^2*x^2+1)^(3/2)/ a/c^5/(-a*x+1)^3
Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.40 \[ \int \frac {e^{\text {arctanh}(a x)}}{(c-a c x)^5} \, dx=\frac {(1+a x)^{3/2} \left (58-33 a x+12 a^2 x^2-2 a^3 x^3\right )}{315 a c^5 (1-a x)^{9/2}} \] Input:
Integrate[E^ArcTanh[a*x]/(c - a*c*x)^5,x]
Output:
((1 + a*x)^(3/2)*(58 - 33*a*x + 12*a^2*x^2 - 2*a^3*x^3))/(315*a*c^5*(1 - a *x)^(9/2))
Time = 0.31 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6677, 27, 461, 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.
| \(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{(c-a c x)^5} \, dx\) |
| \(\Big \downarrow \) 6677 |
| \(\displaystyle c \int \frac {\sqrt {1-a^2 x^2}}{c^6 (1-a x)^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {1-a^2 x^2}}{(1-a x)^6}dx}{c^5}\) |
| \(\Big \downarrow \) 461 |
| \(\displaystyle \frac {\frac {1}{3} \int \frac {\sqrt {1-a^2 x^2}}{(1-a x)^5}dx+\frac {\left (1-a^2 x^2\right )^{3/2}}{9 a (1-a x)^6}}{c^5}\) |
| \(\Big \downarrow \) 461 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {2}{7} \int \frac {\sqrt {1-a^2 x^2}}{(1-a x)^4}dx+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a (1-a x)^5}\right )+\frac {\left (1-a^2 x^2\right )^{3/2}}{9 a (1-a x)^6}}{c^5}\) |
| \(\Big \downarrow \) 461 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {2}{7} \left (\frac {1}{5} \int \frac {\sqrt {1-a^2 x^2}}{(1-a x)^3}dx+\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a (1-a x)^4}\right )+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a (1-a x)^5}\right )+\frac {\left (1-a^2 x^2\right )^{3/2}}{9 a (1-a x)^6}}{c^5}\) |
| \(\Big \downarrow \) 460 |
| \(\displaystyle \frac {\frac {\left (1-a^2 x^2\right )^{3/2}}{9 a (1-a x)^6}+\frac {1}{3} \left (\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a (1-a x)^5}+\frac {2}{7} \left (\frac {\left (1-a^2 x^2\right )^{3/2}}{15 a (1-a x)^3}+\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a (1-a x)^4}\right )\right )}{c^5}\) |
Input:
Int[E^ArcTanh[a*x]/(c - a*c*x)^5,x]
Output:
((1 - a^2*x^2)^(3/2)/(9*a*(1 - a*x)^6) + ((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)/(1 5*a*(1 - a*x)^3)))/7)/3)/c^5
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 = 0.29 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.44
| method | result | size |
| gosper | \(-\frac {\left (2 a^{3} x^{3}-12 a^{2} x^{2}+33 a x -58\right ) \left (a x +1\right )^{2}}{315 \left (a x -1\right )^{4} c^{5} \sqrt {-a^{2} x^{2}+1}\, a}\) | \(57\) |
| trager | \(\frac {\left (2 a^{4} x^{4}-10 a^{3} x^{3}+21 a^{2} x^{2}-25 a x -58\right ) \sqrt {-a^{2} x^{2}+1}}{315 c^{5} \left (a x -1\right )^{5} a}\) | \(58\) |
| orering | \(\frac {\left (2 a^{3} x^{3}-12 a^{2} x^{2}+33 a x -58\right ) \left (a x -1\right ) \left (a x +1\right )^{2}}{315 a \sqrt {-a^{2} x^{2}+1}\, \left (-a c x +c \right )^{5}}\) | \(61\) |
| default | \(-\frac {\frac {\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{7 a \left (x -\frac {1}{a}\right )^{4}}-\frac {3 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )}{7}}{a^{4}}+\frac {\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{9 a \left (x -\frac {1}{a}\right )^{5}}-\frac {8 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{7 a \left (x -\frac {1}{a}\right )^{4}}-\frac {3 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )}{7}\right )}{9}}{a^{5}}}{c^{5}}\) | \(401\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^5,x,method=_RETURNVERBOSE)
Output:
-1/315*(2*a^3*x^3-12*a^2*x^2+33*a*x-58)*(a*x+1)^2/(a*x-1)^4/c^5/(-a^2*x^2+ 1)^(1/2)/a
Time = 0.09 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.12 \[ \int \frac {e^{\text {arctanh}(a x)}}{(c-a c x)^5} \, dx=\frac {58 \, a^{5} x^{5} - 290 \, a^{4} x^{4} + 580 \, a^{3} x^{3} - 580 \, a^{2} x^{2} + 290 \, a x + {\left (2 \, a^{4} x^{4} - 10 \, a^{3} x^{3} + 21 \, a^{2} x^{2} - 25 \, a x - 58\right )} \sqrt {-a^{2} x^{2} + 1} - 58}{315 \, {\left (a^{6} c^{5} x^{5} - 5 \, a^{5} c^{5} x^{4} + 10 \, a^{4} c^{5} x^{3} - 10 \, a^{3} c^{5} x^{2} + 5 \, a^{2} c^{5} x - a c^{5}\right )}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^5,x, algorithm="fricas")
Output:
1/315*(58*a^5*x^5 - 290*a^4*x^4 + 580*a^3*x^3 - 580*a^2*x^2 + 290*a*x + (2 *a^4*x^4 - 10*a^3*x^3 + 21*a^2*x^2 - 25*a*x - 58)*sqrt(-a^2*x^2 + 1) - 58) /(a^6*c^5*x^5 - 5*a^5*c^5*x^4 + 10*a^4*c^5*x^3 - 10*a^3*c^5*x^2 + 5*a^2*c^ 5*x - a*c^5)
\[ \int \frac {e^{\text {arctanh}(a x)}}{(c-a c x)^5} \, dx=- \frac {\int \frac {a x}{a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 10 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 10 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 5 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 10 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 10 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 5 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{5}} \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(-a*c*x+c)**5,x)
Output:
-(Integral(a*x/(a**5*x**5*sqrt(-a**2*x**2 + 1) - 5*a**4*x**4*sqrt(-a**2*x* *2 + 1) + 10*a**3*x**3*sqrt(-a**2*x**2 + 1) - 10*a**2*x**2*sqrt(-a**2*x**2 + 1) + 5*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral( 1/(a**5*x**5*sqrt(-a**2*x**2 + 1) - 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 10* a**3*x**3*sqrt(-a**2*x**2 + 1) - 10*a**2*x**2*sqrt(-a**2*x**2 + 1) + 5*a*x *sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c**5
Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (109) = 218\).
Time = 0.11 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.05 \[ \int \frac {e^{\text {arctanh}(a x)}}{(c-a c x)^5} \, dx=-\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{9 \, {\left (a^{6} c^{5} x^{5} - 5 \, a^{5} c^{5} x^{4} + 10 \, a^{4} c^{5} x^{3} - 10 \, a^{3} c^{5} x^{2} + 5 \, a^{2} c^{5} x - a c^{5}\right )}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{63 \, {\left (a^{5} c^{5} x^{4} - 4 \, a^{4} c^{5} x^{3} + 6 \, a^{3} c^{5} x^{2} - 4 \, a^{2} c^{5} x + a c^{5}\right )}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right )}} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{315 \, {\left (a^{3} c^{5} x^{2} - 2 \, a^{2} c^{5} x + a c^{5}\right )}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{315 \, {\left (a^{2} c^{5} x - a c^{5}\right )}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^5,x, algorithm="maxima")
Output:
-2/9*sqrt(-a^2*x^2 + 1)/(a^6*c^5*x^5 - 5*a^5*c^5*x^4 + 10*a^4*c^5*x^3 - 10 *a^3*c^5*x^2 + 5*a^2*c^5*x - a*c^5) - 1/63*sqrt(-a^2*x^2 + 1)/(a^5*c^5*x^4 - 4*a^4*c^5*x^3 + 6*a^3*c^5*x^2 - 4*a^2*c^5*x + a*c^5) + 1/105*sqrt(-a^2* x^2 + 1)/(a^4*c^5*x^3 - 3*a^3*c^5*x^2 + 3*a^2*c^5*x - a*c^5) - 2/315*sqrt( -a^2*x^2 + 1)/(a^3*c^5*x^2 - 2*a^2*c^5*x + a*c^5) + 2/315*sqrt(-a^2*x^2 + 1)/(a^2*c^5*x - a*c^5)
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.49 \[ \int \frac {e^{\text {arctanh}(a x)}}{(c-a c x)^5} \, dx=\frac {-\frac {16 i \, \mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right )}{c^{3}} - \frac {\frac {35 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{4} \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 180 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{3} \sqrt {-\frac {2 \, c}{a c x - c} - 1} + 378 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{2} \sqrt {-\frac {2 \, c}{a c x - c} - 1} + 420 \, {\left (-\frac {2 \, c}{a c x - c} - 1\right )}^{\frac {3}{2}} + 315 \, \sqrt {-\frac {2 \, c}{a c x - c} - 1}}{c^{3}} + \frac {9 \, {\left (5 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{3} \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 21 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{2} \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 35 \, {\left (-\frac {2 \, c}{a c x - c} - 1\right )}^{\frac {3}{2}} - 35 \, \sqrt {-\frac {2 \, c}{a c x - c} - 1}\right )}}{c^{3}}}{\mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right )}}{2520 \, c^{2} {\left | a \right |}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^5,x, algorithm="giac")
Output:
1/2520*(-16*I*sgn(1/(a*c*x - c))*sgn(a)*sgn(c)/c^3 - ((35*(2*c/(a*c*x - c) + 1)^4*sqrt(-2*c/(a*c*x - c) - 1) - 180*(2*c/(a*c*x - c) + 1)^3*sqrt(-2*c /(a*c*x - c) - 1) + 378*(2*c/(a*c*x - c) + 1)^2*sqrt(-2*c/(a*c*x - c) - 1) + 420*(-2*c/(a*c*x - c) - 1)^(3/2) + 315*sqrt(-2*c/(a*c*x - c) - 1))/c^3 + 9*(5*(2*c/(a*c*x - c) + 1)^3*sqrt(-2*c/(a*c*x - c) - 1) - 21*(2*c/(a*c*x - c) + 1)^2*sqrt(-2*c/(a*c*x - c) - 1) - 35*(-2*c/(a*c*x - c) - 1)^(3/2) - 35*sqrt(-2*c/(a*c*x - c) - 1))/c^3)/(sgn(1/(a*c*x - c))*sgn(a)*sgn(c)))/ (c^2*abs(a))
Time = 14.05 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.44 \[ \int \frac {e^{\text {arctanh}(a x)}}{(c-a c x)^5} \, dx=-\frac {\sqrt {1-a^2\,x^2}\,\left (-2\,a^4\,x^4+10\,a^3\,x^3-21\,a^2\,x^2+25\,a\,x+58\right )}{315\,a\,c^5\,{\left (a\,x-1\right )}^5} \] Input:
int((a*x + 1)/((1 - a^2*x^2)^(1/2)*(c - a*c*x)^5),x)
Output:
-((1 - a^2*x^2)^(1/2)*(25*a*x - 21*a^2*x^2 + 10*a^3*x^3 - 2*a^4*x^4 + 58)) /(315*a*c^5*(a*x - 1)^5)
\[ \int \frac {e^{\text {arctanh}(a x)}}{(c-a c x)^5} \, dx=\int \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}\, \left (-a c x +c \right )^{5}}d x \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^5,x)
Output:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a*c*x+c)^5,x)