Integrand size = 18, antiderivative size = 65 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{(c-a c x)^3} \, dx=\frac {\left (1-a^2 x^2\right )^{5/2}}{7 a c^3 (1-a x)^6}+\frac {\left (1-a^2 x^2\right )^{5/2}}{35 a c^3 (1-a x)^5} \] Output:
1/7*(-a^2*x^2+1)^(5/2)/a/c^3/(-a*x+1)^6+1/35*(-a^2*x^2+1)^(5/2)/a/c^3/(-a* x+1)^5
Time = 0.05 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.52 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{(c-a c x)^3} \, dx=-\frac {(-6+a x) (1+a x)^{5/2}}{35 a c^3 (1-a x)^{7/2}} \] Input:
Integrate[E^(3*ArcTanh[a*x])/(c - a*c*x)^3,x]
Output:
-1/35*((-6 + a*x)*(1 + a*x)^(5/2))/(a*c^3*(1 - a*x)^(7/2))
Time = 0.40 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6677, 27, 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.
\(\displaystyle \int \frac {e^{3 \text {arctanh}(a x)}}{(c-a c x)^3} \, dx\) |
\(\Big \downarrow \) 6677 |
\(\displaystyle c^3 \int \frac {\left (1-a^2 x^2\right )^{3/2}}{c^6 (1-a x)^6}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\left (1-a^2 x^2\right )^{3/2}}{(1-a x)^6}dx}{c^3}\) |
\(\Big \downarrow \) 461 |
\(\displaystyle \frac {\frac {1}{7} \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(1-a x)^5}dx+\frac {\left (1-a^2 x^2\right )^{5/2}}{7 a (1-a x)^6}}{c^3}\) |
\(\Big \downarrow \) 460 |
\(\displaystyle \frac {\frac {\left (1-a^2 x^2\right )^{5/2}}{35 a (1-a x)^5}+\frac {\left (1-a^2 x^2\right )^{5/2}}{7 a (1-a x)^6}}{c^3}\) |
Input:
Int[E^(3*ArcTanh[a*x])/(c - a*c*x)^3,x]
Output:
((1 - a^2*x^2)^(5/2)/(7*a*(1 - a*x)^6) + (1 - a^2*x^2)^(5/2)/(35*a*(1 - a* x)^5))/c^3
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 = 0.17 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.62
method | result | size |
gosper | \(-\frac {\left (a x -6\right ) \left (a x +1\right )^{4}}{35 \left (a x -1\right )^{2} c^{3} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} a}\) | \(40\) |
orering | \(\frac {\left (a x -6\right ) \left (a x -1\right ) \left (a x +1\right )^{4}}{35 a \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \left (-a c x +c \right )^{3}}\) | \(44\) |
trager | \(-\frac {\left (a^{3} x^{3}-4 a^{2} x^{2}-11 a x -6\right ) \sqrt {-a^{2} x^{2}+1}}{35 c^{3} \left (a x -1\right )^{4} a}\) | \(49\) |
default | \(-\frac {\frac {x}{\sqrt {-a^{2} x^{2}+1}}+\frac {\frac {8}{7 a \left (x -\frac {1}{a}\right )^{3} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {32 a \left (\frac {1}{5 a \left (x -\frac {1}{a}\right )^{2} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {3 a \left (\frac {1}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a}{3 a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{5}\right )}{7}}{a^{3}}+\frac {\frac {12}{5 a \left (x -\frac {1}{a}\right )^{2} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {36 a \left (\frac {1}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a}{3 a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{5}}{a^{2}}+\frac {\frac {2}{a \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {2 \left (-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a \right )}{a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}}{a}}{c^{3}}\) | \(441\) |
Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^3,x,method=_RETURNVERBOSE)
Output:
-1/35*(a*x-6)*(a*x+1)^4/(a*x-1)^2/c^3/(-a^2*x^2+1)^(3/2)/a
Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (55) = 110\).
Time = 0.09 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.78 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{(c-a c x)^3} \, dx=\frac {6 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 36 \, a^{2} x^{2} - 24 \, a x - {\left (a^{3} x^{3} - 4 \, a^{2} x^{2} - 11 \, a x - 6\right )} \sqrt {-a^{2} x^{2} + 1} + 6}{35 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^3,x, algorithm="fricas")
Output:
1/35*(6*a^4*x^4 - 24*a^3*x^3 + 36*a^2*x^2 - 24*a*x - (a^3*x^3 - 4*a^2*x^2 - 11*a*x - 6)*sqrt(-a^2*x^2 + 1) + 6)/(a^5*c^3*x^4 - 4*a^4*c^3*x^3 + 6*a^3 *c^3*x^2 - 4*a^2*c^3*x + a*c^3)
\[ \int \frac {e^{3 \text {arctanh}(a x)}}{(c-a c x)^3} \, dx=- \frac {\int \frac {3 a x}{- a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} x^{2}}{- a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{3} x^{3}}{- a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \] Input:
integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(-a*c*x+c)**3,x)
Output:
-(Integral(3*a*x/(-a**5*x**5*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2 *x**2 + 1) - 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x** 2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral (3*a**2*x**2/(-a**5*x**5*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x** 2 + 1) - 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + Integral(a** 3*x**3/(-a**5*x**5*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3 *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) + 3*a**4*x**4*sqrt(-a**2*x**2 + 1) - 2*a**3*x**3 *sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x**2 + 1) + 3*a*x*sqrt(-a** 2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c**3
Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (55) = 110\).
Time = 0.04 (sec) , antiderivative size = 216, normalized size of antiderivative = 3.32 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{(c-a c x)^3} \, dx=-\frac {8}{7 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{4} c^{3} x^{3} - 3 \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{3} x^{2} + 3 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{3} x - \sqrt {-a^{2} x^{2} + 1} a c^{3}\right )}} - \frac {52}{35 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{3} c^{3} x^{2} - 2 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{3} x + \sqrt {-a^{2} x^{2} + 1} a c^{3}\right )}} - \frac {18}{35 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{2} c^{3} x - \sqrt {-a^{2} x^{2} + 1} a c^{3}\right )}} + \frac {x}{35 \, \sqrt {-a^{2} x^{2} + 1} c^{3}} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^3,x, algorithm="maxima")
Output:
-8/7/(sqrt(-a^2*x^2 + 1)*a^4*c^3*x^3 - 3*sqrt(-a^2*x^2 + 1)*a^3*c^3*x^2 + 3*sqrt(-a^2*x^2 + 1)*a^2*c^3*x - sqrt(-a^2*x^2 + 1)*a*c^3) - 52/35/(sqrt(- a^2*x^2 + 1)*a^3*c^3*x^2 - 2*sqrt(-a^2*x^2 + 1)*a^2*c^3*x + sqrt(-a^2*x^2 + 1)*a*c^3) - 18/35/(sqrt(-a^2*x^2 + 1)*a^2*c^3*x - sqrt(-a^2*x^2 + 1)*a*c ^3) + 1/35*x/(sqrt(-a^2*x^2 + 1)*c^3)
Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (55) = 110\).
Time = 0.14 (sec) , antiderivative size = 199, normalized size of antiderivative = 3.06 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{(c-a c x)^3} \, dx=-\frac {2 \, {\left (\frac {7 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - \frac {91 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac {70 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac {140 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} + \frac {35 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5}}{a^{10} x^{5}} - \frac {35 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6}}{a^{12} x^{6}} - 6\right )}}{35 \, c^{3} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{7} {\left | a \right |}} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^3,x, algorithm="giac")
Output:
-2/35*(7*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 91*(sqrt(-a^2*x^2 + 1)* abs(a) + a)^2/(a^4*x^2) + 70*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3/(a^6*x^3) - 140*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4/(a^8*x^4) + 35*(sqrt(-a^2*x^2 + 1)* abs(a) + a)^5/(a^10*x^5) - 35*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^6/(a^12*x^6) - 6)/(c^3*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)^7*abs(a))
Time = 22.48 (sec) , antiderivative size = 299, normalized size of antiderivative = 4.60 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{(c-a c x)^3} \, dx=\frac {8\,a^3\,\sqrt {1-a^2\,x^2}}{35\,\left (a^6\,c^3\,x^2-2\,a^5\,c^3\,x+a^4\,c^3\right )}-\frac {a\,\sqrt {1-a^2\,x^2}}{5\,\left (a^4\,c^3\,x^2-2\,a^3\,c^3\,x+a^2\,c^3\right )}+\frac {4\,a\,\sqrt {1-a^2\,x^2}}{7\,\left (a^6\,c^3\,x^4-4\,a^5\,c^3\,x^3+6\,a^4\,c^3\,x^2-4\,a^3\,c^3\,x+a^2\,c^3\right )}+\frac {\sqrt {1-a^2\,x^2}}{35\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}\right )}-\frac {16\,\sqrt {1-a^2\,x^2}}{35\,\sqrt {-a^2}\,\left (3\,c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}+a^2\,c^3\,x^3\,\sqrt {-a^2}-3\,a\,c^3\,x^2\,\sqrt {-a^2}\right )} \] Input:
int((a*x + 1)^3/((1 - a^2*x^2)^(3/2)*(c - a*c*x)^3),x)
Output:
(8*a^3*(1 - a^2*x^2)^(1/2))/(35*(a^4*c^3 - 2*a^5*c^3*x + a^6*c^3*x^2)) - ( a*(1 - a^2*x^2)^(1/2))/(5*(a^2*c^3 - 2*a^3*c^3*x + a^4*c^3*x^2)) + (4*a*(1 - a^2*x^2)^(1/2))/(7*(a^2*c^3 - 4*a^3*c^3*x + 6*a^4*c^3*x^2 - 4*a^5*c^3*x ^3 + a^6*c^3*x^4)) + (1 - a^2*x^2)^(1/2)/(35*(-a^2)^(1/2)*(c^3*x*(-a^2)^(1 /2) - (c^3*(-a^2)^(1/2))/a)) - (16*(1 - a^2*x^2)^(1/2))/(35*(-a^2)^(1/2)*( 3*c^3*x*(-a^2)^(1/2) - (c^3*(-a^2)^(1/2))/a + a^2*c^3*x^3*(-a^2)^(1/2) - 3 *a*c^3*x^2*(-a^2)^(1/2)))
Time = 0.16 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.11 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{(c-a c x)^3} \, dx=\frac {\sqrt {-a^{2} x^{2}+1}\, \left (-a^{3} x^{3}+4 a^{2} x^{2}+11 a x +6\right )}{35 a \,c^{3} \left (a^{4} x^{4}-4 a^{3} x^{3}+6 a^{2} x^{2}-4 a x +1\right )} \] Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^3,x)
Output:
(sqrt( - a**2*x**2 + 1)*( - a**3*x**3 + 4*a**2*x**2 + 11*a*x + 6))/(35*a*c **3*(a**4*x**4 - 4*a**3*x**3 + 6*a**2*x**2 - 4*a*x + 1))