Integrand size = 18, antiderivative size = 129 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{(c-a c x)^5} \, dx=\frac {\left (1-a^2 x^2\right )^{5/2}}{11 a c^5 (1-a x)^8}+\frac {\left (1-a^2 x^2\right )^{5/2}}{33 a c^5 (1-a x)^7}+\frac {2 \left (1-a^2 x^2\right )^{5/2}}{231 a c^5 (1-a x)^6}+\frac {2 \left (1-a^2 x^2\right )^{5/2}}{1155 a c^5 (1-a x)^5} \] Output:
1/11*(-a^2*x^2+1)^(5/2)/a/c^5/(-a*x+1)^8+1/33*(-a^2*x^2+1)^(5/2)/a/c^5/(-a *x+1)^7+2/231*(-a^2*x^2+1)^(5/2)/a/c^5/(-a*x+1)^6+2/1155*(-a^2*x^2+1)^(5/2 )/a/c^5/(-a*x+1)^5
Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.40 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{(c-a c x)^5} \, dx=-\frac {(1+a x)^{5/2} \left (-152+61 a x-16 a^2 x^2+2 a^3 x^3\right )}{1155 a c^5 (1-a x)^{11/2}} \] Input:
Integrate[E^(3*ArcTanh[a*x])/(c - a*c*x)^5,x]
Output:
-1/1155*((1 + a*x)^(5/2)*(-152 + 61*a*x - 16*a^2*x^2 + 2*a^3*x^3))/(a*c^5* (1 - a*x)^(11/2))
Time = 0.33 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, 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.
| \(\displaystyle \int \frac {e^{3 \text {arctanh}(a x)}}{(c-a c x)^5} \, dx\) |
| \(\Big \downarrow \) 6677 |
| \(\displaystyle c^3 \int \frac {\left (1-a^2 x^2\right )^{3/2}}{c^8 (1-a x)^8}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (1-a^2 x^2\right )^{3/2}}{(1-a x)^8}dx}{c^5}\) |
| \(\Big \downarrow \) 461 |
| \(\displaystyle \frac {\frac {3}{11} \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(1-a x)^7}dx+\frac {\left (1-a^2 x^2\right )^{5/2}}{11 a (1-a x)^8}}{c^5}\) |
| \(\Big \downarrow \) 461 |
| \(\displaystyle \frac {\frac {3}{11} \left (\frac {2}{9} \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(1-a x)^6}dx+\frac {\left (1-a^2 x^2\right )^{5/2}}{9 a (1-a x)^7}\right )+\frac {\left (1-a^2 x^2\right )^{5/2}}{11 a (1-a x)^8}}{c^5}\) |
| \(\Big \downarrow \) 461 |
| \(\displaystyle \frac {\frac {3}{11} \left (\frac {2}{9} \left (\frac {1}{7} \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(1-a x)^5}dx+\frac {\left (1-a^2 x^2\right )^{5/2}}{7 a (1-a x)^6}\right )+\frac {\left (1-a^2 x^2\right )^{5/2}}{9 a (1-a x)^7}\right )+\frac {\left (1-a^2 x^2\right )^{5/2}}{11 a (1-a x)^8}}{c^5}\) |
| \(\Big \downarrow \) 460 |
| \(\displaystyle \frac {\frac {\left (1-a^2 x^2\right )^{5/2}}{11 a (1-a x)^8}+\frac {3}{11} \left (\frac {\left (1-a^2 x^2\right )^{5/2}}{9 a (1-a x)^7}+\frac {2}{9} \left (\frac {\left (1-a^2 x^2\right )^{5/2}}{35 a (1-a x)^5}+\frac {\left (1-a^2 x^2\right )^{5/2}}{7 a (1-a x)^6}\right )\right )}{c^5}\) |
Input:
Int[E^(3*ArcTanh[a*x])/(c - a*c*x)^5,x]
Output:
((1 - a^2*x^2)^(5/2)/(11*a*(1 - a*x)^8) + (3*((1 - a^2*x^2)^(5/2)/(9*a*(1 - a*x)^7) + (2*((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)))/9))/11)/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 = 0.34 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.44
| method | result | size |
| gosper | \(-\frac {\left (2 a^{3} x^{3}-16 a^{2} x^{2}+61 a x -152\right ) \left (a x +1\right )^{4}}{1155 \left (a x -1\right )^{4} c^{5} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} a}\) | \(57\) |
| orering | \(\frac {\left (2 a^{3} x^{3}-16 a^{2} x^{2}+61 a x -152\right ) \left (a x -1\right ) \left (a x +1\right )^{4}}{1155 a \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \left (-a c x +c \right )^{5}}\) | \(61\) |
| trager | \(-\frac {\left (2 a^{5} x^{5}-12 a^{4} x^{4}+31 a^{3} x^{3}-46 a^{2} x^{2}-243 a x -152\right ) \sqrt {-a^{2} x^{2}+1}}{1155 c^{5} \left (a x -1\right )^{6} a}\) | \(66\) |
| default | \(-\frac {\frac {\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}}{a^{2}}+\frac {\frac {8}{11 a \left (x -\frac {1}{a}\right )^{5} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {48 a \left (\frac {1}{9 a \left (x -\frac {1}{a}\right )^{4} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {5 a \left (\frac {1}{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 {4 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}\right )}{9}\right )}{11}}{a^{5}}+\frac {\frac {4}{3 a \left (x -\frac {1}{a}\right )^{4} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {20 a \left (\frac {1}{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 {4 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}\right )}{3}}{a^{4}}+\frac {\frac {6}{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 {24 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}}}{c^{5}}\) | \(836\) |
Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^5,x,method=_RETURNVERBOSE)
Output:
-1/1155*(2*a^3*x^3-16*a^2*x^2+61*a*x-152)*(a*x+1)^4/(a*x-1)^4/c^5/(-a^2*x^ 2+1)^(3/2)/a
Time = 0.10 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.33 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{(c-a c x)^5} \, dx=\frac {152 \, a^{6} x^{6} - 912 \, a^{5} x^{5} + 2280 \, a^{4} x^{4} - 3040 \, a^{3} x^{3} + 2280 \, a^{2} x^{2} - 912 \, a x - {\left (2 \, a^{5} x^{5} - 12 \, a^{4} x^{4} + 31 \, a^{3} x^{3} - 46 \, a^{2} x^{2} - 243 \, a x - 152\right )} \sqrt {-a^{2} x^{2} + 1} + 152}{1155 \, {\left (a^{7} c^{5} x^{6} - 6 \, a^{6} c^{5} x^{5} + 15 \, a^{5} c^{5} x^{4} - 20 \, a^{4} c^{5} x^{3} + 15 \, a^{3} c^{5} x^{2} - 6 \, a^{2} c^{5} x + a c^{5}\right )}} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^5,x, algorithm="fricas")
Output:
1/1155*(152*a^6*x^6 - 912*a^5*x^5 + 2280*a^4*x^4 - 3040*a^3*x^3 + 2280*a^2 *x^2 - 912*a*x - (2*a^5*x^5 - 12*a^4*x^4 + 31*a^3*x^3 - 46*a^2*x^2 - 243*a *x - 152)*sqrt(-a^2*x^2 + 1) + 152)/(a^7*c^5*x^6 - 6*a^6*c^5*x^5 + 15*a^5* c^5*x^4 - 20*a^4*c^5*x^3 + 15*a^3*c^5*x^2 - 6*a^2*c^5*x + a*c^5)
\[ \int \frac {e^{3 \text {arctanh}(a x)}}{(c-a c x)^5} \, dx=- \frac {\int \frac {3 a x}{- a^{7} x^{7} \sqrt {- a^{2} x^{2} + 1} + 5 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} - 9 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 5 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 9 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 5 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} x^{2}}{- a^{7} x^{7} \sqrt {- a^{2} x^{2} + 1} + 5 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} - 9 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 5 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 9 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 5 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{3} x^{3}}{- a^{7} x^{7} \sqrt {- a^{2} x^{2} + 1} + 5 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} - 9 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 5 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 9 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 5 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{7} x^{7} \sqrt {- a^{2} x^{2} + 1} + 5 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} - 9 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 5 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 9 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 5 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{5}} \] Input:
integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(-a*c*x+c)**5,x)
Output:
-(Integral(3*a*x/(-a**7*x**7*sqrt(-a**2*x**2 + 1) + 5*a**6*x**6*sqrt(-a**2 *x**2 + 1) - 9*a**5*x**5*sqrt(-a**2*x**2 + 1) + 5*a**4*x**4*sqrt(-a**2*x** 2 + 1) + 5*a**3*x**3*sqrt(-a**2*x**2 + 1) - 9*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(3*a **2*x**2/(-a**7*x**7*sqrt(-a**2*x**2 + 1) + 5*a**6*x**6*sqrt(-a**2*x**2 + 1) - 9*a**5*x**5*sqrt(-a**2*x**2 + 1) + 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 5*a**3*x**3*sqrt(-a**2*x**2 + 1) - 9*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(a**3*x**3/( -a**7*x**7*sqrt(-a**2*x**2 + 1) + 5*a**6*x**6*sqrt(-a**2*x**2 + 1) - 9*a** 5*x**5*sqrt(-a**2*x**2 + 1) + 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 5*a**3*x* *3*sqrt(-a**2*x**2 + 1) - 9*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**7*x**7*sqrt(-a **2*x**2 + 1) + 5*a**6*x**6*sqrt(-a**2*x**2 + 1) - 9*a**5*x**5*sqrt(-a**2* x**2 + 1) + 5*a**4*x**4*sqrt(-a**2*x**2 + 1) + 5*a**3*x**3*sqrt(-a**2*x**2 + 1) - 9*a**2*x**2*sqrt(-a**2*x**2 + 1) + 5*a*x*sqrt(-a**2*x**2 + 1) - sq rt(-a**2*x**2 + 1)), x))/c**5
Leaf count of result is larger than twice the leaf count of optimal. 462 vs. \(2 (109) = 218\).
Time = 0.04 (sec) , antiderivative size = 462, normalized size of antiderivative = 3.58 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{(c-a c x)^5} \, dx=-\frac {8}{11 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{6} c^{5} x^{5} - 5 \, \sqrt {-a^{2} x^{2} + 1} a^{5} c^{5} x^{4} + 10 \, \sqrt {-a^{2} x^{2} + 1} a^{4} c^{5} x^{3} - 10 \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{5} x^{2} + 5 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{5} x - \sqrt {-a^{2} x^{2} + 1} a c^{5}\right )}} - \frac {28}{33 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{5} c^{5} x^{4} - 4 \, \sqrt {-a^{2} x^{2} + 1} a^{4} c^{5} x^{3} + 6 \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{5} x^{2} - 4 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{5} x + \sqrt {-a^{2} x^{2} + 1} a c^{5}\right )}} - \frac {58}{231 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{4} c^{5} x^{3} - 3 \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{5} x^{2} + 3 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{5} x - \sqrt {-a^{2} x^{2} + 1} a c^{5}\right )}} + \frac {1}{1155 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{3} c^{5} x^{2} - 2 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{5} x + \sqrt {-a^{2} x^{2} + 1} a c^{5}\right )}} - \frac {1}{1155 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{2} c^{5} x - \sqrt {-a^{2} x^{2} + 1} a c^{5}\right )}} + \frac {2 \, x}{1155 \, \sqrt {-a^{2} x^{2} + 1} c^{5}} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^5,x, algorithm="maxima")
Output:
-8/11/(sqrt(-a^2*x^2 + 1)*a^6*c^5*x^5 - 5*sqrt(-a^2*x^2 + 1)*a^5*c^5*x^4 + 10*sqrt(-a^2*x^2 + 1)*a^4*c^5*x^3 - 10*sqrt(-a^2*x^2 + 1)*a^3*c^5*x^2 + 5 *sqrt(-a^2*x^2 + 1)*a^2*c^5*x - sqrt(-a^2*x^2 + 1)*a*c^5) - 28/33/(sqrt(-a ^2*x^2 + 1)*a^5*c^5*x^4 - 4*sqrt(-a^2*x^2 + 1)*a^4*c^5*x^3 + 6*sqrt(-a^2*x ^2 + 1)*a^3*c^5*x^2 - 4*sqrt(-a^2*x^2 + 1)*a^2*c^5*x + sqrt(-a^2*x^2 + 1)* a*c^5) - 58/231/(sqrt(-a^2*x^2 + 1)*a^4*c^5*x^3 - 3*sqrt(-a^2*x^2 + 1)*a^3 *c^5*x^2 + 3*sqrt(-a^2*x^2 + 1)*a^2*c^5*x - sqrt(-a^2*x^2 + 1)*a*c^5) + 1/ 1155/(sqrt(-a^2*x^2 + 1)*a^3*c^5*x^2 - 2*sqrt(-a^2*x^2 + 1)*a^2*c^5*x + sq rt(-a^2*x^2 + 1)*a*c^5) - 1/1155/(sqrt(-a^2*x^2 + 1)*a^2*c^5*x - sqrt(-a^2 *x^2 + 1)*a*c^5) + 2/1155*x/(sqrt(-a^2*x^2 + 1)*c^5)
Result contains complex when optimal does not.
Time = 0.49 (sec) , antiderivative size = 509, normalized size of antiderivative = 3.95 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{(c-a c x)^5} \, dx=\frac {\frac {48 i \, \mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right )}{c^{3}} + \frac {\frac {5 \, {\left (63 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{5} \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 385 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{4} \sqrt {-\frac {2 \, c}{a c x - c} - 1} + 990 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{3} \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 1386 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{2} \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 1155 \, {\left (-\frac {2 \, c}{a c x - c} - 1\right )}^{\frac {3}{2}} - 693 \, \sqrt {-\frac {2 \, c}{a c x - c} - 1}\right )}}{c^{3}} + \frac {22 \, {\left (35 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{4} \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 180 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{3} \sqrt {-\frac {2 \, c}{a c x - c} - 1} + 378 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{2} \sqrt {-\frac {2 \, c}{a c x - c} - 1} + 420 \, {\left (-\frac {2 \, c}{a c x - c} - 1\right )}^{\frac {3}{2}} + 315 \, \sqrt {-\frac {2 \, c}{a c x - c} - 1}\right )}}{c^{3}} + \frac {99 \, {\left (5 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{3} \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 21 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{2} \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 35 \, {\left (-\frac {2 \, c}{a c x - c} - 1\right )}^{\frac {3}{2}} - 35 \, \sqrt {-\frac {2 \, c}{a c x - c} - 1}\right )}}{c^{3}}}{\mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right )}}{27720 \, c^{2} {\left | a \right |}} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^5,x, algorithm="giac")
Output:
1/27720*(48*I*sgn(1/(a*c*x - c))*sgn(a)*sgn(c)/c^3 + (5*(63*(2*c/(a*c*x - c) + 1)^5*sqrt(-2*c/(a*c*x - c) - 1) - 385*(2*c/(a*c*x - c) + 1)^4*sqrt(-2 *c/(a*c*x - c) - 1) + 990*(2*c/(a*c*x - c) + 1)^3*sqrt(-2*c/(a*c*x - c) - 1) - 1386*(2*c/(a*c*x - c) + 1)^2*sqrt(-2*c/(a*c*x - c) - 1) - 1155*(-2*c/ (a*c*x - c) - 1)^(3/2) - 693*sqrt(-2*c/(a*c*x - c) - 1))/c^3 + 22*(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 + 99*(5*(2*c/(a*c*x - c) + 1)^3*sqrt(-2*c/(a*c*x - c) - 1) - 2 1*(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 = 14.22 (sec) , antiderivative size = 604, normalized size of antiderivative = 4.68 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{(c-a c x)^5} \, dx=\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {32\,a^6}{693\,c^5\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}-\frac {16\,a^6}{231\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^3}+\frac {20\,a^6}{99\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^5}+\frac {4\,a^7}{11\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^6\,\sqrt {-a^2}}+\frac {80\,a^9}{693\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^4\,{\left (-a^2\right )}^{3/2}}+\frac {32\,a^{11}}{693\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^2\,{\left (-a^2\right )}^{5/2}}\right )}{a^6\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {32\,a^5}{315\,c^5\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}-\frac {16\,a^5}{105\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^3}+\frac {4\,a^5}{9\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^5}-\frac {16\,a^2\,{\left (-a^2\right )}^{3/2}}{63\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^4}+\frac {32\,a^6}{315\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^2\,\sqrt {-a^2}}\right )}{a^5\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {3\,a^4}{35\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^3}-\frac {2\,a^4}{35\,c^5\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}+\frac {a^5}{7\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^4\,\sqrt {-a^2}}+\frac {2\,a^7}{35\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^2\,{\left (-a^2\right )}^{3/2}}\right )}{a^4\,\sqrt {-a^2}} \] Input:
int((a*x + 1)^3/((1 - a^2*x^2)^(3/2)*(c - a*c*x)^5),x)
Output:
((1 - a^2*x^2)^(1/2)*((32*a^6)/(693*c^5*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)) - (16*a^6)/(231*c^5*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)^3) + (20*a^6)/(99*c ^5*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)^5) + (4*a^7)/(11*c^5*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)^6*(-a^2)^(1/2)) + (80*a^9)/(693*c^5*(x*(-a^2)^(1/2) - (- a^2)^(1/2)/a)^4*(-a^2)^(3/2)) + (32*a^11)/(693*c^5*(x*(-a^2)^(1/2) - (-a^2 )^(1/2)/a)^2*(-a^2)^(5/2))))/(a^6*(-a^2)^(1/2)) - ((1 - a^2*x^2)^(1/2)*((3 2*a^5)/(315*c^5*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)) - (16*a^5)/(105*c^5*(x* (-a^2)^(1/2) - (-a^2)^(1/2)/a)^3) + (4*a^5)/(9*c^5*(x*(-a^2)^(1/2) - (-a^2 )^(1/2)/a)^5) - (16*a^2*(-a^2)^(3/2))/(63*c^5*(x*(-a^2)^(1/2) - (-a^2)^(1/ 2)/a)^4) + (32*a^6)/(315*c^5*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)^2*(-a^2)^(1 /2))))/(a^5*(-a^2)^(1/2)) - ((1 - a^2*x^2)^(1/2)*((3*a^4)/(35*c^5*(x*(-a^2 )^(1/2) - (-a^2)^(1/2)/a)^3) - (2*a^4)/(35*c^5*(x*(-a^2)^(1/2) - (-a^2)^(1 /2)/a)) + a^5/(7*c^5*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)^4*(-a^2)^(1/2)) + ( 2*a^7)/(35*c^5*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)^2*(-a^2)^(3/2))))/(a^4*(- a^2)^(1/2))
Time = 0.16 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.39 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{(c-a c x)^5} \, dx=\frac {-60 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+302 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-611 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+626 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-47 \sqrt {-a^{2} x^{2}+1}\, a x +210 \sqrt {-a^{2} x^{2}+1}-56 a^{6} x^{6}+338 a^{5} x^{5}-851 a^{4} x^{4}+1145 a^{3} x^{3}-1159 a^{2} x^{2}-47 a x -210}{1155 a \,c^{5} \left (\sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}-5 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+10 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-10 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+5 \sqrt {-a^{2} x^{2}+1}\, a x -\sqrt {-a^{2} x^{2}+1}+a^{6} x^{6}-6 a^{5} x^{5}+15 a^{4} x^{4}-20 a^{3} x^{3}+15 a^{2} x^{2}-6 a x +1\right )} \] Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a*c*x+c)^5,x)
Output:
( - 60*sqrt( - a**2*x**2 + 1)*a**5*x**5 + 302*sqrt( - a**2*x**2 + 1)*a**4* x**4 - 611*sqrt( - a**2*x**2 + 1)*a**3*x**3 + 626*sqrt( - a**2*x**2 + 1)*a **2*x**2 - 47*sqrt( - a**2*x**2 + 1)*a*x + 210*sqrt( - a**2*x**2 + 1) - 56 *a**6*x**6 + 338*a**5*x**5 - 851*a**4*x**4 + 1145*a**3*x**3 - 1159*a**2*x* *2 - 47*a*x - 210)/(1155*a*c**5*(sqrt( - a**2*x**2 + 1)*a**5*x**5 - 5*sqrt ( - a**2*x**2 + 1)*a**4*x**4 + 10*sqrt( - a**2*x**2 + 1)*a**3*x**3 - 10*sq rt( - a**2*x**2 + 1)*a**2*x**2 + 5*sqrt( - a**2*x**2 + 1)*a*x - sqrt( - a* *2*x**2 + 1) + a**6*x**6 - 6*a**5*x**5 + 15*a**4*x**4 - 20*a**3*x**3 + 15* a**2*x**2 - 6*a*x + 1))