Integrand size = 18, antiderivative size = 91 \[ \int e^{6 \text {arctanh}(a x)} (c-a c x)^p \, dx=\frac {8 c^2 (c-a c x)^{-2+p}}{a (2-p)}-\frac {12 c (c-a c x)^{-1+p}}{a (1-p)}-\frac {6 (c-a c x)^p}{a p}+\frac {(c-a c x)^{1+p}}{a c (1+p)} \] Output:
8*c^2*(-a*c*x+c)^(-2+p)/a/(2-p)-12*c*(-a*c*x+c)^(-1+p)/a/(1-p)-6*(-a*c*x+c )^p/a/p+(-a*c*x+c)^(p+1)/a/c/(p+1)
Time = 0.14 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.05 \[ \int e^{6 \text {arctanh}(a x)} (c-a c x)^p \, dx=-\frac {(c-a c x)^p \left (12 (-1+a x)^2-3 (-1+a x) (p+a p x)^2+(p+a p x)^3+2 p \left (4-3 a x-6 a^2 x^2+a^3 x^3\right )\right )}{a (-2+p) (-1+p) p (1+p) (-1+a x)^2} \] Input:
Integrate[E^(6*ArcTanh[a*x])*(c - a*c*x)^p,x]
Output:
-(((c - a*c*x)^p*(12*(-1 + a*x)^2 - 3*(-1 + a*x)*(p + a*p*x)^2 + (p + a*p* x)^3 + 2*p*(4 - 3*a*x - 6*a^2*x^2 + a^3*x^3)))/(a*(-2 + p)*(-1 + p)*p*(1 + p)*(-1 + a*x)^2))
Time = 0.49 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6680, 35, 53, 2009}
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.
\(\displaystyle \int e^{6 \text {arctanh}(a x)} (c-a c x)^p \, dx\) |
\(\Big \downarrow \) 6680 |
\(\displaystyle \int \frac {(a x+1)^3 (c-a c x)^p}{(1-a x)^3}dx\) |
\(\Big \downarrow \) 35 |
\(\displaystyle c^3 \int (a x+1)^3 (c-a c x)^{p-3}dx\) |
\(\Big \downarrow \) 53 |
\(\displaystyle c^3 \int \left (8 (c-a c x)^{p-3}-\frac {12 (c-a c x)^{p-2}}{c}+\frac {6 (c-a c x)^{p-1}}{c^2}-\frac {(c-a c x)^p}{c^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle c^3 \left (\frac {(c-a c x)^{p+1}}{a c^4 (p+1)}-\frac {6 (c-a c x)^p}{a c^3 p}-\frac {12 (c-a c x)^{p-1}}{a c^2 (1-p)}+\frac {8 (c-a c x)^{p-2}}{a c (2-p)}\right )\) |
Input:
Int[E^(6*ArcTanh[a*x])*(c - a*c*x)^p,x]
Output:
c^3*((8*(c - a*c*x)^(-2 + p))/(a*c*(2 - p)) - (12*(c - a*c*x)^(-1 + p))/(a *c^2*(1 - p)) - (6*(c - a*c*x)^p)/(a*c^3*p) + (c - a*c*x)^(1 + p)/(a*c^4*( 1 + p)))
Int[(u_.)*((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.), x_Symbol] :> Simp[(b/d)^m Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n} , x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && !(IntegerQ[n] && SimplerQ[a + b*x, c + d*x])
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol ] :> Int[u*(c + d*x)^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c , d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && !(IntegerQ[p] || GtQ[c, 0])
Time = 0.14 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.59
method | result | size |
gosper | \(-\frac {\left (-a c x +c \right )^{p} \left (a^{3} p^{3} x^{3}-3 a^{3} p^{2} x^{3}+2 a^{3} x^{3} p +3 a^{2} p^{3} x^{2}-3 a^{2} p^{2} x^{2}-12 a^{2} x^{2} p +3 a \,p^{3} x +12 a^{2} x^{2}+3 a \,p^{2} x -6 a p x +p^{3}-24 a x +3 p^{2}+8 p +12\right )}{\left (a x -1\right )^{2} a p \left (p^{3}-2 p^{2}-p +2\right )}\) | \(145\) |
risch | \(-\frac {\left (a^{3} p^{3} x^{3}-3 a^{3} p^{2} x^{3}+2 a^{3} x^{3} p +3 a^{2} p^{3} x^{2}-3 a^{2} p^{2} x^{2}-12 a^{2} x^{2} p +3 a \,p^{3} x +12 a^{2} x^{2}+3 a \,p^{2} x -6 a p x +p^{3}-24 a x +3 p^{2}+8 p +12\right ) \left (-a c x +c \right )^{p}}{a p \left (p +1\right ) \left (p -1\right ) \left (a x -1\right )^{2} \left (-2+p \right )}\) | \(145\) |
orering | \(\frac {\left (a^{3} p^{3} x^{3}-3 a^{3} p^{2} x^{3}+2 a^{3} x^{3} p +3 a^{2} p^{3} x^{2}-3 a^{2} p^{2} x^{2}-12 a^{2} x^{2} p +3 a \,p^{3} x +12 a^{2} x^{2}+3 a \,p^{2} x -6 a p x +p^{3}-24 a x +3 p^{2}+8 p +12\right ) \left (a x +1\right )^{3} \left (a x -1\right ) \left (-a c x +c \right )^{p}}{a p \left (p^{3}-2 p^{2}-p +2\right ) \left (-a^{2} x^{2}+1\right )^{3}}\) | \(161\) |
norman | \(\frac {-\frac {a^{4} x^{5} {\mathrm e}^{p \ln \left (-a c x +c \right )}}{p +1}-\frac {\left (5 p^{2}+9 p +10\right ) x \,{\mathrm e}^{p \ln \left (-a c x +c \right )}}{p^{3}-2 p^{2}-p +2}-\frac {2 a^{2} \left (5 p +7\right ) x^{3} {\mathrm e}^{p \ln \left (-a c x +c \right )}}{p^{2}-1}-\frac {\left (p^{3}+3 p^{2}+8 p +12\right ) {\mathrm e}^{p \ln \left (-a c x +c \right )}}{a p \left (p^{3}-2 p^{2}-p +2\right )}-\frac {2 a \left (5 p^{3}+3 p^{2}-8 p -12\right ) x^{2} {\mathrm e}^{p \ln \left (-a c x +c \right )}}{p \left (p^{3}-2 p^{2}-p +2\right )}-\frac {a^{3} \left (5 p +6\right ) x^{4} {\mathrm e}^{p \ln \left (-a c x +c \right )}}{p \left (p +1\right )}}{\left (a^{2} x^{2}-1\right )^{2}}\) | \(237\) |
parallelrisch | \(-\frac {3 \left (-a c x +c \right )^{p} p^{2}+8 \left (-a c x +c \right )^{p} p +12 \left (-a c x +c \right )^{p}+2 x^{3} \left (-a c x +c \right )^{p} a^{3} p +3 x^{2} \left (-a c x +c \right )^{p} a^{2} p^{3}-3 x^{2} \left (-a c x +c \right )^{p} a^{2} p^{2}-12 x^{2} \left (-a c x +c \right )^{p} a^{2} p +3 x \left (-a c x +c \right )^{p} a \,p^{3}+3 x \left (-a c x +c \right )^{p} a \,p^{2}-6 x \left (-a c x +c \right )^{p} a p +x^{3} \left (-a c x +c \right )^{p} a^{3} p^{3}-3 x^{3} \left (-a c x +c \right )^{p} a^{3} p^{2}+12 x^{2} \left (-a c x +c \right )^{p} a^{2}+\left (-a c x +c \right )^{p} p^{3}-24 x \left (-a c x +c \right )^{p} a}{\left (a x -1\right )^{2} a p \left (p^{3}-2 p^{2}-p +2\right )}\) | \(273\) |
Input:
int((a*x+1)^6/(-a^2*x^2+1)^3*(-a*c*x+c)^p,x,method=_RETURNVERBOSE)
Output:
-(-a*c*x+c)^p*(a^3*p^3*x^3-3*a^3*p^2*x^3+2*a^3*p*x^3+3*a^2*p^3*x^2-3*a^2*p ^2*x^2-12*a^2*p*x^2+3*a*p^3*x+12*a^2*x^2+3*a*p^2*x-6*a*p*x+p^3-24*a*x+3*p^ 2+8*p+12)/(a*x-1)^2/a/p/(p^3-2*p^2-p+2)
Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (87) = 174\).
Time = 0.08 (sec) , antiderivative size = 194, normalized size of antiderivative = 2.13 \[ \int e^{6 \text {arctanh}(a x)} (c-a c x)^p \, dx=-\frac {{\left ({\left (a^{3} p^{3} - 3 \, a^{3} p^{2} + 2 \, a^{3} p\right )} x^{3} + p^{3} + 3 \, {\left (a^{2} p^{3} - a^{2} p^{2} - 4 \, a^{2} p + 4 \, a^{2}\right )} x^{2} + 3 \, p^{2} + 3 \, {\left (a p^{3} + a p^{2} - 2 \, a p - 8 \, a\right )} x + 8 \, p + 12\right )} {\left (-a c x + c\right )}^{p}}{a p^{4} - 2 \, a p^{3} - a p^{2} + {\left (a^{3} p^{4} - 2 \, a^{3} p^{3} - a^{3} p^{2} + 2 \, a^{3} p\right )} x^{2} + 2 \, a p - 2 \, {\left (a^{2} p^{4} - 2 \, a^{2} p^{3} - a^{2} p^{2} + 2 \, a^{2} p\right )} x} \] Input:
integrate((a*x+1)^6/(-a^2*x^2+1)^3*(-a*c*x+c)^p,x, algorithm="fricas")
Output:
-((a^3*p^3 - 3*a^3*p^2 + 2*a^3*p)*x^3 + p^3 + 3*(a^2*p^3 - a^2*p^2 - 4*a^2 *p + 4*a^2)*x^2 + 3*p^2 + 3*(a*p^3 + a*p^2 - 2*a*p - 8*a)*x + 8*p + 12)*(- a*c*x + c)^p/(a*p^4 - 2*a*p^3 - a*p^2 + (a^3*p^4 - 2*a^3*p^3 - a^3*p^2 + 2 *a^3*p)*x^2 + 2*a*p - 2*(a^2*p^4 - 2*a^2*p^3 - a^2*p^2 + 2*a^2*p)*x)
Leaf count of result is larger than twice the leaf count of optimal. 2426 vs. \(2 (68) = 136\).
Time = 4.79 (sec) , antiderivative size = 2426, normalized size of antiderivative = 26.66 \[ \int e^{6 \text {arctanh}(a x)} (c-a c x)^p \, dx=\text {Too large to display} \] Input:
integrate((a*x+1)**6/(-a**2*x**2+1)**3*(-a*c*x+c)**p,x)
Output:
Piecewise((c**p*x, Eq(a, 0)), (3*a**3*x**3*log(x - 1/a)/(3*a**4*c*x**3 - 9 *a**3*c*x**2 + 9*a**2*c*x - 3*a*c) - 9*a**2*x**2*log(x - 1/a)/(3*a**4*c*x* *3 - 9*a**3*c*x**2 + 9*a**2*c*x - 3*a*c) - 18*a**2*x**2/(3*a**4*c*x**3 - 9 *a**3*c*x**2 + 9*a**2*c*x - 3*a*c) + 9*a*x*log(x - 1/a)/(3*a**4*c*x**3 - 9 *a**3*c*x**2 + 9*a**2*c*x - 3*a*c) + 18*a*x/(3*a**4*c*x**3 - 9*a**3*c*x**2 + 9*a**2*c*x - 3*a*c) - 3*log(x - 1/a)/(3*a**4*c*x**3 - 9*a**3*c*x**2 + 9 *a**2*c*x - 3*a*c) - 8/(3*a**4*c*x**3 - 9*a**3*c*x**2 + 9*a**2*c*x - 3*a*c ), Eq(p, -1)), (-a**3*x**3/(a**3*x**2 - 2*a**2*x + a) - 6*a**2*x**2*log(x - 1/a)/(a**3*x**2 - 2*a**2*x + a) + 2*a**2*x**2/(a**3*x**2 - 2*a**2*x + a) + 12*a*x*log(x - 1/a)/(a**3*x**2 - 2*a**2*x + a) + 11*a*x/(a**3*x**2 - 2* a**2*x + a) - 6*log(x - 1/a)/(a**3*x**2 - 2*a**2*x + a) - 8/(a**3*x**2 - 2 *a**2*x + a), Eq(p, 0)), (a**3*c*x**3/(2*a**2*x - 2*a) + 9*a**2*c*x**2/(2* a**2*x - 2*a) + 24*a*c*x*log(x - 1/a)/(2*a**2*x - 2*a) - 8*a*c*x/(2*a**2*x - 2*a) - 24*c*log(x - 1/a)/(2*a**2*x - 2*a) - 18*c/(2*a**2*x - 2*a), Eq(p , 1)), (-a**2*c**2*x**3/3 - 2*a*c**2*x**2 - 7*c**2*x - 8*c**2*log(x - 1/a) /a, Eq(p, 2)), (-a**3*p**3*x**3*(-a*c*x + c)**p/(a**3*p**4*x**2 - 2*a**3*p **3*x**2 - a**3*p**2*x**2 + 2*a**3*p*x**2 - 2*a**2*p**4*x + 4*a**2*p**3*x + 2*a**2*p**2*x - 4*a**2*p*x + a*p**4 - 2*a*p**3 - a*p**2 + 2*a*p) + 3*a** 3*p**2*x**3*(-a*c*x + c)**p/(a**3*p**4*x**2 - 2*a**3*p**3*x**2 - a**3*p**2 *x**2 + 2*a**3*p*x**2 - 2*a**2*p**4*x + 4*a**2*p**3*x + 2*a**2*p**2*x -...
Time = 0.08 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.77 \[ \int e^{6 \text {arctanh}(a x)} (c-a c x)^p \, dx=-\frac {{\left ({\left (p^{3} - 3 \, p^{2} + 2 \, p\right )} a^{3} c^{p} x^{3} + 3 \, {\left (p^{3} - p^{2} - 4 \, p + 4\right )} a^{2} c^{p} x^{2} + 3 \, {\left (p^{3} + p^{2} - 2 \, p - 8\right )} a c^{p} x + {\left (p^{3} + 3 \, p^{2} + 8 \, p + 12\right )} c^{p}\right )} {\left (-a x + 1\right )}^{p}}{{\left (p^{4} - 2 \, p^{3} - p^{2} + 2 \, p\right )} a^{3} x^{2} - 2 \, {\left (p^{4} - 2 \, p^{3} - p^{2} + 2 \, p\right )} a^{2} x + {\left (p^{4} - 2 \, p^{3} - p^{2} + 2 \, p\right )} a} \] Input:
integrate((a*x+1)^6/(-a^2*x^2+1)^3*(-a*c*x+c)^p,x, algorithm="maxima")
Output:
-((p^3 - 3*p^2 + 2*p)*a^3*c^p*x^3 + 3*(p^3 - p^2 - 4*p + 4)*a^2*c^p*x^2 + 3*(p^3 + p^2 - 2*p - 8)*a*c^p*x + (p^3 + 3*p^2 + 8*p + 12)*c^p)*(-a*x + 1) ^p/((p^4 - 2*p^3 - p^2 + 2*p)*a^3*x^2 - 2*(p^4 - 2*p^3 - p^2 + 2*p)*a^2*x + (p^4 - 2*p^3 - p^2 + 2*p)*a)
\[ \int e^{6 \text {arctanh}(a x)} (c-a c x)^p \, dx=\int { -\frac {{\left (a x + 1\right )}^{6} {\left (-a c x + c\right )}^{p}}{{\left (a^{2} x^{2} - 1\right )}^{3}} \,d x } \] Input:
integrate((a*x+1)^6/(-a^2*x^2+1)^3*(-a*c*x+c)^p,x, algorithm="giac")
Output:
integrate(-(a*x + 1)^6*(-a*c*x + c)^p/(a^2*x^2 - 1)^3, x)
Time = 23.53 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.92 \[ \int e^{6 \text {arctanh}(a x)} (c-a c x)^p \, dx=-\frac {12\,{\left (c-a\,c\,x\right )}^p}{a\,\left (a\,x-1\right )\,\left (p-1\right )}-\frac {8\,{\left (c-a\,c\,x\right )}^p}{a\,{\left (a\,x-1\right )}^2\,\left (p-2\right )}-\frac {{\left (c-a\,c\,x\right )}^p\,\left (5\,p+a\,p\,x+6\right )}{a\,p\,\left (p+1\right )} \] Input:
int(-((c - a*c*x)^p*(a*x + 1)^6)/(a^2*x^2 - 1)^3,x)
Output:
- (12*(c - a*c*x)^p)/(a*(a*x - 1)*(p - 1)) - (8*(c - a*c*x)^p)/(a*(a*x - 1 )^2*(p - 2)) - ((c - a*c*x)^p*(5*p + a*p*x + 6))/(a*p*(p + 1))
Time = 0.16 (sec) , antiderivative size = 200, normalized size of antiderivative = 2.20 \[ \int e^{6 \text {arctanh}(a x)} (c-a c x)^p \, dx=\frac {\left (-a c x +c \right )^{p} \left (-a^{3} p^{3} x^{3}+3 a^{3} p^{2} x^{3}-2 a^{3} p \,x^{3}-3 a^{2} p^{3} x^{2}+3 a^{2} p^{2} x^{2}+12 a^{2} p \,x^{2}-3 a \,p^{3} x -12 a^{2} x^{2}-3 a \,p^{2} x +6 a p x -p^{3}+24 a x -3 p^{2}-8 p -12\right )}{a p \left (a^{2} p^{3} x^{2}-2 a^{2} p^{2} x^{2}-a^{2} p \,x^{2}-2 a \,p^{3} x +2 a^{2} x^{2}+4 a \,p^{2} x +2 a p x +p^{3}-4 a x -2 p^{2}-p +2\right )} \] Input:
int((a*x+1)^6/(-a^2*x^2+1)^3*(-a*c*x+c)^p,x)
Output:
(( - a*c*x + c)**p*( - a**3*p**3*x**3 + 3*a**3*p**2*x**3 - 2*a**3*p*x**3 - 3*a**2*p**3*x**2 + 3*a**2*p**2*x**2 + 12*a**2*p*x**2 - 12*a**2*x**2 - 3*a *p**3*x - 3*a*p**2*x + 6*a*p*x + 24*a*x - p**3 - 3*p**2 - 8*p - 12))/(a*p* (a**2*p**3*x**2 - 2*a**2*p**2*x**2 - a**2*p*x**2 + 2*a**2*x**2 - 2*a*p**3* x + 4*a*p**2*x + 2*a*p*x - 4*a*x + p**3 - 2*p**2 - p + 2))