Integrand size = 17, antiderivative size = 88 \[ \int e^{-6 \text {arctanh}(a x)} (c+a c x)^p \, dx=-\frac {8 c^2 (c+a c x)^{-2+p}}{a (2-p)}+\frac {12 c (c+a c x)^{-1+p}}{a (1-p)}+\frac {6 (c+a c x)^p}{a p}-\frac {(c+a c x)^{1+p}}{a c (1+p)} \] 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 = 0.04 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.14 \[ \int e^{-6 \text {arctanh}(a x)} (c+a c x)^p \, dx=\frac {(c+a c x)^p \left (-p^3 (-1+a x)^3+3 p^2 (-1+a x)^2 (1+a x)+12 (1+a x)^2-2 p \left (-4-3 a x+6 a^2 x^2+a^3 x^3\right )\right )}{a (-2+p) (-1+p) p (1+p) (1+a x)^2} \] Input:
Integrate[(c + a*c*x)^p/E^(6*ArcTanh[a*x]),x]
Output:
((c + a*c*x)^p*(-(p^3*(-1 + a*x)^3) + 3*p^2*(-1 + a*x)^2*(1 + a*x) + 12*(1 + a*x)^2 - 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 = 0.31 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, 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.
\(\displaystyle \int e^{-6 \text {arctanh}(a x)} (a c x+c)^p \, dx\) |
\(\Big \downarrow \) 6680 |
\(\displaystyle \int \frac {(1-a x)^3 (a c x+c)^p}{(a x+1)^3}dx\) |
\(\Big \downarrow \) 35 |
\(\displaystyle c^3 \int (1-a x)^3 (a x c+c)^{p-3}dx\) |
\(\Big \downarrow \) 53 |
\(\displaystyle c^3 \int \left (8 (a x c+c)^{p-3}-\frac {12 (a x c+c)^{p-2}}{c}+\frac {6 (a x c+c)^{p-1}}{c^2}-\frac {(a x c+c)^p}{c^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle c^3 \left (-\frac {(a c x+c)^{p+1}}{a c^4 (p+1)}+\frac {6 (a c x+c)^p}{a c^3 p}+\frac {12 (a c x+c)^{p-1}}{a c^2 (1-p)}-\frac {8 (a c x+c)^{p-2}}{a c (2-p)}\right )\) |
Input:
Int[(c + a*c*x)^p/E^(6*ArcTanh[a*x]),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 = 0.22 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.66
method | result | size |
gosper | \(-\frac {\left (a c x +c \right )^{p} \left (a^{3} p^{3} x^{3}-3 a^{3} p^{2} x^{3}+2 a^{3} x^{3} p -3 a^{2} p^{3} x^{2}+3 a^{2} p^{2} x^{2}+12 a^{2} x^{2} p +3 a \,p^{3} x -12 a^{2} x^{2}+3 a \,p^{2} x -6 a p x -p^{3}-24 a x -3 p^{2}-8 p -12\right )}{\left (p^{3}-2 p^{2}-p +2\right ) a p \left (a x +1\right )^{2}}\) | \(146\) |
risch | \(-\frac {\left (a^{3} p^{3} x^{3}-3 a^{3} p^{2} x^{3}+2 a^{3} x^{3} p -3 a^{2} p^{3} x^{2}+3 a^{2} p^{2} x^{2}+12 a^{2} x^{2} p +3 a \,p^{3} x -12 a^{2} x^{2}+3 a \,p^{2} x -6 a p x -p^{3}-24 a x -3 p^{2}-8 p -12\right ) \left (a c x +c \right )^{p}}{a p \left (p +1\right ) \left (p -1\right ) \left (a x +1\right )^{2} \left (-2+p \right )}\) | \(146\) |
orering | \(\frac {\left (a^{3} p^{3} x^{3}-3 a^{3} p^{2} x^{3}+2 a^{3} x^{3} p -3 a^{2} p^{3} x^{2}+3 a^{2} p^{2} x^{2}+12 a^{2} x^{2} p +3 a \,p^{3} x -12 a^{2} x^{2}+3 a \,p^{2} x -6 a p x -p^{3}-24 a x -3 p^{2}-8 p -12\right ) \left (a c x +c \right )^{p} \left (-a^{2} x^{2}+1\right )^{3}}{a p \left (p^{3}-2 p^{2}-p +2\right ) \left (a x +1\right )^{5} \left (a x -1\right )^{3}}\) | \(164\) |
parallelrisch | \(-\frac {-3 \left (a c x +c \right )^{p} p^{2}-\left (a c x +c \right )^{p} p^{3}+3 x^{2} \left (a c x +c \right )^{p} a^{2} p^{2}+12 x^{2} \left (a c x +c \right )^{p} a^{2} p -6 a \left (a c x +c \right )^{p} x p -12 \left (a c x +c \right )^{p}+3 x \left (a c x +c \right )^{p} a \,p^{2}-24 a \left (a c x +c \right )^{p} x -12 x^{2} \left (a c x +c \right )^{p} a^{2}+x^{3} \left (a c x +c \right )^{p} a^{3} p^{3}-3 x^{3} \left (a c x +c \right )^{p} a^{3} p^{2}+2 x^{3} \left (a c x +c \right )^{p} a^{3} p -3 x^{2} \left (a c x +c \right )^{p} a^{2} p^{3}-8 \left (a c x +c \right )^{p} p +3 x \left (a c x +c \right )^{p} a \,p^{3}}{\left (a x +1\right )^{2} a p \left (p^{3}-2 p^{2}-p +2\right )}\) | \(259\) |
norman | \(\frac {\frac {\left (p^{3}+3 p^{2}+8 p +12\right ) {\mathrm e}^{p \ln \left (a c x +c \right )}}{a p \left (p^{3}-2 p^{2}-p +2\right )}-\frac {a^{5} x^{6} {\mathrm e}^{p \ln \left (a c x +c \right )}}{p +1}+\frac {6 a^{4} x^{5} {\mathrm e}^{p \ln \left (a c x +c \right )}}{p \left (p +1\right )}+\frac {6 \left (p^{2}+5 p +10\right ) x \,{\mathrm e}^{p \ln \left (a c x +c \right )}}{p \left (p^{3}-2 p^{2}-p +2\right )}-\frac {3 a \left (p^{3}+p^{2}-10 p -40\right ) x^{2} {\mathrm e}^{p \ln \left (a c x +c \right )}}{p \left (p^{3}-2 p^{2}-p +2\right )}-\frac {12 a^{2} \left (p^{2}+p -10\right ) x^{3} {\mathrm e}^{p \ln \left (a c x +c \right )}}{p \left (p^{3}-2 p^{2}-p +2\right )}+\frac {3 a^{3} \left (p^{2}+p -10\right ) x^{4} {\mathrm e}^{p \ln \left (a c x +c \right )}}{p \left (p^{2}-1\right )}}{\left (a x +1\right )^{5}}\) | \(264\) |
Input:
int((a*c*x+c)^p/(a*x+1)^6*(-a^2*x^2+1)^3,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)/(p^3-2*p^2-p+2)/a/p/(a*x+1)^2
Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (84) = 168\).
Time = 0.09 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.22 \[ \int e^{-6 \text {arctanh}(a x)} (c+a c x)^p \, dx=-\frac {{\left ({\left (a^{3} p^{3} - 3 \, a^{3} p^{2} + 2 \, a^{3} p\right )} x^{3} - p^{3} - 3 \, {\left (a^{2} p^{3} - a^{2} p^{2} - 4 \, a^{2} p + 4 \, a^{2}\right )} x^{2} - 3 \, p^{2} + 3 \, {\left (a p^{3} + a p^{2} - 2 \, a p - 8 \, a\right )} x - 8 \, p - 12\right )} {\left (a c x + c\right )}^{p}}{a p^{4} - 2 \, a p^{3} - a p^{2} + {\left (a^{3} p^{4} - 2 \, a^{3} p^{3} - a^{3} p^{2} + 2 \, a^{3} p\right )} x^{2} + 2 \, a p + 2 \, {\left (a^{2} p^{4} - 2 \, a^{2} p^{3} - a^{2} p^{2} + 2 \, a^{2} p\right )} x} \] Input:
integrate((a*c*x+c)^p/(a*x+1)^6*(-a^2*x^2+1)^3,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 = 1.86 (sec) , antiderivative size = 2426, normalized size of antiderivative = 27.57 \[ \int e^{-6 \text {arctanh}(a x)} (c+a c x)^p \, dx=\text {Too large to display} \] Input:
integrate((a*c*x+c)**p/(a*x+1)**6*(-a**2*x**2+1)**3,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) + 3*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) + 21*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) + 13/(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) - 30*a*c*x/(2*a* *2*x + 2*a) - 24*c*log(x + 1/a)/(2*a**2*x + 2*a) - 56*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...
Leaf count of result is larger than twice the leaf count of optimal. 928 vs. \(2 (84) = 168\).
Time = 0.09 (sec) , antiderivative size = 928, normalized size of antiderivative = 10.55 \[ \int e^{-6 \text {arctanh}(a x)} (c+a c x)^p \, dx=\text {Too large to display} \] Input:
integrate((a*c*x+c)^p/(a*x+1)^6*(-a^2*x^2+1)^3,x, algorithm="maxima")
Output:
-((p^6 - 15*p^5 + 85*p^4 - 225*p^3 + 274*p^2 - 120*p)*a^6*c^p*x^6 - 6*(p^5 - 15*p^4 + 85*p^3 - 225*p^2 + 274*p - 120)*a^5*c^p*x^5 + 30*(p^4 - 14*p^3 + 71*p^2 - 154*p + 120)*a^4*c^p*x^4 - 120*(p^3 - 12*p^2 + 47*p - 60)*a^3* c^p*x^3 + 360*(p^2 - 9*p + 20)*a^2*c^p*x^2 - 720*a*c^p*(p - 5)*x + 720*c^p )*(a*x + 1)^p*a^6/((p^7 - 14*p^6 + 70*p^5 - 140*p^4 + 49*p^3 + 154*p^2 - 1 20*p)*a^12*x^5 + 5*(p^7 - 14*p^6 + 70*p^5 - 140*p^4 + 49*p^3 + 154*p^2 - 1 20*p)*a^11*x^4 + 10*(p^7 - 14*p^6 + 70*p^5 - 140*p^4 + 49*p^3 + 154*p^2 - 120*p)*a^10*x^3 + 10*(p^7 - 14*p^6 + 70*p^5 - 140*p^4 + 49*p^3 + 154*p^2 - 120*p)*a^9*x^2 + 5*(p^7 - 14*p^6 + 70*p^5 - 140*p^4 + 49*p^3 + 154*p^2 - 120*p)*a^8*x + (p^7 - 14*p^6 + 70*p^5 - 140*p^4 + 49*p^3 + 154*p^2 - 120*p )*a^7) + 3*((p^4 - 14*p^3 + 71*p^2 - 154*p + 120)*a^4*c^p*x^4 - 4*(p^3 - 1 2*p^2 + 47*p - 60)*a^3*c^p*x^3 + 12*(p^2 - 9*p + 20)*a^2*c^p*x^2 - 24*a*c^ p*(p - 5)*x + 24*c^p)*(a*x + 1)^p*a^4/((p^5 - 15*p^4 + 85*p^3 - 225*p^2 + 274*p - 120)*a^10*x^5 + 5*(p^5 - 15*p^4 + 85*p^3 - 225*p^2 + 274*p - 120)* a^9*x^4 + 10*(p^5 - 15*p^4 + 85*p^3 - 225*p^2 + 274*p - 120)*a^8*x^3 + 10* (p^5 - 15*p^4 + 85*p^3 - 225*p^2 + 274*p - 120)*a^7*x^2 + 5*(p^5 - 15*p^4 + 85*p^3 - 225*p^2 + 274*p - 120)*a^6*x + (p^5 - 15*p^4 + 85*p^3 - 225*p^2 + 274*p - 120)*a^5) - 3*((p^2 - 9*p + 20)*a^2*c^p*x^2 - 2*a*c^p*(p - 5)*x + 2*c^p)*(a*x + 1)^p*a^2/((p^3 - 12*p^2 + 47*p - 60)*a^8*x^5 + 5*(p^3 - 1 2*p^2 + 47*p - 60)*a^7*x^4 + 10*(p^3 - 12*p^2 + 47*p - 60)*a^6*x^3 + 10...
\[ \int e^{-6 \text {arctanh}(a x)} (c+a c x)^p \, dx=\int { -\frac {{\left (a^{2} x^{2} - 1\right )}^{3} {\left (a c x + c\right )}^{p}}{{\left (a x + 1\right )}^{6}} \,d x } \] Input:
integrate((a*c*x+c)^p/(a*x+1)^6*(-a^2*x^2+1)^3,x, algorithm="giac")
Output:
integrate(-(a^2*x^2 - 1)^3*(a*c*x + c)^p/(a*x + 1)^6, x)
Time = 14.34 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.92 \[ \int e^{-6 \text {arctanh}(a x)} (c+a c x)^p \, dx=\frac {8\,{\left (c+a\,c\,x\right )}^p}{a\,{\left (a\,x+1\right )}^2\,\left (p-2\right )}-\frac {12\,{\left (c+a\,c\,x\right )}^p}{a\,\left (a\,x+1\right )\,\left (p-1\right )}+\frac {{\left (c+a\,c\,x\right )}^p\,\left (5\,p-a\,p\,x+6\right )}{a\,p\,\left (p+1\right )} \] Input:
int(-((a^2*x^2 - 1)^3*(c + a*c*x)^p)/(a*x + 1)^6,x)
Output:
(8*(c + a*c*x)^p)/(a*(a*x + 1)^2*(p - 2)) - (12*(c + a*c*x)^p)/(a*(a*x + 1 )*(p - 1)) + ((c + a*c*x)^p*(5*p - a*p*x + 6))/(a*p*(p + 1))
Time = 0.15 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.24 \[ \int e^{-6 \text {arctanh}(a x)} (c+a c x)^p \, dx=\frac {\left (a c x +c \right )^{p} \left (-a^{3} p^{3} x^{3}+3 a^{3} p^{2} x^{3}-2 a^{3} p \,x^{3}+3 a^{2} p^{3} x^{2}-3 a^{2} p^{2} x^{2}-12 a^{2} p \,x^{2}-3 a \,p^{3} x +12 a^{2} x^{2}-3 a \,p^{2} x +6 a p x +p^{3}+24 a x +3 p^{2}+8 p +12\right )}{a p \left (a^{2} p^{3} x^{2}-2 a^{2} p^{2} x^{2}-a^{2} p \,x^{2}+2 a \,p^{3} x +2 a^{2} x^{2}-4 a \,p^{2} x -2 a p x +p^{3}+4 a x -2 p^{2}-p +2\right )} \] Input:
int((a*c*x+c)^p/(a*x+1)^6*(-a^2*x^2+1)^3,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))