Integrand size = 22, antiderivative size = 63 \[ \int e^{4 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {2^{2+p} c (1+a x)^{1-p} \left (c-a^2 c x^2\right )^{-1+p} \operatorname {Hypergeometric2F1}\left (-2-p,-1+p,p,\frac {1}{2} (1-a x)\right )}{a (1-p)} \] Output:
2^(2+p)*c*(a*x+1)^(1-p)*(-a^2*c*x^2+c)^(-1+p)*hypergeom([-2-p, -1+p],[p],- 1/2*a*x+1/2)/a/(1-p)
Time = 0.02 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.14 \[ \int e^{4 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=-\frac {2^{2+p} (1-a x)^{-1+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-2-p,-1+p,p,\frac {1}{2} (1-a x)\right )}{a (-1+p)} \] Input:
Integrate[E^(4*ArcTanh[a*x])*(c - a^2*c*x^2)^p,x]
Output:
-((2^(2 + p)*(1 - a*x)^(-1 + p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[-2 - p , -1 + p, p, (1 - a*x)/2])/(a*(-1 + p)*(1 - a^2*x^2)^p))
Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6691, 473, 79}
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.
| \(\displaystyle \int e^{4 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx\) |
| \(\Big \downarrow \) 6691 |
| \(\displaystyle c^2 \int (a x+1)^4 \left (c-a^2 c x^2\right )^{p-2}dx\) |
| \(\Big \downarrow \) 473 |
| \(\displaystyle c^2 (a x+1)^{1-p} (c-a c x)^{1-p} \left (c-a^2 c x^2\right )^{p-1} \int (a x+1)^{p+2} (c-a c x)^{p-2}dx\) |
| \(\Big \downarrow \) 79 |
| \(\displaystyle \frac {c 2^{p+2} (a x+1)^{1-p} \left (c-a^2 c x^2\right )^{p-1} \operatorname {Hypergeometric2F1}\left (-p-2,p-1,p,\frac {1}{2} (1-a x)\right )}{a (1-p)}\) |
Input:
Int[E^(4*ArcTanh[a*x])*(c - a^2*c*x^2)^p,x]
Output:
(2^(2 + p)*c*(1 + a*x)^(1 - p)*(c - a^2*c*x^2)^(-1 + p)*Hypergeometric2F1[ -2 - p, -1 + p, p, (1 - a*x)/2])/(a*(1 - p))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ c^(n - 1)*((a + b*x^2)^(p + 1)/((1 + d*(x/c))^(p + 1)*(a/c + (b*x)/d)^(p + 1))) Int[(1 + d*(x/c))^(n + p)*(a/c + (b/d)*x)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[n] || GtQ[c, 0]) && !Gt Q[a, 0] && !(IntegerQ[n] && (IntegerQ[3*p] || IntegerQ[4*p]))
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[c^(n/2) Int[(c + d*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c , d, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ[c, 0]) && IGtQ[n/ 2, 0]
\[\int \frac {\left (a x +1\right )^{4} \left (-a^{2} c \,x^{2}+c \right )^{p}}{\left (-a^{2} x^{2}+1\right )^{2}}d x\]
Input:
int((a*x+1)^4/(-a^2*x^2+1)^2*(-a^2*c*x^2+c)^p,x)
Output:
int((a*x+1)^4/(-a^2*x^2+1)^2*(-a^2*c*x^2+c)^p,x)
\[ \int e^{4 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int { \frac {{\left (a x + 1\right )}^{4} {\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (a^{2} x^{2} - 1\right )}^{2}} \,d x } \] Input:
integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a^2*c*x^2+c)^p,x, algorithm="fricas")
Output:
integral((a^2*x^2 + 2*a*x + 1)*(-a^2*c*x^2 + c)^p/(a^2*x^2 - 2*a*x + 1), x )
\[ \int e^{4 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int \frac {\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{p} \left (a x + 1\right )^{2}}{\left (a x - 1\right )^{2}}\, dx \] Input:
integrate((a*x+1)**4/(-a**2*x**2+1)**2*(-a**2*c*x**2+c)**p,x)
Output:
Integral((-c*(a*x - 1)*(a*x + 1))**p*(a*x + 1)**2/(a*x - 1)**2, x)
\[ \int e^{4 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int { \frac {{\left (a x + 1\right )}^{4} {\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (a^{2} x^{2} - 1\right )}^{2}} \,d x } \] Input:
integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a^2*c*x^2+c)^p,x, algorithm="maxima")
Output:
integrate((a*x + 1)^4*(-a^2*c*x^2 + c)^p/(a^2*x^2 - 1)^2, x)
\[ \int e^{4 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int { \frac {{\left (a x + 1\right )}^{4} {\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (a^{2} x^{2} - 1\right )}^{2}} \,d x } \] Input:
integrate((a*x+1)^4/(-a^2*x^2+1)^2*(-a^2*c*x^2+c)^p,x, algorithm="giac")
Output:
integrate((a*x + 1)^4*(-a^2*c*x^2 + c)^p/(a^2*x^2 - 1)^2, x)
Timed out. \[ \int e^{4 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^p\,{\left (a\,x+1\right )}^4}{{\left (a^2\,x^2-1\right )}^2} \,d x \] Input:
int(((c - a^2*c*x^2)^p*(a*x + 1)^4)/(a^2*x^2 - 1)^2,x)
Output:
int(((c - a^2*c*x^2)^p*(a*x + 1)^4)/(a^2*x^2 - 1)^2, x)
\[ \int e^{4 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {\left (-a^{2} c \,x^{2}+c \right )^{p} a^{2} p \,x^{2}+3 \left (-a^{2} c \,x^{2}+c \right )^{p} a p x +2 \left (-a^{2} c \,x^{2}+c \right )^{p} a x -2 \left (-a^{2} c \,x^{2}+c \right )^{p} p^{2}-4 \left (-a^{2} c \,x^{2}+c \right )^{p} p -2 \left (-a^{2} c \,x^{2}+c \right )^{p}+8 \left (\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p} x}{2 a^{3} p \,x^{3}+a^{3} x^{3}-2 a^{2} p \,x^{2}-a^{2} x^{2}-2 a p x -a x +2 p +1}d x \right ) a^{3} p^{4} x +28 \left (\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p} x}{2 a^{3} p \,x^{3}+a^{3} x^{3}-2 a^{2} p \,x^{2}-a^{2} x^{2}-2 a p x -a x +2 p +1}d x \right ) a^{3} p^{3} x +28 \left (\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p} x}{2 a^{3} p \,x^{3}+a^{3} x^{3}-2 a^{2} p \,x^{2}-a^{2} x^{2}-2 a p x -a x +2 p +1}d x \right ) a^{3} p^{2} x +8 \left (\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p} x}{2 a^{3} p \,x^{3}+a^{3} x^{3}-2 a^{2} p \,x^{2}-a^{2} x^{2}-2 a p x -a x +2 p +1}d x \right ) a^{3} p x -8 \left (\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p} x}{2 a^{3} p \,x^{3}+a^{3} x^{3}-2 a^{2} p \,x^{2}-a^{2} x^{2}-2 a p x -a x +2 p +1}d x \right ) a^{2} p^{4}-28 \left (\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p} x}{2 a^{3} p \,x^{3}+a^{3} x^{3}-2 a^{2} p \,x^{2}-a^{2} x^{2}-2 a p x -a x +2 p +1}d x \right ) a^{2} p^{3}-28 \left (\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p} x}{2 a^{3} p \,x^{3}+a^{3} x^{3}-2 a^{2} p \,x^{2}-a^{2} x^{2}-2 a p x -a x +2 p +1}d x \right ) a^{2} p^{2}-8 \left (\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p} x}{2 a^{3} p \,x^{3}+a^{3} x^{3}-2 a^{2} p \,x^{2}-a^{2} x^{2}-2 a p x -a x +2 p +1}d x \right ) a^{2} p}{a p \left (2 a p x +a x -2 p -1\right )} \] Input:
int((a*x+1)^4/(-a^2*x^2+1)^2*(-a^2*c*x^2+c)^p,x)
Output:
(( - a**2*c*x**2 + c)**p*a**2*p*x**2 + 3*( - a**2*c*x**2 + c)**p*a*p*x + 2 *( - a**2*c*x**2 + c)**p*a*x - 2*( - a**2*c*x**2 + c)**p*p**2 - 4*( - a**2 *c*x**2 + c)**p*p - 2*( - a**2*c*x**2 + c)**p + 8*int((( - a**2*c*x**2 + c )**p*x)/(2*a**3*p*x**3 + a**3*x**3 - 2*a**2*p*x**2 - a**2*x**2 - 2*a*p*x - a*x + 2*p + 1),x)*a**3*p**4*x + 28*int((( - a**2*c*x**2 + c)**p*x)/(2*a** 3*p*x**3 + a**3*x**3 - 2*a**2*p*x**2 - a**2*x**2 - 2*a*p*x - a*x + 2*p + 1 ),x)*a**3*p**3*x + 28*int((( - a**2*c*x**2 + c)**p*x)/(2*a**3*p*x**3 + a** 3*x**3 - 2*a**2*p*x**2 - a**2*x**2 - 2*a*p*x - a*x + 2*p + 1),x)*a**3*p**2 *x + 8*int((( - a**2*c*x**2 + c)**p*x)/(2*a**3*p*x**3 + a**3*x**3 - 2*a**2 *p*x**2 - a**2*x**2 - 2*a*p*x - a*x + 2*p + 1),x)*a**3*p*x - 8*int((( - a* *2*c*x**2 + c)**p*x)/(2*a**3*p*x**3 + a**3*x**3 - 2*a**2*p*x**2 - a**2*x** 2 - 2*a*p*x - a*x + 2*p + 1),x)*a**2*p**4 - 28*int((( - a**2*c*x**2 + c)** p*x)/(2*a**3*p*x**3 + a**3*x**3 - 2*a**2*p*x**2 - a**2*x**2 - 2*a*p*x - a* x + 2*p + 1),x)*a**2*p**3 - 28*int((( - a**2*c*x**2 + c)**p*x)/(2*a**3*p*x **3 + a**3*x**3 - 2*a**2*p*x**2 - a**2*x**2 - 2*a*p*x - a*x + 2*p + 1),x)* a**2*p**2 - 8*int((( - a**2*c*x**2 + c)**p*x)/(2*a**3*p*x**3 + a**3*x**3 - 2*a**2*p*x**2 - a**2*x**2 - 2*a*p*x - a*x + 2*p + 1),x)*a**2*p)/(a*p*(2*a *p*x + a*x - 2*p - 1))