Integrand size = 20, antiderivative size = 57 \[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{2}-p,-2 p,\frac {2}{1+a x}\right )}{1+2 p} \] Output:
(1-1/a^2/x^2)^(1/2)*x*(-a^2*c*x^2+c)^p*hypergeom([1, -1/2-p],[-2*p],2/(a*x +1))/(1+2*p)
Leaf count is larger than twice the leaf count of optimal. \(122\) vs. \(2(57)=114\).
Time = 0.33 (sec) , antiderivative size = 122, normalized size of antiderivative = 2.14 \[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=-\frac {4^{1+p} e^{3 \coth ^{-1}(a x)} \left (1-e^{2 \coth ^{-1}(a x)}\right )^{2 p} \left (\frac {e^{\coth ^{-1}(a x)}}{-1+e^{2 \coth ^{-1}(a x)}}\right )^{2 p} \left (a \sqrt {1-\frac {1}{a^2 x^2}} x\right )^{-2 p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {3}{2}+p,2+2 p,\frac {5}{2}+p,e^{2 \coth ^{-1}(a x)}\right )}{3 a+2 a p} \] Input:
Integrate[E^ArcCoth[a*x]*(c - a^2*c*x^2)^p,x]
Output:
-((4^(1 + p)*E^(3*ArcCoth[a*x])*(1 - E^(2*ArcCoth[a*x]))^(2*p)*(E^ArcCoth[ a*x]/(-1 + E^(2*ArcCoth[a*x])))^(2*p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[ 3/2 + p, 2 + 2*p, 5/2 + p, E^(2*ArcCoth[a*x])])/((3*a + 2*a*p)*(a*Sqrt[1 - 1/(a^2*x^2)]*x)^(2*p)))
Leaf count is larger than twice the leaf count of optimal. \(118\) vs. \(2(57)=114\).
Time = 0.67 (sec) , antiderivative size = 118, normalized size of antiderivative = 2.07, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6746, 6750, 142}
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.
\(\displaystyle \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx\) |
\(\Big \downarrow \) 6746 |
\(\displaystyle x^{-2 p} \left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (c-a^2 c x^2\right )^p \int e^{\coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^p x^{2 p}dx\) |
\(\Big \downarrow \) 6750 |
\(\displaystyle \left (\frac {1}{x}\right )^{2 p} \left (-\left (1-\frac {1}{a^2 x^2}\right )^{-p}\right ) \left (c-a^2 c x^2\right )^p \int \left (1-\frac {1}{a x}\right )^{p-\frac {1}{2}} \left (1+\frac {1}{a x}\right )^{p+\frac {1}{2}} \left (\frac {1}{x}\right )^{-2 (p+1)}d\frac {1}{x}\) |
\(\Big \downarrow \) 142 |
\(\displaystyle \frac {x \left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1}{2}-p} \left (1-\frac {1}{a x}\right )^{p-\frac {1}{2}} \left (\frac {1}{a x}+1\right )^{p+\frac {3}{2}} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-2 p-1,\frac {1}{2}-p,-2 p,\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{2 p+1}\) |
Input:
Int[E^ArcCoth[a*x]*(c - a^2*c*x^2)^p,x]
Output:
(((a - x^(-1))/(a + x^(-1)))^(1/2 - p)*(1 - 1/(a*x))^(-1/2 + p)*(1 + 1/(a* x))^(3/2 + p)*x*(c - a^2*c*x^2)^p*Hypergeometric2F1[-1 - 2*p, 1/2 - p, -2* p, 2/((a + x^(-1))*x)])/((1 + 2*p)*(1 - 1/(a^2*x^2))^p)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((b*e - a*f)*(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f))*((a + b*x)/((b*c - a*d)*(e + f*x)))])/((b*e - a*f)*((c + d*x)/((b*c - a*d)*(e + f *x))))^n, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[m + n + p + 2, 0] && !IntegerQ[n]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[(c + d*x^2)^p/(x^(2*p)*(1 - 1/(a^2*x^2))^p) Int[u*x^(2*p)*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n/2] && !IntegerQ[p]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_)^2)^(p_.)*(x_)^(m_), x_ Symbol] :> Simp[(-c^p)*x^m*(1/x)^m Subst[Int[(1 - x/a)^(p - n/2)*((1 + x/ a)^(p + n/2)/x^(m + 2)), x], x, 1/x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && !In tegersQ[2*p, p + n/2] && !IntegerQ[m]
\[\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p}}{\sqrt {\frac {a x -1}{a x +1}}}d x\]
Input:
int(1/((a*x-1)/(a*x+1))^(1/2)*(-a^2*c*x^2+c)^p,x)
Output:
int(1/((a*x-1)/(a*x+1))^(1/2)*(-a^2*c*x^2+c)^p,x)
\[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int { \frac {{\left (-a^{2} c x^{2} + c\right )}^{p}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(-a^2*c*x^2+c)^p,x, algorithm="fricas" )
Output:
integral((a*x + 1)*(-a^2*c*x^2 + c)^p*sqrt((a*x - 1)/(a*x + 1))/(a*x - 1), x)
\[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int \frac {\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{p}}{\sqrt {\frac {a x - 1}{a x + 1}}}\, dx \] Input:
integrate(1/((a*x-1)/(a*x+1))**(1/2)*(-a**2*c*x**2+c)**p,x)
Output:
Integral((-c*(a*x - 1)*(a*x + 1))**p/sqrt((a*x - 1)/(a*x + 1)), x)
\[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int { \frac {{\left (-a^{2} c x^{2} + c\right )}^{p}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(-a^2*c*x^2+c)^p,x, algorithm="maxima" )
Output:
integrate((-a^2*c*x^2 + c)^p/sqrt((a*x - 1)/(a*x + 1)), x)
\[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int { \frac {{\left (-a^{2} c x^{2} + c\right )}^{p}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(-a^2*c*x^2+c)^p,x, algorithm="giac")
Output:
integrate((-a^2*c*x^2 + c)^p/sqrt((a*x - 1)/(a*x + 1)), x)
Timed out. \[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^p}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \] Input:
int((c - a^2*c*x^2)^p/((a*x - 1)/(a*x + 1))^(1/2),x)
Output:
int((c - a^2*c*x^2)^p/((a*x - 1)/(a*x + 1))^(1/2), x)
\[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int \frac {\sqrt {a x +1}\, \left (-a^{2} c \,x^{2}+c \right )^{p}}{\sqrt {a x -1}}d x \] Input:
int(1/((a*x-1)/(a*x+1))^(1/2)*(-a^2*c*x^2+c)^p,x)
Output:
int((sqrt(a*x + 1)*( - a**2*c*x**2 + c)**p)/sqrt(a*x - 1),x)