Integrand size = 22, antiderivative size = 75 \[ \int \frac {e^{\text {arctanh}(a x)} \left (1-a^2 x^2\right )^p}{x^2} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{2}-p,\frac {1}{2},a^2 x^2\right )}{x}-\frac {a \left (1-a^2 x^2\right )^{\frac {1}{2}+p} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+p,\frac {3}{2}+p,1-a^2 x^2\right )}{1+2 p} \] Output:
-hypergeom([-1/2, 1/2-p],[1/2],a^2*x^2)/x-a*(-a^2*x^2+1)^(1/2+p)*hypergeom ([1, 1/2+p],[3/2+p],-a^2*x^2+1)/(1+2*p)
Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.03 \[ \int \frac {e^{\text {arctanh}(a x)} \left (1-a^2 x^2\right )^p}{x^2} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{2}-p,\frac {1}{2},a^2 x^2\right )}{x}-\frac {a \left (1-a^2 x^2\right )^{\frac {1}{2}+p} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+p,\frac {3}{2}+p,1-a^2 x^2\right )}{2 \left (\frac {1}{2}+p\right )} \] Input:
Integrate[(E^ArcTanh[a*x]*(1 - a^2*x^2)^p)/x^2,x]
Output:
-(Hypergeometric2F1[-1/2, 1/2 - p, 1/2, a^2*x^2]/x) - (a*(1 - a^2*x^2)^(1/ 2 + p)*Hypergeometric2F1[1, 1/2 + p, 3/2 + p, 1 - a^2*x^2])/(2*(1/2 + p))
Time = 0.50 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6698, 542, 243, 75, 278}
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.
\(\displaystyle \int \frac {e^{\text {arctanh}(a x)} \left (1-a^2 x^2\right )^p}{x^2} \, dx\) |
\(\Big \downarrow \) 6698 |
\(\displaystyle \int \frac {(a x+1) \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{x^2}dx\) |
\(\Big \downarrow \) 542 |
\(\displaystyle \int \frac {\left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{x^2}dx+a \int \frac {\left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{x}dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \int \frac {\left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{x^2}dx+\frac {1}{2} a \int \frac {\left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{x^2}dx^2\) |
\(\Big \downarrow \) 75 |
\(\displaystyle \int \frac {\left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{x^2}dx-\frac {a \left (1-a^2 x^2\right )^{p+\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (1,p+\frac {1}{2},p+\frac {3}{2},1-a^2 x^2\right )}{2 p+1}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle -\frac {a \left (1-a^2 x^2\right )^{p+\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (1,p+\frac {1}{2},p+\frac {3}{2},1-a^2 x^2\right )}{2 p+1}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{2}-p,\frac {1}{2},a^2 x^2\right )}{x}\) |
Input:
Int[(E^ArcTanh[a*x]*(1 - a^2*x^2)^p)/x^2,x]
Output:
-(Hypergeometric2F1[-1/2, 1/2 - p, 1/2, a^2*x^2]/x) - (a*(1 - a^2*x^2)^(1/ 2 + p)*Hypergeometric2F1[1, 1/2 + p, 3/2 + p, 1 - a^2*x^2])/(1 + 2*p)
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x )^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[n] && (IntegerQ[m] || GtQ[-d/(b*c), 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c Int[x^m*(a + b*x^2)^p, x], x] + Simp[d Int[x^(m + 1)*(a + b*x^2 )^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && IntegerQ[m] && !IntegerQ[2*p]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[c^p Int[x^m*(1 - a^2*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0]) && IGtQ[(n + 1)/2, 0] && !IntegerQ[p - n/2]
Time = 0.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.24
method | result | size |
meijerg | \(\frac {a \left (\left (\Psi \left (\frac {1}{2}-p \right )+\gamma +2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \Gamma \left (\frac {1}{2}-p \right )+\Gamma \left (\frac {3}{2}-p \right ) a^{2} x^{2} \operatorname {hypergeom}\left (\left [1, 1, \frac {3}{2}-p \right ], \left [2, 2\right ], a^{2} x^{2}\right )\right )}{2 \Gamma \left (\frac {1}{2}-p \right )}-\frac {\operatorname {hypergeom}\left (\left [-\frac {1}{2}, \frac {1}{2}-p \right ], \left [\frac {1}{2}\right ], a^{2} x^{2}\right )}{x}\) | \(93\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*x^2+1)^p/x^2,x,method=_RETURNVERBOSE)
Output:
1/2*a*((Psi(1/2-p)+gamma+2*ln(x)+ln(-a^2))*GAMMA(1/2-p)+GAMMA(3/2-p)*a^2*x ^2*hypergeom([1,1,3/2-p],[2,2],a^2*x^2))/GAMMA(1/2-p)-hypergeom([-1/2,1/2- p],[1/2],a^2*x^2)/x
\[ \int \frac {e^{\text {arctanh}(a x)} \left (1-a^2 x^2\right )^p}{x^2} \, dx=\int { \frac {{\left (a x + 1\right )} {\left (-a^{2} x^{2} + 1\right )}^{p}}{\sqrt {-a^{2} x^{2} + 1} x^{2}} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*x^2+1)^p/x^2,x, algorithm="fric as")
Output:
integral(-sqrt(-a^2*x^2 + 1)*(-a^2*x^2 + 1)^p/(a*x^3 - x^2), x)
Result contains complex when optimal does not.
Time = 7.77 (sec) , antiderivative size = 284, normalized size of antiderivative = 3.79 \[ \int \frac {e^{\text {arctanh}(a x)} \left (1-a^2 x^2\right )^p}{x^2} \, dx=- \frac {a a^{2 p} x^{2 p} e^{i \pi p} \Gamma \left (- p\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{2}, 1, p \\ p + 1, p + 1 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt {\pi } \Gamma \left (1 - p\right ) \Gamma \left (p + 1\right )} - \frac {a a^{2 p} x^{2 p} e^{i \pi p} \Gamma \left (- p\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} 1, - p, - p \\ \frac {1}{2}, 1 - p \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 \sqrt {\pi } \Gamma \left (1 - p\right ) \Gamma \left (p + 1\right )} - \frac {a a^{2 p - 1} x^{2 p - 1} e^{i \pi p} \Gamma \left (\frac {1}{2} - p\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{2}, 1, p - \frac {1}{2} \\ p + \frac {1}{2}, p + 1 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt {\pi } \Gamma \left (\frac {3}{2} - p\right ) \Gamma \left (p + 1\right )} - \frac {a^{2 p} x^{2 p - 1} e^{i \pi p} \Gamma \left (\frac {1}{2} - p\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} 1, - p, \frac {1}{2} - p \\ \frac {1}{2}, \frac {3}{2} - p \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 \sqrt {\pi } \Gamma \left (\frac {3}{2} - p\right ) \Gamma \left (p + 1\right )} \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a**2*x**2+1)**p/x**2,x)
Output:
-a*a**(2*p)*x**(2*p)*exp(I*pi*p)*gamma(-p)*gamma(p + 1/2)*hyper((1/2, 1, p ), (p + 1, p + 1), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*gamma(1 - p)*g amma(p + 1)) - a*a**(2*p)*x**(2*p)*exp(I*pi*p)*gamma(-p)*gamma(p + 1/2)*hy per((1, -p, -p), (1/2, 1 - p), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(1 - p)*gam ma(p + 1)) - a*a**(2*p - 1)*x**(2*p - 1)*exp(I*pi*p)*gamma(1/2 - p)*gamma( p + 1/2)*hyper((1/2, 1, p - 1/2), (p + 1/2, p + 1), a**2*x**2*exp_polar(2* I*pi))/(2*sqrt(pi)*gamma(3/2 - p)*gamma(p + 1)) - a**(2*p)*x**(2*p - 1)*ex p(I*pi*p)*gamma(1/2 - p)*gamma(p + 1/2)*hyper((1, -p, 1/2 - p), (1/2, 3/2 - p), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(3/2 - p)*gamma(p + 1))
\[ \int \frac {e^{\text {arctanh}(a x)} \left (1-a^2 x^2\right )^p}{x^2} \, dx=\int { \frac {{\left (a x + 1\right )} {\left (-a^{2} x^{2} + 1\right )}^{p}}{\sqrt {-a^{2} x^{2} + 1} x^{2}} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*x^2+1)^p/x^2,x, algorithm="maxi ma")
Output:
integrate((a*x + 1)*(-a^2*x^2 + 1)^(p - 1/2)/x^2, x)
\[ \int \frac {e^{\text {arctanh}(a x)} \left (1-a^2 x^2\right )^p}{x^2} \, dx=\int { \frac {{\left (a x + 1\right )} {\left (-a^{2} x^{2} + 1\right )}^{p}}{\sqrt {-a^{2} x^{2} + 1} x^{2}} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*x^2+1)^p/x^2,x, algorithm="giac ")
Output:
integrate((a*x + 1)*(-a^2*x^2 + 1)^p/(sqrt(-a^2*x^2 + 1)*x^2), x)
Timed out. \[ \int \frac {e^{\text {arctanh}(a x)} \left (1-a^2 x^2\right )^p}{x^2} \, dx=\int \frac {{\left (1-a^2\,x^2\right )}^p\,\left (a\,x+1\right )}{x^2\,\sqrt {1-a^2\,x^2}} \,d x \] Input:
int(((1 - a^2*x^2)^p*(a*x + 1))/(x^2*(1 - a^2*x^2)^(1/2)),x)
Output:
int(((1 - a^2*x^2)^p*(a*x + 1))/(x^2*(1 - a^2*x^2)^(1/2)), x)
\[ \int \frac {e^{\text {arctanh}(a x)} \left (1-a^2 x^2\right )^p}{x^2} \, dx=\int \frac {\left (-a^{2} x^{2}+1\right )^{p}}{\sqrt {-a^{2} x^{2}+1}\, x^{2}}d x +\left (\int \frac {\left (-a^{2} x^{2}+1\right )^{p}}{\sqrt {-a^{2} x^{2}+1}\, x}d x \right ) a \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*x^2+1)^p/x^2,x)
Output:
int(( - a**2*x**2 + 1)**p/(sqrt( - a**2*x**2 + 1)*x**2),x) + int(( - a**2* x**2 + 1)**p/(sqrt( - a**2*x**2 + 1)*x),x)*a