Integrand size = 23, antiderivative size = 110 \[ \int \frac {e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x} \, dx=a x \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-p,\frac {3}{2},a^2 x^2\right )-\frac {\sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+p,\frac {3}{2}+p,1-a^2 x^2\right )}{1+2 p} \]
a*x*(-a^2*c*x^2+c)^p*hypergeom([1/2, 1/2-p],[3/2],a^2*x^2)/((-a^2*x^2+1)^p )-(-a^2*c*x^2+c)^p*hypergeom([1, 1/2+p],[3/2+p],-a^2*x^2+1)*(-a^2*x^2+1)^( 1/2)/(1+2*p)
Time = 0.03 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.93 \[ \int \frac {e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x} \, dx=\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (a x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-p,\frac {3}{2},a^2 x^2\right )-\frac {\left (1-a^2 x^2\right )^{\frac {1}{2}+p} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+p,\frac {3}{2}+p,1-a^2 x^2\right )}{2 \left (\frac {1}{2}+p\right )}\right ) \]
((c - a^2*c*x^2)^p*(a*x*Hypergeometric2F1[1/2, 1/2 - p, 3/2, a^2*x^2] - (( 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))))/(1 - a^2*x^2)^p
Time = 0.41 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.91, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6703, 6698, 542, 237, 243, 75}
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.
\(\displaystyle \int \frac {e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x} \, dx\) |
\(\Big \downarrow \) 6703 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \int \frac {e^{\text {arctanh}(a x)} \left (1-a^2 x^2\right )^p}{x}dx\) |
\(\Big \downarrow \) 6698 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \int \frac {(a x+1) \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{x}dx\) |
\(\Big \downarrow \) 542 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (a \int \left (1-a^2 x^2\right )^{p-\frac {1}{2}}dx+\int \frac {\left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{x}dx\right )\) |
\(\Big \downarrow \) 237 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\int \frac {\left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{x}dx+a x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-p,\frac {3}{2},a^2 x^2\right )\right )\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {1}{2} \int \frac {\left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{x^2}dx^2+a x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-p,\frac {3}{2},a^2 x^2\right )\right )\) |
\(\Big \downarrow \) 75 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (a x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-p,\frac {3}{2},a^2 x^2\right )-\frac {\left (1-a^2 x^2\right )^{p+\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (1,p+\frac {1}{2},p+\frac {3}{2},1-a^2 x^2\right )}{2 p+1}\right )\) |
((c - a^2*c*x^2)^p*(a*x*Hypergeometric2F1[1/2, 1/2 - p, 3/2, a^2*x^2] - (( 1 - a^2*x^2)^(1/2 + p)*Hypergeometric2F1[1, 1/2 + p, 3/2 + p, 1 - a^2*x^2] )/(1 + 2*p)))/(1 - a^2*x^2)^p
3.11.16.3.1 Defintions of rubi rules used
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[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[- p, 1/2, 1/2 + 1, (-b)*(x^2/a)], x] /; FreeQ[{a, b, p}, x] && !IntegerQ[2*p ] && GtQ[a, 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[(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]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_), x_ Symbol] :> Simp[c^IntPart[p]*((c + d*x^2)^FracPart[p]/(1 - a^2*x^2)^FracPar t[p]) Int[x^m*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ[c, 0]) && !I ntegerQ[n/2]
\[\int \frac {\left (a x +1\right ) \left (-a^{2} c \,x^{2}+c \right )^{p}}{\sqrt {-a^{2} x^{2}+1}\, x}d x\]
\[ \int \frac {e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x} \, dx=\int { \frac {{\left (a x + 1\right )} {\left (-a^{2} c x^{2} + c\right )}^{p}}{\sqrt {-a^{2} x^{2} + 1} x} \,d x } \]
Result contains complex when optimal does not.
Time = 9.87 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.72 \[ \int \frac {e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x} \, dx=- \frac {a a^{2 p} c^{p} x^{2 p + 1} e^{i \pi p} \Gamma \left (- p - \frac {1}{2}\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} 1, - p, - p - \frac {1}{2} \\ \frac {1}{2}, \frac {1}{2} - p \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 \sqrt {\pi } \Gamma \left (\frac {1}{2} - p\right ) \Gamma \left (p + 1\right )} - \frac {a^{2 p} c^{p} x^{2 p} e^{i \pi p} \Gamma \left (- p\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{2}, 1, p \\ p + 1, p + 1 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt {\pi } \Gamma \left (1 - p\right ) \Gamma \left (p + 1\right )} - \frac {a^{2 p} c^{p} x^{2 p} e^{i \pi p} \Gamma \left (- p\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} 1, - p, - p \\ \frac {1}{2}, 1 - p \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 \sqrt {\pi } \Gamma \left (1 - p\right ) \Gamma \left (p + 1\right )} - \frac {a^{2 p + 1} c^{p} x^{2 p + 1} e^{i \pi p} \Gamma \left (- p - \frac {1}{2}\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{2}, 1, p + \frac {1}{2} \\ p + 1, p + \frac {3}{2} \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt {\pi } \Gamma \left (\frac {1}{2} - p\right ) \Gamma \left (p + 1\right )} \]
-a*a**(2*p)*c**p*x**(2*p + 1)*exp(I*pi*p)*gamma(-p - 1/2)*gamma(p + 1/2)*h yper((1, -p, -p - 1/2), (1/2, 1/2 - p), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(1 /2 - p)*gamma(p + 1)) - a**(2*p)*c**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)*gamma(p + 1)) - a**(2*p)*c**p*x**(2*p)*exp(I*pi*p )*gamma(-p)*gamma(p + 1/2)*hyper((1, -p, -p), (1/2, 1 - p), 1/(a**2*x**2)) /(2*sqrt(pi)*gamma(1 - p)*gamma(p + 1)) - a**(2*p + 1)*c**p*x**(2*p + 1)*e xp(I*pi*p)*gamma(-p - 1/2)*gamma(p + 1/2)*hyper((1/2, 1, p + 1/2), (p + 1, p + 3/2), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*gamma(1/2 - p)*gamma(p + 1))
\[ \int \frac {e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x} \, dx=\int { \frac {{\left (a x + 1\right )} {\left (-a^{2} c x^{2} + c\right )}^{p}}{\sqrt {-a^{2} x^{2} + 1} x} \,d x } \]
\[ \int \frac {e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x} \, dx=\int { \frac {{\left (a x + 1\right )} {\left (-a^{2} c x^{2} + c\right )}^{p}}{\sqrt {-a^{2} x^{2} + 1} x} \,d x } \]
Timed out. \[ \int \frac {e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x} \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^p\,\left (a\,x+1\right )}{x\,\sqrt {1-a^2\,x^2}} \,d x \]