Integrand size = 23, antiderivative size = 136 \[ \int e^{\text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^p \, dx=\frac {x^{1+m} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},\frac {1}{2}-p,\frac {3+m}{2},a^2 x^2\right )}{1+m}+\frac {a x^{2+m} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},\frac {1}{2}-p,\frac {4+m}{2},a^2 x^2\right )}{2+m} \]
x^(1+m)*(-a^2*c*x^2+c)^p*hypergeom([1/2-p, 1/2+1/2*m],[3/2+1/2*m],a^2*x^2) /(1+m)/((-a^2*x^2+1)^p)+a*x^(2+m)*(-a^2*c*x^2+c)^p*hypergeom([1+1/2*m, 1/2 -p],[2+1/2*m],a^2*x^2)/(2+m)/((-a^2*x^2+1)^p)
Time = 0.03 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.84 \[ \int e^{\text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^p \, dx=\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},\frac {1}{2}-p,1+\frac {1+m}{2},a^2 x^2\right )}{1+m}+\frac {a x^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},\frac {1}{2}-p,1+\frac {2+m}{2},a^2 x^2\right )}{2+m}\right ) \]
((c - a^2*c*x^2)^p*((x^(1 + m)*Hypergeometric2F1[(1 + m)/2, 1/2 - p, 1 + ( 1 + m)/2, a^2*x^2])/(1 + m) + (a*x^(2 + m)*Hypergeometric2F1[(2 + m)/2, 1/ 2 - p, 1 + (2 + m)/2, a^2*x^2])/(2 + m)))/(1 - a^2*x^2)^p
Time = 0.43 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.81, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6703, 6698, 557, 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.
\(\displaystyle \int x^m e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx\) |
\(\Big \downarrow \) 6703 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \int e^{\text {arctanh}(a x)} x^m \left (1-a^2 x^2\right )^pdx\) |
\(\Big \downarrow \) 6698 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \int x^m (a x+1) \left (1-a^2 x^2\right )^{p-\frac {1}{2}}dx\) |
\(\Big \downarrow \) 557 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (a \int x^{m+1} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}dx+\int x^m \left (1-a^2 x^2\right )^{p-\frac {1}{2}}dx\right )\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {x^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},\frac {1}{2}-p,\frac {m+3}{2},a^2 x^2\right )}{m+1}+\frac {a x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {m+2}{2},\frac {1}{2}-p,\frac {m+4}{2},a^2 x^2\right )}{m+2}\right )\) |
((c - a^2*c*x^2)^p*((x^(1 + m)*Hypergeometric2F1[(1 + m)/2, 1/2 - p, (3 + m)/2, a^2*x^2])/(1 + m) + (a*x^(2 + m)*Hypergeometric2F1[(2 + m)/2, 1/2 - p, (4 + m)/2, a^2*x^2])/(2 + m)))/(1 - a^2*x^2)^p
3.11.4.3.1 Defintions of rubi rules used
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[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[c Int[(e*x)^m*(a + b*x^2)^p, x], x] + Simp[d/e Int[(e*x)^( m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x]
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]
\[\int \frac {\left (a x +1\right ) x^{m} \left (-a^{2} c \,x^{2}+c \right )^{p}}{\sqrt {-a^{2} x^{2}+1}}d x\]
\[ \int e^{\text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^p \, dx=\int { \frac {{\left (a x + 1\right )} {\left (-a^{2} c x^{2} + c\right )}^{p} x^{m}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \]
Result contains complex when optimal does not.
Time = 39.88 (sec) , antiderivative size = 391, normalized size of antiderivative = 2.88 \[ \int e^{\text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^p \, dx=- \frac {a a^{2 p} c^{p} x^{m + 2 p + 2} e^{i \pi p} \Gamma \left (p + \frac {1}{2}\right ) \Gamma \left (- \frac {m}{2} - p - 1\right ) {{}_{3}F_{2}\left (\begin {matrix} 1, - p, - \frac {m}{2} - p - 1 \\ \frac {1}{2}, - \frac {m}{2} - p \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 \sqrt {\pi } \Gamma \left (- \frac {m}{2} - p\right ) \Gamma \left (p + 1\right )} - \frac {a a^{- m - 2} a^{m + 2 p + 2} c^{p} x^{m + 2 p + 2} e^{i \pi p} \Gamma \left (p + \frac {1}{2}\right ) \Gamma \left (- \frac {m}{2} - p - 1\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{2}, 1, \frac {m}{2} + p + 1 \\ p + 1, \frac {m}{2} + p + 2 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt {\pi } \Gamma \left (- \frac {m}{2} - p\right ) \Gamma \left (p + 1\right )} - \frac {a^{2 p} c^{p} x^{m + 2 p + 1} e^{i \pi p} \Gamma \left (p + \frac {1}{2}\right ) \Gamma \left (- \frac {m}{2} - p - \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} 1, - p, - \frac {m}{2} - p - \frac {1}{2} \\ \frac {1}{2}, - \frac {m}{2} - p + \frac {1}{2} \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 \sqrt {\pi } \Gamma \left (p + 1\right ) \Gamma \left (- \frac {m}{2} - p + \frac {1}{2}\right )} - \frac {a^{- m - 1} a^{m + 2 p + 1} c^{p} x^{m + 2 p + 1} e^{i \pi p} \Gamma \left (p + \frac {1}{2}\right ) \Gamma \left (- \frac {m}{2} - p - \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{2}, 1, \frac {m}{2} + p + \frac {1}{2} \\ p + 1, \frac {m}{2} + p + \frac {3}{2} \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt {\pi } \Gamma \left (p + 1\right ) \Gamma \left (- \frac {m}{2} - p + \frac {1}{2}\right )} \]
-a*a**(2*p)*c**p*x**(m + 2*p + 2)*exp(I*pi*p)*gamma(p + 1/2)*gamma(-m/2 - p - 1)*hyper((1, -p, -m/2 - p - 1), (1/2, -m/2 - p), 1/(a**2*x**2))/(2*sqr t(pi)*gamma(-m/2 - p)*gamma(p + 1)) - a*a**(-m - 2)*a**(m + 2*p + 2)*c**p* x**(m + 2*p + 2)*exp(I*pi*p)*gamma(p + 1/2)*gamma(-m/2 - p - 1)*hyper((1/2 , 1, m/2 + p + 1), (p + 1, m/2 + p + 2), a**2*x**2*exp_polar(2*I*pi))/(2*s qrt(pi)*gamma(-m/2 - p)*gamma(p + 1)) - a**(2*p)*c**p*x**(m + 2*p + 1)*exp (I*pi*p)*gamma(p + 1/2)*gamma(-m/2 - p - 1/2)*hyper((1, -p, -m/2 - p - 1/2 ), (1/2, -m/2 - p + 1/2), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(p + 1)*gamma(-m /2 - p + 1/2)) - a**(-m - 1)*a**(m + 2*p + 1)*c**p*x**(m + 2*p + 1)*exp(I* pi*p)*gamma(p + 1/2)*gamma(-m/2 - p - 1/2)*hyper((1/2, 1, m/2 + p + 1/2), (p + 1, m/2 + p + 3/2), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*gamma(p + 1)*gamma(-m/2 - p + 1/2))
\[ \int e^{\text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^p \, dx=\int { \frac {{\left (a x + 1\right )} {\left (-a^{2} c x^{2} + c\right )}^{p} x^{m}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \]
\[ \int e^{\text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^p \, dx=\int { \frac {{\left (a x + 1\right )} {\left (-a^{2} c x^{2} + c\right )}^{p} x^{m}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \]
Timed out. \[ \int e^{\text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^p \, dx=\int \frac {x^m\,{\left (c-a^2\,c\,x^2\right )}^p\,\left (a\,x+1\right )}{\sqrt {1-a^2\,x^2}} \,d x \]