\(\int e^{\text {arctanh}(a x)} (c-a^2 c x^2)^p \, dx\) [1037]

Optimal result

Integrand size = 20, antiderivative size = 90 \[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=-\frac {\sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^p}{a (1+2 p)}+x \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-p,\frac {3}{2},a^2 x^2\right ) \] Output:

-(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^p/a/(1+2*p)+x*(-a^2*c*x^2+c)^p*hypergeo 
m([1/2, 1/2-p],[3/2],a^2*x^2)/((-a^2*x^2+1)^p)
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.96 \[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=-\frac {2^{\frac {1}{2}+p} (1-a x)^{\frac {1}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}-p,\frac {1}{2}+p,\frac {3}{2}+p,\frac {1}{2} (1-a x)\right )}{a \left (\frac {1}{2}+p\right )} \] Input:

Integrate[E^ArcTanh[a*x]*(c - a^2*c*x^2)^p,x]
 

Output:

-((2^(1/2 + p)*(1 - a*x)^(1/2 + p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[-1/ 
2 - p, 1/2 + p, 3/2 + p, (1 - a*x)/2])/(a*(1/2 + p)*(1 - a^2*x^2)^p))
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6693, 6690, 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.

\(\displaystyle \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx\)

\(\Big \downarrow \) 6693

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \int e^{\text {arctanh}(a x)} \left (1-a^2 x^2\right )^pdx\)

\(\Big \downarrow \) 6690

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \int (1-a x)^{p-\frac {1}{2}} (a x+1)^{p+\frac {1}{2}}dx\)

\(\Big \downarrow \) 79

\(\displaystyle -\frac {2^{p+\frac {3}{2}} (1-a x)^{p+\frac {1}{2}} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-p-\frac {1}{2},p+\frac {1}{2},p+\frac {3}{2},\frac {1}{2} (1-a x)\right )}{a (2 p+1)}\)

Input:

Int[E^ArcTanh[a*x]*(c - a^2*c*x^2)^p,x]
 

Output:

-((2^(3/2 + p)*(1 - a*x)^(1/2 + p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[-1/ 
2 - p, 1/2 + p, 3/2 + p, (1 - a*x)/2])/(a*(1 + 2*p)*(1 - a^2*x^2)^p))
 

Defintions of rubi rules used

rule 79

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]))
 

rule 6690

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> 
 Simp[c^p   Int[(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a 
, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
 

rule 6693

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[c^IntPart[p]*((c + d*x^2)^FracPart[p]/(1 - a^2*x^2)^FracPart[p])   Int 
[(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && 
 EqQ[a^2*c + d, 0] &&  !(IntegerQ[p] || GtQ[c, 0])
 

Maple [F]

\[\int \frac {\left (a x +1\right ) \left (-a^{2} c \,x^{2}+c \right )^{p}}{\sqrt {-a^{2} x^{2}+1}}d x\]

Input:

int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^p,x)
 

Output:

int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^p,x)
 

Fricas [F]

\[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int { \frac {{\left (a x + 1\right )} {\left (-a^{2} c x^{2} + c\right )}^{p}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^p,x, algorithm="fricas 
")
 

Output:

integral(-sqrt(-a^2*x^2 + 1)*(-a^2*c*x^2 + c)^p/(a*x - 1), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 6.37 (sec) , antiderivative size = 306, normalized size of antiderivative = 3.40 \[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=- \frac {a a^{2 p} c^{p} x^{2 p + 2} e^{i \pi p} \Gamma \left (- p - 1\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, 1 \\ p + 2 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt {\pi } \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} - \frac {a a^{2 p} c^{p} x^{2 p + 2} e^{i \pi p} \Gamma \left (- p - 1\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} 1, - p - 1 \\ \frac {1}{2} \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 \sqrt {\pi } \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} - \frac {a^{2 p} c^{p} x^{2 p + 1} e^{i \pi p} \Gamma \left (- p - \frac {1}{2}\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} 1, - p, - p - \frac {1}{2} \\ \frac {1}{2}, \frac {1}{2} - p \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 \sqrt {\pi } \Gamma \left (\frac {1}{2} - p\right ) \Gamma \left (p + 1\right )} - \frac {a^{2 p + 1} c^{p} x^{2 p + 1} e^{i \pi p} \Gamma \left (- p - \frac {1}{2}\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{2}, 1, p + \frac {1}{2} \\ p + 1, p + \frac {3}{2} \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt {\pi } a \Gamma \left (\frac {1}{2} - p\right ) \Gamma \left (p + 1\right )} \] Input:

integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a**2*c*x**2+c)**p,x)
 

Output:

-a*a**(2*p)*c**p*x**(2*p + 2)*exp(I*pi*p)*gamma(-p - 1)*gamma(p + 1/2)*hyp 
er((1/2, 1), (p + 2,), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*gamma(-p)* 
gamma(p + 1)) - a*a**(2*p)*c**p*x**(2*p + 2)*exp(I*pi*p)*gamma(-p - 1)*gam 
ma(p + 1/2)*hyper((1, -p - 1), (1/2,), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(-p 
)*gamma(p + 1)) - a**(2*p)*c**p*x**(2*p + 1)*exp(I*pi*p)*gamma(-p - 1/2)*g 
amma(p + 1/2)*hyper((1, -p, -p - 1/2), (1/2, 1/2 - p), 1/(a**2*x**2))/(2*s 
qrt(pi)*gamma(1/2 - p)*gamma(p + 1)) - a**(2*p + 1)*c**p*x**(2*p + 1)*exp( 
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)*a*gamma(1/2 - p)*gamma(p 
+ 1))
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int { \frac {{\left (a x + 1\right )} {\left (-a^{2} c x^{2} + c\right )}^{p}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^p,x, algorithm="maxima 
")
 

Output:

integrate((a*x + 1)*(-a^2*c*x^2 + c)^p/sqrt(-a^2*x^2 + 1), x)
 

Giac [F]

\[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int { \frac {{\left (a x + 1\right )} {\left (-a^{2} c x^{2} + c\right )}^{p}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^p,x, algorithm="giac")
 

Output:

integrate((a*x + 1)*(-a^2*c*x^2 + c)^p/sqrt(-a^2*x^2 + 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^p\,\left (a\,x+1\right )}{\sqrt {1-a^2\,x^2}} \,d x \] Input:

int(((c - a^2*c*x^2)^p*(a*x + 1))/(1 - a^2*x^2)^(1/2),x)
 

Output:

int(((c - a^2*c*x^2)^p*(a*x + 1))/(1 - a^2*x^2)^(1/2), x)
 

Reduce [F]

\[ \int e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {-c^{p} \left (-a^{2} x^{2}+1\right )^{p +\frac {1}{2}}+2 \left (\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p}}{\sqrt {-a^{2} x^{2}+1}}d x \right ) a p +\left (\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p}}{\sqrt {-a^{2} x^{2}+1}}d x \right ) a}{a \left (2 p +1\right )} \] Input:

int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^p,x)
 

Output:

( - c**p*( - a**2*x**2 + 1)**((2*p + 1)/2) + 2*int(( - a**2*c*x**2 + c)**p 
/sqrt( - a**2*x**2 + 1),x)*a*p + int(( - a**2*c*x**2 + c)**p/sqrt( - a**2* 
x**2 + 1),x)*a)/(a*(2*p + 1))