\(\int e^{n \text {arctanh}(a x)} x (c-a^2 c x^2)^p \, dx\) [1384]

Optimal result

Integrand size = 23, antiderivative size = 177 \[ \int e^{n \text {arctanh}(a x)} x \left (c-a^2 c x^2\right )^p \, dx=-\frac {(1-a x)^{1-\frac {n}{2}+p} (1+a x)^{1+\frac {n}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p}{2 a^2 (1+p)}-\frac {2^{\frac {n}{2}+p} n (1-a x)^{1-\frac {n}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-\frac {n}{2}-p,1-\frac {n}{2}+p,2-\frac {n}{2}+p,\frac {1}{2} (1-a x)\right )}{a^2 (1+p) (2-n+2 p)} \] Output:

-1/2*(-a*x+1)^(1-1/2*n+p)*(a*x+1)^(1+1/2*n+p)*(-a^2*c*x^2+c)^p/a^2/(p+1)/( 
(-a^2*x^2+1)^p)-2^(1/2*n+p)*n*(-a*x+1)^(1-1/2*n+p)*(-a^2*c*x^2+c)^p*hyperg 
eom([-1/2*n-p, 1-1/2*n+p],[2-1/2*n+p],-1/2*a*x+1/2)/a^2/(p+1)/(2-n+2*p)/(( 
-a^2*x^2+1)^p)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.77 \[ \int e^{n \text {arctanh}(a x)} x \left (c-a^2 c x^2\right )^p \, dx=-\frac {(1-a x)^{1-\frac {n}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (-\left ((n-2 (1+p)) (1+a x)^{1+\frac {n}{2}+p}\right )+2^{1+\frac {n}{2}+p} n \operatorname {Hypergeometric2F1}\left (-\frac {n}{2}-p,1-\frac {n}{2}+p,2-\frac {n}{2}+p,\frac {1}{2} (1-a x)\right )\right )}{2 a^2 (1+p) (2-n+2 p)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.85, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6703, 6700, 90, 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.

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

\(\Big \downarrow \) 6703

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

\(\Big \downarrow \) 6700

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 79

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

Input:

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

Output:

((c - a^2*c*x^2)^p*(-1/2*((1 - a*x)^(1 - n/2 + p)*(1 + a*x)^(1 + n/2 + p)) 
/(a^2*(1 + p)) - (2^(n/2 + p)*n*(1 - a*x)^(1 - n/2 + p)*Hypergeometric2F1[ 
-1/2*n - p, 1 - n/2 + p, 2 - n/2 + p, (1 - a*x)/2])/(a^2*(1 + p)*(2 - n + 
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 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 6700

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

rule 6703

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]
 

Maple [F]

\[\int {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} x \left (-a^{2} c \,x^{2}+c \right )^{p}d x\]

Input:

int(exp(n*arctanh(a*x))*x*(-a^2*c*x^2+c)^p,x)
 

Output:

int(exp(n*arctanh(a*x))*x*(-a^2*c*x^2+c)^p,x)
 

Fricas [F]

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

integrate(exp(n*arctanh(a*x))*x*(-a^2*c*x^2+c)^p,x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate(exp(n*atanh(a*x))*x*(-a**2*c*x**2+c)**p,x)
 

Output:

Integral(x*(-c*(a*x - 1)*(a*x + 1))**p*exp(n*atanh(a*x)), x)
 

Maxima [F]

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

integrate(exp(n*arctanh(a*x))*x*(-a^2*c*x^2+c)^p,x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(exp(n*arctanh(a*x))*x*(-a^2*c*x^2+c)^p,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int(x*exp(n*atanh(a*x))*(c - a^2*c*x^2)^p,x)
 

Output:

int(x*exp(n*atanh(a*x))*(c - a^2*c*x^2)^p, x)
 

Reduce [F]

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

int(exp(n*atanh(a*x))*x*(-a^2*c*x^2+c)^p,x)
 

Output:

int(e**(atanh(a*x)*n)*( - a**2*c*x**2 + c)**p*x,x)