\(\int \frac {e^{n \text {arctanh}(a x)} x^4}{c-a^2 c x^2} \, dx\) [1334]

Optimal result

Integrand size = 25, antiderivative size = 226 \[ \int \frac {e^{n \text {arctanh}(a x)} x^4}{c-a^2 c x^2} \, dx=-\frac {(1-a x)^{1-\frac {n}{2}} (1+a x)^{n/2}}{3 a^5 c}+\frac {\left (6+2 n+n^2\right ) (1-a x)^{-n/2} (1+a x)^{n/2}}{6 a^5 c n}-\frac {n x^2 (1-a x)^{-n/2} (1+a x)^{n/2}}{6 a^3 c}-\frac {x^3 (1-a x)^{-n/2} (1+a x)^{n/2}}{3 a^2 c}+\frac {2^{n/2} \left (8+n^2\right ) (1-a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{3 a^5 c (2-n)} \] Output:

-1/3*(-a*x+1)^(1-1/2*n)*(a*x+1)^(1/2*n)/a^5/c+1/6*(n^2+2*n+6)*(a*x+1)^(1/2 
*n)/a^5/c/n/((-a*x+1)^(1/2*n))-1/6*n*x^2*(a*x+1)^(1/2*n)/a^3/c/((-a*x+1)^( 
1/2*n))-1/3*x^3*(a*x+1)^(1/2*n)/a^2/c/((-a*x+1)^(1/2*n))+1/3*2^(1/2*n)*(n^ 
2+8)*(-a*x+1)^(1-1/2*n)*hypergeom([-1/2*n, 1-1/2*n],[2-1/2*n],-1/2*a*x+1/2 
)/a^5/c/(2-n)
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.76 \[ \int \frac {e^{n \text {arctanh}(a x)} x^4}{c-a^2 c x^2} \, dx=\frac {(1-a x)^{-n/2} \left (2^{1+\frac {n}{2}} n \left (2-2 n+n^2\right ) (-1+a x) \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )-(-2+n) \left ((1+a x)^{n/2} \left (-6+(n+a n x)^2+2 n \left (3+2 a x+a^3 x^3\right )\right )-2^{2+\frac {n}{2}} n (3+n) \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},-\frac {n}{2},1-\frac {n}{2},\frac {1}{2} (1-a x)\right )\right )\right )}{6 a^5 c (-2+n) n} \] Input:

Integrate[(E^(n*ArcTanh[a*x])*x^4)/(c - a^2*c*x^2),x]
 

Output:

(2^(1 + n/2)*n*(2 - 2*n + n^2)*(-1 + a*x)*Hypergeometric2F1[1 - n/2, -1/2* 
n, 2 - n/2, (1 - a*x)/2] - (-2 + n)*((1 + a*x)^(n/2)*(-6 + (n + a*n*x)^2 + 
 2*n*(3 + 2*a*x + a^3*x^3)) - 2^(2 + n/2)*n*(3 + n)*Hypergeometric2F1[-1/2 
*n, -1/2*n, 1 - n/2, (1 - a*x)/2]))/(6*a^5*c*(-2 + n)*n*(1 - a*x)^(n/2))
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.93, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {6700, 111, 25, 170, 25, 27, 160, 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.

\(\displaystyle \int \frac {x^4 e^{n \text {arctanh}(a x)}}{c-a^2 c x^2} \, dx\)

\(\Big \downarrow \) 6700

\(\displaystyle \frac {\int x^4 (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}}dx}{c}\)

\(\Big \downarrow \) 111

\(\displaystyle \frac {-\frac {\int -x^2 (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}} (a n x+3)dx}{3 a^2}-\frac {x^3 (a x+1)^{n/2} (1-a x)^{-n/2}}{3 a^2}}{c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int x^2 (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}} (a n x+3)dx}{3 a^2}-\frac {x^3 (1-a x)^{-n/2} (a x+1)^{n/2}}{3 a^2}}{c}\)

\(\Big \downarrow \) 170

\(\displaystyle \frac {\frac {-\frac {\int -a x (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}} \left (2 n+a \left (n^2+6\right ) x\right )dx}{2 a^2}-\frac {n x^2 (a x+1)^{n/2} (1-a x)^{-n/2}}{2 a}}{3 a^2}-\frac {x^3 (1-a x)^{-n/2} (a x+1)^{n/2}}{3 a^2}}{c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\int a x (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}} \left (2 n+a \left (n^2+6\right ) x\right )dx}{2 a^2}-\frac {n x^2 (1-a x)^{-n/2} (a x+1)^{n/2}}{2 a}}{3 a^2}-\frac {x^3 (1-a x)^{-n/2} (a x+1)^{n/2}}{3 a^2}}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int x (1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n-2}{2}} \left (2 n+a \left (n^2+6\right ) x\right )dx}{2 a}-\frac {n x^2 (1-a x)^{-n/2} (a x+1)^{n/2}}{2 a}}{3 a^2}-\frac {x^3 (1-a x)^{-n/2} (a x+1)^{n/2}}{3 a^2}}{c}\)

\(\Big \downarrow \) 160

\(\displaystyle \frac {\frac {\frac {\frac {(1-a x)^{-n/2} (a x+1)^{n/2} \left (-a \left (n^2+6\right ) n x+n^3+n^2+8 n+6\right )}{a^2 n}-\frac {n \left (n^2+8\right ) \int (1-a x)^{-n/2} (a x+1)^{\frac {n-2}{2}}dx}{a}}{2 a}-\frac {n x^2 (1-a x)^{-n/2} (a x+1)^{n/2}}{2 a}}{3 a^2}-\frac {x^3 (1-a x)^{-n/2} (a x+1)^{n/2}}{3 a^2}}{c}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {\frac {\frac {\frac {2^{n/2} n \left (n^2+8\right ) (1-a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {2-n}{2},1-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a^2 (2-n)}+\frac {(a x+1)^{n/2} \left (-a \left (n^2+6\right ) n x+n^3+n^2+8 n+6\right ) (1-a x)^{-n/2}}{a^2 n}}{2 a}-\frac {n x^2 (1-a x)^{-n/2} (a x+1)^{n/2}}{2 a}}{3 a^2}-\frac {x^3 (1-a x)^{-n/2} (a x+1)^{n/2}}{3 a^2}}{c}\)

Input:

Int[(E^(n*ArcTanh[a*x])*x^4)/(c - a^2*c*x^2),x]
 

Output:

(-1/3*(x^3*(1 + a*x)^(n/2))/(a^2*(1 - a*x)^(n/2)) + (-1/2*(n*x^2*(1 + a*x) 
^(n/2))/(a*(1 - a*x)^(n/2)) + (((1 + a*x)^(n/2)*(6 + 8*n + n^2 + n^3 - a*n 
*(6 + n^2)*x))/(a^2*n*(1 - a*x)^(n/2)) + (2^(n/2)*n*(8 + n^2)*(1 - a*x)^(1 
 - n/2)*Hypergeometric2F1[(2 - n)/2, 1 - n/2, 2 - n/2, (1 - a*x)/2])/(a^2* 
(2 - n)))/(2*a))/(3*a^2))/c
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 111

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

rule 160

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) 
)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b^2*d*e*g - a^2*d*f*h*m - a*b*(d*(f*g 
+ e*h) - c*f*h*(m + 1)) + b*f*h*(b*c - a*d)*(m + 1)*x)*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d*(b*c - a*d)*(m + 1))), x] + Simp[(a*d*f*h*m + b*(d* 
(f*g + e*h) - c*f*h*(m + 2)))/(b^2*d)   Int[(a + b*x)^(m + 1)*(c + d*x)^n, 
x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[m + n + 2, 0] && 
NeQ[m, -1] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])
 

rule 170

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) 
  Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 
) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) 
+ h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] 
 && IntegerQ[m]
 

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])
 

Maple [F]

\[\int \frac {{\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} x^{4}}{-a^{2} c \,x^{2}+c}d x\]

Input:

int(exp(n*arctanh(a*x))*x^4/(-a^2*c*x^2+c),x)
 

Output:

int(exp(n*arctanh(a*x))*x^4/(-a^2*c*x^2+c),x)
 

Fricas [F]

\[ \int \frac {e^{n \text {arctanh}(a x)} x^4}{c-a^2 c x^2} \, dx=\int { -\frac {x^{4} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} c x^{2} - c} \,d x } \] Input:

integrate(exp(n*arctanh(a*x))*x^4/(-a^2*c*x^2+c),x, algorithm="fricas")
 

Output:

integral(-x^4*(-(a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c*x^2 - c), x)
 

Sympy [F]

\[ \int \frac {e^{n \text {arctanh}(a x)} x^4}{c-a^2 c x^2} \, dx=- \frac {\int \frac {x^{4} e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{2} x^{2} - 1}\, dx}{c} \] Input:

integrate(exp(n*atanh(a*x))*x**4/(-a**2*c*x**2+c),x)
 

Output:

-Integral(x**4*exp(n*atanh(a*x))/(a**2*x**2 - 1), x)/c
 

Maxima [F]

\[ \int \frac {e^{n \text {arctanh}(a x)} x^4}{c-a^2 c x^2} \, dx=\int { -\frac {x^{4} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} c x^{2} - c} \,d x } \] Input:

integrate(exp(n*arctanh(a*x))*x^4/(-a^2*c*x^2+c),x, algorithm="maxima")
 

Output:

-integrate(x^4*(-(a*x + 1)/(a*x - 1))^(1/2*n)/(a^2*c*x^2 - c), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{n \text {arctanh}(a x)} x^4}{c-a^2 c x^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(exp(n*arctanh(a*x))*x^4/(-a^2*c*x^2+c),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{1,[0,1,2,2,0]%%%}+%%%{1,[0,1,0,0,0]%%%} / %%%{1,[0,0,4,0,1 
]%%%} Err
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{n \text {arctanh}(a x)} x^4}{c-a^2 c x^2} \, dx=\int \frac {x^4\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{c-a^2\,c\,x^2} \,d x \] Input:

int((x^4*exp(n*atanh(a*x)))/(c - a^2*c*x^2),x)
 

Output:

int((x^4*exp(n*atanh(a*x)))/(c - a^2*c*x^2), x)
 

Reduce [F]

\[ \int \frac {e^{n \text {arctanh}(a x)} x^4}{c-a^2 c x^2} \, dx=-\frac {\int \frac {e^{\mathit {atanh} \left (a x \right ) n} x^{4}}{a^{2} x^{2}-1}d x}{c} \] Input:

int(exp(n*atanh(a*x))*x^4/(-a^2*c*x^2+c),x)
 

Output:

( - int((e**(atanh(a*x)*n)*x**4)/(a**2*x**2 - 1),x))/c