\(\int e^{-\frac {1}{2} \text {arctanh}(a x)} (c x)^m \, dx\) [143]

Optimal result

Integrand size = 16, antiderivative size = 36 \[ \int e^{-\frac {1}{2} \text {arctanh}(a x)} (c x)^m \, dx=\frac {(c x)^{1+m} \operatorname {AppellF1}\left (1+m,-\frac {1}{4},\frac {1}{4},2+m,a x,-a x\right )}{c (1+m)} \] Output:

(c*x)^(1+m)*AppellF1(1+m,-1/4,1/4,2+m,a*x,-a*x)/c/(1+m)
 

Mathematica [F]

\[ \int e^{-\frac {1}{2} \text {arctanh}(a x)} (c x)^m \, dx=\int e^{-\frac {1}{2} \text {arctanh}(a x)} (c x)^m \, dx \] Input:

Integrate[(c*x)^m/E^(ArcTanh[a*x]/2),x]
 

Output:

Integrate[(c*x)^m/E^(ArcTanh[a*x]/2), x]
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6676, 150}

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.

\(\displaystyle \int e^{-\frac {1}{2} \text {arctanh}(a x)} (c x)^m \, dx\)

\(\Big \downarrow \) 6676

\(\displaystyle \int \frac {\sqrt [4]{1-a x} (c x)^m}{\sqrt [4]{a x+1}}dx\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {(c x)^{m+1} \operatorname {AppellF1}\left (m+1,-\frac {1}{4},\frac {1}{4},m+2,a x,-a x\right )}{c (m+1)}\)

Input:

Int[(c*x)^m/E^(ArcTanh[a*x]/2),x]
 

Output:

((c*x)^(1 + m)*AppellF1[1 + m, -1/4, 1/4, 2 + m, a*x, -(a*x)])/(c*(1 + m))
 

Defintions of rubi rules used

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 6676

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_.)*(x_))^(m_.), x_Symbol] :> Int[(c*x) 
^m*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c, m, n}, x] &&  !Int 
egerQ[(n - 1)/2]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((c*x)**m/sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int e^{-\frac {1}{2} \text {arctanh}(a x)} (c x)^m \, dx =\text {Too large to display} \] Input:

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

Output:

(c**m*(4*x**m*sqrt(a*x + 1)*( - a**2*x**2 + 1)**(1/4)*m + 4*int((x**m*(a*x 
 + 1)**(3/4)*( - a*x + 1)**(1/4)*x)/(4*a**2*m**2*x**2 + 4*a**2*m*x**2 + a* 
*2*x**2 - 4*m**2 - 4*m - 1),x)*a**2*m**2 + 4*int((x**m*(a*x + 1)**(3/4)*( 
- a*x + 1)**(1/4)*x)/(4*a**2*m**2*x**2 + 4*a**2*m*x**2 + a**2*x**2 - 4*m** 
2 - 4*m - 1),x)*a**2*m + int((x**m*(a*x + 1)**(3/4)*( - a*x + 1)**(1/4)*x) 
/(4*a**2*m**2*x**2 + 4*a**2*m*x**2 + a**2*x**2 - 4*m**2 - 4*m - 1),x)*a**2 
 + 16*int((x**m*(a*x + 1)**(3/4)*( - a*x + 1)**(1/4))/(4*a**2*m**2*x**3 + 
4*a**2*m*x**3 + a**2*x**3 - 4*m**2*x - 4*m*x - x),x)*m**4 + 16*int((x**m*( 
a*x + 1)**(3/4)*( - a*x + 1)**(1/4))/(4*a**2*m**2*x**3 + 4*a**2*m*x**3 + a 
**2*x**3 - 4*m**2*x - 4*m*x - x),x)*m**3 + 4*int((x**m*(a*x + 1)**(3/4)*( 
- a*x + 1)**(1/4))/(4*a**2*m**2*x**3 + 4*a**2*m*x**3 + a**2*x**3 - 4*m**2* 
x - 4*m*x - x),x)*m**2 - 16*int((x**m*(a*x + 1)**(3/4)*( - a*x + 1)**(1/4) 
)/(4*a**2*m**2*x**2 + 4*a**2*m*x**2 + a**2*x**2 - 4*m**2 - 4*m - 1),x)*a*m 
**4 - 24*int((x**m*(a*x + 1)**(3/4)*( - a*x + 1)**(1/4))/(4*a**2*m**2*x**2 
 + 4*a**2*m*x**2 + a**2*x**2 - 4*m**2 - 4*m - 1),x)*a*m**3 - 16*int((x**m* 
(a*x + 1)**(3/4)*( - a*x + 1)**(1/4))/(4*a**2*m**2*x**2 + 4*a**2*m*x**2 + 
a**2*x**2 - 4*m**2 - 4*m - 1),x)*a*m**2 - 6*int((x**m*(a*x + 1)**(3/4)*( - 
 a*x + 1)**(1/4))/(4*a**2*m**2*x**2 + 4*a**2*m*x**2 + a**2*x**2 - 4*m**2 - 
 4*m - 1),x)*a*m - int((x**m*(a*x + 1)**(3/4)*( - a*x + 1)**(1/4))/(4*a**2 
*m**2*x**2 + 4*a**2*m*x**2 + a**2*x**2 - 4*m**2 - 4*m - 1),x)*a))/(a*(4...