\(\int \frac {e^{\text {arctanh}(a x)}}{(c-a^2 c x^2)^3} \, dx\) [951]

Optimal result

Integrand size = 20, antiderivative size = 74 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {1+a x}{5 a c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 x}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {8 x}{15 c^3 \sqrt {1-a^2 x^2}} \] Output:

1/5*(a*x+1)/a/c^3/(-a^2*x^2+1)^(5/2)+4/15*x/c^3/(-a^2*x^2+1)^(3/2)+8/15*x/ 
c^3/(-a^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.80 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {3+12 a x-12 a^2 x^2-8 a^3 x^3+8 a^4 x^4}{15 a c^3 (1-a x)^{5/2} (1+a x)^{3/2}} \] Input:

Integrate[E^ArcTanh[a*x]/(c - a^2*c*x^2)^3,x]
 

Output:

(3 + 12*a*x - 12*a^2*x^2 - 8*a^3*x^3 + 8*a^4*x^4)/(15*a*c^3*(1 - a*x)^(5/2 
)*(1 + a*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6688, 454, 209, 208}

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.

\(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx\)

\(\Big \downarrow \) 6688

\(\displaystyle \frac {\int \frac {a x+1}{\left (1-a^2 x^2\right )^{7/2}}dx}{c^3}\)

\(\Big \downarrow \) 454

\(\displaystyle \frac {\frac {4}{5} \int \frac {1}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {a x+1}{5 a \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 209

\(\displaystyle \frac {\frac {4}{5} \left (\frac {2}{3} \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {a x+1}{5 a \left (1-a^2 x^2\right )^{5/2}}}{c^3}\)

\(\Big \downarrow \) 208

\(\displaystyle \frac {\frac {a x+1}{5 a \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {2 x}{3 \sqrt {1-a^2 x^2}}+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )}{c^3}\)

Input:

Int[E^ArcTanh[a*x]/(c - a^2*c*x^2)^3,x]
 

Output:

((1 + a*x)/(5*a*(1 - a^2*x^2)^(5/2)) + (4*(x/(3*(1 - a^2*x^2)^(3/2)) + (2* 
x)/(3*Sqrt[1 - a^2*x^2])))/5)/c^3
 

Defintions of rubi rules used

rule 208

Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), 
x] /; FreeQ[{a, b}, x]
 

rule 209

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
 

rule 454

Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*d 
 - b*c*x)/(2*a*b*(p + 1)))*(a + b*x^2)^(p + 1), x] + Simp[c*((2*p + 3)/(2*a 
*(p + 1)))   Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d}, x] && L 
tQ[p, -1] && NeQ[p, -3/2]
 

rule 6688

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

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.78

method	result	size
gosper	\(-\frac {8 a^{4} x^{4}-8 a^{3} x^{3}-12 a^{2} x^{2}+12 a x +3}{15 a \,c^{3} \left (a x -1\right ) \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\)	\(58\)
trager	\(-\frac {\left (8 a^{4} x^{4}-8 a^{3} x^{3}-12 a^{2} x^{2}+12 a x +3\right ) \sqrt {-a^{2} x^{2}+1}}{15 c^{3} \left (a x -1\right )^{3} \left (a x +1\right )^{2} a}\)	\(65\)
orering	\(-\frac {\left (8 a^{4} x^{4}-8 a^{3} x^{3}-12 a^{2} x^{2}+12 a x +3\right ) \left (a x -1\right ) \left (a x +1\right )^{2}}{15 a \sqrt {-a^{2} x^{2}+1}\, \left (-a^{2} c \,x^{2}+c \right )^{3}}\)	\(73\)
default	\(-\frac {\frac {\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{4 a^{3}}-\frac {\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{4 a^{2}}+\frac {3 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{16 a^{2} \left (x -\frac {1}{a}\right )}-\frac {-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 a \left (x +\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{3 \left (x +\frac {1}{a}\right )}}{8 a^{2}}+\frac {3 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{16 a^{2} \left (x +\frac {1}{a}\right )}}{c^{3}}\)	\(371\)

Input:

int((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/15/a/c^3/(a*x-1)/(-a^2*x^2+1)^(3/2)*(8*a^4*x^4-8*a^3*x^3-12*a^2*x^2+12* 
a*x+3)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (62) = 124\).

Time = 0.09 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.95 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {3 \, a^{5} x^{5} - 3 \, a^{4} x^{4} - 6 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + 3 \, a x - {\left (8 \, a^{4} x^{4} - 8 \, a^{3} x^{3} - 12 \, a^{2} x^{2} + 12 \, a x + 3\right )} \sqrt {-a^{2} x^{2} + 1} - 3}{15 \, {\left (a^{6} c^{3} x^{5} - a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} + 2 \, a^{3} c^{3} x^{2} + a^{2} c^{3} x - a c^{3}\right )}} \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^3,x, algorithm="fricas 
")
 

Output:

1/15*(3*a^5*x^5 - 3*a^4*x^4 - 6*a^3*x^3 + 6*a^2*x^2 + 3*a*x - (8*a^4*x^4 - 
 8*a^3*x^3 - 12*a^2*x^2 + 12*a*x + 3)*sqrt(-a^2*x^2 + 1) - 3)/(a^6*c^3*x^5 
 - a^5*c^3*x^4 - 2*a^4*c^3*x^3 + 2*a^3*c^3*x^2 + a^2*c^3*x - a*c^3)
 

Sympy [F]

\[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {\int \frac {a x}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \] Input:

integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(-a**2*c*x**2+c)**3,x)
 

Output:

(Integral(a*x/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x* 
*2 + 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + I 
ntegral(1/(-a**6*x**6*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a**2*x**2 + 
 1) - 3*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**3
 

Maxima [F]

\[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\int { -\frac {a x + 1}{{\left (a^{2} c x^{2} - c\right )}^{3} \sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^3,x, algorithm="maxima 
")
 

Output:

-integrate((a*x + 1)/((a^2*c*x^2 - c)^3*sqrt(-a^2*x^2 + 1)), x)
 

Giac [F]

\[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\int { -\frac {a x + 1}{{\left (a^{2} c x^{2} - c\right )}^{3} \sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^3,x, algorithm="giac")
 

Output:

integrate(-(a*x + 1)/((a^2*c*x^2 - c)^3*sqrt(-a^2*x^2 + 1)), x)
 

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 279, normalized size of antiderivative = 3.77 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {7\,a\,\sqrt {1-a^2\,x^2}}{60\,\left (a^4\,c^3\,x^2-2\,a^3\,c^3\,x+a^2\,c^3\right )}-\frac {a\,\sqrt {1-a^2\,x^2}}{24\,\left (a^4\,c^3\,x^2+2\,a^3\,c^3\,x+a^2\,c^3\right )}+\frac {11\,\sqrt {1-a^2\,x^2}}{48\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}+\frac {c^3\,\sqrt {-a^2}}{a}\right )}+\frac {73\,\sqrt {1-a^2\,x^2}}{240\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}\right )}+\frac {\sqrt {1-a^2\,x^2}}{20\,\sqrt {-a^2}\,\left (3\,c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}+a^2\,c^3\,x^3\,\sqrt {-a^2}-3\,a\,c^3\,x^2\,\sqrt {-a^2}\right )} \] Input:

int((a*x + 1)/((c - a^2*c*x^2)^3*(1 - a^2*x^2)^(1/2)),x)
 

Output:

(7*a*(1 - a^2*x^2)^(1/2))/(60*(a^2*c^3 - 2*a^3*c^3*x + a^4*c^3*x^2)) - (a* 
(1 - a^2*x^2)^(1/2))/(24*(a^2*c^3 + 2*a^3*c^3*x + a^4*c^3*x^2)) + (11*(1 - 
 a^2*x^2)^(1/2))/(48*(-a^2)^(1/2)*(c^3*x*(-a^2)^(1/2) + (c^3*(-a^2)^(1/2)) 
/a)) + (73*(1 - a^2*x^2)^(1/2))/(240*(-a^2)^(1/2)*(c^3*x*(-a^2)^(1/2) - (c 
^3*(-a^2)^(1/2))/a)) + (1 - a^2*x^2)^(1/2)/(20*(-a^2)^(1/2)*(3*c^3*x*(-a^2 
)^(1/2) - (c^3*(-a^2)^(1/2))/a + a^2*c^3*x^3*(-a^2)^(1/2) - 3*a*c^3*x^2*(- 
a^2)^(1/2)))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.89 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx=\frac {-12 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+12 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+12 \sqrt {-a^{2} x^{2}+1}\, a x -12 \sqrt {-a^{2} x^{2}+1}+8 a^{4} x^{4}-8 a^{3} x^{3}-12 a^{2} x^{2}+12 a x +3}{15 \sqrt {-a^{2} x^{2}+1}\, a \,c^{3} \left (a^{3} x^{3}-a^{2} x^{2}-a x +1\right )} \] Input:

int((a*x+1)/(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^3,x)
 

Output:

( - 12*sqrt( - a**2*x**2 + 1)*a**3*x**3 + 12*sqrt( - a**2*x**2 + 1)*a**2*x 
**2 + 12*sqrt( - a**2*x**2 + 1)*a*x - 12*sqrt( - a**2*x**2 + 1) + 8*a**4*x 
**4 - 8*a**3*x**3 - 12*a**2*x**2 + 12*a*x + 3)/(15*sqrt( - a**2*x**2 + 1)* 
a*c**3*(a**3*x**3 - a**2*x**2 - a*x + 1))