\(\int \frac {e^{-\text {arctanh}(a x)}}{(c-a^2 c x^2)^5} \, dx\) [1237]

Optimal result

Integrand size = 22, antiderivative size = 120 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^5} \, dx=\frac {8 x}{63 c^5 \left (1-a^2 x^2\right )^{7/2}}-\frac {1}{9 a c^5 (1+a x) \left (1-a^2 x^2\right )^{7/2}}+\frac {16 x}{105 c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {64 x}{315 c^5 \left (1-a^2 x^2\right )^{3/2}}+\frac {128 x}{315 c^5 \sqrt {1-a^2 x^2}} \] Output:

8/63*x/c^5/(-a^2*x^2+1)^(7/2)-1/9/a/c^5/(a*x+1)/(-a^2*x^2+1)^(7/2)+16/105* 
x/c^5/(-a^2*x^2+1)^(5/2)+64/315*x/c^5/(-a^2*x^2+1)^(3/2)+128/315*x/c^5/(-a 
^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.76 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^5} \, dx=-\frac {35-280 a x-280 a^2 x^2+560 a^3 x^3+560 a^4 x^4-448 a^5 x^5-448 a^6 x^6+128 a^7 x^7+128 a^8 x^8}{315 a c^5 (1-a x)^{7/2} (1+a x)^{9/2}} \] Input:

Integrate[1/(E^ArcTanh[a*x]*(c - a^2*c*x^2)^5),x]
 

Output:

-1/315*(35 - 280*a*x - 280*a^2*x^2 + 560*a^3*x^3 + 560*a^4*x^4 - 448*a^5*x 
^5 - 448*a^6*x^6 + 128*a^7*x^7 + 128*a^8*x^8)/(a*c^5*(1 - a*x)^(7/2)*(1 + 
a*x)^(9/2))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6689, 454, 209, 209, 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.

\(\displaystyle \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^5} \, dx\)

\(\Big \downarrow \) 6689

\(\displaystyle \frac {\int \frac {1-a x}{\left (1-a^2 x^2\right )^{11/2}}dx}{c^5}\)

\(\Big \downarrow \) 454

\(\displaystyle \frac {\frac {8}{9} \int \frac {1}{\left (1-a^2 x^2\right )^{9/2}}dx-\frac {1-a x}{9 a \left (1-a^2 x^2\right )^{9/2}}}{c^5}\)

\(\Big \downarrow \) 209

\(\displaystyle \frac {\frac {8}{9} \left (\frac {6}{7} \int \frac {1}{\left (1-a^2 x^2\right )^{7/2}}dx+\frac {x}{7 \left (1-a^2 x^2\right )^{7/2}}\right )-\frac {1-a x}{9 a \left (1-a^2 x^2\right )^{9/2}}}{c^5}\)

\(\Big \downarrow \) 209

\(\displaystyle \frac {\frac {8}{9} \left (\frac {6}{7} \left (\frac {4}{5} \int \frac {1}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {x}{7 \left (1-a^2 x^2\right )^{7/2}}\right )-\frac {1-a x}{9 a \left (1-a^2 x^2\right )^{9/2}}}{c^5}\)

\(\Big \downarrow \) 209

\(\displaystyle \frac {\frac {8}{9} \left (\frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {x}{7 \left (1-a^2 x^2\right )^{7/2}}\right )-\frac {1-a x}{9 a \left (1-a^2 x^2\right )^{9/2}}}{c^5}\)

\(\Big \downarrow \) 208

\(\displaystyle \frac {\frac {8}{9} \left (\frac {x}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac {6}{7} \left (\frac {x}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5} \left (\frac {2 x}{3 \sqrt {1-a^2 x^2}}+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )\right )\right )-\frac {1-a x}{9 a \left (1-a^2 x^2\right )^{9/2}}}{c^5}\)

Input:

Int[1/(E^ArcTanh[a*x]*(c - a^2*c*x^2)^5),x]
 

Output:

(-1/9*(1 - a*x)/(a*(1 - a^2*x^2)^(9/2)) + (8*(x/(7*(1 - a^2*x^2)^(7/2)) + 
(6*(x/(5*(1 - a^2*x^2)^(5/2)) + (4*(x/(3*(1 - a^2*x^2)^(3/2)) + (2*x)/(3*S 
qrt[1 - a^2*x^2])))/5))/7))/9)/c^5
 

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 6689

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] && ILtQ[(n - 1)/2, 0] &&  !In 
tegerQ[p - n/2]
 

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.75

method	result	size
gosper	\(-\frac {128 a^{8} x^{8}+128 a^{7} x^{7}-448 x^{6} a^{6}-448 a^{5} x^{5}+560 a^{4} x^{4}+560 a^{3} x^{3}-280 a^{2} x^{2}-280 a x +35}{315 a \,c^{5} \left (a x +1\right ) \left (-a^{2} x^{2}+1\right )^{\frac {7}{2}}}\)	\(90\)
trager	\(-\frac {\left (128 a^{8} x^{8}+128 a^{7} x^{7}-448 x^{6} a^{6}-448 a^{5} x^{5}+560 a^{4} x^{4}+560 a^{3} x^{3}-280 a^{2} x^{2}-280 a x +35\right ) \sqrt {-a^{2} x^{2}+1}}{315 c^{5} \left (a x +1\right )^{5} \left (a x -1\right )^{4} a}\)	\(97\)
orering	\(\frac {\left (128 a^{8} x^{8}+128 a^{7} x^{7}-448 x^{6} a^{6}-448 a^{5} x^{5}+560 a^{4} x^{4}+560 a^{3} x^{3}-280 a^{2} x^{2}-280 a x +35\right ) \left (a x -1\right ) \sqrt {-a^{2} x^{2}+1}}{315 a \left (-a^{2} c \,x^{2}+c \right )^{5}}\)	\(98\)
default	\(\text {Expression too large to display}\)	\(1002\)

Input:

int(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^5,x,method=_RETURNVERBOSE)
 

Output:

-1/315/a/c^5/(a*x+1)/(-a^2*x^2+1)^(7/2)*(128*a^8*x^8+128*a^7*x^7-448*a^6*x 
^6-448*a^5*x^5+560*a^4*x^4+560*a^3*x^3-280*a^2*x^2-280*a*x+35)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 249 vs. \(2 (100) = 200\).

Time = 0.21 (sec) , antiderivative size = 249, normalized size of antiderivative = 2.08 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^5} \, dx=-\frac {35 \, a^{9} x^{9} + 35 \, a^{8} x^{8} - 140 \, a^{7} x^{7} - 140 \, a^{6} x^{6} + 210 \, a^{5} x^{5} + 210 \, a^{4} x^{4} - 140 \, a^{3} x^{3} - 140 \, a^{2} x^{2} + 35 \, a x + {\left (128 \, a^{8} x^{8} + 128 \, a^{7} x^{7} - 448 \, a^{6} x^{6} - 448 \, a^{5} x^{5} + 560 \, a^{4} x^{4} + 560 \, a^{3} x^{3} - 280 \, a^{2} x^{2} - 280 \, a x + 35\right )} \sqrt {-a^{2} x^{2} + 1} + 35}{315 \, {\left (a^{10} c^{5} x^{9} + a^{9} c^{5} x^{8} - 4 \, a^{8} c^{5} x^{7} - 4 \, a^{7} c^{5} x^{6} + 6 \, a^{6} c^{5} x^{5} + 6 \, a^{5} c^{5} x^{4} - 4 \, a^{4} c^{5} x^{3} - 4 \, a^{3} c^{5} x^{2} + a^{2} c^{5} x + a c^{5}\right )}} \] Input:

integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^5,x, algorithm="fric 
as")
 

Output:

-1/315*(35*a^9*x^9 + 35*a^8*x^8 - 140*a^7*x^7 - 140*a^6*x^6 + 210*a^5*x^5 
+ 210*a^4*x^4 - 140*a^3*x^3 - 140*a^2*x^2 + 35*a*x + (128*a^8*x^8 + 128*a^ 
7*x^7 - 448*a^6*x^6 - 448*a^5*x^5 + 560*a^4*x^4 + 560*a^3*x^3 - 280*a^2*x^ 
2 - 280*a*x + 35)*sqrt(-a^2*x^2 + 1) + 35)/(a^10*c^5*x^9 + a^9*c^5*x^8 - 4 
*a^8*c^5*x^7 - 4*a^7*c^5*x^6 + 6*a^6*c^5*x^5 + 6*a^5*c^5*x^4 - 4*a^4*c^5*x 
^3 - 4*a^3*c^5*x^2 + a^2*c^5*x + a*c^5)
 

Sympy [F]

\[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^5} \, dx=\frac {\int \frac {1}{a^{9} x^{9} \sqrt {- a^{2} x^{2} + 1} + a^{8} x^{8} \sqrt {- a^{2} x^{2} + 1} - 4 a^{7} x^{7} \sqrt {- a^{2} x^{2} + 1} - 4 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 6 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 6 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 4 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{5}} \] Input:

integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(-a**2*c*x**2+c)**5,x)
 

Output:

Integral(1/(a**9*x**9*sqrt(-a**2*x**2 + 1) + a**8*x**8*sqrt(-a**2*x**2 + 1 
) - 4*a**7*x**7*sqrt(-a**2*x**2 + 1) - 4*a**6*x**6*sqrt(-a**2*x**2 + 1) + 
6*a**5*x**5*sqrt(-a**2*x**2 + 1) + 6*a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a* 
*3*x**3*sqrt(-a**2*x**2 + 1) - 4*a**2*x**2*sqrt(-a**2*x**2 + 1) + a*x*sqrt 
(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x)/c**5
 

Maxima [F]

\[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^5} \, dx=\int { -\frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a^{2} c x^{2} - c\right )}^{5} {\left (a x + 1\right )}} \,d x } \] Input:

integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^5,x, algorithm="maxi 
ma")
 

Output:

-integrate(sqrt(-a^2*x^2 + 1)/((a^2*c*x^2 - c)^5*(a*x + 1)), x)
 

Giac [F]

\[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^5} \, dx=\int { -\frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a^{2} c x^{2} - c\right )}^{5} {\left (a x + 1\right )}} \,d x } \] Input:

integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^5,x, algorithm="giac 
")
 

Output:

integrate(-sqrt(-a^2*x^2 + 1)/((a^2*c*x^2 - c)^5*(a*x + 1)), x)
 

Mupad [B] (verification not implemented)

Time = 23.41 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.48 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^5} \, dx=\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {53\,x}{252\,c^5}-\frac {5}{36\,a\,c^5}\right )}{{\left (a\,x-1\right )}^4\,{\left (a\,x+1\right )}^4}-\frac {\sqrt {1-a^2\,x^2}}{144\,a\,c^5\,{\left (a\,x+1\right )}^5}-\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {733\,x}{5040\,c^5}+\frac {5}{144\,a\,c^5}\right )}{{\left (a\,x-1\right )}^3\,{\left (a\,x+1\right )}^3}-\frac {128\,x\,\sqrt {1-a^2\,x^2}}{315\,c^5\,\left (a\,x-1\right )\,\left (a\,x+1\right )}+\frac {64\,x\,\sqrt {1-a^2\,x^2}}{315\,c^5\,{\left (a\,x-1\right )}^2\,{\left (a\,x+1\right )}^2} \] Input:

int((1 - a^2*x^2)^(1/2)/((c - a^2*c*x^2)^5*(a*x + 1)),x)
 

Output:

((1 - a^2*x^2)^(1/2)*((53*x)/(252*c^5) - 5/(36*a*c^5)))/((a*x - 1)^4*(a*x 
+ 1)^4) - (1 - a^2*x^2)^(1/2)/(144*a*c^5*(a*x + 1)^5) - ((1 - a^2*x^2)^(1/ 
2)*((733*x)/(5040*c^5) + 5/(144*a*c^5)))/((a*x - 1)^3*(a*x + 1)^3) - (128* 
x*(1 - a^2*x^2)^(1/2))/(315*c^5*(a*x - 1)*(a*x + 1)) + (64*x*(1 - a^2*x^2) 
^(1/2))/(315*c^5*(a*x - 1)^2*(a*x + 1)^2)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.32 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^5} \, dx=\frac {280 \sqrt {-a^{2} x^{2}+1}\, a^{7} x^{7}+280 \sqrt {-a^{2} x^{2}+1}\, a^{6} x^{6}-840 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}-840 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+840 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+840 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-280 \sqrt {-a^{2} x^{2}+1}\, a x -280 \sqrt {-a^{2} x^{2}+1}+128 a^{8} x^{8}+128 a^{7} x^{7}-448 a^{6} x^{6}-448 a^{5} x^{5}+560 a^{4} x^{4}+560 a^{3} x^{3}-280 a^{2} x^{2}-280 a x +35}{315 \sqrt {-a^{2} x^{2}+1}\, a \,c^{5} \left (a^{7} x^{7}+a^{6} x^{6}-3 a^{5} x^{5}-3 a^{4} x^{4}+3 a^{3} x^{3}+3 a^{2} x^{2}-a x -1\right )} \] Input:

int(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^5,x)
 

Output:

(280*sqrt( - a**2*x**2 + 1)*a**7*x**7 + 280*sqrt( - a**2*x**2 + 1)*a**6*x* 
*6 - 840*sqrt( - a**2*x**2 + 1)*a**5*x**5 - 840*sqrt( - a**2*x**2 + 1)*a** 
4*x**4 + 840*sqrt( - a**2*x**2 + 1)*a**3*x**3 + 840*sqrt( - a**2*x**2 + 1) 
*a**2*x**2 - 280*sqrt( - a**2*x**2 + 1)*a*x - 280*sqrt( - a**2*x**2 + 1) + 
 128*a**8*x**8 + 128*a**7*x**7 - 448*a**6*x**6 - 448*a**5*x**5 + 560*a**4* 
x**4 + 560*a**3*x**3 - 280*a**2*x**2 - 280*a*x + 35)/(315*sqrt( - a**2*x** 
2 + 1)*a*c**5*(a**7*x**7 + a**6*x**6 - 3*a**5*x**5 - 3*a**4*x**4 + 3*a**3* 
x**3 + 3*a**2*x**2 - a*x - 1))