\(\int \frac {e^{\text {arctanh}(a x)}}{(c-a^2 c x^2)^5} \, dx\) [941]

Optimal result

Integrand size = 20, antiderivative size = 118 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^5} \, dx=\frac {1+a x}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}}+\frac {8 x}{63 c^5 \left (1-a^2 x^2\right )^{7/2}}+\frac {16 x}{105 c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {64 x}{315 c^5 \left (1-a^2 x^2\right )^{3/2}}+\frac {128 x}{315 c^5 \sqrt {1-a^2 x^2}} \] Output:

1/9*(a*x+1)/a/c^5/(-a^2*x^2+1)^(9/2)+8/63*x/c^5/(-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 = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.77 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^5} \, dx=\frac {35+280 a x-280 a^2 x^2-560 a^3 x^3+560 a^4 x^4+448 a^5 x^5-448 a^6 x^6-128 a^7 x^7+128 a^8 x^8}{315 a c^5 (1-a x)^{9/2} (1+a x)^{7/2}} \] Input:

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

Output:

(35 + 280*a*x - 280*a^2*x^2 - 560*a^3*x^3 + 560*a^4*x^4 + 448*a^5*x^5 - 44 
8*a^6*x^6 - 128*a^7*x^7 + 128*a^8*x^8)/(315*a*c^5*(1 - a*x)^(9/2)*(1 + a*x 
)^(7/2))
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6688, 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.

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

\(\Big \downarrow \) 6688

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

\(\Big \downarrow \) 454

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

\(\Big \downarrow \) 209

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

\(\Big \downarrow \) 209

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

\(\Big \downarrow \) 209

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

\(\Big \downarrow \) 208

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

Input:

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

Output:

((1 + a*x)/(9*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*Sqrt 
[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 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 = 0.42 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.76

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	\(-\frac {\left (128 a^{8} x^{8}-128 a^{7} x^{7}-448 x^{6} a^{6}+448 a^{5} x^{5}+560 a^{4} x^{4}-560 a^{3} x^{3}-280 a^{2} x^{2}+280 a x +35\right ) \left (a x -1\right ) \left (a x +1\right )^{2}}{315 a \sqrt {-a^{2} x^{2}+1}\, \left (-a^{2} c \,x^{2}+c \right )^{5}}\)	\(105\)
default	\(\text {Expression too large to display}\)	\(1029\)

Input:

int((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. 252 vs. \(2 (98) = 196\).

Time = 0.16 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.14 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^5} \, dx=\frac {35 \, a^{9} x^{9} - 35 \, a^{8} x^{8} - 140 \, a^{7} x^{7} + 140 \, a^{6} x^{6} + 210 \, a^{5} x^{5} - 210 \, a^{4} x^{4} - 140 \, a^{3} x^{3} + 140 \, a^{2} x^{2} + 35 \, a x - {\left (128 \, a^{8} x^{8} - 128 \, a^{7} x^{7} - 448 \, a^{6} x^{6} + 448 \, a^{5} x^{5} + 560 \, a^{4} x^{4} - 560 \, a^{3} x^{3} - 280 \, a^{2} x^{2} + 280 \, a x + 35\right )} \sqrt {-a^{2} x^{2} + 1} - 35}{315 \, {\left (a^{10} c^{5} x^{9} - a^{9} c^{5} x^{8} - 4 \, a^{8} c^{5} x^{7} + 4 \, a^{7} c^{5} x^{6} + 6 \, a^{6} c^{5} x^{5} - 6 \, a^{5} c^{5} x^{4} - 4 \, a^{4} c^{5} x^{3} + 4 \, a^{3} c^{5} x^{2} + a^{2} c^{5} x - a c^{5}\right )}} \] Input:

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

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]

\[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^5} \, dx=\frac {\int \frac {a x}{- a^{10} x^{10} \sqrt {- a^{2} x^{2} + 1} + 5 a^{8} x^{8} \sqrt {- a^{2} x^{2} + 1} - 10 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 10 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 5 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{10} x^{10} \sqrt {- a^{2} x^{2} + 1} + 5 a^{8} x^{8} \sqrt {- a^{2} x^{2} + 1} - 10 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 10 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 5 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{5}} \] Input:

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

Output:

(Integral(a*x/(-a**10*x**10*sqrt(-a**2*x**2 + 1) + 5*a**8*x**8*sqrt(-a**2* 
x**2 + 1) - 10*a**6*x**6*sqrt(-a**2*x**2 + 1) + 10*a**4*x**4*sqrt(-a**2*x* 
*2 + 1) - 5*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + I 
ntegral(1/(-a**10*x**10*sqrt(-a**2*x**2 + 1) + 5*a**8*x**8*sqrt(-a**2*x**2 
 + 1) - 10*a**6*x**6*sqrt(-a**2*x**2 + 1) + 10*a**4*x**4*sqrt(-a**2*x**2 + 
 1) - 5*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**5
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 14.31 (sec) , antiderivative size = 783, normalized size of antiderivative = 6.64 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^5} \, dx=\frac {751\,a\,\sqrt {1-a^2\,x^2}}{10080\,\left (a^4\,c^5\,x^2-2\,a^3\,c^5\,x+a^2\,c^5\right )}-\frac {19\,a\,\sqrt {1-a^2\,x^2}}{384\,\left (a^4\,c^5\,x^2+2\,a^3\,c^5\,x+a^2\,c^5\right )}+\frac {\sqrt {1-a^2\,x^2}}{144\,\sqrt {-a^2}\,\left (5\,c^5\,x\,\sqrt {-a^2}-\frac {c^5\,\sqrt {-a^2}}{a}+10\,a^2\,c^5\,x^3\,\sqrt {-a^2}-5\,a^3\,c^5\,x^4\,\sqrt {-a^2}+a^4\,c^5\,x^5\,\sqrt {-a^2}-10\,a\,c^5\,x^2\,\sqrt {-a^2}\right )}+\frac {a^3\,\sqrt {1-a^2\,x^2}}{140\,\left (a^6\,c^5\,x^2-2\,a^5\,c^5\,x+a^4\,c^5\right )}-\frac {a^3\,\sqrt {1-a^2\,x^2}}{560\,\left (a^6\,c^5\,x^2+2\,a^5\,c^5\,x+a^4\,c^5\right )}+\frac {a\,\sqrt {1-a^2\,x^2}}{56\,\left (a^6\,c^5\,x^4-4\,a^5\,c^5\,x^3+6\,a^4\,c^5\,x^2-4\,a^3\,c^5\,x+a^2\,c^5\right )}-\frac {a\,\sqrt {1-a^2\,x^2}}{224\,\left (a^6\,c^5\,x^4+4\,a^5\,c^5\,x^3+6\,a^4\,c^5\,x^2+4\,a^3\,c^5\,x+a^2\,c^5\right )}+\frac {5053\,\sqrt {1-a^2\,x^2}}{26880\,\sqrt {-a^2}\,\left (c^5\,x\,\sqrt {-a^2}+\frac {c^5\,\sqrt {-a^2}}{a}\right )}+\frac {17609\,\sqrt {1-a^2\,x^2}}{80640\,\sqrt {-a^2}\,\left (c^5\,x\,\sqrt {-a^2}-\frac {c^5\,\sqrt {-a^2}}{a}\right )}+\frac {41\,\sqrt {1-a^2\,x^2}}{2240\,\sqrt {-a^2}\,\left (3\,c^5\,x\,\sqrt {-a^2}+\frac {c^5\,\sqrt {-a^2}}{a}+a^2\,c^5\,x^3\,\sqrt {-a^2}+3\,a\,c^5\,x^2\,\sqrt {-a^2}\right )}+\frac {149\,\sqrt {1-a^2\,x^2}}{3360\,\sqrt {-a^2}\,\left (3\,c^5\,x\,\sqrt {-a^2}-\frac {c^5\,\sqrt {-a^2}}{a}+a^2\,c^5\,x^3\,\sqrt {-a^2}-3\,a\,c^5\,x^2\,\sqrt {-a^2}\right )}+\frac {a^2\,\sqrt {1-a^2\,x^2}}{252\,\left (a^7\,c^5\,x^4-4\,a^6\,c^5\,x^3+6\,a^5\,c^5\,x^2-4\,a^4\,c^5\,x+a^3\,c^5\right )} \] Input:

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

Output:

(751*a*(1 - a^2*x^2)^(1/2))/(10080*(a^2*c^5 - 2*a^3*c^5*x + a^4*c^5*x^2)) 
- (19*a*(1 - a^2*x^2)^(1/2))/(384*(a^2*c^5 + 2*a^3*c^5*x + a^4*c^5*x^2)) + 
 (1 - a^2*x^2)^(1/2)/(144*(-a^2)^(1/2)*(5*c^5*x*(-a^2)^(1/2) - (c^5*(-a^2) 
^(1/2))/a + 10*a^2*c^5*x^3*(-a^2)^(1/2) - 5*a^3*c^5*x^4*(-a^2)^(1/2) + a^4 
*c^5*x^5*(-a^2)^(1/2) - 10*a*c^5*x^2*(-a^2)^(1/2))) + (a^3*(1 - a^2*x^2)^( 
1/2))/(140*(a^4*c^5 - 2*a^5*c^5*x + a^6*c^5*x^2)) - (a^3*(1 - a^2*x^2)^(1/ 
2))/(560*(a^4*c^5 + 2*a^5*c^5*x + a^6*c^5*x^2)) + (a*(1 - a^2*x^2)^(1/2))/ 
(56*(a^2*c^5 - 4*a^3*c^5*x + 6*a^4*c^5*x^2 - 4*a^5*c^5*x^3 + a^6*c^5*x^4)) 
 - (a*(1 - a^2*x^2)^(1/2))/(224*(a^2*c^5 + 4*a^3*c^5*x + 6*a^4*c^5*x^2 + 4 
*a^5*c^5*x^3 + a^6*c^5*x^4)) + (5053*(1 - a^2*x^2)^(1/2))/(26880*(-a^2)^(1 
/2)*(c^5*x*(-a^2)^(1/2) + (c^5*(-a^2)^(1/2))/a)) + (17609*(1 - a^2*x^2)^(1 
/2))/(80640*(-a^2)^(1/2)*(c^5*x*(-a^2)^(1/2) - (c^5*(-a^2)^(1/2))/a)) + (4 
1*(1 - a^2*x^2)^(1/2))/(2240*(-a^2)^(1/2)*(3*c^5*x*(-a^2)^(1/2) + (c^5*(-a 
^2)^(1/2))/a + a^2*c^5*x^3*(-a^2)^(1/2) + 3*a*c^5*x^2*(-a^2)^(1/2))) + (14 
9*(1 - a^2*x^2)^(1/2))/(3360*(-a^2)^(1/2)*(3*c^5*x*(-a^2)^(1/2) - (c^5*(-a 
^2)^(1/2))/a + a^2*c^5*x^3*(-a^2)^(1/2) - 3*a*c^5*x^2*(-a^2)^(1/2))) + (a^ 
2*(1 - a^2*x^2)^(1/2))/(252*(a^3*c^5 - 4*a^4*c^5*x + 6*a^5*c^5*x^2 - 4*a^6 
*c^5*x^3 + a^7*c^5*x^4))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.37 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^5} \, dx=\frac {-280 \sqrt {-a^{2} x^{2}+1}\, a^{7} x^{7}+280 \sqrt {-a^{2} x^{2}+1}\, a^{6} x^{6}+840 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}-840 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-840 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+840 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+280 \sqrt {-a^{2} x^{2}+1}\, a x -280 \sqrt {-a^{2} x^{2}+1}+128 a^{8} x^{8}-128 a^{7} x^{7}-448 a^{6} x^{6}+448 a^{5} x^{5}+560 a^{4} x^{4}-560 a^{3} x^{3}-280 a^{2} x^{2}+280 a x +35}{315 \sqrt {-a^{2} x^{2}+1}\, a \,c^{5} \left (a^{7} x^{7}-a^{6} x^{6}-3 a^{5} x^{5}+3 a^{4} x^{4}+3 a^{3} x^{3}-3 a^{2} x^{2}-a x +1\right )} \] Input:

int((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))