\(\int \frac {e^{-3 \text {arctanh}(a x)}}{(c-\frac {c}{a x})^5} \, dx\) [541]

Optimal result

Integrand size = 22, antiderivative size = 129 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=-\frac {1}{5 a c^5 (1-a x)^2 \sqrt {1-a^2 x^2}}+\frac {22}{15 a c^5 (1-a x) \sqrt {1-a^2 x^2}}-\frac {2 (30+23 a x)}{15 a c^5 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a c^5}+\frac {2 \arcsin (a x)}{a c^5} \] Output:

-1/5/a/c^5/(-a*x+1)^2/(-a^2*x^2+1)^(1/2)+22/15/a/c^5/(-a*x+1)/(-a^2*x^2+1) 
^(1/2)-2/15*(23*a*x+30)/a/c^5/(-a^2*x^2+1)^(1/2)-(-a^2*x^2+1)^(1/2)/a/c^5+ 
2*arcsin(a*x)/a/c^5
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.59 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=\frac {\frac {\sqrt {1-a^2 x^2} \left (56-82 a x-32 a^2 x^2+76 a^3 x^3-15 a^4 x^4\right )}{(-1+a x)^3 (1+a x)}+30 \arcsin (a x)}{15 a c^5} \] Input:

Integrate[1/(E^(3*ArcTanh[a*x])*(c - c/(a*x))^5),x]
 

Output:

((Sqrt[1 - a^2*x^2]*(56 - 82*a*x - 32*a^2*x^2 + 76*a^3*x^3 - 15*a^4*x^4))/ 
((-1 + a*x)^3*(1 + a*x)) + 30*ArcSin[a*x])/(15*a*c^5)
 

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6681, 6678, 570, 529, 2166, 2345, 27, 455, 223}

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.

\(\displaystyle \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx\)

\(\Big \downarrow \) 6681

\(\displaystyle -\frac {a^5 \int \frac {e^{-3 \text {arctanh}(a x)} x^5}{(1-a x)^5}dx}{c^5}\)

\(\Big \downarrow \) 6678

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

\(\Big \downarrow \) 570

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

\(\Big \downarrow \) 529

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

\(\Big \downarrow \) 2166

\(\displaystyle -\frac {a^5 \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {\frac {15 x^3}{a^2}+\frac {30 x^2}{a^3}+\frac {45 x}{a^4}+\frac {16}{a^5}}{\left (1-a^2 x^2\right )^{3/2}}dx-\frac {22 (a x+1)}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {(a x+1)^2}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}\right )}{c^5}\)

\(\Big \downarrow \) 2345

\(\displaystyle -\frac {a^5 \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 (23 a x+30)}{a^6 \sqrt {1-a^2 x^2}}-\int \frac {15 (a x+2)}{a^5 \sqrt {1-a^2 x^2}}dx\right )-\frac {22 (a x+1)}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {(a x+1)^2}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}\right )}{c^5}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a^5 \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 (23 a x+30)}{a^6 \sqrt {1-a^2 x^2}}-\frac {15 \int \frac {a x+2}{\sqrt {1-a^2 x^2}}dx}{a^5}\right )-\frac {22 (a x+1)}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {(a x+1)^2}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}\right )}{c^5}\)

\(\Big \downarrow \) 455

\(\displaystyle -\frac {a^5 \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 (23 a x+30)}{a^6 \sqrt {1-a^2 x^2}}-\frac {15 \left (2 \int \frac {1}{\sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a^5}\right )-\frac {22 (a x+1)}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {(a x+1)^2}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}\right )}{c^5}\)

\(\Big \downarrow \) 223

\(\displaystyle -\frac {a^5 \left (\frac {(a x+1)^2}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}+\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 (23 a x+30)}{a^6 \sqrt {1-a^2 x^2}}-\frac {15 \left (\frac {2 \arcsin (a x)}{a}-\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a^5}\right )-\frac {22 (a x+1)}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}\right )\right )}{c^5}\)

Input:

Int[1/(E^(3*ArcTanh[a*x])*(c - c/(a*x))^5),x]
 

Output:

-((a^5*((1 + a*x)^2/(5*a^6*(1 - a^2*x^2)^(5/2)) + ((-22*(1 + a*x))/(3*a^6* 
(1 - a^2*x^2)^(3/2)) + ((2*(30 + 23*a*x))/(a^6*Sqrt[1 - a^2*x^2]) - (15*(- 
(Sqrt[1 - a^2*x^2]/a) + (2*ArcSin[a*x])/a))/a^5)/3)/5))/c^5)
 

Defintions of rubi rules used

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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 455

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

rule 529

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{Qx = PolynomialQuotient[x^m, a*d + b*c*x, x], R = PolynomialRem 
ainder[x^m, a*d + b*c*x, x]}, Simp[(-c)*R*(c + d*x)^n*((a + b*x^2)^(p + 1)/ 
(2*a*d*(p + 1))), x] + Simp[c/(2*a*(p + 1))   Int[(c + d*x)^(n - 1)*(a + b* 
x^2)^(p + 1)*ExpandToSum[2*a*d*(p + 1)*Qx + R*(n + 2*p + 2), x], x], x]] /; 
 FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 1] && LtQ[p, -1] && EqQ[b* 
c^2 + a*d^2, 0]
 

rule 570

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
x_Symbol] :> Simp[c^(2*n)/a^n   Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ 
n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I 
LtQ[n, -1] &&  !(IGtQ[m, 0] && ILtQ[m + n, 0] &&  !GtQ[p, 1])
 

rule 2166

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Pq, a*e + b*d*x, x], R = PolynomialRemainde 
r[Pq, a*e + b*d*x, x]}, Simp[(-d)*R*(d + e*x)^m*((a + b*x^2)^(p + 1)/(2*a*e 
*(p + 1))), x] + Simp[d/(2*a*(p + 1))   Int[(d + e*x)^(m - 1)*(a + b*x^2)^( 
p + 1)*ExpandToSum[2*a*e*(p + 1)*Qx + R*(m + 2*p + 2), x], x], x]] /; FreeQ 
[{a, b, d, e}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 0] && ILtQ[p + 1/2, 
 0] && GtQ[m, 0]
 

rule 2345

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot 
ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 
 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b 
*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   In 
t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] 
/; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
 

rule 6678

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

rule 6681

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

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.76

method	result	size
risch	\(\frac {a^{2} x^{2}-1}{a \sqrt {-a^{2} x^{2}+1}\, c^{5}}-\frac {\left (-\frac {2 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{5} \sqrt {a^{2}}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{10 a^{9} \left (x -\frac {1}{a}\right )^{3}}-\frac {41 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{60 a^{8} \left (x -\frac {1}{a}\right )^{2}}-\frac {383 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{120 a^{7} \left (x -\frac {1}{a}\right )}+\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{8 a^{7} \left (x +\frac {1}{a}\right )}\right ) a^{5}}{c^{5}}\)	\(227\)
default	\(\text {Expression too large to display}\)	\(1291\)

Input:

int(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^5,x,method=_RETURNVERBOSE)
 

Output:

1/a*(a^2*x^2-1)/(-a^2*x^2+1)^(1/2)/c^5-(-2/a^5/(a^2)^(1/2)*arctan((a^2)^(1 
/2)*x/(-a^2*x^2+1)^(1/2))-1/10/a^9/(x-1/a)^3*(-(x-1/a)^2*a^2-2*a*(x-1/a))^ 
(1/2)-41/60/a^8/(x-1/a)^2*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)-383/120/a^7/( 
x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)+1/8/a^7/(x+1/a)*(-a^2*(x+1/a)^2+ 
2*a*(x+1/a))^(1/2))*a^5/c^5
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.17 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=-\frac {56 \, a^{4} x^{4} - 112 \, a^{3} x^{3} + 112 \, a x + 60 \, {\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (15 \, a^{4} x^{4} - 76 \, a^{3} x^{3} + 32 \, a^{2} x^{2} + 82 \, a x - 56\right )} \sqrt {-a^{2} x^{2} + 1} - 56}{15 \, {\left (a^{5} c^{5} x^{4} - 2 \, a^{4} c^{5} x^{3} + 2 \, a^{2} c^{5} x - a c^{5}\right )}} \] Input:

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^5,x, algorithm="fricas" 
)
 

Output:

-1/15*(56*a^4*x^4 - 112*a^3*x^3 + 112*a*x + 60*(a^4*x^4 - 2*a^3*x^3 + 2*a* 
x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (15*a^4*x^4 - 76*a^3*x^3 + 
 32*a^2*x^2 + 82*a*x - 56)*sqrt(-a^2*x^2 + 1) - 56)/(a^5*c^5*x^4 - 2*a^4*c 
^5*x^3 + 2*a^2*c^5*x - a*c^5)
 

Sympy [F]

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

integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(c-c/a/x)**5,x)
 

Output:

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

Maxima [F]

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

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^5,x, algorithm="maxima" 
)
 

Output:

integrate((-a^2*x^2 + 1)^(3/2)/((a*x + 1)^3*(c - c/(a*x))^5), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^5,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [B] (verification not implemented)

Time = 23.03 (sec) , antiderivative size = 275, normalized size of antiderivative = 2.13 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=\frac {41\,a\,\sqrt {1-a^2\,x^2}}{60\,\left (a^4\,c^5\,x^2-2\,a^3\,c^5\,x+a^2\,c^5\right )}+\frac {2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^5\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a\,c^5}+\frac {\sqrt {1-a^2\,x^2}}{8\,\sqrt {-a^2}\,\left (c^5\,x\,\sqrt {-a^2}+\frac {c^5\,\sqrt {-a^2}}{a}\right )}-\frac {383\,\sqrt {1-a^2\,x^2}}{120\,\sqrt {-a^2}\,\left (c^5\,x\,\sqrt {-a^2}-\frac {c^5\,\sqrt {-a^2}}{a}\right )}-\frac {\sqrt {1-a^2\,x^2}}{10\,\sqrt {-a^2}\,\left (3\,c^5\,x\,\sqrt {-a^2}-\frac {c^5\,\sqrt {-a^2}}{a}+a^2\,c^5\,x^3\,\sqrt {-a^2}-3\,a\,c^5\,x^2\,\sqrt {-a^2}\right )} \] Input:

int((1 - a^2*x^2)^(3/2)/((c - c/(a*x))^5*(a*x + 1)^3),x)
 

Output:

(41*a*(1 - a^2*x^2)^(1/2))/(60*(a^2*c^5 - 2*a^3*c^5*x + a^4*c^5*x^2)) + (2 
*asinh(x*(-a^2)^(1/2)))/(c^5*(-a^2)^(1/2)) - (1 - a^2*x^2)^(1/2)/(a*c^5) + 
 (1 - a^2*x^2)^(1/2)/(8*(-a^2)^(1/2)*(c^5*x*(-a^2)^(1/2) + (c^5*(-a^2)^(1/ 
2))/a)) - (383*(1 - a^2*x^2)^(1/2))/(120*(-a^2)^(1/2)*(c^5*x*(-a^2)^(1/2) 
- (c^5*(-a^2)^(1/2))/a)) - (1 - a^2*x^2)^(1/2)/(10*(-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)))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.33 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx=\frac {30 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a^{2} x^{2}-60 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a x +30 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )-41 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+82 \sqrt {-a^{2} x^{2}+1}\, a x -41 \sqrt {-a^{2} x^{2}+1}+15 a^{4} x^{4}-76 a^{3} x^{3}+32 a^{2} x^{2}+82 a x -56}{15 \sqrt {-a^{2} x^{2}+1}\, a \,c^{5} \left (a^{2} x^{2}-2 a x +1\right )} \] Input:

int(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^5,x)
 

Output:

(30*sqrt( - a**2*x**2 + 1)*asin(a*x)*a**2*x**2 - 60*sqrt( - a**2*x**2 + 1) 
*asin(a*x)*a*x + 30*sqrt( - a**2*x**2 + 1)*asin(a*x) - 41*sqrt( - a**2*x** 
2 + 1)*a**2*x**2 + 82*sqrt( - a**2*x**2 + 1)*a*x - 41*sqrt( - a**2*x**2 + 
1) + 15*a**4*x**4 - 76*a**3*x**3 + 32*a**2*x**2 + 82*a*x - 56)/(15*sqrt( - 
 a**2*x**2 + 1)*a*c**5*(a**2*x**2 - 2*a*x + 1))