\(\int e^{3 \text {arctanh}(a x)} (c-\frac {c}{a x})^5 \, dx\) [487]

Optimal result

Integrand size = 22, antiderivative size = 131 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^5 \, dx=\frac {c^5 (16+9 a x) \sqrt {1-a^2 x^2}}{8 a^2 x}-\frac {c^5 (16-9 a x) \left (1-a^2 x^2\right )^{3/2}}{24 a^4 x^3}+\frac {c^5 \left (1-a^2 x^2\right )^{5/2}}{4 a^5 x^4}+\frac {2 c^5 \arcsin (a x)}{a}-\frac {9 c^5 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{8 a} \] Output:

1/8*c^5*(9*a*x+16)*(-a^2*x^2+1)^(1/2)/a^2/x-1/24*c^5*(-9*a*x+16)*(-a^2*x^2 
+1)^(3/2)/a^4/x^3+1/4*c^5*(-a^2*x^2+1)^(5/2)/a^5/x^4+2*c^5*arcsin(a*x)/a-9 
/8*c^5*arctanh((-a^2*x^2+1)^(1/2))/a
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.71 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^5 \, dx=\frac {c^5 \left (\frac {\sqrt {1-a^2 x^2} \left (6-16 a x-3 a^2 x^2+64 a^3 x^3+24 a^4 x^4\right )}{a^4 x^4}+48 \arcsin (a x)+27 \log (a x)-27 \log \left (1+\sqrt {1-a^2 x^2}\right )\right )}{24 a} \] Input:

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

Output:

(c^5*((Sqrt[1 - a^2*x^2]*(6 - 16*a*x - 3*a^2*x^2 + 64*a^3*x^3 + 24*a^4*x^4 
))/(a^4*x^4) + 48*ArcSin[a*x] + 27*Log[a*x] - 27*Log[1 + Sqrt[1 - a^2*x^2] 
]))/(24*a)
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.92, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {6681, 6678, 540, 27, 537, 25, 536, 538, 223, 243, 73, 221}

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.

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

\(\Big \downarrow \) 6681

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

\(\Big \downarrow \) 6678

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

\(\Big \downarrow \) 540

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 537

\(\displaystyle -\frac {c^5 \left (-\frac {1}{4} a \left (\frac {1}{2} a^2 \int -\frac {(16-9 a x) \sqrt {1-a^2 x^2}}{x^2}dx-\frac {(16-9 a x) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^5}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {c^5 \left (-\frac {1}{4} a \left (-\frac {1}{2} a^2 \int \frac {(16-9 a x) \sqrt {1-a^2 x^2}}{x^2}dx-\frac {(16-9 a x) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^5}\)

\(\Big \downarrow \) 536

\(\displaystyle -\frac {c^5 \left (-\frac {1}{4} a \left (-\frac {1}{2} a^2 \left (\int \frac {-16 x a^2-9 a}{x \sqrt {1-a^2 x^2}}dx-\frac {(9 a x+16) \sqrt {1-a^2 x^2}}{x}\right )-\frac {(16-9 a x) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^5}\)

\(\Big \downarrow \) 538

\(\displaystyle -\frac {c^5 \left (-\frac {1}{4} a \left (-\frac {1}{2} a^2 \left (-16 a^2 \int \frac {1}{\sqrt {1-a^2 x^2}}dx-9 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {(9 a x+16) \sqrt {1-a^2 x^2}}{x}\right )-\frac {(16-9 a x) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^5}\)

\(\Big \downarrow \) 223

\(\displaystyle -\frac {c^5 \left (-\frac {1}{4} a \left (-\frac {1}{2} a^2 \left (-9 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} (9 a x+16)}{x}-16 a \arcsin (a x)\right )-\frac {(16-9 a x) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^5}\)

\(\Big \downarrow \) 243

\(\displaystyle -\frac {c^5 \left (-\frac {1}{4} a \left (-\frac {1}{2} a^2 \left (-\frac {9}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2} (9 a x+16)}{x}-16 a \arcsin (a x)\right )-\frac {(16-9 a x) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^5}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {c^5 \left (-\frac {1}{4} a \left (-\frac {1}{2} a^2 \left (\frac {9 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {\sqrt {1-a^2 x^2} (9 a x+16)}{x}-16 a \arcsin (a x)\right )-\frac {(16-9 a x) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^5}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {c^5 \left (-\frac {1}{4} a \left (-\frac {1}{2} a^2 \left (9 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2} (9 a x+16)}{x}-16 a \arcsin (a x)\right )-\frac {(16-9 a x) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )-\frac {\left (1-a^2 x^2\right )^{5/2}}{4 x^4}\right )}{a^5}\)

Input:

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

Output:

-((c^5*(-1/4*(1 - a^2*x^2)^(5/2)/x^4 - (a*(-1/6*((16 - 9*a*x)*(1 - a^2*x^2 
)^(3/2))/x^3 - (a^2*(-(((16 + 9*a*x)*Sqrt[1 - a^2*x^2])/x) - 16*a*ArcSin[a 
*x] + 9*a*ArcTanh[Sqrt[1 - a^2*x^2]]))/2))/4))/a^5)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

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

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 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 536

Int[(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_)^2, x_Symbol] :> S 
imp[(-(2*c*p - d*x))*((a + b*x^2)^p/(2*p*x)), x] + Int[(a*d + 2*b*c*p*x)*(( 
a + b*x^2)^(p - 1)/x), x] /; FreeQ[{a, b, c, d}, x] && GtQ[p, 0] && Integer 
Q[2*p]
 

rule 537

Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[x^(m + 1)*(c*(m + 2) + d*(m + 1)*x)*((a + b*x^2)^p/((m + 1)*(m + 2))), 
 x] - Simp[2*b*(p/((m + 1)*(m + 2)))   Int[x^(m + 2)*(c*(m + 2) + d*(m + 1) 
*x)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, -2] && 
 GtQ[p, 0] &&  !ILtQ[m + 2*p + 3, 0] && IntegerQ[2*p]
 

rule 538

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

rule 540

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol 
] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemain 
der[(c + d*x)^n, x, x]}, Simp[R*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))) 
, x] + Simp[1/(a*(m + 1))   Int[x^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*(m + 
1)*Qx - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IG 
tQ[n, 1] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
 

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 = 0.38 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.05

method	result	size
risch	\(-\frac {\left (64 a^{5} x^{5}-3 a^{4} x^{4}-80 a^{3} x^{3}+9 a^{2} x^{2}+16 a x -6\right ) c^{5}}{24 x^{4} \sqrt {-a^{2} x^{2}+1}\, a^{5}}-\frac {\left (\frac {9 a^{4} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{8}-\frac {2 a^{5} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-a^{4} \sqrt {-a^{2} x^{2}+1}\right ) c^{5}}{a^{5}}\)	\(137\)
default	\(\frac {c^{5} \left (a^{8} \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )+\frac {6 a^{5} x}{\sqrt {-a^{2} x^{2}+1}}+\frac {1}{4 x^{4} \sqrt {-a^{2} x^{2}+1}}+\frac {3 a^{2} \left (-\frac {1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 a^{2} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )}{4}+2 a \left (-\frac {1}{3 x^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {4 a^{2} \left (-\frac {1}{x \sqrt {-a^{2} x^{2}+1}}+\frac {2 a^{2} x}{\sqrt {-a^{2} x^{2}+1}}\right )}{3}\right )-6 a^{3} \left (-\frac {1}{x \sqrt {-a^{2} x^{2}+1}}+\frac {2 a^{2} x}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {2 a^{4}}{\sqrt {-a^{2} x^{2}+1}}-2 a^{7} \left (\frac {x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}\right )\right )}{a^{5}}\)	\(315\)
meijerg	\(\frac {6 c^{5} x}{\sqrt {-a^{2} x^{2}+1}}+\frac {6 c^{5} \left (-2 a^{2} x^{2}+1\right )}{a^{2} x \sqrt {-a^{2} x^{2}+1}}-\frac {2 c^{5} \left (\frac {\sqrt {\pi }}{2 x^{2} a^{2}}-\frac {3 \left (\frac {5}{3}-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{4}-\frac {\sqrt {\pi }\, \left (-20 a^{2} x^{2}+8\right )}{16 a^{2} x^{2}}+\frac {\sqrt {\pi }\, \left (-24 a^{2} x^{2}+8\right )}{16 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )}{2}\right )}{a \sqrt {\pi }}-\frac {2 c^{5} \left (-8 a^{4} x^{4}+4 a^{2} x^{2}+1\right )}{3 a^{4} x^{3} \sqrt {-a^{2} x^{2}+1}}-\frac {c^{5} \left (-\frac {\sqrt {\pi }}{4 x^{4} a^{4}}-\frac {3 \sqrt {\pi }}{4 x^{2} a^{2}}+\frac {15 \left (\frac {47}{30}-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{16}+\frac {\sqrt {\pi }\, \left (-47 a^{4} x^{4}+24 a^{2} x^{2}+8\right )}{32 a^{4} x^{4}}-\frac {\sqrt {\pi }\, \left (-60 a^{4} x^{4}+20 a^{2} x^{2}+8\right )}{32 a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {15 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )}{8}\right )}{a \sqrt {\pi }}+\frac {c^{5} \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{4 \sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}+\frac {2 c^{5} \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {2 c^{5} \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}\)	\(498\)

Input:

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

Output:

-1/24*(64*a^5*x^5-3*a^4*x^4-80*a^3*x^3+9*a^2*x^2+16*a*x-6)/x^4/(-a^2*x^2+1 
)^(1/2)*c^5/a^5-(9/8*a^4*arctanh(1/(-a^2*x^2+1)^(1/2))-2*a^5/(a^2)^(1/2)*a 
rctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))-a^4*(-a^2*x^2+1)^(1/2))*c^5/a^5
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.09 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^5 \, dx=-\frac {96 \, a^{4} c^{5} x^{4} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - 27 \, a^{4} c^{5} x^{4} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - 24 \, a^{4} c^{5} x^{4} - {\left (24 \, a^{4} c^{5} x^{4} + 64 \, a^{3} c^{5} x^{3} - 3 \, a^{2} c^{5} x^{2} - 16 \, a c^{5} x + 6 \, c^{5}\right )} \sqrt {-a^{2} x^{2} + 1}}{24 \, a^{5} x^{4}} \] Input:

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

Output:

-1/24*(96*a^4*c^5*x^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - 27*a^4*c^5* 
x^4*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 24*a^4*c^5*x^4 - (24*a^4*c^5*x^4 + 6 
4*a^3*c^5*x^3 - 3*a^2*c^5*x^2 - 16*a*c^5*x + 6*c^5)*sqrt(-a^2*x^2 + 1))/(a 
^5*x^4)
 

Sympy [A] (verification not implemented)

Time = 10.26 (sec) , antiderivative size = 529, normalized size of antiderivative = 4.04 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^5 \, dx =\text {Too large to display} \] Input:

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

Output:

-a*c**5*Piecewise((-sqrt(-a**2*x**2 + 1)/a**2, Ne(a**2, 0)), (x**2/2, True 
)) + 2*c**5*Piecewise((log(-2*a**2*x + 2*sqrt(-a**2)*sqrt(-a**2*x**2 + 1)) 
/sqrt(-a**2), Ne(a**2, 0)), (x, True)) + c**5*Piecewise((-acosh(1/(a*x)), 
1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))/a - 4*c**5*Piecewise((-I*s 
qrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True) 
)/a**2 + c**5*Piecewise((-a**2*acosh(1/(a*x))/2 + a/(2*x*sqrt(-1 + 1/(a**2 
*x**2))) - 1/(2*a*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), ( 
I*a**2*asin(1/(a*x))/2 - I*a*sqrt(1 - 1/(a**2*x**2))/(2*x), True))/a**3 + 
2*c**5*Piecewise((-2*I*a**2*sqrt(a**2*x**2 - 1)/(3*x) - I*sqrt(a**2*x**2 - 
 1)/(3*x**3), Abs(a**2*x**2) > 1), (-2*a**2*sqrt(-a**2*x**2 + 1)/(3*x) - s 
qrt(-a**2*x**2 + 1)/(3*x**3), True))/a**4 - c**5*Piecewise((-3*a**4*acosh( 
1/(a*x))/8 + 3*a**3/(8*x*sqrt(-1 + 1/(a**2*x**2))) - a/(8*x**3*sqrt(-1 + 1 
/(a**2*x**2))) - 1/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 
 1), (3*I*a**4*asin(1/(a*x))/8 - 3*I*a**3/(8*x*sqrt(1 - 1/(a**2*x**2))) + 
I*a/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I/(4*a*x**5*sqrt(1 - 1/(a**2*x**2)) 
), True))/a**5
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 394 vs. \(2 (115) = 230\).

Time = 0.12 (sec) , antiderivative size = 394, normalized size of antiderivative = 3.01 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^5 \, dx=-a^{3} c^{5} {\left (\frac {x^{2}}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {2}{\sqrt {-a^{2} x^{2} + 1} a^{4}}\right )} - 2 \, a^{2} c^{5} {\left (\frac {x}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {\arcsin \left (a x\right )}{a^{3}}\right )} + \frac {6 \, c^{5} x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {6 \, {\left (\frac {2 \, a^{2} x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {1}{\sqrt {-a^{2} x^{2} + 1} x}\right )} c^{5}}{a^{2}} - \frac {2 \, c^{5}}{\sqrt {-a^{2} x^{2} + 1} a} - \frac {{\left (3 \, a^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {3 \, a^{2}}{\sqrt {-a^{2} x^{2} + 1}} + \frac {1}{\sqrt {-a^{2} x^{2} + 1} x^{2}}\right )} c^{5}}{a^{3}} + \frac {2 \, {\left (\frac {8 \, a^{4} x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {4 \, a^{2}}{\sqrt {-a^{2} x^{2} + 1} x} - \frac {1}{\sqrt {-a^{2} x^{2} + 1} x^{3}}\right )} c^{5}}{3 \, a^{4}} + \frac {{\left (15 \, a^{4} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {15 \, a^{4}}{\sqrt {-a^{2} x^{2} + 1}} + \frac {5 \, a^{2}}{\sqrt {-a^{2} x^{2} + 1} x^{2}} + \frac {2}{\sqrt {-a^{2} x^{2} + 1} x^{4}}\right )} c^{5}}{8 \, a^{5}} \] Input:

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

Output:

-a^3*c^5*(x^2/(sqrt(-a^2*x^2 + 1)*a^2) - 2/(sqrt(-a^2*x^2 + 1)*a^4)) - 2*a 
^2*c^5*(x/(sqrt(-a^2*x^2 + 1)*a^2) - arcsin(a*x)/a^3) + 6*c^5*x/sqrt(-a^2* 
x^2 + 1) - 6*(2*a^2*x/sqrt(-a^2*x^2 + 1) - 1/(sqrt(-a^2*x^2 + 1)*x))*c^5/a 
^2 - 2*c^5/(sqrt(-a^2*x^2 + 1)*a) - (3*a^2*log(2*sqrt(-a^2*x^2 + 1)/abs(x) 
 + 2/abs(x)) - 3*a^2/sqrt(-a^2*x^2 + 1) + 1/(sqrt(-a^2*x^2 + 1)*x^2))*c^5/ 
a^3 + 2/3*(8*a^4*x/sqrt(-a^2*x^2 + 1) - 4*a^2/(sqrt(-a^2*x^2 + 1)*x) - 1/( 
sqrt(-a^2*x^2 + 1)*x^3))*c^5/a^4 + 1/8*(15*a^4*log(2*sqrt(-a^2*x^2 + 1)/ab 
s(x) + 2/abs(x)) - 15*a^4/sqrt(-a^2*x^2 + 1) + 5*a^2/(sqrt(-a^2*x^2 + 1)*x 
^2) + 2/(sqrt(-a^2*x^2 + 1)*x^4))*c^5/a^5
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (115) = 230\).

Time = 0.15 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.04 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^5 \, dx=-\frac {{\left (3 \, c^{5} - \frac {16 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{5}}{a^{2} x} + \frac {240 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{5}}{a^{6} x^{3}}\right )} a^{8} x^{4}}{192 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} {\left | a \right |}} + \frac {2 \, c^{5} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} - \frac {9 \, c^{5} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{8 \, {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{5}}{a} + \frac {\frac {240 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{5} {\left | a \right |}}{x} - \frac {16 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{5} {\left | a \right |}}{a^{4} x^{3}} + \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{5} {\left | a \right |}}{a^{6} x^{4}}}{192 \, a^{4}} \] Input:

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

Output:

-1/192*(3*c^5 - 16*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^5/(a^2*x) + 240*(sqrt 
(-a^2*x^2 + 1)*abs(a) + a)^3*c^5/(a^6*x^3))*a^8*x^4/((sqrt(-a^2*x^2 + 1)*a 
bs(a) + a)^4*abs(a)) + 2*c^5*arcsin(a*x)*sgn(a)/abs(a) - 9/8*c^5*log(1/2*a 
bs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) + sqrt(-a^2*x^ 
2 + 1)*c^5/a + 1/192*(240*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^5*abs(a)/x - 1 
6*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*c^5*abs(a)/(a^4*x^3) + 3*(sqrt(-a^2*x^ 
2 + 1)*abs(a) + a)^4*c^5*abs(a)/(a^6*x^4))/a^4
 

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.21 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^5 \, dx=\frac {2\,c^5\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}+\frac {c^5\,\sqrt {1-a^2\,x^2}}{a}+\frac {8\,c^5\,\sqrt {1-a^2\,x^2}}{3\,a^2\,x}-\frac {c^5\,\sqrt {1-a^2\,x^2}}{8\,a^3\,x^2}-\frac {2\,c^5\,\sqrt {1-a^2\,x^2}}{3\,a^4\,x^3}+\frac {c^5\,\sqrt {1-a^2\,x^2}}{4\,a^5\,x^4}+\frac {c^5\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,9{}\mathrm {i}}{8\,a} \] Input:

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

Output:

(2*c^5*asinh(x*(-a^2)^(1/2)))/(-a^2)^(1/2) + (c^5*atan((1 - a^2*x^2)^(1/2) 
*1i)*9i)/(8*a) + (c^5*(1 - a^2*x^2)^(1/2))/a + (8*c^5*(1 - a^2*x^2)^(1/2)) 
/(3*a^2*x) - (c^5*(1 - a^2*x^2)^(1/2))/(8*a^3*x^2) - (2*c^5*(1 - a^2*x^2)^ 
(1/2))/(3*a^4*x^3) + (c^5*(1 - a^2*x^2)^(1/2))/(4*a^5*x^4)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.02 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^5 \, dx=\frac {c^{5} \left (48 \mathit {asin} \left (a x \right ) a^{4} x^{4}+24 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+64 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-3 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-16 \sqrt {-a^{2} x^{2}+1}\, a x +6 \sqrt {-a^{2} x^{2}+1}+27 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{4} x^{4}-24 a^{4} x^{4}\right )}{24 a^{5} x^{4}} \] Input:

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

Output:

(c**5*(48*asin(a*x)*a**4*x**4 + 24*sqrt( - a**2*x**2 + 1)*a**4*x**4 + 64*s 
qrt( - a**2*x**2 + 1)*a**3*x**3 - 3*sqrt( - a**2*x**2 + 1)*a**2*x**2 - 16* 
sqrt( - a**2*x**2 + 1)*a*x + 6*sqrt( - a**2*x**2 + 1) + 27*log(tan(asin(a* 
x)/2))*a**4*x**4 - 24*a**4*x**4))/(24*a**5*x**4)