\(\int e^{3 \text {arctanh}(a x)} (c-\frac {c}{a x})^4 \, dx\) [488]

Optimal result

Integrand size = 22, antiderivative size = 103 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {c^4 (2+3 a x) \sqrt {1-a^2 x^2}}{2 a^2 x}-\frac {c^4 (2-3 a x) \left (1-a^2 x^2\right )^{3/2}}{6 a^4 x^3}+\frac {c^4 \arcsin (a x)}{a}-\frac {3 c^4 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{2 a} \] Output:

1/2*c^4*(3*a*x+2)*(-a^2*x^2+1)^(1/2)/a^2/x-1/6*c^4*(-3*a*x+2)*(-a^2*x^2+1) 
^(3/2)/a^4/x^3+c^4*arcsin(a*x)/a-3/2*c^4*arctanh((-a^2*x^2+1)^(1/2))/a
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.62 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {c^4 \left (-16+24 a x+80 a^2 x^2+24 a^3 x^3-64 a^4 x^4-48 a^5 x^5+45 a^3 x^3 \sqrt {1-a^2 x^2} \arcsin (a x)-6 a^3 x^3 \sqrt {1-a^2 x^2} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )-72 a^3 x^3 \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )}{48 a^4 x^3 \sqrt {1-a^2 x^2}} \] Input:

Integrate[E^(3*ArcTanh[a*x])*(c - c/(a*x))^4,x]
 

Output:

(c^4*(-16 + 24*a*x + 80*a^2*x^2 + 24*a^3*x^3 - 64*a^4*x^4 - 48*a^5*x^5 + 4 
5*a^3*x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x] - 6*a^3*x^3*Sqrt[1 - a^2*x^2]*ArcS 
in[Sqrt[1 - a*x]/Sqrt[2]] - 72*a^3*x^3*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - 
a^2*x^2]]))/(48*a^4*x^3*Sqrt[1 - a^2*x^2])
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.90, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {6681, 6678, 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.

\(\displaystyle \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx\)

\(\Big \downarrow \) 6681

\(\displaystyle \frac {c^4 \int \frac {e^{3 \text {arctanh}(a x)} (1-a x)^4}{x^4}dx}{a^4}\)

\(\Big \downarrow \) 6678

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

\(\Big \downarrow \) 537

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 536

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

Int[E^(3*ArcTanh[a*x])*(c - c/(a*x))^4,x]
 

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 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 = 129, normalized size of antiderivative = 1.25

method	result	size
risch	\(-\frac {\left (8 a^{4} x^{4}+3 a^{3} x^{3}-10 a^{2} x^{2}-3 a x +2\right ) c^{4}}{6 x^{3} \sqrt {-a^{2} x^{2}+1}\, a^{4}}-\frac {\left (\frac {3 a^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-\frac {a^{4} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-a^{3} \sqrt {-a^{2} x^{2}+1}\right ) c^{4}}{a^{4}}\)	\(129\)
default	\(\frac {c^{4} \left (-\frac {1}{3 x^{3} \sqrt {-a^{2} x^{2}+1}}-\frac {5 a^{2} \left (-\frac {1}{x \sqrt {-a^{2} x^{2}+1}}+\frac {2 a^{2} x}{\sqrt {-a^{2} x^{2}+1}}\right )}{3}+a^{7} \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )+\frac {3 a^{4} x}{\sqrt {-a^{2} x^{2}+1}}-a \left (-\frac {1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 a^{2} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )+3 a^{3} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )-a^{6} \left (\frac {x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}\right )-\frac {3 a^{3}}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{4}}\)	\(284\)
meijerg	\(\frac {3 c^{4} x}{\sqrt {-a^{2} x^{2}+1}}+\frac {3 c^{4} \left (\frac {\left (2-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}-\sqrt {\pi }+\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}-\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )\right )}{\sqrt {\pi }\, a}+\frac {3 c^{4} \left (-2 a^{2} x^{2}+1\right )}{a^{2} x \sqrt {-a^{2} x^{2}+1}}+\frac {c^{4} \left (\frac {\sqrt {\pi }}{2 x^{2} a^{2}}-\frac {3 \left (\frac {5}{3}-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{4}-\frac {\sqrt {\pi }\, \left (-20 a^{2} x^{2}+8\right )}{16 a^{2} x^{2}}+\frac {\sqrt {\pi }\, \left (-24 a^{2} x^{2}+8\right )}{16 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )}{2}\right )}{a \sqrt {\pi }}-\frac {c^{4} \left (-8 a^{4} x^{4}+4 a^{2} x^{2}+1\right )}{3 a^{4} x^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {c^{4} \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{4 \sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}+\frac {c^{4} \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {3 c^{4} \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}\)	\(425\)

Input:

int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^4,x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

-1/6*(8*a^4*x^4+3*a^3*x^3-10*a^2*x^2-3*a*x+2)/x^3/(-a^2*x^2+1)^(1/2)*c^4/a 
^4-(3/2*a^3*arctanh(1/(-a^2*x^2+1)^(1/2))-a^4/(a^2)^(1/2)*arctan((a^2)^(1/ 
2)*x/(-a^2*x^2+1)^(1/2))-a^3*(-a^2*x^2+1)^(1/2))*c^4/a^4
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.28 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=-\frac {12 \, a^{3} c^{4} x^{3} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - 9 \, a^{3} c^{4} x^{3} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - 6 \, a^{3} c^{4} x^{3} - {\left (6 \, a^{3} c^{4} x^{3} + 8 \, a^{2} c^{4} x^{2} + 3 \, a c^{4} x - 2 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, a^{4} x^{3}} \] Input:

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^4,x, algorithm="fricas")
 

Output:

-1/6*(12*a^3*c^4*x^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - 9*a^3*c^4*x^ 
3*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 6*a^3*c^4*x^3 - (6*a^3*c^4*x^3 + 8*a^2 
*c^4*x^2 + 3*a*c^4*x - 2*c^4)*sqrt(-a^2*x^2 + 1))/(a^4*x^3)
 

Sympy [A] (verification not implemented)

Time = 8.22 (sec) , antiderivative size = 350, normalized size of antiderivative = 3.40 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=- a c^{4} \left (\begin {cases} - \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{2}}{2} & \text {otherwise} \end {cases}\right ) + c^{4} \left (\begin {cases} \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{\sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\x & \text {otherwise} \end {cases}\right ) + \frac {2 c^{4} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right )}{a} - \frac {2 c^{4} \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right )}{a^{2}} - \frac {c^{4} \left (\begin {cases} - \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right )}{a^{3}} + \frac {c^{4} \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right )}{a^{4}} \] Input:

integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a/x)**4,x)
 

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (91) = 182\).

Time = 0.12 (sec) , antiderivative size = 349, normalized size of antiderivative = 3.39 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=-a^{3} c^{4} {\left (\frac {x^{2}}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {2}{\sqrt {-a^{2} x^{2} + 1} a^{4}}\right )} - a^{2} c^{4} {\left (\frac {x}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {\arcsin \left (a x\right )}{a^{3}}\right )} + \frac {3 \, c^{4} x}{\sqrt {-a^{2} x^{2} + 1}} + \frac {3 \, c^{4} {\left (\frac {1}{\sqrt {-a^{2} x^{2} + 1}} - \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right )\right )}}{a} - \frac {3 \, {\left (\frac {2 \, a^{2} x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {1}{\sqrt {-a^{2} x^{2} + 1} x}\right )} c^{4}}{a^{2}} - \frac {3 \, c^{4}}{\sqrt {-a^{2} x^{2} + 1} a} + \frac {{\left (3 \, a^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {3 \, a^{2}}{\sqrt {-a^{2} x^{2} + 1}} + \frac {1}{\sqrt {-a^{2} x^{2} + 1} x^{2}}\right )} c^{4}}{2 \, a^{3}} + \frac {{\left (\frac {8 \, a^{4} x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {4 \, a^{2}}{\sqrt {-a^{2} x^{2} + 1} x} - \frac {1}{\sqrt {-a^{2} x^{2} + 1} x^{3}}\right )} c^{4}}{3 \, a^{4}} \] Input:

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^4,x, algorithm="maxima")
 

Output:

-a^3*c^4*(x^2/(sqrt(-a^2*x^2 + 1)*a^2) - 2/(sqrt(-a^2*x^2 + 1)*a^4)) - a^2 
*c^4*(x/(sqrt(-a^2*x^2 + 1)*a^2) - arcsin(a*x)/a^3) + 3*c^4*x/sqrt(-a^2*x^ 
2 + 1) + 3*c^4*(1/sqrt(-a^2*x^2 + 1) - log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2 
/abs(x)))/a - 3*(2*a^2*x/sqrt(-a^2*x^2 + 1) - 1/(sqrt(-a^2*x^2 + 1)*x))*c^ 
4/a^2 - 3*c^4/(sqrt(-a^2*x^2 + 1)*a) + 1/2*(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^4/a^3 + 1/3*(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^4/a^4
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (91) = 182\).

Time = 0.17 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.54 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {{\left (c^{4} - \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{4}}{a^{2} x} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{4}}{a^{4} x^{2}}\right )} a^{6} x^{3}}{24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} {\left | a \right |}} + \frac {c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} - \frac {3 \, c^{4} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{4}}{a} + \frac {\frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{4}}{x} + \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{4}}{a^{2} x^{2}} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{4}}{a^{4} x^{3}}}{24 \, a^{2} {\left | a \right |}} \] Input:

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^4,x, algorithm="giac")
 

Output:

1/24*(c^4 - 3*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^4/(a^2*x) - 15*(sqrt(-a^2* 
x^2 + 1)*abs(a) + a)^2*c^4/(a^4*x^2))*a^6*x^3/((sqrt(-a^2*x^2 + 1)*abs(a) 
+ a)^3*abs(a)) + c^4*arcsin(a*x)*sgn(a)/abs(a) - 3/2*c^4*log(1/2*abs(-2*sq 
rt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) + sqrt(-a^2*x^2 + 1)*c 
^4/a + 1/24*(15*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^4/x + 3*(sqrt(-a^2*x^2 + 
 1)*abs(a) + a)^2*c^4/(a^2*x^2) - (sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*c^4/(a 
^4*x^3))/(a^2*abs(a))
 

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.31 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}+\frac {c^4\,\sqrt {1-a^2\,x^2}}{a}+\frac {4\,c^4\,\sqrt {1-a^2\,x^2}}{3\,a^2\,x}+\frac {c^4\,\sqrt {1-a^2\,x^2}}{2\,a^3\,x^2}-\frac {c^4\,\sqrt {1-a^2\,x^2}}{3\,a^4\,x^3}+\frac {c^4\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2\,a} \] Input:

int(((c - c/(a*x))^4*(a*x + 1)^3)/(1 - a^2*x^2)^(3/2),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.30 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {c^{4} \left (12 \mathit {asin} \left (a x \right ) a^{3} x^{3}+12 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+16 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+6 \sqrt {-a^{2} x^{2}+1}\, a x -4 \sqrt {-a^{2} x^{2}+1}+9 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}-1\right ) a^{3} x^{3}-9 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}+1\right ) a^{3} x^{3}\right )}{12 a^{4} x^{3}} \] Input:

int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^4,x)
 

Output:

(c**4*(12*asin(a*x)*a**3*x**3 + 12*sqrt( - a**2*x**2 + 1)*a**3*x**3 + 16*s 
qrt( - a**2*x**2 + 1)*a**2*x**2 + 6*sqrt( - a**2*x**2 + 1)*a*x - 4*sqrt( - 
 a**2*x**2 + 1) + 9*log(sqrt( - a**2*x**2 + 1) - 1)*a**3*x**3 - 9*log(sqrt 
( - a**2*x**2 + 1) + 1)*a**3*x**3))/(12*a**4*x**3)