\(\int e^{\text {arctanh}(a x)} (c-\frac {c}{a x})^4 \, dx\) [472]

Optimal result

Integrand size = 20, antiderivative size = 141 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {c^4 \sqrt {1-a^2 x^2}}{2 a}-\frac {3 c^4 \sqrt {1-a^2 x^2}}{a^2 x}-\frac {c^4 \left (1-a^2 x^2\right )^{3/2}}{3 a^4 x^3}+\frac {3 c^4 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}-\frac {3 c^4 \arcsin (a x)}{a}-\frac {c^4 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{2 a} \] Output:

1/2*c^4*(-a^2*x^2+1)^(1/2)/a-3*c^4*(-a^2*x^2+1)^(1/2)/a^2/x-1/3*c^4*(-a^2* 
x^2+1)^(3/2)/a^4/x^3+3/2*c^4*(-a^2*x^2+1)^(3/2)/a^3/x^2-3*c^4*arcsin(a*x)/ 
a-1/2*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 = 1.18 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {c^4 \left (-4+18 a x-28 a^2 x^2-30 a^3 x^3+32 a^4 x^4+12 a^5 x^5+3 a^3 x^3 \sqrt {1-a^2 x^2} \arcsin (a x)+78 a^3 x^3 \sqrt {1-a^2 x^2} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )-6 a^3 x^3 \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )}{12 a^4 x^3 \sqrt {1-a^2 x^2}} \] Input:

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

Output:

(c^4*(-4 + 18*a*x - 28*a^2*x^2 - 30*a^3*x^3 + 32*a^4*x^4 + 12*a^5*x^5 + 3* 
a^3*x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x] + 78*a^3*x^3*Sqrt[1 - a^2*x^2]*ArcSi 
n[Sqrt[1 - a*x]/Sqrt[2]] - 6*a^3*x^3*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^ 
2*x^2]]))/(12*a^4*x^3*Sqrt[1 - a^2*x^2])
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.77, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6681, 6678, 540, 27, 2338, 27, 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.

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

\(\Big \downarrow \) 6681

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

\(\Big \downarrow \) 6678

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

\(\Big \downarrow \) 540

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 536

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 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 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 2338

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ 
Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S 
imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( 
m + 1))   Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( 
m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt 
Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 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 = 0.30 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.91

method	result	size
risch	\(\frac {\left (16 a^{4} x^{4}-9 a^{3} x^{3}-14 a^{2} x^{2}+9 a x -2\right ) c^{4}}{6 x^{3} \sqrt {-a^{2} x^{2}+1}\, a^{4}}+\frac {\left (-\frac {a^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-\frac {3 a^{4} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-a^{3} \sqrt {-a^{2} x^{2}+1}\right ) c^{4}}{a^{4}}\)	\(128\)
default	\(\frac {c^{4} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}-\frac {8 a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}-a^{3} \sqrt {-a^{2} x^{2}+1}-\frac {3 a^{4} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-3 a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )-2 a^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{a^{4}}\)	\(150\)
meijerg	\(-\frac {c^{4} \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{2 a \sqrt {\pi }}-\frac {3 c^{4} \arcsin \left (a x \right )}{a}+\frac {c^{4} \left (\left (-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )\right )}{a \sqrt {\pi }}-\frac {2 c^{4} \sqrt {-a^{2} x^{2}+1}}{a^{2} x}+\frac {3 c^{4} \left (\frac {\sqrt {\pi }}{x^{2} a^{2}}-\frac {\left (1-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}-\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{8 a^{2} x^{2}}+\frac {\sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}}{a^{2} x^{2}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )\right )}{2 a \sqrt {\pi }}-\frac {c^{4} \left (2 a^{2} x^{2}+1\right ) \sqrt {-a^{2} x^{2}+1}}{3 a^{4} x^{3}}\)	\(263\)

Input:

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

Output:

1/6*(16*a^4*x^4-9*a^3*x^3-14*a^2*x^2+9*a*x-2)/x^3/(-a^2*x^2+1)^(1/2)*c^4/a 
^4+(-1/2*a^3*arctanh(1/(-a^2*x^2+1)^(1/2))-3*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 = 0.09 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.94 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {36 \, a^{3} c^{4} x^{3} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + 3 \, a^{3} c^{4} x^{3} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - 6 \, a^{3} c^{4} x^{3} - {\left (6 \, a^{3} c^{4} x^{3} + 16 \, a^{2} c^{4} x^{2} - 9 \, a c^{4} x + 2 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, a^{4} x^{3}} \] Input:

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

Output:

1/6*(36*a^3*c^4*x^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + 3*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 + 16*a^2 
*c^4*x^2 - 9*a*c^4*x + 2*c^4)*sqrt(-a^2*x^2 + 1))/(a^4*x^3)
 

Sympy [A] (verification not implemented)

Time = 5.35 (sec) , antiderivative size = 354, normalized size of antiderivative = 2.51 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=a c^{4} \left (\begin {cases} - \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{2}}{2} & \text {otherwise} \end {cases}\right ) - 3 c^{4} \left (\begin {cases} \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{\sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\x & \text {otherwise} \end {cases}\right ) + \frac {2 c^{4} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right )}{a} + \frac {2 c^{4} \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right )}{a^{2}} - \frac {3 c^{4} \left (\begin {cases} - \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right )}{a^{3}} + \frac {c^{4} \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right )}{a^{4}} \] Input:

integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(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) 
) - 3*c**4*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)) + 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* 
sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True 
))/a**2 - 3*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) - 
sqrt(-a**2*x**2 + 1)/(3*x**3), True))/a**4
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.35 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=-\frac {3 \, c^{4} \arcsin \left (a x\right )}{a} - \frac {2 \, c^{4} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right )}{a} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{4}}{a} + \frac {3 \, {\left (a^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\sqrt {-a^{2} x^{2} + 1}}{x^{2}}\right )} c^{4}}{2 \, a^{3}} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} c^{4}}{a^{2} x} - \frac {{\left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a^{2}}{x} + \frac {\sqrt {-a^{2} x^{2} + 1}}{x^{3}}\right )} c^{4}}{3 \, a^{4}} \] Input:

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

Output:

-3*c^4*arcsin(a*x)/a - 2*c^4*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x))/a 
 - sqrt(-a^2*x^2 + 1)*c^4/a + 3/2*(a^2*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2 
/abs(x)) + sqrt(-a^2*x^2 + 1)/x^2)*c^4/a^3 - 2*sqrt(-a^2*x^2 + 1)*c^4/(a^2 
*x) - 1/3*(2*sqrt(-a^2*x^2 + 1)*a^2/x + 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. 263 vs. \(2 (123) = 246\).

Time = 0.17 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.87 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {{\left (c^{4} - \frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{4}}{a^{2} x} + \frac {33 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{4}}{a^{4} x^{2}}\right )} a^{6} x^{3}}{24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} {\left | a \right |}} - \frac {3 \, c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} - \frac {c^{4} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{4}}{a} - \frac {\frac {33 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{4}}{x} - \frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{4}}{a^{2} x^{2}} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{4}}{a^{4} x^{3}}}{24 \, a^{2} {\left | a \right |}} \] Input:

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

Output:

1/24*(c^4 - 9*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^4/(a^2*x) + 33*(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)) - 3*c^4*arcsin(a*x)*sgn(a)/abs(a) - 1/2*c^4*log(1/2*abs(-2* 
sqrt(-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*(33*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^4/x - 9*(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 = 137, normalized size of antiderivative = 0.97 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {3\,c^4\,\sqrt {1-a^2\,x^2}}{2\,a^3\,x^2}-\frac {c^4\,\sqrt {1-a^2\,x^2}}{a}-\frac {8\,c^4\,\sqrt {1-a^2\,x^2}}{3\,a^2\,x}-\frac {3\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {c^4\,\sqrt {1-a^2\,x^2}}{3\,a^4\,x^3}+\frac {c^4\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,a} \] Input:

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

Output:

(c^4*atan((1 - a^2*x^2)^(1/2)*1i)*1i)/(2*a) - (3*c^4*asinh(x*(-a^2)^(1/2)) 
)/(-a^2)^(1/2) - (c^4*(1 - a^2*x^2)^(1/2))/a - (8*c^4*(1 - a^2*x^2)^(1/2)) 
/(3*a^2*x) + (3*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 = 0.95 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx=\frac {c^{4} \left (-36 \mathit {asin} \left (a x \right ) a^{3} x^{3}-12 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-32 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+18 \sqrt {-a^{2} x^{2}+1}\, a x -4 \sqrt {-a^{2} x^{2}+1}+3 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}-1\right ) a^{3} x^{3}-3 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}+1\right ) a^{3} x^{3}\right )}{12 a^{4} x^{3}} \] Input:

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

Output:

(c**4*( - 36*asin(a*x)*a**3*x**3 - 12*sqrt( - a**2*x**2 + 1)*a**3*x**3 - 3 
2*sqrt( - a**2*x**2 + 1)*a**2*x**2 + 18*sqrt( - a**2*x**2 + 1)*a*x - 4*sqr 
t( - a**2*x**2 + 1) + 3*log(sqrt( - a**2*x**2 + 1) - 1)*a**3*x**3 - 3*log( 
sqrt( - a**2*x**2 + 1) + 1)*a**3*x**3))/(12*a**4*x**3)