\(\int e^{-\coth ^{-1}(a x)} (c-\frac {c}{a x})^3 \, dx\) [429]

Optimal result

Integrand size = 22, antiderivative size = 106 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=-\frac {4 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{a}+\frac {c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^3 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {13 c^3 \csc ^{-1}(a x)}{2 a}-\frac {4 c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \] Output:

-4*c^3*(1-1/a^2/x^2)^(1/2)/a+1/2*c^3*(1-1/a^2/x^2)^(1/2)/a^2/x+c^3*(1-1/a^ 
2/x^2)^(1/2)*x-13/2*c^3*arccsc(a*x)/a-4*c^3*arctanh((1-1/a^2/x^2)^(1/2))/a
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.58 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {c^3 \left (-1+8 a x-a^2 x^2-8 a^3 x^3+2 a^4 x^4+10 a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3 \arcsin \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {2}}\right )-8 a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3 \arcsin \left (\frac {1}{a x}\right )-8 a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{2 a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^3} \] Input:

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

Output:

(c^3*(-1 + 8*a*x - a^2*x^2 - 8*a^3*x^3 + 2*a^4*x^4 + 10*a^3*Sqrt[1 - 1/(a^ 
2*x^2)]*x^3*ArcSin[Sqrt[1 - 1/(a*x)]/Sqrt[2]] - 8*a^3*Sqrt[1 - 1/(a^2*x^2) 
]*x^3*ArcSin[1/(a*x)] - 8*a^3*Sqrt[1 - 1/(a^2*x^2)]*x^3*ArcTanh[Sqrt[1 - 1 
/(a^2*x^2)]]))/(2*a^4*Sqrt[1 - 1/(a^2*x^2)]*x^3)
 

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.01, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {6731, 27, 540, 2340, 25, 2340, 25, 27, 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.

\(\displaystyle \int \left (c-\frac {c}{a x}\right )^3 e^{-\coth ^{-1}(a x)} \, dx\)

\(\Big \downarrow \) 6731

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 540

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

\(\Big \downarrow \) 2340

\(\displaystyle -\frac {c^3 \left (\frac {1}{2} a^2 \int -\frac {\left (8 a-\frac {13}{x}+\frac {8}{x^2 a}\right ) x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}+a^4 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{a^4}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {c^3 \left (-\frac {1}{2} a^2 \int \frac {\left (8 a-\frac {13}{x}+\frac {8}{x^2 a}\right ) x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}+a^4 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{a^4}\)

\(\Big \downarrow \) 2340

\(\displaystyle -\frac {c^3 \left (-\frac {1}{2} a^2 \left (a^2 \left (-\int -\frac {\left (8 a-\frac {13}{x}\right ) x}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\right )-8 a \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}+a^4 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{a^4}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {c^3 \left (-\frac {1}{2} a^2 \left (a^2 \int \frac {\left (8 a-\frac {13}{x}\right ) x}{a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-8 a \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}+a^4 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{a^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {c^3 \left (-\frac {1}{2} a^2 \left (\int \frac {\left (8 a-\frac {13}{x}\right ) x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-8 a \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}+a^4 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{a^4}\)

\(\Big \downarrow \) 538

\(\displaystyle -\frac {c^3 \left (-\frac {1}{2} a^2 \left (8 a \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-13 \int \frac {1}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-8 a \sqrt {1-\frac {1}{a^2 x^2}}\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}+a^4 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{a^4}\)

\(\Big \downarrow \) 223

\(\displaystyle -\frac {c^3 \left (-\frac {1}{2} a^2 \left (8 a \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}-8 a \sqrt {1-\frac {1}{a^2 x^2}}-13 a \arcsin \left (\frac {1}{a x}\right )\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}+a^4 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{a^4}\)

\(\Big \downarrow \) 243

\(\displaystyle -\frac {c^3 \left (-\frac {1}{2} a^2 \left (4 a \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}-8 a \sqrt {1-\frac {1}{a^2 x^2}}-13 a \arcsin \left (\frac {1}{a x}\right )\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}+a^4 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{a^4}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {c^3 \left (-\frac {1}{2} a^2 \left (-8 a^3 \int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}-8 a \sqrt {1-\frac {1}{a^2 x^2}}-13 a \arcsin \left (\frac {1}{a x}\right )\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}+a^4 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{a^4}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {c^3 \left (-\frac {1}{2} a^2 \left (-8 a \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-8 a \sqrt {1-\frac {1}{a^2 x^2}}-13 a \arcsin \left (\frac {1}{a x}\right )\right )-\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}+a^4 x \left (-\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{a^4}\)

Input:

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

Output:

-((c^3*(-1/2*(a^2*Sqrt[1 - 1/(a^2*x^2)])/x - a^4*Sqrt[1 - 1/(a^2*x^2)]*x - 
 (a^2*(-8*a*Sqrt[1 - 1/(a^2*x^2)] - 13*a*ArcSin[1/(a*x)] - 8*a*ArcTanh[Sqr 
t[1 - 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 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 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 2340

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 
)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m 
+ q + 2*p + 1))   Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) 
*Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; 
GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ 
[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
 

rule 6731

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.), x_Symbol] :> S 
imp[-c^n   Subst[Int[(c + d*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^2), x], x, 1/ 
x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] && IntegerQ[(n - 1)/2] 
&& IntegerQ[2*p]
 

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.41

method	result	size
risch	\(-\frac {\left (a x +1\right ) \left (8 a x -1\right ) c^{3} \sqrt {\frac {a x -1}{a x +1}}}{2 x^{2} a^{3}}+\frac {\left (-\frac {13 a^{2} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{2}-\frac {4 a^{3} \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}+a^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\right ) c^{3} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a^{3} \left (a x -1\right )}\)	\(149\)
default	\(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) c^{3} \left (-8 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{3} x^{3}+16 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}+8 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x -13 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{2} x^{2}-13 a^{2} \sqrt {a^{2}}\, x^{2} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+8 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-16 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{2 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{2} \sqrt {a^{2}}}\)	\(266\)

Input:

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

Output:

-1/2*(a*x+1)*(8*a*x-1)/x^2*c^3/a^3*((a*x-1)/(a*x+1))^(1/2)+(-13/2*a^2*arct 
an(1/(a^2*x^2-1)^(1/2))-4*a^3*ln(a^2*x/(a^2)^(1/2)+(a^2*x^2-1)^(1/2))/(a^2 
)^(1/2)+a^2*((a*x-1)*(a*x+1))^(1/2))*c^3/a^3/(a*x-1)*((a*x-1)/(a*x+1))^(1/ 
2)*((a*x-1)*(a*x+1))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.35 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {26 \, a^{2} c^{3} x^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - 8 \, a^{2} c^{3} x^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 8 \, a^{2} c^{3} x^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (2 \, a^{3} c^{3} x^{3} - 6 \, a^{2} c^{3} x^{2} - 7 \, a c^{3} x + c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, a^{3} x^{2}} \] Input:

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

Output:

1/2*(26*a^2*c^3*x^2*arctan(sqrt((a*x - 1)/(a*x + 1))) - 8*a^2*c^3*x^2*log( 
sqrt((a*x - 1)/(a*x + 1)) + 1) + 8*a^2*c^3*x^2*log(sqrt((a*x - 1)/(a*x + 1 
)) - 1) + (2*a^3*c^3*x^3 - 6*a^2*c^3*x^2 - 7*a*c^3*x + c^3)*sqrt((a*x - 1) 
/(a*x + 1)))/(a^3*x^2)
 

Sympy [F]

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

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

Output:

c**3*(Integral(a**3*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)), x) + Integral(-sqrt 
(a*x/(a*x + 1) - 1/(a*x + 1))/x**3, x) + Integral(3*a*sqrt(a*x/(a*x + 1) - 
 1/(a*x + 1))/x**2, x) + Integral(-3*a**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1) 
)/x, x))/a**3
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (94) = 188\).

Time = 0.13 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.90 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx={\left (\frac {13 \, c^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} - \frac {4 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {4 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} + \frac {11 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 2 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 5 \, c^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - \frac {{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} + a^{2}}\right )} a \] Input:

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

Output:

(13*c^3*arctan(sqrt((a*x - 1)/(a*x + 1)))/a^2 - 4*c^3*log(sqrt((a*x - 1)/( 
a*x + 1)) + 1)/a^2 + 4*c^3*log(sqrt((a*x - 1)/(a*x + 1)) - 1)/a^2 + (11*c^ 
3*((a*x - 1)/(a*x + 1))^(5/2) + 2*c^3*((a*x - 1)/(a*x + 1))^(3/2) - 5*c^3* 
sqrt((a*x - 1)/(a*x + 1)))/((a*x - 1)*a^2/(a*x + 1) - (a*x - 1)^2*a^2/(a*x 
 + 1)^2 - (a*x - 1)^3*a^2/(a*x + 1)^3 + a^2))*a
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (94) = 188\).

Time = 0.13 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.19 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {13 \, c^{3} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right ) \mathrm {sgn}\left (a x + 1\right )}{a} + \frac {4 \, c^{3} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{{\left | a \right |}} + \frac {\sqrt {a^{2} x^{2} - 1} c^{3} \mathrm {sgn}\left (a x + 1\right )}{a} - \frac {{\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{3} c^{3} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) + 8 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} a c^{3} \mathrm {sgn}\left (a x + 1\right ) - {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )} c^{3} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) + 8 \, a c^{3} \mathrm {sgn}\left (a x + 1\right )}{{\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{2} a {\left | a \right |}} \] Input:

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

Output:

13*c^3*arctan(-x*abs(a) + sqrt(a^2*x^2 - 1))*sgn(a*x + 1)/a + 4*c^3*log(ab 
s(-x*abs(a) + sqrt(a^2*x^2 - 1)))*sgn(a*x + 1)/abs(a) + sqrt(a^2*x^2 - 1)* 
c^3*sgn(a*x + 1)/a - ((x*abs(a) - sqrt(a^2*x^2 - 1))^3*c^3*abs(a)*sgn(a*x 
+ 1) + 8*(x*abs(a) - sqrt(a^2*x^2 - 1))^2*a*c^3*sgn(a*x + 1) - (x*abs(a) - 
 sqrt(a^2*x^2 - 1))*c^3*abs(a)*sgn(a*x + 1) + 8*a*c^3*sgn(a*x + 1))/(((x*a 
bs(a) - sqrt(a^2*x^2 - 1))^2 + 1)^2*a*abs(a))
 

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.54 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {2\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}-5\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}+11\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{a+\frac {a\,\left (a\,x-1\right )}{a\,x+1}-\frac {a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}}+\frac {13\,c^3\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}-\frac {8\,c^3\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \] Input:

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

Output:

(2*c^3*((a*x - 1)/(a*x + 1))^(3/2) - 5*c^3*((a*x - 1)/(a*x + 1))^(1/2) + 1 
1*c^3*((a*x - 1)/(a*x + 1))^(5/2))/(a + (a*(a*x - 1))/(a*x + 1) - (a*(a*x 
- 1)^2)/(a*x + 1)^2 - (a*(a*x - 1)^3)/(a*x + 1)^3) + (13*c^3*atan(((a*x - 
1)/(a*x + 1))^(1/2)))/a - (8*c^3*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/a
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.26 \[ \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx=\frac {c^{3} \left (26 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}-1\right ) a^{2} x^{2}-26 \mathit {atan} \left (\sqrt {a x -1}+\sqrt {a x +1}+1\right ) a^{2} x^{2}+2 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}-8 \sqrt {a x +1}\, \sqrt {a x -1}\, a x +\sqrt {a x +1}\, \sqrt {a x -1}-16 \,\mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a^{2} x^{2}\right )}{2 a^{3} x^{2}} \] Input:

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

Output:

(c**3*(26*atan(sqrt(a*x - 1) + sqrt(a*x + 1) - 1)*a**2*x**2 - 26*atan(sqrt 
(a*x - 1) + sqrt(a*x + 1) + 1)*a**2*x**2 + 2*sqrt(a*x + 1)*sqrt(a*x - 1)*a 
**2*x**2 - 8*sqrt(a*x + 1)*sqrt(a*x - 1)*a*x + sqrt(a*x + 1)*sqrt(a*x - 1) 
 - 16*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a**2*x**2))/(2*a**3*x** 
2)