\(\int \frac {e^{\text {arctanh}(a x)} (c-a c x)^4}{x} \, dx\) [341]

Optimal result

Integrand size = 19, antiderivative size = 115 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^4}{x} \, dx=c^4 \sqrt {1-a^2 x^2}-\frac {13}{8} a c^4 x \sqrt {1-a^2 x^2}-c^4 \left (1-a^2 x^2\right )^{3/2}+\frac {1}{4} a c^4 x \left (1-a^2 x^2\right )^{3/2}-\frac {13}{8} c^4 \arcsin (a x)-c^4 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \] Output:

c^4*(-a^2*x^2+1)^(1/2)-13/8*a*c^4*x*(-a^2*x^2+1)^(1/2)-c^4*(-a^2*x^2+1)^(3 
/2)+1/4*a*c^4*x*(-a^2*x^2+1)^(3/2)-13/8*c^4*arcsin(a*x)-c^4*arctanh((-a^2* 
x^2+1)^(1/2))
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.23 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^4}{x} \, dx=\frac {c^4 \left (-11 a x+8 a^2 x^2+9 a^3 x^3-8 a^4 x^4+2 a^5 x^5+4 \sqrt {1-a^2 x^2} \arcsin (a x)+34 \sqrt {1-a^2 x^2} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )-8 \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )}{8 \sqrt {1-a^2 x^2}} \] Input:

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

Output:

(c^4*(-11*a*x + 8*a^2*x^2 + 9*a^3*x^3 - 8*a^4*x^4 + 2*a^5*x^5 + 4*Sqrt[1 - 
 a^2*x^2]*ArcSin[a*x] + 34*Sqrt[1 - a^2*x^2]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]] 
 - 8*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]]))/(8*Sqrt[1 - a^2*x^2])
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.95, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {6678, 27, 541, 25, 2340, 27, 535, 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.

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

\(\Big \downarrow \) 6678

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 541

\(\displaystyle c^4 \left (\frac {1}{4} a x \left (1-a^2 x^2\right )^{3/2}-\frac {\int -\frac {\sqrt {1-a^2 x^2} \left (12 x^2 a^4-13 x a^3+4 a^2\right )}{x}dx}{4 a^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle c^4 \left (\frac {\int \frac {\sqrt {1-a^2 x^2} \left (12 x^2 a^4-13 x a^3+4 a^2\right )}{x}dx}{4 a^2}+\frac {1}{4} a x \left (1-a^2 x^2\right )^{3/2}\right )\)

\(\Big \downarrow \) 2340

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 535

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 535

Int[(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_), x_Symbol] :> Sim 
p[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^p/(2*p*(2*p + 1))), x] + Simp[a/(2*p 
 + 1)   Int[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^(p - 1)/x), x], x] /; Free 
Q[{a, b, c, d}, x] && GtQ[p, 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 541

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Simp[d^n*x^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*(m + n + 2*p + 1))), x 
] + Simp[1/(b*(m + n + 2*p + 1))   Int[x^m*(a + b*x^2)^p*ExpandToSum[b*(m + 
 n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1)*x^n - a*d^n*(m + n - 1) 
*x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, m, p}, x] && IGtQ[n, 1] && IGt 
Q[m, -2] && 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 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(238\) vs. \(2(99)=198\).

Time = 0.26 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.08

method	result	size
default	\(c^{4} \left (-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+a^{5} \left (-\frac {x^{3} \sqrt {-a^{2} x^{2}+1}}{4 a^{2}}+\frac {-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )-\frac {3 a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-2 \sqrt {-a^{2} x^{2}+1}+2 a^{3} \left (-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )-3 a^{4} \left (-\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}\right )\right )\)	\(239\)
meijerg	\(\frac {a \,c^{4} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (10 a^{2} x^{2}+15\right ) \sqrt {-a^{2} x^{2}+1}}{20 a^{4}}+\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{4 a^{5}}\right )}{2 \sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {3 c^{4} \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{6}\right )}{2 \sqrt {\pi }}-\frac {a \,c^{4} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}} \sqrt {-a^{2} x^{2}+1}}{a^{2}}+\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {c^{4} \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{\sqrt {\pi }}-3 c^{4} \arcsin \left (a x \right )+\frac {c^{4} \left (\left (-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )\right )}{2 \sqrt {\pi }}\)	\(270\)

Input:

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

Output:

c^4*(-arctanh(1/(-a^2*x^2+1)^(1/2))+a^5*(-1/4*x^3*(-a^2*x^2+1)^(1/2)/a^2+3 
/4/a^2*(-1/2*x*(-a^2*x^2+1)^(1/2)/a^2+1/2/a^2/(a^2)^(1/2)*arctan((a^2)^(1/ 
2)*x/(-a^2*x^2+1)^(1/2))))-3*a/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+ 
1)^(1/2))-2*(-a^2*x^2+1)^(1/2)+2*a^3*(-1/2*x*(-a^2*x^2+1)^(1/2)/a^2+1/2/a^ 
2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2)))-3*a^4*(-1/3*x^2/a^ 
2*(-a^2*x^2+1)^(1/2)-2/3*(-a^2*x^2+1)^(1/2)/a^4))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.83 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^4}{x} \, dx=\frac {13}{4} \, c^{4} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + c^{4} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - \frac {1}{8} \, {\left (2 \, a^{3} c^{4} x^{3} - 8 \, a^{2} c^{4} x^{2} + 11 \, a c^{4} x\right )} \sqrt {-a^{2} x^{2} + 1} \] Input:

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

Output:

13/4*c^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + c^4*log((sqrt(-a^2*x^2 + 
 1) - 1)/x) - 1/8*(2*a^3*c^4*x^3 - 8*a^2*c^4*x^2 + 11*a*c^4*x)*sqrt(-a^2*x 
^2 + 1)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 5.30 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.98 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^4}{x} \, dx=a^{5} c^{4} \left (\begin {cases} - \frac {x^{3} \sqrt {- a^{2} x^{2} + 1}}{4 a^{2}} - \frac {3 x \sqrt {- a^{2} x^{2} + 1}}{8 a^{4}} + \frac {3 \log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{8 a^{4} \sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{5}}{5} & \text {otherwise} \end {cases}\right ) - 3 a^{4} c^{4} \left (\begin {cases} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{3 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{3 a^{4}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) + 2 a^{3} c^{4} \left (\begin {cases} - \frac {x \sqrt {- a^{2} x^{2} + 1}}{2 a^{2}} + \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{2 a^{2} \sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{3}}{3} & \text {otherwise} \end {cases}\right ) + 2 a^{2} c^{4} \left (\begin {cases} - \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{2}}{2} & \text {otherwise} \end {cases}\right ) - 3 a c^{4} \left (\begin {cases} \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{\sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\x & \text {otherwise} \end {cases}\right ) + c^{4} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right ) \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.91 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^4}{x} \, dx=-\frac {1}{4} \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{4} x^{3} + \sqrt {-a^{2} x^{2} + 1} a^{2} c^{4} x^{2} - \frac {11}{8} \, \sqrt {-a^{2} x^{2} + 1} a c^{4} x - \frac {13}{8} \, c^{4} \arcsin \left (a x\right ) - c^{4} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \] Input:

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

Output:

-1/4*sqrt(-a^2*x^2 + 1)*a^3*c^4*x^3 + sqrt(-a^2*x^2 + 1)*a^2*c^4*x^2 - 11/ 
8*sqrt(-a^2*x^2 + 1)*a*c^4*x - 13/8*c^4*arcsin(a*x) - c^4*log(2*sqrt(-a^2* 
x^2 + 1)/abs(x) + 2/abs(x))
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.87 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^4}{x} \, dx=-\frac {13 \, a c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{8 \, {\left | a \right |}} - \frac {a c^{4} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} - \frac {1}{8} \, {\left (11 \, a c^{4} + 2 \, {\left (a^{3} c^{4} x - 4 \, a^{2} c^{4}\right )} x\right )} \sqrt {-a^{2} x^{2} + 1} x \] Input:

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

Output:

-13/8*a*c^4*arcsin(a*x)*sgn(a)/abs(a) - a*c^4*log(1/2*abs(-2*sqrt(-a^2*x^2 
 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) - 1/8*(11*a*c^4 + 2*(a^3*c^4*x - 
4*a^2*c^4)*x)*sqrt(-a^2*x^2 + 1)*x
 

Mupad [B] (verification not implemented)

Time = 14.09 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.96 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^4}{x} \, dx=a^2\,c^4\,x^2\,\sqrt {1-a^2\,x^2}-\frac {a^3\,c^4\,x^3\,\sqrt {1-a^2\,x^2}}{4}-\frac {11\,a\,c^4\,x\,\sqrt {1-a^2\,x^2}}{8}-\frac {13\,a\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,\sqrt {-a^2}}+c^4\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i} \] Input:

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

Output:

c^4*atan((1 - a^2*x^2)^(1/2)*1i)*1i + a^2*c^4*x^2*(1 - a^2*x^2)^(1/2) - (a 
^3*c^4*x^3*(1 - a^2*x^2)^(1/2))/4 - (11*a*c^4*x*(1 - a^2*x^2)^(1/2))/8 - ( 
13*a*c^4*asinh(x*(-a^2)^(1/2)))/(8*(-a^2)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.65 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^4}{x} \, dx=\frac {c^{4} \left (-13 \mathit {asin} \left (a x \right )-2 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+8 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-11 \sqrt {-a^{2} x^{2}+1}\, a x +8 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right )\right )}{8} \] Input:

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

Output:

(c**4*( - 13*asin(a*x) - 2*sqrt( - a**2*x**2 + 1)*a**3*x**3 + 8*sqrt( - a* 
*2*x**2 + 1)*a**2*x**2 - 11*sqrt( - a**2*x**2 + 1)*a*x + 8*log(tan(asin(a* 
x)/2))))/8