\(\int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^5} \, dx\) [324]

Optimal result

Integrand size = 19, antiderivative size = 102 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^5} \, dx=-\frac {c^2 \sqrt {1-a^2 x^2}}{4 x^4}+\frac {a^2 c^2 \sqrt {1-a^2 x^2}}{8 x^2}+\frac {a c^2 \left (1-a^2 x^2\right )^{3/2}}{3 x^3}+\frac {1}{8} a^4 c^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.97 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^5} \, dx=\frac {c^2 \left (-6+8 a x+9 a^2 x^2-16 a^3 x^3-3 a^4 x^4+8 a^5 x^5+3 a^4 x^4 \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )}{24 x^4 \sqrt {1-a^2 x^2}} \] Input:

Integrate[(E^ArcTanh[a*x]*(c - a*c*x)^2)/x^5,x]
 

Output:

(c^2*(-6 + 8*a*x + 9*a^2*x^2 - 16*a^3*x^3 - 3*a^4*x^4 + 8*a^5*x^5 + 3*a^4* 
x^4*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]]))/(24*x^4*Sqrt[1 - a^2*x^ 
2])
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.95, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {6678, 27, 539, 27, 534, 243, 51, 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.

\(\displaystyle \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^5} \, dx\)

\(\Big \downarrow \) 6678

\(\displaystyle c \int \frac {c (1-a x) \sqrt {1-a^2 x^2}}{x^5}dx\)

\(\Big \downarrow \) 27

\(\displaystyle c^2 \int \frac {(1-a x) \sqrt {1-a^2 x^2}}{x^5}dx\)

\(\Big \downarrow \) 539

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 534

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 51

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

Int[(E^ArcTanh[a*x]*(c - a*c*x)^2)/x^5,x]
 

Output:

c^2*(-1/4*(1 - a^2*x^2)^(3/2)/x^4 - (a*((-4*(1 - a^2*x^2)^(3/2))/(3*x^3) - 
 (a*(-(Sqrt[1 - a^2*x^2]/x^2) + a^2*ArcTanh[Sqrt[1 - a^2*x^2]]))/2))/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 51

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

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 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 534

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

rule 539

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

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.79

method	result	size
risch	\(\frac {\left (8 a^{5} x^{5}-3 a^{4} x^{4}-16 a^{3} x^{3}+9 a^{2} x^{2}+8 a x -6\right ) c^{2}}{24 x^{4} \sqrt {-a^{2} x^{2}+1}}+\frac {a^{4} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{2}}{8}\)	\(81\)
default	\(c^{2} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{4 x^{4}}-\frac {a^{2} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )}{4}-\frac {a^{3} \sqrt {-a^{2} x^{2}+1}}{x}-a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}-\frac {2 a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}\right )\right )\)	\(125\)
meijerg	\(-\frac {a^{3} c^{2} \sqrt {-a^{2} x^{2}+1}}{x}+\frac {a^{4} c^{2} \left (\frac {\sqrt {\pi }}{x^{2} a^{2}}-\frac {\left (1-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}-\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{8 a^{2} x^{2}}+\frac {\sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}}{a^{2} x^{2}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )\right )}{2 \sqrt {\pi }}+\frac {a \,c^{2} \left (2 a^{2} x^{2}+1\right ) \sqrt {-a^{2} x^{2}+1}}{3 x^{3}}+\frac {a^{4} c^{2} \left (-\frac {\sqrt {\pi }}{2 x^{4} a^{4}}-\frac {\sqrt {\pi }}{2 x^{2} a^{2}}+\frac {3 \left (\frac {7}{6}-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{8}+\frac {\sqrt {\pi }\, \left (-7 a^{4} x^{4}+8 a^{2} x^{2}+8\right )}{16 a^{4} x^{4}}-\frac {\sqrt {\pi }\, \left (12 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{16 a^{4} x^{4}}-\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )}{4}\right )}{2 \sqrt {\pi }}\)	\(302\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.82 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^5} \, dx=-\frac {3 \, a^{4} c^{2} x^{4} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + {\left (8 \, a^{3} c^{2} x^{3} - 3 \, a^{2} c^{2} x^{2} - 8 \, a c^{2} x + 6 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{24 \, x^{4}} \] Input:

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

Output:

-1/24*(3*a^4*c^2*x^4*log((sqrt(-a^2*x^2 + 1) - 1)/x) + (8*a^3*c^2*x^3 - 3* 
a^2*c^2*x^2 - 8*a*c^2*x + 6*c^2)*sqrt(-a^2*x^2 + 1))/x^4
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.53 (sec) , antiderivative size = 413, normalized size of antiderivative = 4.05 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^5} \, dx=a^{3} c^{2} \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right ) - a^{2} c^{2} \left (\begin {cases} - \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right ) - a c^{2} \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right ) + c^{2} \left (\begin {cases} - \frac {3 a^{4} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} + \frac {3 a^{3}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {a}{8 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{4 a x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {3 i a^{4} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} - \frac {3 i a^{3}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i a}{8 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{4 a x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right ) \] Input:

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

Output:

a**3*c**2*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*c**2*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*c**2*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)) + c**2*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/A 
bs(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))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.20 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^5} \, dx=\frac {1}{8} \, a^{4} c^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {\sqrt {-a^{2} x^{2} + 1} a^{3} c^{2}}{3 \, x} + \frac {\sqrt {-a^{2} x^{2} + 1} a^{2} c^{2}}{8 \, x^{2}} + \frac {\sqrt {-a^{2} x^{2} + 1} a c^{2}}{3 \, x^{3}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{4 \, x^{4}} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (86) = 172\).

Time = 0.13 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.35 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^5} \, dx=\frac {{\left (3 \, a^{5} c^{2} - \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{3} c^{2}}{x} + \frac {24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{2}}{a x^{3}}\right )} a^{8} x^{4}}{192 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} {\left | a \right |}} + \frac {a^{5} c^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{8 \, {\left | a \right |}} - \frac {\frac {24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{5} c^{2} {\left | a \right |}}{x} - \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} a c^{2} {\left | a \right |}}{x^{3}} + \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{2} {\left | a \right |}}{a x^{4}}}{192 \, a^{4}} \] Input:

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

Output:

1/192*(3*a^5*c^2 - 8*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^3*c^2/x + 24*(sqrt( 
-a^2*x^2 + 1)*abs(a) + a)^3*c^2/(a*x^3))*a^8*x^4/((sqrt(-a^2*x^2 + 1)*abs( 
a) + a)^4*abs(a)) + 1/8*a^5*c^2*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 
 2*a)/(a^2*abs(x)))/abs(a) - 1/192*(24*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^5 
*c^2*abs(a)/x - 8*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*a*c^2*abs(a)/x^3 + 3*( 
sqrt(-a^2*x^2 + 1)*abs(a) + a)^4*c^2*abs(a)/(a*x^4))/a^4
 

Mupad [B] (verification not implemented)

Time = 14.21 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.11 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^5} \, dx=\frac {a\,c^2\,\sqrt {1-a^2\,x^2}}{3\,x^3}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{4\,x^4}+\frac {a^2\,c^2\,\sqrt {1-a^2\,x^2}}{8\,x^2}-\frac {a^3\,c^2\,\sqrt {1-a^2\,x^2}}{3\,x}-\frac {a^4\,c^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{8} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.89 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^5} \, dx=\frac {c^{2} \left (-8 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+3 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+8 \sqrt {-a^{2} x^{2}+1}\, a x -6 \sqrt {-a^{2} x^{2}+1}-3 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{4} x^{4}\right )}{24 x^{4}} \] Input:

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

Output:

(c**2*( - 8*sqrt( - a**2*x**2 + 1)*a**3*x**3 + 3*sqrt( - a**2*x**2 + 1)*a* 
*2*x**2 + 8*sqrt( - a**2*x**2 + 1)*a*x - 6*sqrt( - a**2*x**2 + 1) - 3*log( 
tan(asin(a*x)/2))*a**4*x**4))/(24*x**4)