\(\int e^{-\text {arctanh}(a x)} (c-\frac {c}{a^2 x^2})^2 \, dx\) [705]

Optimal result

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

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.68 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=\frac {c^2 \left (-\frac {5 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},a^2 x^2\right )}{x^3}+3 a^3 \left (1-a^2 x^2\right )^{5/2} \operatorname {Hypergeometric2F1}\left (2,\frac {5}{2},\frac {7}{2},1-a^2 x^2\right )\right )}{15 a^4} \] Input:

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

Output:

(c^2*((-5*Hypergeometric2F1[-3/2, -3/2, -1/2, a^2*x^2])/x^3 + 3*a^3*(1 - a 
^2*x^2)^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, 1 - a^2*x^2]))/(15*a^4)
 

Rubi [A] (verified)

Time = 0.69 (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 = {6707, 6699, 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.

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

\(\Big \downarrow \) 6707

\(\displaystyle \frac {c^2 \int \frac {e^{-\text {arctanh}(a x)} \left (1-a^2 x^2\right )^2}{x^4}dx}{a^4}\)

\(\Big \downarrow \) 6699

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

\(\Big \downarrow \) 537

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 536

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(c^2*(-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 6699

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_ 
Symbol] :> Simp[c^p   Int[x^m*((1 - a^2*x^2)^(p + n/2)/(1 - a*x)^n), x], x] 
 /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c 
, 0]) && ILtQ[(n - 1)/2, 0] &&  !IntegerQ[p - n/2]
 

rule 6707

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symb 
ol] :> Simp[d^p   Int[(u/x^(2*p))*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x 
] /; FreeQ[{a, c, d, n}, x] && EqQ[c + a^2*d, 0] && IntegerQ[p]
 

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.25

method	result	size
risch	\(-\frac {\left (8 a^{4} x^{4}+3 a^{3} x^{3}-10 a^{2} x^{2}-3 a x +2\right ) c^{2}}{6 x^{3} \sqrt {-a^{2} x^{2}+1}\, a^{4}}-\frac {\left (\frac {3 a^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-\frac {a^{4} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-a^{3} \sqrt {-a^{2} x^{2}+1}\right ) c^{2}}{a^{4}}\)	\(129\)
default	\(\frac {c^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 x^{3}}+a^{3} \left (\sqrt {-a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )-a \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{2 x^{2}}-\frac {a^{2} \left (\sqrt {-a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )-a^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{x}-2 a^{2} \left (\frac {x \sqrt {-a^{2} x^{2}+1}}{2}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )}{a^{4}}\)	\(183\)

Input:

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

Fricas [A] (verification not implemented)

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.66 (sec) , antiderivative size = 381, normalized size of antiderivative = 3.70 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=\frac {c^{2} \left (\begin {cases} i \sqrt {a^{2} x^{2} - 1} - \log {\left (a x \right )} + \frac {\log {\left (a^{2} x^{2} \right )}}{2} + i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\sqrt {- a^{2} x^{2} + 1} + \frac {\log {\left (a^{2} x^{2} \right )}}{2} - \log {\left (\sqrt {- a^{2} x^{2} + 1} + 1 \right )} & \text {otherwise} \end {cases}\right )}{a} - \frac {c^{2} \left (\begin {cases} - \frac {i a^{2} x}{\sqrt {a^{2} x^{2} - 1}} + i a \operatorname {acosh}{\left (a x \right )} + \frac {i}{x \sqrt {a^{2} x^{2} - 1}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {a^{2} x}{\sqrt {- a^{2} x^{2} + 1}} - a \operatorname {asin}{\left (a x \right )} - \frac {1}{x \sqrt {- a^{2} x^{2} + 1}} & \text {otherwise} \end {cases}\right )}{a^{2}} - \frac {c^{2} \left (\begin {cases} \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right )}{a^{3}} + \frac {c^{2} \left (\begin {cases} \frac {a^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{3} - \frac {a \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{3 x^{2}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{3} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{3 x^{2}} & \text {otherwise} \end {cases}\right )}{a^{4}} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

c^2*(arcsin(a*x)/a + sqrt(-a^2*x^2 + 1)/a) - integrate((2*a^2*c^2*x^2 - c^ 
2)*sqrt(a*x + 1)*sqrt(-a*x + 1)/(a^5*x^5 + a^4*x^4), x)
 

Giac [B] (verification not implemented)

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

Time = 0.13 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.54 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=\frac {{\left (c^{2} - \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{2}}{a^{2} x} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{2}}{a^{4} x^{2}}\right )} a^{6} x^{3}}{24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} {\left | a \right |}} + \frac {c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} - \frac {3 \, c^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{a} + \frac {\frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{2}}{x} + \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{2}}{a^{2} x^{2}} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{2}}{a^{4} x^{3}}}{24 \, a^{2} {\left | a \right |}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.31 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=\frac {c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}+\frac {c^2\,\sqrt {1-a^2\,x^2}}{a}+\frac {4\,c^2\,\sqrt {1-a^2\,x^2}}{3\,a^2\,x}+\frac {c^2\,\sqrt {1-a^2\,x^2}}{2\,a^3\,x^2}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{3\,a^4\,x^3}+\frac {c^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2\,a} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.30 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=\frac {c^{2} \left (12 \mathit {asin} \left (a x \right ) a^{3} x^{3}+12 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+16 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+6 \sqrt {-a^{2} x^{2}+1}\, a x -4 \sqrt {-a^{2} x^{2}+1}+9 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}-1\right ) a^{3} x^{3}-9 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}+1\right ) a^{3} x^{3}\right )}{12 a^{4} x^{3}} \] Input:

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

Output:

(c**2*(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)