\(\int \frac {e^{2 \text {arctanh}(a x)} (c-a^2 c x^2)^{3/2}}{x^2} \, dx\) [1128]

Optimal result

Integrand size = 27, antiderivative size = 112 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^2} \, dx=\frac {1}{2} a c (4-a x) \sqrt {c-a^2 c x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{x}-\frac {1}{2} a c^{3/2} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )-2 a c^{3/2} \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.11 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^2} \, dx=\frac {c \left (-2+4 a x+a^2 x^2\right ) \sqrt {c-a^2 c x^2}}{2 x}+\frac {1}{2} a c^{3/2} \arctan \left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (-1+a^2 x^2\right )}\right )+2 a c^{3/2} \log (x)-2 a c^{3/2} \log \left (c+\sqrt {c} \sqrt {c-a^2 c x^2}\right ) \] Input:

Integrate[(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2))/x^2,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {6701, 540, 25, 27, 535, 538, 224, 216, 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.

\(\displaystyle \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^2} \, dx\)

\(\Big \downarrow \) 6701

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

\(\Big \downarrow \) 540

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 535

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

Int[(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2))/x^2,x]
 

Output:

c*(-((c - a^2*c*x^2)^(3/2)/(c*x)) + a*(((4 - a*x)*Sqrt[c - a^2*c*x^2])/2 + 
 (c*(-(ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]]/Sqrt[c]) - (4*ArcTanh[Sqr 
t[c - a^2*c*x^2]/Sqrt[c]])/Sqrt[c]))/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 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

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 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

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 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 6701

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_ 
Symbol] :> Simp[c^(n/2)   Int[x^m*(c + d*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]) && IGtQ[n/2, 0]
 

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.32

method	result	size
risch	\(\frac {\left (a^{2} x^{2}-1\right ) c^{2}}{x \sqrt {-c \left (a^{2} x^{2}-1\right )}}-\left (-\frac {a^{2} x \sqrt {-a^{2} c \,x^{2}+c}}{2 c}+\frac {a^{2} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}+\frac {2 a \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{\sqrt {c}}-\frac {2 a \sqrt {-c \left (a^{2} x^{2}-1\right )}}{c}\right ) c^{2}\)	\(148\)
default	\(-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{c x}-4 a^{2} \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )+2 a \left (\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )\right )-2 a \left (\frac {\left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}{3}-a c \left (-\frac {\left (-2 \left (x -\frac {1}{a}\right ) a^{2} c -2 a c \right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}\right )}{2 \sqrt {a^{2} c}}\right )\right )\)	\(309\)

Input:

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

Output:

(a^2*x^2-1)/x/(-c*(a^2*x^2-1))^(1/2)*c^2-(-1/2*a^2*x/c*(-a^2*c*x^2+c)^(1/2 
)+1/2*a^2/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-a^2*c*x^2+c)^(1/2))+2*a/c 
^(1/2)*ln((2*c+2*c^(1/2)*(-a^2*c*x^2+c)^(1/2))/x)-2*a/c*(-c*(a^2*x^2-1))^( 
1/2))*c^2
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.12 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^2} \, dx=\left [\frac {a c^{\frac {3}{2}} x \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) + 2 \, a c^{\frac {3}{2}} x \log \left (-\frac {a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) + {\left (a^{2} c x^{2} + 4 \, a c x - 2 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{2 \, x}, \frac {8 \, a \sqrt {-c} c x \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{c}\right ) + a \sqrt {-c} c x \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) + 2 \, {\left (a^{2} c x^{2} + 4 \, a c x - 2 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{4 \, x}\right ] \] Input:

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

Output:

[1/2*(a*c^(3/2)*x*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) 
 + 2*a*c^(3/2)*x*log(-(a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x 
^2) + (a^2*c*x^2 + 4*a*c*x - 2*c)*sqrt(-a^2*c*x^2 + c))/x, 1/4*(8*a*sqrt(- 
c)*c*x*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/c) + a*sqrt(-c)*c*x*log(2*a^2* 
c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) + 2*(a^2*c*x^2 + 4*a*c*x 
- 2*c)*sqrt(-a^2*c*x^2 + c))/x]
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 5.63 (sec) , antiderivative size = 340, normalized size of antiderivative = 3.04 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^2} \, dx=a^{2} c \left (\begin {cases} \frac {c \left (\begin {cases} \frac {\log {\left (- 2 a^{2} c x + 2 \sqrt {- a^{2} c} \sqrt {- a^{2} c x^{2} + c} \right )}}{\sqrt {- a^{2} c}} & \text {for}\: c \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- a^{2} c x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {- a^{2} c x^{2} + c}}{2} & \text {for}\: a^{2} c \neq 0 \\\sqrt {c} x & \text {otherwise} \end {cases}\right ) + 2 a c \left (\begin {cases} i \sqrt {c} \sqrt {a^{2} x^{2} - 1} - \sqrt {c} \log {\left (a x \right )} + \frac {\sqrt {c} \log {\left (a^{2} x^{2} \right )}}{2} + i \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\sqrt {c} \sqrt {- a^{2} x^{2} + 1} + \frac {\sqrt {c} \log {\left (a^{2} x^{2} \right )}}{2} - \sqrt {c} \log {\left (\sqrt {- a^{2} x^{2} + 1} + 1 \right )} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} - \frac {i a^{2} \sqrt {c} x}{\sqrt {a^{2} x^{2} - 1}} + i a \sqrt {c} \operatorname {acosh}{\left (a x \right )} + \frac {i \sqrt {c}}{x \sqrt {a^{2} x^{2} - 1}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {a^{2} \sqrt {c} x}{\sqrt {- a^{2} x^{2} + 1}} - a \sqrt {c} \operatorname {asin}{\left (a x \right )} - \frac {\sqrt {c}}{x \sqrt {- a^{2} x^{2} + 1}} & \text {otherwise} \end {cases}\right ) \] Input:

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

Output:

a**2*c*Piecewise((c*Piecewise((log(-2*a**2*c*x + 2*sqrt(-a**2*c)*sqrt(-a** 
2*c*x**2 + c))/sqrt(-a**2*c), Ne(c, 0)), (x*log(x)/sqrt(-a**2*c*x**2), Tru 
e))/2 + x*sqrt(-a**2*c*x**2 + c)/2, Ne(a**2*c, 0)), (sqrt(c)*x, True)) + 2 
*a*c*Piecewise((I*sqrt(c)*sqrt(a**2*x**2 - 1) - sqrt(c)*log(a*x) + sqrt(c) 
*log(a**2*x**2)/2 + I*sqrt(c)*asin(1/(a*x)), Abs(a**2*x**2) > 1), (sqrt(c) 
*sqrt(-a**2*x**2 + 1) + sqrt(c)*log(a**2*x**2)/2 - sqrt(c)*log(sqrt(-a**2* 
x**2 + 1) + 1), True)) + c*Piecewise((-I*a**2*sqrt(c)*x/sqrt(a**2*x**2 - 1 
) + I*a*sqrt(c)*acosh(a*x) + I*sqrt(c)/(x*sqrt(a**2*x**2 - 1)), Abs(a**2*x 
**2) > 1), (a**2*sqrt(c)*x/sqrt(-a**2*x**2 + 1) - a*sqrt(c)*asin(a*x) - sq 
rt(c)/(x*sqrt(-a**2*x**2 + 1)), True))
 

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.47 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^2} \, dx=\frac {4 \, a c^{2} \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - \frac {a^{2} \sqrt {-c} c \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{2 \, {\left | a \right |}} + \frac {2 \, a^{2} \sqrt {-c} c^{2}}{{\left ({\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} - c\right )} {\left | a \right |}} + \frac {1}{2} \, \sqrt {-a^{2} c x^{2} + c} {\left (a^{2} c x + 4 \, a c\right )} \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^2} \, dx=-\int \frac {{\left (c-a^2\,c\,x^2\right )}^{3/2}\,{\left (a\,x+1\right )}^2}{x^2\,\left (a^2\,x^2-1\right )} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.71 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^2} \, dx=\frac {\sqrt {c}\, c \left (-\mathit {asin} \left (a x \right ) a x +\sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+4 \sqrt {-a^{2} x^{2}+1}\, a x -2 \sqrt {-a^{2} x^{2}+1}+4 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a x -4 a x \right )}{2 x} \] Input:

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

Output:

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