\(\int \frac {e^{2 \text {arctanh}(a x)} (c-a^2 c x^2)^{3/2}}{x^4} \, dx\) [1130]

Optimal result

Integrand size = 27, antiderivative size = 115 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^4} \, dx=-\frac {a c (1+a x) \sqrt {c-a^2 c x^2}}{x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{3 x^3}-a^3 c^{3/2} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )+a^3 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.10 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^4} \, dx=-\frac {c \left (1+3 a x+2 a^2 x^2\right ) \sqrt {c-a^2 c x^2}}{3 x^3}+a^3 c^{3/2} \arctan \left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (-1+a^2 x^2\right )}\right )-a^3 c^{3/2} \log (x)+a^3 c^{3/2} \log \left (c+\sqrt {c} \sqrt {c-a^2 c x^2}\right ) \] Input:

Integrate[(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2))/x^4,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {6701, 540, 27, 537, 27, 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.

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

\(\Big \downarrow \) 6701

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

\(\Big \downarrow \) 540

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 537

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

Int[(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2))/x^4,x]
 

Output:

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

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 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 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 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 = 0.20 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.12

method	result	size
risch	\(\frac {\left (2 a^{4} x^{4}+3 a^{3} x^{3}-a^{2} x^{2}-3 a x -1\right ) c^{2}}{3 x^{3} \sqrt {-c \left (a^{2} x^{2}-1\right )}}-\left (-\frac {a^{3} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{\sqrt {c}}+\frac {a^{4} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{\sqrt {a^{2} c}}\right ) c^{2}\)	\(129\)
default	\(-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{3 c \,x^{3}}+\frac {4 a^{2} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{c x}-4 a^{2} \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )\right )}{3}+2 a \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{2 c \,x^{2}}-\frac {3 a^{2} \left (\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )\right )}{2}\right )+2 a^{3} \left (\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )\right )-2 a^{3} \left (\frac {\left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}{3}-a c \left (-\frac {\left (-2 \left (x -\frac {1}{a}\right ) a^{2} c -2 a c \right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}\right )}{2 \sqrt {a^{2} c}}\right )\right )\)	\(434\)

Input:

int((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(3/2)/x^4,x,method=_RETURNVERBOS 
E)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 254, normalized size of antiderivative = 2.21 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^4} \, dx=\left [\frac {6 \, a^{3} c^{\frac {3}{2}} x^{3} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) + 3 \, a^{3} c^{\frac {3}{2}} x^{3} \log \left (-\frac {a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) - 2 \, {\left (2 \, a^{2} c x^{2} + 3 \, a c x + c\right )} \sqrt {-a^{2} c x^{2} + c}}{6 \, x^{3}}, -\frac {6 \, a^{3} \sqrt {-c} c x^{3} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{c}\right ) - 3 \, a^{3} \sqrt {-c} c x^{3} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) + 2 \, {\left (2 \, a^{2} c x^{2} + 3 \, a c x + c\right )} \sqrt {-a^{2} c x^{2} + c}}{6 \, x^{3}}\right ] \] Input:

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(3/2)/x^4,x, algorithm="fr 
icas")
 

Output:

[1/6*(6*a^3*c^(3/2)*x^3*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 
 - c)) + 3*a^3*c^(3/2)*x^3*log(-(a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*sqrt(c 
) - 2*c)/x^2) - 2*(2*a^2*c*x^2 + 3*a*c*x + c)*sqrt(-a^2*c*x^2 + c))/x^3, - 
1/6*(6*a^3*sqrt(-c)*c*x^3*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/c) - 3*a^3* 
sqrt(-c)*c*x^3*log(2*a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) 
+ 2*(2*a^2*c*x^2 + 3*a*c*x + c)*sqrt(-a^2*c*x^2 + c))/x^3]
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 7.34 (sec) , antiderivative size = 359, normalized size of antiderivative = 3.12 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^4} \, dx=a^{2} c \left (\begin {cases} - \frac {i a^{2} \sqrt {c} x}{\sqrt {a^{2} x^{2} - 1}} + i a \sqrt {c} \operatorname {acosh}{\left (a x \right )} + \frac {i \sqrt {c}}{x \sqrt {a^{2} x^{2} - 1}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {a^{2} \sqrt {c} x}{\sqrt {- a^{2} x^{2} + 1}} - a \sqrt {c} \operatorname {asin}{\left (a x \right )} - \frac {\sqrt {c}}{x \sqrt {- a^{2} x^{2} + 1}} & \text {otherwise} \end {cases}\right ) + 2 a c \left (\begin {cases} \frac {a^{2} \sqrt {c} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a \sqrt {c}}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {\sqrt {c}}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {i a^{2} \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} \frac {a^{3} \sqrt {c} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{3} - \frac {a \sqrt {c} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{3 x^{2}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{3} \sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{3} - \frac {i a \sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{3 x^{2}} & \text {otherwise} \end {cases}\right ) \] Input:

integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**(3/2)/x**4,x)
 

Output:

a**2*c*Piecewise((-I*a**2*sqrt(c)*x/sqrt(a**2*x**2 - 1) + I*a*sqrt(c)*acos 
h(a*x) + I*sqrt(c)/(x*sqrt(a**2*x**2 - 1)), Abs(a**2*x**2) > 1), (a**2*sqr 
t(c)*x/sqrt(-a**2*x**2 + 1) - a*sqrt(c)*asin(a*x) - sqrt(c)/(x*sqrt(-a**2* 
x**2 + 1)), True)) + 2*a*c*Piecewise((a**2*sqrt(c)*acosh(1/(a*x))/2 + a*sq 
rt(c)/(2*x*sqrt(-1 + 1/(a**2*x**2))) - sqrt(c)/(2*a*x**3*sqrt(-1 + 1/(a**2 
*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**2*sqrt(c)*asin(1/(a*x))/2 - I*a*sq 
rt(c)*sqrt(1 - 1/(a**2*x**2))/(2*x), True)) + c*Piecewise((a**3*sqrt(c)*sq 
rt(-1 + 1/(a**2*x**2))/3 - a*sqrt(c)*sqrt(-1 + 1/(a**2*x**2))/(3*x**2), 1/ 
Abs(a**2*x**2) > 1), (I*a**3*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/3 - I*a*sqrt( 
c)*sqrt(1 - 1/(a**2*x**2))/(3*x**2), True))
 

Maxima [F]

\[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^4} \, dx=\int { -\frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} {\left (a x + 1\right )}^{2}}{{\left (a^{2} x^{2} - 1\right )} x^{4}} \,d x } \] Input:

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(3/2)/x^4,x, algorithm="ma 
xima")
 

Output:

a^2*c^(3/2)*integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)/x^2, x) - 1/2*(a^4*c^(5 
/2)*log((sqrt(-a^2*c*x^2 + c) - sqrt(c))/(sqrt(-a^2*c*x^2 + c) + sqrt(c))) 
 + 2*sqrt(-a^2*c*x^2 + c)*a^2*c^2/x^2)/(a*c) + 1/3*(a^2*c^(3/2)*x^2 - c^(3 
/2))*sqrt(a*x + 1)*sqrt(-a*x + 1)/x^3
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (97) = 194\).

Time = 0.15 (sec) , antiderivative size = 259, normalized size of antiderivative = 2.25 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^4} \, dx=-\frac {2 \, a^{3} c^{2} \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - \frac {a^{4} \sqrt {-c} c \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{{\left | a \right |}} - \frac {2 \, {\left (3 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{5} a^{3} c^{2} {\left | a \right |} + 6 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} a^{4} \sqrt {-c} c^{3} - 3 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )} a^{3} c^{4} {\left | a \right |} - 2 \, a^{4} \sqrt {-c} c^{4}\right )}}{3 \, {\left ({\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{3} {\left | a \right |}} \] Input:

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(3/2)/x^4,x, algorithm="gi 
ac")
 

Output:

-2*a^3*c^2*arctan(-(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))/sqrt(-c))/sqrt( 
-c) - a^4*sqrt(-c)*c*log(abs(-sqrt(-a^2*c)*x + sqrt(-a^2*c*x^2 + c)))/abs( 
a) - 2/3*(3*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^5*a^3*c^2*abs(a) + 6*( 
sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^2*a^4*sqrt(-c)*c^3 - 3*(sqrt(-a^2*c 
)*x - sqrt(-a^2*c*x^2 + c))*a^3*c^4*abs(a) - 2*a^4*sqrt(-c)*c^4)/(((sqrt(- 
a^2*c)*x - sqrt(-a^2*c*x^2 + c))^2 - c)^3*abs(a))
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^4} \, dx=-\int \frac {{\left (c-a^2\,c\,x^2\right )}^{3/2}\,{\left (a\,x+1\right )}^2}{x^4\,\left (a^2\,x^2-1\right )} \,d x \] Input:

int(-((c - a^2*c*x^2)^(3/2)*(a*x + 1)^2)/(x^4*(a^2*x^2 - 1)),x)
 

Output:

-int(((c - a^2*c*x^2)^(3/2)*(a*x + 1)^2)/(x^4*(a^2*x^2 - 1)), x)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^4} \, dx=\frac {\sqrt {c}\, c \left (-6 \mathit {asin} \left (a x \right ) a^{3} x^{3}-4 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-6 \sqrt {-a^{2} x^{2}+1}\, a x -2 \sqrt {-a^{2} x^{2}+1}-3 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}-1\right ) a^{3} x^{3}+3 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}+1\right ) a^{3} x^{3}\right )}{6 x^{3}} \] Input:

int((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(3/2)/x^4,x)
 

Output:

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