\(\int e^{2 \text {arctanh}(a x)} (c-\frac {c}{a^2 x^2})^{3/2} \, dx\) [722]

Optimal result

Integrand size = 24, antiderivative size = 123 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=\frac {c \sqrt {c-\frac {c}{a^2 x^2}} \left (4 a-\frac {1}{x}\right )}{2 a^2}-\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x-\frac {c^{3/2} \arctan \left (\frac {\sqrt {c}}{a \sqrt {c-\frac {c}{a^2 x^2}} x}\right )}{2 a}-\frac {2 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {c}}\right )}{a} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.93 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=-\frac {c \sqrt {c-\frac {c}{a^2 x^2}} \left (\sqrt {-1+a^2 x^2} \left (-1-4 a x+2 a^2 x^2\right )+a^2 x^2 \arctan \left (\frac {1}{\sqrt {-1+a^2 x^2}}\right )+4 a^2 x^2 \log \left (a x+\sqrt {-1+a^2 x^2}\right )\right )}{2 a^2 x \sqrt {-1+a^2 x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.90, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6709, 540, 25, 27, 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.

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

\(\Big \downarrow \) 6709

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

\(\Big \downarrow \) 540

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 536

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 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 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 6709

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

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.65

method	result	size
risch	\(-\frac {\left (2 a^{4} x^{4}-4 a^{3} x^{3}-3 a^{2} x^{2}+4 a x +1\right ) c \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{2 x \,a^{2} \left (a^{2} x^{2}-1\right )}-\frac {\left (\frac {a^{2} \ln \left (\frac {-2 c +2 \sqrt {-c}\, \sqrt {a^{2} c \,x^{2}-c}}{x}\right )}{2 \sqrt {-c}}+\frac {2 a^{3} \ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right )}{\sqrt {a^{2} c}}\right ) c \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \sqrt {c \left (a^{2} x^{2}-1\right )}}{a^{2} \left (a^{2} x^{2}-1\right )}\)	\(203\)
default	\(\frac {{\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {3}{2}} x \left (-12 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{5} c \,x^{3}+12 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{5} x +{\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\, a^{4} c \,x^{2}-4 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}\right )^{\frac {3}{2}} a^{4} c \,x^{2}+3 a^{4} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {5}{2}} \sqrt {-\frac {c}{a^{2}}}+18 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \sqrt {-\frac {c}{a^{2}}}\, a^{3} c^{2} x^{3}-6 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a^{3} c^{2} x^{3}-3 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \sqrt {-\frac {c}{a^{2}}}\, a^{2} c^{2} x^{2}-18 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {5}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) a \,x^{2}+6 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {5}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}+x c}{\sqrt {c}}\right ) a \,x^{2}-3 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) c^{3} x^{2}\right )}{6 a^{2} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\, c}\)	\(454\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.46 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=\left [\frac {8 \, a \sqrt {-c} c x \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + a \sqrt {-c} c x \log \left (-\frac {a^{2} c x^{2} + 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) - 2 \, {\left (2 \, a^{2} c x^{2} - 4 \, a c x - c\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, a^{2} x}, \frac {a c^{\frac {3}{2}} x \arctan \left (\frac {a x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{\sqrt {c}}\right ) + 2 \, a c^{\frac {3}{2}} x \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) - {\left (2 \, a^{2} c x^{2} - 4 \, a c x - c\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, a^{2} x}\right ] \] Input:

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 6.92 (sec) , antiderivative size = 377, normalized size of antiderivative = 3.07 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=- c \left (\begin {cases} \frac {\sqrt {c} \sqrt {a^{2} x^{2} - 1}}{a} - \frac {i \sqrt {c} \log {\left (a x \right )}}{a} + \frac {i \sqrt {c} \log {\left (a^{2} x^{2} \right )}}{2 a} + \frac {\sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{a} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {i \sqrt {c} \sqrt {- a^{2} x^{2} + 1}}{a} + \frac {i \sqrt {c} \log {\left (a^{2} x^{2} \right )}}{2 a} - \frac {i \sqrt {c} \log {\left (\sqrt {- a^{2} x^{2} + 1} + 1 \right )}}{a} & \text {otherwise} \end {cases}\right ) - \frac {2 c \left (\begin {cases} - \frac {a \sqrt {c} x}{\sqrt {a^{2} x^{2} - 1}} + \sqrt {c} \operatorname {acosh}{\left (a x \right )} + \frac {\sqrt {c}}{a x \sqrt {a^{2} x^{2} - 1}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {i a \sqrt {c} x}{\sqrt {- a^{2} x^{2} + 1}} - i \sqrt {c} \operatorname {asin}{\left (a x \right )} - \frac {i \sqrt {c}}{a x \sqrt {- a^{2} x^{2} + 1}} & \text {otherwise} \end {cases}\right )}{a} - \frac {c \left (\begin {cases} \frac {i a \sqrt {c} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {i \sqrt {c}}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {i \sqrt {c}}{2 a^{2} x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {a \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {\sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right )}{a^{2}} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (103) = 206\).

Time = 0.18 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.15 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx={\left (\frac {c^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\left (x\right )}{a^{2}} + \frac {2 \, c^{\frac {3}{2}} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\left (x\right )}{a {\left | a \right |}} - \frac {\sqrt {a^{2} c x^{2} - c} c \mathrm {sgn}\left (x\right )}{a^{2}} - \frac {{\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} c^{2} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 4 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} a c^{\frac {5}{2}} \mathrm {sgn}\left (x\right ) - {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} c^{3} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 4 \, a c^{\frac {7}{2}} \mathrm {sgn}\left (x\right )}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{2} a^{2} {\left | a \right |}}\right )} {\left | a \right |} \] Input:

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

Output:

(c^(3/2)*arctan(-(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))/sqrt(c))*sgn(x)/a^2 
 + 2*c^(3/2)*log(abs(-sqrt(a^2*c)*x + sqrt(a^2*c*x^2 - c)))*sgn(x)/(a*abs( 
a)) - sqrt(a^2*c*x^2 - c)*c*sgn(x)/a^2 - ((sqrt(a^2*c)*x - sqrt(a^2*c*x^2 
- c))^3*c^2*abs(a)*sgn(x) - 4*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2*a*c^ 
(5/2)*sgn(x) - (sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))*c^3*abs(a)*sgn(x) - 4 
*a*c^(7/2)*sgn(x))/(((sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2 + c)^2*a^2*ab 
s(a)))*abs(a)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.81 \[ \int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx=\frac {\sqrt {c}\, c \left (2 \mathit {atan} \left (\sqrt {a^{2} x^{2}-1}+a x \right ) a^{2} x^{2}-2 \sqrt {a^{2} x^{2}-1}\, a^{2} x^{2}+4 \sqrt {a^{2} x^{2}-1}\, a x +\sqrt {a^{2} x^{2}-1}-4 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right ) a^{2} x^{2}\right )}{2 a^{3} x^{2}} \] Input:

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

Output:

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