\(\int e^{3 \text {arctanh}(a x)} (c-\frac {c}{a^2 x^2})^2 \, dx\) [688]

Optimal result

Integrand size = 22, antiderivative size = 124 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=-\frac {c^2 (6+a x) \sqrt {1-a^2 x^2}}{2 a^2 x}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a^4 x^3}-\frac {3 c^2 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}-\frac {3 c^2 \arcsin (a x)}{a}+\frac {c^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{2 a} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.03 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=-\frac {c^2 \left (2+9 a x+14 a^2 x^2-15 a^3 x^3-16 a^4 x^4+6 a^5 x^5+18 a^3 x^3 \sqrt {1-a^2 x^2} \arcsin (a x)-3 a^3 x^3 \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )}{6 a^4 x^3 \sqrt {1-a^2 x^2}} \] Input:

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

Output:

-1/6*(c^2*(2 + 9*a*x + 14*a^2*x^2 - 15*a^3*x^3 - 16*a^4*x^4 + 6*a^5*x^5 + 
18*a^3*x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x] - 3*a^3*x^3*Sqrt[1 - a^2*x^2]*Arc 
Tanh[Sqrt[1 - a^2*x^2]]))/(a^4*x^3*Sqrt[1 - a^2*x^2])
 

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.86, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {6707, 6698, 540, 27, 2338, 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.

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

\(\Big \downarrow \) 6707

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

\(\Big \downarrow \) 6698

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

\(\Big \downarrow \) 540

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2338

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 536

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(c^2*(-1/3*(1 - a^2*x^2)^(3/2)/x^3 - (3*a*(1 - a^2*x^2)^(3/2))/(2*x^2) + ( 
a^2*(-(((6 + a*x)*Sqrt[1 - a^2*x^2])/x) - 6*a*ArcSin[a*x] + 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 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 2338

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ 
Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S 
imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( 
m + 1))   Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( 
m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt 
Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
 

rule 6698

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]) && IGtQ[(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 = 0.26 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.02

method	result	size
risch	\(\frac {\left (16 a^{4} x^{4}+9 a^{3} x^{3}-14 a^{2} x^{2}-9 a x -2\right ) c^{2}}{6 x^{3} \sqrt {-a^{2} x^{2}+1}\, a^{4}}+\frac {\left (\frac {a^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-\frac {3 a^{4} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+a^{3} \sqrt {-a^{2} x^{2}+1}\right ) c^{2}}{a^{4}}\)	\(127\)
default	\(\frac {c^{2} \left (-\frac {1}{3 x^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {7 a^{2} \left (-\frac {1}{x \sqrt {-a^{2} x^{2}+1}}+\frac {2 a^{2} x}{\sqrt {-a^{2} x^{2}+1}}\right )}{3}+\frac {a^{3}}{\sqrt {-a^{2} x^{2}+1}}+a^{7} \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )-\frac {5 a^{4} x}{\sqrt {-a^{2} x^{2}+1}}+3 a \left (-\frac {1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 a^{2} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )-5 a^{3} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )+3 a^{6} \left (\frac {x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}\right )\right )}{a^{4}}\)	\(283\)
meijerg	\(-\frac {5 c^{2} x}{\sqrt {-a^{2} x^{2}+1}}-\frac {c^{2} \left (-2 a^{2} x^{2}+1\right )}{a^{2} x \sqrt {-a^{2} x^{2}+1}}-\frac {c^{2} \left (-8 a^{4} x^{4}+4 a^{2} x^{2}+1\right )}{3 a^{4} x^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {c^{2} \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{4 \sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}-\frac {c^{2} \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}-\frac {5 c^{2} \left (\frac {\left (2-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}-\sqrt {\pi }+\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}-\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )\right )}{a \sqrt {\pi }}-\frac {3 c^{2} \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {3 c^{2} \left (\frac {\sqrt {\pi }}{2 x^{2} a^{2}}-\frac {3 \left (\frac {5}{3}-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{4}-\frac {\sqrt {\pi }\, \left (-20 a^{2} x^{2}+8\right )}{16 a^{2} x^{2}}+\frac {\sqrt {\pi }\, \left (-24 a^{2} x^{2}+8\right )}{16 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )}{2}\right )}{a \sqrt {\pi }}\)	\(427\)

Input:

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

Output:

1/6*(16*a^4*x^4+9*a^3*x^3-14*a^2*x^2-9*a*x-2)/x^3/(-a^2*x^2+1)^(1/2)*c^2/a 
^4+(1/2*a^3*arctanh(1/(-a^2*x^2+1)^(1/2))-3*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 = 0.10 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.06 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=\frac {36 \, a^{3} c^{2} x^{3} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - 3 \, a^{3} c^{2} x^{3} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + 6 \, a^{3} c^{2} x^{3} + {\left (6 \, a^{3} c^{2} x^{3} - 16 \, a^{2} c^{2} x^{2} - 9 \, a c^{2} x - 2 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, a^{4} x^{3}} \] Input:

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

Output:

1/6*(36*a^3*c^2*x^3*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - 3*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 - 16*a^2 
*c^2*x^2 - 9*a*c^2*x - 2*c^2)*sqrt(-a^2*x^2 + 1))/(a^4*x^3)
 

Sympy [A] (verification not implemented)

Time = 8.43 (sec) , antiderivative size = 354, normalized size of antiderivative = 2.85 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=- a c^{2} \left (\begin {cases} - \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{2}}{2} & \text {otherwise} \end {cases}\right ) - 3 c^{2} \left (\begin {cases} \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{\sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\x & \text {otherwise} \end {cases}\right ) - \frac {2 c^{2} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right )}{a} + \frac {2 c^{2} \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right )}{a^{2}} + \frac {3 c^{2} \left (\begin {cases} - \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right )}{a^{3}} + \frac {c^{2} \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right )}{a^{4}} \] Input:

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

Output:

-a*c**2*Piecewise((-sqrt(-a**2*x**2 + 1)/a**2, Ne(a**2, 0)), (x**2/2, True 
)) - 3*c**2*Piecewise((log(-2*a**2*x + 2*sqrt(-a**2)*sqrt(-a**2*x**2 + 1)) 
/sqrt(-a**2), Ne(a**2, 0)), (x, True)) - 2*c**2*Piecewise((-acosh(1/(a*x)) 
, 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))/a + 2*c**2*Piecewise((-I 
*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, Tru 
e))/a**2 + 3*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((-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))/a**4
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (108) = 216\).

Time = 0.18 (sec) , antiderivative size = 347, normalized size of antiderivative = 2.80 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=-a^{3} c^{2} {\left (\frac {x^{2}}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {2}{\sqrt {-a^{2} x^{2} + 1} a^{4}}\right )} + 3 \, a^{2} c^{2} {\left (\frac {x}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {\arcsin \left (a x\right )}{a^{3}}\right )} - \frac {5 \, c^{2} x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {5 \, c^{2} {\left (\frac {1}{\sqrt {-a^{2} x^{2} + 1}} - \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right )\right )}}{a} + \frac {{\left (\frac {2 \, a^{2} x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {1}{\sqrt {-a^{2} x^{2} + 1} x}\right )} c^{2}}{a^{2}} + \frac {c^{2}}{\sqrt {-a^{2} x^{2} + 1} a} - \frac {3 \, {\left (3 \, a^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {3 \, a^{2}}{\sqrt {-a^{2} x^{2} + 1}} + \frac {1}{\sqrt {-a^{2} x^{2} + 1} x^{2}}\right )} c^{2}}{2 \, a^{3}} + \frac {{\left (\frac {8 \, a^{4} x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {4 \, a^{2}}{\sqrt {-a^{2} x^{2} + 1} x} - \frac {1}{\sqrt {-a^{2} x^{2} + 1} x^{3}}\right )} c^{2}}{3 \, a^{4}} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (108) = 216\).

Time = 0.14 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.11 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=\frac {{\left (c^{2} + \frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{2}}{a^{2} x} + \frac {33 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{2}}{a^{4} x^{2}}\right )} a^{6} x^{3}}{24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} {\left | a \right |}} - \frac {3 \, c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} + \frac {c^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{a} - \frac {\frac {33 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{2}}{x} + \frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{2}}{a^{2} x^{2}} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{2}}{a^{4} x^{3}}}{24 \, a^{2} {\left | a \right |}} \] Input:

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

Output:

1/24*(c^2 + 9*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^2/(a^2*x) + 33*(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)) - 3*c^2*arcsin(a*x)*sgn(a)/abs(a) + 1/2*c^2*log(1/2*abs(-2* 
sqrt(-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*(33*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c^2/x + 9*(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 = 0.04 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.10 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=\frac {c^2\,\sqrt {1-a^2\,x^2}}{a}-\frac {3\,c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {8\,c^2\,\sqrt {1-a^2\,x^2}}{3\,a^2\,x}-\frac {3\,c^2\,\sqrt {1-a^2\,x^2}}{2\,a^3\,x^2}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{3\,a^4\,x^3}-\frac {c^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,a} \] Input:

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

Output:

(c^2*(1 - a^2*x^2)^(1/2))/a - (c^2*atan((1 - a^2*x^2)^(1/2)*1i)*1i)/(2*a) 
- (3*c^2*asinh(x*(-a^2)^(1/2)))/(-a^2)^(1/2) - (8*c^2*(1 - a^2*x^2)^(1/2)) 
/(3*a^2*x) - (3*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 = 0.15 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.08 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=\frac {c^{2} \left (-36 \mathit {asin} \left (a x \right ) a^{3} x^{3}+12 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-32 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-18 \sqrt {-a^{2} x^{2}+1}\, a x -4 \sqrt {-a^{2} x^{2}+1}-3 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}-1\right ) a^{3} x^{3}+3 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}+1\right ) a^{3} x^{3}\right )}{12 a^{4} x^{3}} \] Input:

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

Output:

(c**2*( - 36*asin(a*x)*a**3*x**3 + 12*sqrt( - a**2*x**2 + 1)*a**3*x**3 - 3 
2*sqrt( - a**2*x**2 + 1)*a**2*x**2 - 18*sqrt( - a**2*x**2 + 1)*a*x - 4*sqr 
t( - 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))/(12*a**4*x**3)