Integrand size = 20, 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
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.03 (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[E^ArcTanh[a*x]*(c - c/(a^2*x^2))^2,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)
Time = 0.71 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.90, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6707, 6698, 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 |
| \(\displaystyle \frac {c^2 \int \frac {e^{\text {arctanh}(a x)} \left (1-a^2 x^2\right )^2}{x^4}dx}{a^4}\) |
| \(\Big \downarrow \) 6698 |
| \(\displaystyle \frac {c^2 \int \frac {(a x+1) \left (1-a^2 x^2\right )^{3/2}}{x^4}dx}{a^4}\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle \frac {c^2 \left (\frac {1}{2} a^2 \int -\frac {(3 a x+2) \sqrt {1-a^2 x^2}}{x^2}dx-\frac {(3 a x+2) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )}{a^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {c^2 \left (-\frac {1}{2} a^2 \int \frac {(3 a x+2) \sqrt {1-a^2 x^2}}{x^2}dx-\frac {(3 a x+2) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )}{a^4}\) |
| \(\Big \downarrow \) 536 |
| \(\displaystyle \frac {c^2 \left (-\frac {1}{2} a^2 \left (\int \frac {3 a-2 a^2 x}{x \sqrt {1-a^2 x^2}}dx-\frac {(2-3 a x) \sqrt {1-a^2 x^2}}{x}\right )-\frac {(3 a x+2) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )}{a^4}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle \frac {c^2 \left (-\frac {1}{2} a^2 \left (-2 a^2 \int \frac {1}{\sqrt {1-a^2 x^2}}dx+3 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {(2-3 a x) \sqrt {1-a^2 x^2}}{x}\right )-\frac {(3 a x+2) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )}{a^4}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {c^2 \left (-\frac {1}{2} a^2 \left (3 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} (2-3 a x)}{x}-2 a \arcsin (a x)\right )-\frac {(3 a x+2) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )}{a^4}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {c^2 \left (-\frac {1}{2} a^2 \left (\frac {3}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2} (2-3 a x)}{x}-2 a \arcsin (a x)\right )-\frac {(3 a x+2) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )}{a^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {c^2 \left (-\frac {1}{2} a^2 \left (-\frac {3 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {\sqrt {1-a^2 x^2} (2-3 a x)}{x}-2 a \arcsin (a x)\right )-\frac {(3 a x+2) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )}{a^4}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {c^2 \left (-\frac {1}{2} a^2 \left (-3 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2} (2-3 a x)}{x}-2 a \arcsin (a x)\right )-\frac {(3 a x+2) \left (1-a^2 x^2\right )^{3/2}}{6 x^3}\right )}{a^4}\) |
Input:
Int[E^ArcTanh[a*x]*(c - c/(a^2*x^2))^2,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
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]
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]
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]
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]
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]
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]
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]
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]
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]
Time = 0.18 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.23
| method | result | size |
| risch | \(-\frac {\left (8 a^{4} x^{4}-3 a^{3} x^{3}-10 a^{2} x^{2}+3 a x +2\right ) c^{2}}{6 x^{3} \sqrt {-a^{2} x^{2}+1}\, a^{4}}+\frac {\left (\frac {3 a^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\frac {a^{4} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-a^{3} \sqrt {-a^{2} x^{2}+1}\right ) c^{2}}{a^{4}}\) | \(127\) |
| default | \(\frac {c^{2} \left (\frac {a^{4} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}+\frac {4 a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}+a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )-a^{3} \sqrt {-a^{2} x^{2}+1}+2 a^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{a^{4}}\) | \(148\) |
| meijerg | \(-\frac {c^{2} \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{2 a \sqrt {\pi }}-\frac {c^{2} \left (\left (-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )\right )}{a \sqrt {\pi }}-\frac {c^{2} \left (\frac {\sqrt {\pi }}{x^{2} a^{2}}-\frac {\left (1-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}-\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{8 a^{2} x^{2}}+\frac {\sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}}{a^{2} x^{2}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )\right )}{2 a \sqrt {\pi }}+\frac {c^{2} \arcsin \left (a x \right )}{a}+\frac {2 c^{2} \sqrt {-a^{2} x^{2}+1}}{a^{2} x}-\frac {c^{2} \left (2 a^{2} x^{2}+1\right ) \sqrt {-a^{2} x^{2}+1}}{3 a^{4} x^{3}}\) | \(263\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a^2/x^2)^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
Time = 0.11 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.27 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=-\frac {12 \, a^{3} c^{2} x^{3} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + 9 \, a^{3} c^{2} x^{3} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + 6 \, a^{3} c^{2} x^{3} + {\left (6 \, a^{3} c^{2} x^{3} - 8 \, a^{2} c^{2} x^{2} + 3 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, a^{4} x^{3}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a^2/x^2)^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)
Time = 5.73 (sec) , antiderivative size = 350, normalized size of antiderivative = 3.40 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=a c^{2} \left (\begin {cases} - \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{2}}{2} & \text {otherwise} \end {cases}\right ) + c^{2} \left (\begin {cases} \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{\sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\x & \text {otherwise} \end {cases}\right ) - \frac {2 c^{2} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right )}{a} - \frac {2 c^{2} \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right )}{a^{2}} + \frac {c^{2} \left (\begin {cases} - \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right )}{a^{3}} + \frac {c^{2} \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right )}{a^{4}} \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(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) ) + c**2*Piecewise((log(-2*a**2*x + 2*sqrt(-a**2)*sqrt(-a**2*x**2 + 1))/sq rt(-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*sq rt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, 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((-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
Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (91) = 182\).
Time = 0.17 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.83 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=\frac {c^{2} \arcsin \left (a x\right )}{a} + \frac {2 \, c^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right )}{a} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{a} - \frac {{\left (a^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\sqrt {-a^{2} x^{2} + 1}}{x^{2}}\right )} c^{2}}{2 \, a^{3}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} c^{2}}{a^{2} x} - \frac {{\left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a^{2}}{x} + \frac {\sqrt {-a^{2} x^{2} + 1}}{x^{3}}\right )} c^{2}}{3 \, a^{4}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a^2/x^2)^2,x, algorithm="maxima" )
Output:
c^2*arcsin(a*x)/a + 2*c^2*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x))/a - sqrt(-a^2*x^2 + 1)*c^2/a - 1/2*(a^2*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/ab s(x)) + sqrt(-a^2*x^2 + 1)/x^2)*c^2/a^3 + 2*sqrt(-a^2*x^2 + 1)*c^2/(a^2*x) - 1/3*(2*sqrt(-a^2*x^2 + 1)*a^2/x + sqrt(-a^2*x^2 + 1)/x^3)*c^2/a^4
Leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (91) = 182\).
Time = 0.15 (sec) , antiderivative size = 263, normalized size of antiderivative = 2.55 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=\frac {{\left (c^{2} + \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{2}}{a^{2} x} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{2}}{a^{4} x^{2}}\right )} a^{6} x^{3}}{24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} {\left | a \right |}} + \frac {c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} + \frac {3 \, c^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{a} + \frac {\frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{2}}{x} - \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{2}}{a^{2} x^{2}} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{2}}{a^{4} x^{3}}}{24 \, a^{2} {\left | a \right |}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a^2/x^2)^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))
Time = 0.04 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.32 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^2 \, dx=\frac {c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{a}+\frac {4\,c^2\,\sqrt {1-a^2\,x^2}}{3\,a^2\,x}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{2\,a^3\,x^2}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{3\,a^4\,x^3}-\frac {c^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2\,a} \] Input:
int(((c - c/(a^2*x^2))^2*(a*x + 1))/(1 - a^2*x^2)^(1/2),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)
Time = 0.15 (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((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a^2/x^2)^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)