Integrand size = 19, antiderivative size = 58 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^2} \, dx=-\frac {c^2 (1+a x) \sqrt {1-a^2 x^2}}{x}-a c^2 \arcsin (a x)+a c^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \] Output:
-c^2*(a*x+1)*(-a^2*x^2+1)^(1/2)/x-a*c^2*arcsin(a*x)+a*c^2*arctanh((-a^2*x^ 2+1)^(1/2))
Time = 0.17 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.45 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^2} \, dx=\frac {1}{2} c^2 \left (\frac {2 (-1+a x) (1+a x)^2}{x \sqrt {1-a^2 x^2}}-a \arcsin (a x)+2 a \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )+2 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right ) \] Input:
Integrate[(E^ArcTanh[a*x]*(c - a*c*x)^2)/x^2,x]
Output:
(c^2*((2*(-1 + a*x)*(1 + a*x)^2)/(x*Sqrt[1 - a^2*x^2]) - a*ArcSin[a*x] + 2 *a*ArcSin[Sqrt[1 - a*x]/Sqrt[2]] + 2*a*ArcTanh[Sqrt[1 - a^2*x^2]]))/2
Time = 0.31 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.91, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {6678, 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.
\(\displaystyle \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^2} \, dx\) |
\(\Big \downarrow \) 6678 |
\(\displaystyle c \int \frac {c (1-a x) \sqrt {1-a^2 x^2}}{x^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle c^2 \int \frac {(1-a x) \sqrt {1-a^2 x^2}}{x^2}dx\) |
\(\Big \downarrow \) 536 |
\(\displaystyle c^2 \left (\int \frac {-x a^2-a}{x \sqrt {1-a^2 x^2}}dx-\frac {(a x+1) \sqrt {1-a^2 x^2}}{x}\right )\) |
\(\Big \downarrow \) 538 |
\(\displaystyle c^2 \left (a^2 \left (-\int \frac {1}{\sqrt {1-a^2 x^2}}dx\right )-a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {(a x+1) \sqrt {1-a^2 x^2}}{x}\right )\) |
\(\Big \downarrow \) 223 |
\(\displaystyle c^2 \left (-a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} (a x+1)}{x}-a \arcsin (a x)\right )\) |
\(\Big \downarrow \) 243 |
\(\displaystyle c^2 \left (-\frac {1}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2} (a x+1)}{x}-a \arcsin (a x)\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle c^2 \left (\frac {\int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {\sqrt {1-a^2 x^2} (a x+1)}{x}-a \arcsin (a x)\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle c^2 \left (a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2} (a x+1)}{x}-a \arcsin (a x)\right )\) |
Input:
Int[(E^ArcTanh[a*x]*(c - a*c*x)^2)/x^2,x]
Output:
c^2*(-(((1 + a*x)*Sqrt[1 - a^2*x^2])/x) - a*ArcSin[a*x] + a*ArcTanh[Sqrt[1 - a^2*x^2]])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((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_.))*((c_) + (d_.)*(x_))^(p_.)*((e_.) + (f_.)* (x_))^(m_.), x_Symbol] :> Simp[c^n Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1 , 0]) && IntegerQ[2*p]
Time = 0.23 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.43
method | result | size |
default | \(c^{2} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{x}-a \sqrt {-a^{2} x^{2}+1}-\frac {a^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+a \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\) | \(83\) |
risch | \(\frac {\left (a^{2} x^{2}-1\right ) c^{2}}{x \sqrt {-a^{2} x^{2}+1}}+a \left (-\sqrt {-a^{2} x^{2}+1}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}\right ) c^{2}\) | \(91\) |
meijerg | \(-\frac {a \,c^{2} \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{2 \sqrt {\pi }}-a \,c^{2} \arcsin \left (a x \right )-\frac {a \,c^{2} \left (\left (-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )\right )}{2 \sqrt {\pi }}-\frac {c^{2} \sqrt {-a^{2} x^{2}+1}}{x}\) | \(115\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2/x^2,x,method=_RETURNVERBOSE)
Output:
c^2*(-(-a^2*x^2+1)^(1/2)/x-a*(-a^2*x^2+1)^(1/2)-a^2/(a^2)^(1/2)*arctan((a^ 2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+a*arctanh(1/(-a^2*x^2+1)^(1/2)))
Time = 0.10 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.57 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^2} \, dx=\frac {2 \, a c^{2} x \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - a c^{2} x \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - a c^{2} x - {\left (a c^{2} x + c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{x} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2/x^2,x, algorithm="fricas ")
Output:
(2*a*c^2*x*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - a*c^2*x*log((sqrt(-a^2 *x^2 + 1) - 1)/x) - a*c^2*x - (a*c^2*x + c^2)*sqrt(-a^2*x^2 + 1))/x
Result contains complex when optimal does not.
Time = 1.74 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.60 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^2} \, dx=a^{3} c^{2} \left (\begin {cases} - \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{2}}{2} & \text {otherwise} \end {cases}\right ) - a^{2} c^{2} \left (\begin {cases} \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{\sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\x & \text {otherwise} \end {cases}\right ) - a c^{2} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right ) + c^{2} \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right ) \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**2/x**2,x)
Output:
a**3*c**2*Piecewise((-sqrt(-a**2*x**2 + 1)/a**2, Ne(a**2, 0)), (x**2/2, Tr ue)) - a**2*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)) - a*c**2*Piecewise((-acosh(1/( a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True)) + c**2*Piecewise((- I*sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, Tr ue))
Time = 0.12 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.38 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^2} \, dx=-a c^{2} \arcsin \left (a x\right ) + a c^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \sqrt {-a^{2} x^{2} + 1} a c^{2} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{x} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2/x^2,x, algorithm="maxima ")
Output:
-a*c^2*arcsin(a*x) + a*c^2*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs(x)) - s qrt(-a^2*x^2 + 1)*a*c^2 - sqrt(-a^2*x^2 + 1)*c^2/x
Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (54) = 108\).
Time = 0.13 (sec) , antiderivative size = 140, normalized size of antiderivative = 2.41 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^2} \, dx=\frac {a^{4} c^{2} x}{2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} - \frac {a^{2} c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} + \frac {a^{2} c^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} - \sqrt {-a^{2} x^{2} + 1} a c^{2} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{2}}{2 \, x {\left | a \right |}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2/x^2,x, algorithm="giac")
Output:
1/2*a^4*c^2*x/((sqrt(-a^2*x^2 + 1)*abs(a) + a)*abs(a)) - a^2*c^2*arcsin(a* x)*sgn(a)/abs(a) + a^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)*a*c^2 - 1/2*(sqrt(-a^2*x^2 + 1) *abs(a) + a)*c^2/(x*abs(a))
Time = 0.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.50 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^2} \, dx=-a\,c^2\,\sqrt {1-a^2\,x^2}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{x}-\frac {a^2\,c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-a\,c^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i} \] Input:
int(((c - a*c*x)^2*(a*x + 1))/(x^2*(1 - a^2*x^2)^(1/2)),x)
Output:
- a*c^2*(1 - a^2*x^2)^(1/2) - (c^2*(1 - a^2*x^2)^(1/2))/x - a*c^2*atan((1 - a^2*x^2)^(1/2)*1i)*1i - (a^2*c^2*asinh(x*(-a^2)^(1/2)))/(-a^2)^(1/2)
Time = 0.15 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.38 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^2} \, dx=\frac {c^{2} \left (-2 \mathit {asin} \left (a x \right ) a x -2 \sqrt {-a^{2} x^{2}+1}\, a x -2 \sqrt {-a^{2} x^{2}+1}-\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}-1\right ) a x +\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}+1\right ) a x \right )}{2 x} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2/x^2,x)
Output:
(c**2*( - 2*asin(a*x)*a*x - 2*sqrt( - a**2*x**2 + 1)*a*x - 2*sqrt( - a**2* x**2 + 1) - log(sqrt( - a**2*x**2 + 1) - 1)*a*x + log(sqrt( - a**2*x**2 + 1) + 1)*a*x))/(2*x)