Integrand size = 19, antiderivative size = 67 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^3} \, dx=-\frac {c^2 (1-2 a x) \sqrt {1-a^2 x^2}}{2 x^2}+a^2 c^2 \arcsin (a x)+\frac {1}{2} a^2 c^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \] Output:
-1/2*c^2*(-2*a*x+1)*(-a^2*x^2+1)^(1/2)/x^2+a^2*c^2*arcsin(a*x)+1/2*a^2*c^2 *arctanh((-a^2*x^2+1)^(1/2))
Leaf count is larger than twice the leaf count of optimal. \(147\) vs. \(2(67)=134\).
Time = 0.09 (sec) , antiderivative size = 147, normalized size of antiderivative = 2.19 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^3} \, dx=-\frac {c^2 \left (2-4 a x-2 a^2 x^2+4 a^3 x^3+a^2 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)+10 a^2 x^2 \sqrt {1-a^2 x^2} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )-2 a^2 x^2 \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )}{4 x^2 \sqrt {1-a^2 x^2}} \] Input:
Integrate[(E^ArcTanh[a*x]*(c - a*c*x)^2)/x^3,x]
Output:
-1/4*(c^2*(2 - 4*a*x - 2*a^2*x^2 + 4*a^3*x^3 + a^2*x^2*Sqrt[1 - a^2*x^2]*A rcSin[a*x] + 10*a^2*x^2*Sqrt[1 - a^2*x^2]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]] - 2*a^2*x^2*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]]))/(x^2*Sqrt[1 - a^2 *x^2])
Time = 0.31 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.94, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {6678, 27, 537, 25, 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.
\(\displaystyle \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^3} \, dx\) |
\(\Big \downarrow \) 6678 |
\(\displaystyle c \int \frac {c (1-a x) \sqrt {1-a^2 x^2}}{x^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle c^2 \int \frac {(1-a x) \sqrt {1-a^2 x^2}}{x^3}dx\) |
\(\Big \downarrow \) 537 |
\(\displaystyle c^2 \left (\frac {1}{2} a^2 \int -\frac {1-2 a x}{x \sqrt {1-a^2 x^2}}dx-\frac {(1-2 a x) \sqrt {1-a^2 x^2}}{2 x^2}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle c^2 \left (-\frac {1}{2} a^2 \int \frac {1-2 a x}{x \sqrt {1-a^2 x^2}}dx-\frac {(1-2 a x) \sqrt {1-a^2 x^2}}{2 x^2}\right )\) |
\(\Big \downarrow \) 538 |
\(\displaystyle c^2 \left (-\frac {1}{2} a^2 \left (\int \frac {1}{x \sqrt {1-a^2 x^2}}dx-2 a \int \frac {1}{\sqrt {1-a^2 x^2}}dx\right )-\frac {(1-2 a x) \sqrt {1-a^2 x^2}}{2 x^2}\right )\) |
\(\Big \downarrow \) 223 |
\(\displaystyle c^2 \left (-\frac {1}{2} a^2 \left (\int \frac {1}{x \sqrt {1-a^2 x^2}}dx-2 \arcsin (a x)\right )-\frac {(1-2 a x) \sqrt {1-a^2 x^2}}{2 x^2}\right )\) |
\(\Big \downarrow \) 243 |
\(\displaystyle c^2 \left (-\frac {1}{2} a^2 \left (\frac {1}{2} \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-2 \arcsin (a x)\right )-\frac {(1-2 a x) \sqrt {1-a^2 x^2}}{2 x^2}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle c^2 \left (-\frac {1}{2} a^2 \left (-\frac {\int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a^2}-2 \arcsin (a x)\right )-\frac {(1-2 a x) \sqrt {1-a^2 x^2}}{2 x^2}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle c^2 \left (-\frac {1}{2} a^2 \left (-\text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-2 \arcsin (a x)\right )-\frac {(1-2 a x) \sqrt {1-a^2 x^2}}{2 x^2}\right )\) |
Input:
Int[(E^ArcTanh[a*x]*(c - a*c*x)^2)/x^3,x]
Output:
c^2*(-1/2*((1 - 2*a*x)*Sqrt[1 - a^2*x^2])/x^2 - (a^2*(-2*ArcSin[a*x] - Arc Tanh[Sqrt[1 - a^2*x^2]]))/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[(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_.))*((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 = 0.24 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.30
method | result | size |
default | \(c^{2} \left (\frac {a^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}+\frac {a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\frac {a \sqrt {-a^{2} x^{2}+1}}{x}\right )\) | \(87\) |
risch | \(-\frac {\left (2 a^{3} x^{3}-a^{2} x^{2}-2 a x +1\right ) c^{2}}{2 x^{2} \sqrt {-a^{2} x^{2}+1}}+\left (\frac {a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\frac {a^{3} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}\right ) c^{2}\) | \(96\) |
meijerg | \(a^{2} c^{2} \arcsin \left (a x \right )-\frac {a^{2} c^{2} \left (\left (-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )\right )}{2 \sqrt {\pi }}+\frac {a \,c^{2} \sqrt {-a^{2} x^{2}+1}}{x}-\frac {a^{2} c^{2} \left (\frac {\sqrt {\pi }}{x^{2} a^{2}}-\frac {\left (1-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}-\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{8 a^{2} x^{2}}+\frac {\sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}}{a^{2} x^{2}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )\right )}{2 \sqrt {\pi }}\) | \(193\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2/x^3,x,method=_RETURNVERBOSE)
Output:
c^2*(a^3/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))-1/2*(-a^2*x^ 2+1)^(1/2)/x^2+1/2*a^2*arctanh(1/(-a^2*x^2+1)^(1/2))+a*(-a^2*x^2+1)^(1/2)/ x)
Time = 0.19 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.42 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^3} \, dx=-\frac {4 \, a^{2} c^{2} x^{2} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + a^{2} c^{2} x^{2} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (2 \, a c^{2} x - c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{2 \, x^{2}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2/x^3,x, algorithm="fricas ")
Output:
-1/2*(4*a^2*c^2*x^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + a^2*c^2*x^2*l og((sqrt(-a^2*x^2 + 1) - 1)/x) - (2*a*c^2*x - c^2)*sqrt(-a^2*x^2 + 1))/x^2
Result contains complex when optimal does not.
Time = 2.62 (sec) , antiderivative size = 223, normalized size of antiderivative = 3.33 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^3} \, dx=a^{3} c^{2} \left (\begin {cases} \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{\sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\x & \text {otherwise} \end {cases}\right ) - a^{2} c^{2} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right ) - a c^{2} \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right ) + c^{2} \left (\begin {cases} - \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right ) \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**2/x**3,x)
Output:
a**3*c**2*Piecewise((log(-2*a**2*x + 2*sqrt(-a**2)*sqrt(-a**2*x**2 + 1))/s qrt(-a**2), Ne(a**2, 0)), (x, True)) - a**2*c**2*Piecewise((-acosh(1/(a*x) ), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True)) - a*c**2*Piecewise((-I* sqrt(a**2*x**2 - 1)/x, Abs(a**2*x**2) > 1), (-sqrt(-a**2*x**2 + 1)/x, True )) + 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))
Time = 0.11 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.28 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^3} \, dx=a^{2} c^{2} \arcsin \left (a x\right ) + \frac {1}{2} \, a^{2} c^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\sqrt {-a^{2} x^{2} + 1} a c^{2}}{x} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{2 \, x^{2}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2/x^3,x, algorithm="maxima ")
Output:
a^2*c^2*arcsin(a*x) + 1/2*a^2*c^2*log(2*sqrt(-a^2*x^2 + 1)/abs(x) + 2/abs( x)) + sqrt(-a^2*x^2 + 1)*a*c^2/x - 1/2*sqrt(-a^2*x^2 + 1)*c^2/x^2
Leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (59) = 118\).
Time = 0.13 (sec) , antiderivative size = 192, normalized size of antiderivative = 2.87 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^3} \, dx=\frac {a^{3} c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} + \frac {a^{3} c^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, {\left | a \right |}} + \frac {{\left (a^{3} c^{2} - \frac {4 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a c^{2}}{x}\right )} a^{4} x^{2}}{8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} {\left | a \right |}} + \frac {\frac {4 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a c^{2} {\left | a \right |}}{x} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{2} {\left | a \right |}}{a x^{2}}}{8 \, a^{2}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2/x^3,x, algorithm="giac")
Output:
a^3*c^2*arcsin(a*x)*sgn(a)/abs(a) + 1/2*a^3*c^2*log(1/2*abs(-2*sqrt(-a^2*x ^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) + 1/8*(a^3*c^2 - 4*(sqrt(-a^2*x ^2 + 1)*abs(a) + a)*a*c^2/x)*a^4*x^2/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*ab s(a)) + 1/8*(4*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a*c^2*abs(a)/x - (sqrt(-a^2 *x^2 + 1)*abs(a) + a)^2*c^2*abs(a)/(a*x^2))/a^2
Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.34 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^3} \, dx=\frac {a\,c^2\,\sqrt {1-a^2\,x^2}}{x}-\frac {c^2\,\sqrt {1-a^2\,x^2}}{2\,x^2}+\frac {a^3\,c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {a^2\,c^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} \] Input:
int(((c - a*c*x)^2*(a*x + 1))/(x^3*(1 - a^2*x^2)^(1/2)),x)
Output:
(a*c^2*(1 - a^2*x^2)^(1/2))/x - (c^2*(1 - a^2*x^2)^(1/2))/(2*x^2) - (a^2*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)
Time = 0.14 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.97 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x^3} \, dx=\frac {c^{2} \left (2 \mathit {asin} \left (a x \right ) a^{2} x^{2}+2 \sqrt {-a^{2} x^{2}+1}\, a x -\sqrt {-a^{2} x^{2}+1}-\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{2} x^{2}\right )}{2 x^{2}} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2/x^3,x)
Output:
(c**2*(2*asin(a*x)*a**2*x**2 + 2*sqrt( - a**2*x**2 + 1)*a*x - sqrt( - a**2 *x**2 + 1) - log(tan(asin(a*x)/2))*a**2*x**2))/(2*x**2)