Integrand size = 19, antiderivative size = 59 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x} \, dx=\frac {1}{2} c^2 (2-a x) \sqrt {1-a^2 x^2}-\frac {1}{2} c^2 \arcsin (a x)-c^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \] Output:
1/2*c^2*(-a*x+2)*(-a^2*x^2+1)^(1/2)-1/2*c^2*arcsin(a*x)-c^2*arctanh((-a^2* x^2+1)^(1/2))
Leaf count is larger than twice the leaf count of optimal. \(125\) vs. \(2(59)=118\).
Time = 0.07 (sec) , antiderivative size = 125, normalized size of antiderivative = 2.12 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x} \, dx=\frac {c^2 \left (2-a x-2 a^2 x^2+a^3 x^3+\sqrt {1-a^2 x^2} \arcsin (a x)+4 \sqrt {1-a^2 x^2} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )-2 \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )}{2 \sqrt {1-a^2 x^2}} \] Input:
Integrate[(E^ArcTanh[a*x]*(c - a*c*x)^2)/x,x]
Output:
(c^2*(2 - a*x - 2*a^2*x^2 + a^3*x^3 + Sqrt[1 - a^2*x^2]*ArcSin[a*x] + 4*Sq rt[1 - a^2*x^2]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]] - 2*Sqrt[1 - a^2*x^2]*ArcTan h[Sqrt[1 - a^2*x^2]]))/(2*Sqrt[1 - a^2*x^2])
Time = 0.30 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.97, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {6678, 27, 535, 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.
\(\displaystyle \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x} \, dx\) |
\(\Big \downarrow \) 6678 |
\(\displaystyle c \int \frac {c (1-a x) \sqrt {1-a^2 x^2}}{x}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle c^2 \int \frac {(1-a x) \sqrt {1-a^2 x^2}}{x}dx\) |
\(\Big \downarrow \) 535 |
\(\displaystyle c^2 \left (\frac {1}{2} \int \frac {2-a x}{x \sqrt {1-a^2 x^2}}dx+\frac {1}{2} \sqrt {1-a^2 x^2} (2-a x)\right )\) |
\(\Big \downarrow \) 538 |
\(\displaystyle c^2 \left (\frac {1}{2} \left (2 \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-a \int \frac {1}{\sqrt {1-a^2 x^2}}dx\right )+\frac {1}{2} \sqrt {1-a^2 x^2} (2-a x)\right )\) |
\(\Big \downarrow \) 223 |
\(\displaystyle c^2 \left (\frac {1}{2} \left (2 \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\arcsin (a x)\right )+\frac {1}{2} \sqrt {1-a^2 x^2} (2-a x)\right )\) |
\(\Big \downarrow \) 243 |
\(\displaystyle c^2 \left (\frac {1}{2} \left (\int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\arcsin (a x)\right )+\frac {1}{2} \sqrt {1-a^2 x^2} (2-a x)\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle c^2 \left (\frac {1}{2} \left (-\frac {2 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a^2}-\arcsin (a x)\right )+\frac {1}{2} \sqrt {1-a^2 x^2} (2-a x)\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle c^2 \left (\frac {1}{2} \left (-2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\arcsin (a x)\right )+\frac {1}{2} \sqrt {1-a^2 x^2} (2-a x)\right )\) |
Input:
Int[(E^ArcTanh[a*x]*(c - a*c*x)^2)/x,x]
Output:
c^2*(((2 - a*x)*Sqrt[1 - a^2*x^2])/2 + (-ArcSin[a*x] - 2*ArcTanh[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[(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_), x_Symbol] :> Sim p[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^p/(2*p*(2*p + 1))), x] + Simp[a/(2*p + 1) Int[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^(p - 1)/x), x], x] /; Free Q[{a, b, c, d}, x] && GtQ[p, 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]
Leaf count of result is larger than twice the leaf count of optimal. \(113\) vs. \(2(51)=102\).
Time = 0.22 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.93
method | result | size |
default | \(c^{2} \left (-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+a^{3} \left (-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )-\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\sqrt {-a^{2} x^{2}+1}\right )\) | \(114\) |
meijerg | \(-\frac {a \,c^{2} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}} \sqrt {-a^{2} x^{2}+1}}{a^{2}}+\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{2 \sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {c^{2} \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{2 \sqrt {\pi }}-c^{2} \arcsin \left (a x \right )+\frac {c^{2} \left (\left (-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )\right )}{2 \sqrt {\pi }}\) | \(155\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2/x,x,method=_RETURNVERBOSE)
Output:
c^2*(-arctanh(1/(-a^2*x^2+1)^(1/2))+a^3*(-1/2*x*(-a^2*x^2+1)^(1/2)/a^2+1/2 /a^2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2)))-a/(a^2)^(1/2)*a rctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+(-a^2*x^2+1)^(1/2))
Time = 0.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.29 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x} \, dx=c^{2} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + c^{2} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - \frac {1}{2} \, {\left (a c^{2} x - 2 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2/x,x, algorithm="fricas")
Output:
c^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + c^2*log((sqrt(-a^2*x^2 + 1) - 1)/x) - 1/2*(a*c^2*x - 2*c^2)*sqrt(-a^2*x^2 + 1)
Result contains complex when optimal does not.
Time = 4.19 (sec) , antiderivative size = 182, normalized size of antiderivative = 3.08 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x} \, dx=a^{3} c^{2} \left (\begin {cases} - \frac {x \sqrt {- a^{2} x^{2} + 1}}{2 a^{2}} + \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{2 a^{2} \sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{3}}{3} & \text {otherwise} \end {cases}\right ) - a^{2} c^{2} \left (\begin {cases} - \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{2}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{2}}{2} & \text {otherwise} \end {cases}\right ) - a c^{2} \left (\begin {cases} \frac {\log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{\sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\x & \text {otherwise} \end {cases}\right ) + c^{2} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{a x} \right )} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {otherwise} \end {cases}\right ) \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a*c*x+c)**2/x,x)
Output:
a**3*c**2*Piecewise((-x*sqrt(-a**2*x**2 + 1)/(2*a**2) + log(-2*a**2*x + 2* sqrt(-a**2)*sqrt(-a**2*x**2 + 1))/(2*a**2*sqrt(-a**2)), Ne(a**2, 0)), (x** 3/3, True)) - a**2*c**2*Piecewise((-sqrt(-a**2*x**2 + 1)/a**2, Ne(a**2, 0) ), (x**2/2, True)) - a*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)) + c**2*Piecewise((- acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True))
Time = 0.11 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.29 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x} \, dx=-\frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} a c^{2} x - \frac {1}{2} \, c^{2} \arcsin \left (a x\right ) - c^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \sqrt {-a^{2} x^{2} + 1} c^{2} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2/x,x, algorithm="maxima")
Output:
-1/2*sqrt(-a^2*x^2 + 1)*a*c^2*x - 1/2*c^2*arcsin(a*x) - c^2*log(2*sqrt(-a^ 2*x^2 + 1)/abs(x) + 2/abs(x)) + sqrt(-a^2*x^2 + 1)*c^2
Time = 0.13 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.42 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x} \, dx=-\frac {a c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{2 \, {\left | a \right |}} - \frac {a c^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} - \frac {1}{2} \, {\left (a c^{2} x - 2 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2/x,x, algorithm="giac")
Output:
-1/2*a*c^2*arcsin(a*x)*sgn(a)/abs(a) - a*c^2*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) - 1/2*(a*c^2*x - 2*c^2)*sqrt(-a^2* x^2 + 1)
Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.31 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x} \, dx=c^2\,\sqrt {1-a^2\,x^2}-c^2\,\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right )-\frac {a\,c^2\,x\,\sqrt {1-a^2\,x^2}}{2}-\frac {a\,c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,\sqrt {-a^2}} \] Input:
int(((c - a*c*x)^2*(a*x + 1))/(x*(1 - a^2*x^2)^(1/2)),x)
Output:
c^2*(1 - a^2*x^2)^(1/2) - c^2*atanh((1 - a^2*x^2)^(1/2)) - (a*c^2*x*(1 - a ^2*x^2)^(1/2))/2 - (a*c^2*asinh(x*(-a^2)^(1/2)))/(2*(-a^2)^(1/2))
Time = 0.14 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.86 \[ \int \frac {e^{\text {arctanh}(a x)} (c-a c x)^2}{x} \, dx=\frac {c^{2} \left (-\mathit {asin} \left (a x \right )-\sqrt {-a^{2} x^{2}+1}\, a x +2 \sqrt {-a^{2} x^{2}+1}+2 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right )-2\right )}{2} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^2/x,x)
Output:
(c**2*( - asin(a*x) - sqrt( - a**2*x**2 + 1)*a*x + 2*sqrt( - a**2*x**2 + 1 ) + 2*log(tan(asin(a*x)/2)) - 2))/2