3.1241 \(\int \frac{e^{-2 \tanh ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^2} \, dx\)

Optimal. Leaf size=82 \[ -\frac{\sqrt{c-a^2 c x^2}}{x}+a \sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )+2 a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right ) \]

[Out]

-(Sqrt[c - a^2*c*x^2]/x) + a*Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]] + 2*a*Sqrt[c]*ArcTanh[Sqrt[c -
a^2*c*x^2]/Sqrt[c]]

________________________________________________________________________________________

Rubi [A]  time = 0.248316, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {6152, 1807, 844, 217, 203, 266, 63, 208} \[ -\frac{\sqrt{c-a^2 c x^2}}{x}+a \sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )+2 a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[c - a^2*c*x^2]/x) + a*Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]] + 2*a*Sqrt[c]*ArcTanh[Sqrt[c -
a^2*c*x^2]/Sqrt[c]]

Rule 6152

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/c^(n/2), Int[(x^m
*(c + d*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]) && ILtQ[n/2, 0]

Rule 1807

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]}, Simp[(R*(c*x)^(m + 1)*(a + b*x^2)^(p + 1))/(a*c*(m + 1)), x] + Dist[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] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{e^{-2 \tanh ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^2} \, dx &=c \int \frac{(1-a x)^2}{x^2 \sqrt{c-a^2 c x^2}} \, dx\\ &=-\frac{\sqrt{c-a^2 c x^2}}{x}-\int \frac{2 a c-a^2 c x}{x \sqrt{c-a^2 c x^2}} \, dx\\ &=-\frac{\sqrt{c-a^2 c x^2}}{x}-(2 a c) \int \frac{1}{x \sqrt{c-a^2 c x^2}} \, dx+\left (a^2 c\right ) \int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx\\ &=-\frac{\sqrt{c-a^2 c x^2}}{x}-(a c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-a^2 c x}} \, dx,x,x^2\right )+\left (a^2 c\right ) \operatorname{Subst}\left (\int \frac{1}{1+a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c-a^2 c x^2}}\right )\\ &=-\frac{\sqrt{c-a^2 c x^2}}{x}+a \sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )+\frac{2 \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2 c}} \, dx,x,\sqrt{c-a^2 c x^2}\right )}{a}\\ &=-\frac{\sqrt{c-a^2 c x^2}}{x}+a \sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )+2 a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0887855, size = 106, normalized size = 1.29 \[ -\frac{\sqrt{c-a^2 c x^2}}{x}+2 a \sqrt{c} \log \left (\sqrt{c} \sqrt{c-a^2 c x^2}+c\right )-a \sqrt{c} \tan ^{-1}\left (\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{c} \left (a^2 x^2-1\right )}\right )-2 a \sqrt{c} \log (x) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(Sqrt[c - a^2*c*x^2]/x) - a*Sqrt[c]*ArcTan[(a*x*Sqrt[c - a^2*c*x^2])/(Sqrt[c]*(-1 + a^2*x^2))] - 2*a*Sqrt[c]*
Log[x] + 2*a*Sqrt[c]*Log[c + Sqrt[c]*Sqrt[c - a^2*c*x^2]]

________________________________________________________________________________________

Maple [B]  time = 0.04, size = 203, normalized size = 2.5 \begin{align*} -{\frac{1}{cx} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{a}^{2}x\sqrt{-{a}^{2}c{x}^{2}+c}-{{a}^{2}c\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-{a}^{2}c{x}^{2}+c}}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}}+2\,\ln \left ({\frac{2\,c+2\,\sqrt{c}\sqrt{-{a}^{2}c{x}^{2}+c}}{x}} \right ) \sqrt{c}a-2\,\sqrt{-{a}^{2}c{x}^{2}+c}a+2\,a\sqrt{-c{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,ac \left ( x+{a}^{-1} \right ) }+2\,{\frac{{a}^{2}c}{\sqrt{{a}^{2}c}}\arctan \left ({\frac{\sqrt{{a}^{2}c}x}{\sqrt{-c{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,ac \left ( x+{a}^{-1} \right ) }}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/c/x*(-a^2*c*x^2+c)^(3/2)-a^2*x*(-a^2*c*x^2+c)^(1/2)-a^2*c/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-a^2*c*x^2+
c)^(1/2))+2*ln((2*c+2*c^(1/2)*(-a^2*c*x^2+c)^(1/2))/x)*c^(1/2)*a-2*(-a^2*c*x^2+c)^(1/2)*a+2*a*(-c*a^2*(x+1/a)^
2+2*a*c*(x+1/a))^(1/2)+2*a^2*c/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-c*a^2*(x+1/a)^2+2*a*c*(x+1/a))^(1/2))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{\sqrt{-a^{2} c x^{2} + c}{\left (a^{2} x^{2} - 1\right )}}{{\left (a x + 1\right )}^{2} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.54189, size = 478, normalized size = 5.83 \begin{align*} \left [-\frac{a \sqrt{c} x \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right ) - a \sqrt{c} x \log \left (-\frac{a^{2} c x^{2} - 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right ) + \sqrt{-a^{2} c x^{2} + c}}{x}, \frac{4 \, a \sqrt{-c} x \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) + a \sqrt{-c} x \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right ) - 2 \, \sqrt{-a^{2} c x^{2} + c}}{2 \, x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-(a*sqrt(c)*x*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) - a*sqrt(c)*x*log(-(a^2*c*x^2 - 2*sqrt
(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) + sqrt(-a^2*c*x^2 + c))/x, 1/2*(4*a*sqrt(-c)*x*arctan(sqrt(-a^2*c*x^2 + c
)*sqrt(-c)/(a^2*c*x^2 - c)) + a*sqrt(-c)*x*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) - 2*sqrt
(-a^2*c*x^2 + c))/x]

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{\sqrt{- a^{2} c x^{2} + c}}{a x^{3} + x^{2}}\, dx - \int \frac{a x \sqrt{- a^{2} c x^{2} + c}}{a x^{3} + x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-sqrt(-a**2*c*x**2 + c)/(a*x**3 + x**2), x) - Integral(a*x*sqrt(-a**2*c*x**2 + c)/(a*x**3 + x**2), x
)

________________________________________________________________________________________

Giac [B]  time = 1.19943, size = 227, normalized size = 2.77 \begin{align*} -{\left (\frac{4 \, c \arctan \left (\frac{\sqrt{-c + \frac{2 \, c}{a x + 1}}}{\sqrt{-c}}\right ) \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right )}{\sqrt{-c}} + 2 \, \sqrt{c} \arctan \left (\frac{\sqrt{-c + \frac{2 \, c}{a x + 1}}}{\sqrt{c}}\right ) \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) - \frac{{\left (\pi c + 2 \, \sqrt{-c} \sqrt{c} \arctan \left (\frac{\sqrt{-c}}{\sqrt{c}}\right ) - c\right )} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right )}{\sqrt{-c}} + \frac{c \sqrt{-c + \frac{2 \, c}{a x + 1}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right )}{c - \frac{c}{a x + 1}}\right )}{\left | a \right |} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(4*c*arctan(sqrt(-c + 2*c/(a*x + 1))/sqrt(-c))*sgn(1/(a*x + 1))*sgn(a)/sqrt(-c) + 2*sqrt(c)*arctan(sqrt(-c +
2*c/(a*x + 1))/sqrt(c))*sgn(1/(a*x + 1))*sgn(a) - (pi*c + 2*sqrt(-c)*sqrt(c)*arctan(sqrt(-c)/sqrt(c)) - c)*sgn
(1/(a*x + 1))*sgn(a)/sqrt(-c) + c*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1))*sgn(a)/(c - c/(a*x + 1)))*abs(a)