3.156 \(\int \frac{1}{x^3 (1-a x) \sqrt{1-a^2 x^2}} \, dx\)

Optimal. Leaf size=90 \[ -\frac{2 a \sqrt{1-a^2 x^2}}{x}+\frac{\sqrt{1-a^2 x^2}}{x^2 (1-a x)}-\frac{3 \sqrt{1-a^2 x^2}}{2 x^2}-\frac{3}{2} a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]

[Out]

(-3*Sqrt[1 - a^2*x^2])/(2*x^2) - (2*a*Sqrt[1 - a^2*x^2])/x + Sqrt[1 - a^2*x^2]/(x^2*(1 - a*x)) - (3*a^2*ArcTan
h[Sqrt[1 - a^2*x^2]])/2

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Rubi [A]  time = 0.0792337, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {857, 835, 807, 266, 63, 208} \[ -\frac{2 a \sqrt{1-a^2 x^2}}{x}+\frac{\sqrt{1-a^2 x^2}}{x^2 (1-a x)}-\frac{3 \sqrt{1-a^2 x^2}}{2 x^2}-\frac{3}{2} a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(1 - a*x)*Sqrt[1 - a^2*x^2]),x]

[Out]

(-3*Sqrt[1 - a^2*x^2])/(2*x^2) - (2*a*Sqrt[1 - a^2*x^2])/x + Sqrt[1 - a^2*x^2]/(x^2*(1 - a*x)) - (3*a^2*ArcTan
h[Sqrt[1 - a^2*x^2]])/2

Rule 857

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

Rule 835

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((e*f - d*g)
*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(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]
&& EqQ[Simplify[m + 2*p + 3], 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{1}{x^3 (1-a x) \sqrt{1-a^2 x^2}} \, dx &=\frac{\sqrt{1-a^2 x^2}}{x^2 (1-a x)}-\frac{\int \frac{-3 a^2-2 a^3 x}{x^3 \sqrt{1-a^2 x^2}} \, dx}{a^2}\\ &=-\frac{3 \sqrt{1-a^2 x^2}}{2 x^2}+\frac{\sqrt{1-a^2 x^2}}{x^2 (1-a x)}+\frac{\int \frac{4 a^3+3 a^4 x}{x^2 \sqrt{1-a^2 x^2}} \, dx}{2 a^2}\\ &=-\frac{3 \sqrt{1-a^2 x^2}}{2 x^2}-\frac{2 a \sqrt{1-a^2 x^2}}{x}+\frac{\sqrt{1-a^2 x^2}}{x^2 (1-a x)}+\frac{1}{2} \left (3 a^2\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{3 \sqrt{1-a^2 x^2}}{2 x^2}-\frac{2 a \sqrt{1-a^2 x^2}}{x}+\frac{\sqrt{1-a^2 x^2}}{x^2 (1-a x)}+\frac{1}{4} \left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{3 \sqrt{1-a^2 x^2}}{2 x^2}-\frac{2 a \sqrt{1-a^2 x^2}}{x}+\frac{\sqrt{1-a^2 x^2}}{x^2 (1-a x)}-\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )\\ &=-\frac{3 \sqrt{1-a^2 x^2}}{2 x^2}-\frac{2 a \sqrt{1-a^2 x^2}}{x}+\frac{\sqrt{1-a^2 x^2}}{x^2 (1-a x)}-\frac{3}{2} a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0588669, size = 63, normalized size = 0.7 \[ \frac{1}{2} \left (\frac{\left (-4 a^2 x^2+a x+1\right ) \sqrt{1-a^2 x^2}}{x^2 (a x-1)}-3 a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(1 - a*x)*Sqrt[1 - a^2*x^2]),x]

[Out]

(((1 + a*x - 4*a^2*x^2)*Sqrt[1 - a^2*x^2])/(x^2*(-1 + a*x)) - 3*a^2*ArcTanh[Sqrt[1 - a^2*x^2]])/2

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Maple [A]  time = 0.057, size = 94, normalized size = 1. \begin{align*} -{\frac{3\,{a}^{2}}{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }-{a\sqrt{- \left ( x-{a}^{-1} \right ) ^{2}{a}^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}}-{\frac{1}{2\,{x}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{a}{x}\sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(-a*x+1)/(-a^2*x^2+1)^(1/2),x)

[Out]

-3/2*a^2*arctanh(1/(-a^2*x^2+1)^(1/2))-a/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)-1/2*(-a^2*x^2+1)^(1/2)/x^2
-a*(-a^2*x^2+1)^(1/2)/x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{1}{\sqrt{-a^{2} x^{2} + 1}{\left (a x - 1\right )} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(-a*x+1)/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

-integrate(1/(sqrt(-a^2*x^2 + 1)*(a*x - 1)*x^3), x)

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Fricas [A]  time = 1.63976, size = 192, normalized size = 2.13 \begin{align*} \frac{2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} + 3 \,{\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) -{\left (4 \, a^{2} x^{2} - a x - 1\right )} \sqrt{-a^{2} x^{2} + 1}}{2 \,{\left (a x^{3} - x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(-a*x+1)/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

1/2*(2*a^3*x^3 - 2*a^2*x^2 + 3*(a^3*x^3 - a^2*x^2)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (4*a^2*x^2 - a*x - 1)*sqr
t(-a^2*x^2 + 1))/(a*x^3 - x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{a x^{4} \sqrt{- a^{2} x^{2} + 1} - x^{3} \sqrt{- a^{2} x^{2} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(-a*x+1)/(-a**2*x**2+1)**(1/2),x)

[Out]

-Integral(1/(a*x**4*sqrt(-a**2*x**2 + 1) - x**3*sqrt(-a**2*x**2 + 1)), x)

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Giac [B]  time = 1.26761, size = 288, normalized size = 3.2 \begin{align*} -\frac{{\left (a^{3} + \frac{3 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a}{x} - \frac{20 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a x^{2}}\right )} a^{4} x^{2}}{8 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}{\left | a \right |}} - \frac{3 \, a^{3} \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{\frac{4 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a{\left | a \right |}}{x} + \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}{\left | a \right |}}{a x^{2}}}{8 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(-a*x+1)/(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

-1/8*(a^3 + 3*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a/x - 20*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/(a*x^2))*a^4*x^2/((sq
rt(-a^2*x^2 + 1)*abs(a) + a)^2*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)*abs(a)) - 3/2*a^3*log(1/2*abs(-2*
sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/abs(a) - 1/8*(4*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a*abs(a)/x + (s
qrt(-a^2*x^2 + 1)*abs(a) + a)^2*abs(a)/(a*x^2))/a^2