\(\int \frac {1}{x^3 (1-a x) \sqrt {1-a^2 x^2}} \, dx\) [156]

Optimal result

Integrand size = 26, antiderivative size = 90 \[ \int \frac {1}{x^3 (1-a x) \sqrt {1-a^2 x^2}} \, dx=-\frac {3 \sqrt {1-a^2 x^2}}{2 x^2}-\frac {2 a \sqrt {1-a^2 x^2}}{x}+\frac {\sqrt {1-a^2 x^2}}{x^2 (1-a x)}-\frac {3}{2} a^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {871, 849, 821, 272, 65, 214} \[ \int \frac {1}{x^3 (1-a x) \sqrt {1-a^2 x^2}} \, dx=-\frac {3}{2} a^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {2 a \sqrt {1-a^2 x^2}}{x}+\frac {\sqrt {1-a^2 x^2}}{x^2 (1-a x)}-\frac {3 \sqrt {1-a^2 x^2}}{2 x^2} \]

[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 65

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 214

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]

Rule 272

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 821

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 849

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 871

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]

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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} \text {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] (verified)

Time = 0.17 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x^3 (1-a x) \sqrt {1-a^2 x^2}} \, dx=\frac {1}{2} \left (\frac {\left (1+a x-4 a^2 x^2\right ) \sqrt {1-a^2 x^2}}{x^2 (-1+a x)}-3 a^2 \log (x)+3 a^2 \log \left (-1+\sqrt {1-a^2 x^2}\right )\right ) \]

[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*Log[x] + 3*a^2*Log[-1 + Sqrt[1 - a^2*x^2]]
)/2

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.04

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {3 a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-\frac {a \sqrt {-a^{2} x^{2}+1}}{x}-\frac {a \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{x -\frac {1}{a}}\)	\(94\)
risch	\(\frac {2 a^{3} x^{3}+a^{2} x^{2}-2 a x -1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {a^{2} \left (-3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{a \left (x -\frac {1}{a}\right )}\right )}{2}\)	\(102\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^3 (1-a x) \sqrt {1-a^2 x^2}} \, dx=\frac {2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} + 3 \, {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (4 \, a^{2} x^{2} - a x - 1\right )} \sqrt {-a^{2} x^{2} + 1}}{2 \, {\left (a x^{3} - x^{2}\right )}} \]

[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)

Sympy [F]

\[ \int \frac {1}{x^3 (1-a x) \sqrt {1-a^2 x^2}} \, dx=- \int \frac {1}{a x^{4} \sqrt {- a^{2} x^{2} + 1} - x^{3} \sqrt {- a^{2} x^{2} + 1}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {1}{x^3 (1-a x) \sqrt {1-a^2 x^2}} \, dx=\int { -\frac {1}{\sqrt {-a^{2} x^{2} + 1} {\left (a x - 1\right )} x^{3}} \,d x } \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (78) = 156\).

Time = 0.29 (sec) , antiderivative size = 213, normalized size of antiderivative = 2.37 \[ \int \frac {1}{x^3 (1-a x) \sqrt {1-a^2 x^2}} \, dx=-\frac {{\left (a^{3} + \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a}{x} - \frac {20 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a x^{2}}\right )} a^{4} x^{2}}{8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} - \frac {3 \, a^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, {\left | a \right |}} - \frac {\frac {4 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a {\left | a \right |}}{x} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} {\left | a \right |}}{a x^{2}}}{8 \, a^{2}} \]

[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

Mupad [B] (verification not implemented)

Time = 11.44 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x^3 (1-a x) \sqrt {1-a^2 x^2}} \, dx=\frac {a^3\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {a\,\sqrt {1-a^2\,x^2}}{x}-\frac {\sqrt {1-a^2\,x^2}}{2\,x^2}+\frac {a^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2} \]

[In]

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

[Out]

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