3.1.63 \(\int \frac {(1+x)^2}{x^6 \sqrt {1-x^2}} \, dx\)

Optimal. Leaf size=107 \[ -\frac {6 \sqrt {1-x^2}}{5 x}-\frac {3 \sqrt {1-x^2}}{4 x^2}-\frac {3}{4} \tanh ^{-1}\left (\sqrt {1-x^2}\right )-\frac {\sqrt {1-x^2}}{5 x^5}-\frac {\sqrt {1-x^2}}{2 x^4}-\frac {3 \sqrt {1-x^2}}{5 x^3} \]

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Rubi [A]  time = 0.10, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1807, 835, 807, 266, 63, 206} \begin {gather*} -\frac {6 \sqrt {1-x^2}}{5 x}-\frac {3 \sqrt {1-x^2}}{4 x^2}-\frac {3 \sqrt {1-x^2}}{5 x^3}-\frac {\sqrt {1-x^2}}{2 x^4}-\frac {\sqrt {1-x^2}}{5 x^5}-\frac {3}{4} \tanh ^{-1}\left (\sqrt {1-x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x)^2/(x^6*Sqrt[1 - x^2]),x]

[Out]

-Sqrt[1 - x^2]/(5*x^5) - Sqrt[1 - x^2]/(2*x^4) - (3*Sqrt[1 - x^2])/(5*x^3) - (3*Sqrt[1 - x^2])/(4*x^2) - (6*Sq
rt[1 - x^2])/(5*x) - (3*ArcTanh[Sqrt[1 - x^2]])/4

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 206

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

Rubi steps

\begin {align*} \int \frac {(1+x)^2}{x^6 \sqrt {1-x^2}} \, dx &=-\frac {\sqrt {1-x^2}}{5 x^5}-\frac {1}{5} \int \frac {-10-9 x}{x^5 \sqrt {1-x^2}} \, dx\\ &=-\frac {\sqrt {1-x^2}}{5 x^5}-\frac {\sqrt {1-x^2}}{2 x^4}+\frac {1}{20} \int \frac {36+30 x}{x^4 \sqrt {1-x^2}} \, dx\\ &=-\frac {\sqrt {1-x^2}}{5 x^5}-\frac {\sqrt {1-x^2}}{2 x^4}-\frac {3 \sqrt {1-x^2}}{5 x^3}-\frac {1}{60} \int \frac {-90-72 x}{x^3 \sqrt {1-x^2}} \, dx\\ &=-\frac {\sqrt {1-x^2}}{5 x^5}-\frac {\sqrt {1-x^2}}{2 x^4}-\frac {3 \sqrt {1-x^2}}{5 x^3}-\frac {3 \sqrt {1-x^2}}{4 x^2}+\frac {1}{120} \int \frac {144+90 x}{x^2 \sqrt {1-x^2}} \, dx\\ &=-\frac {\sqrt {1-x^2}}{5 x^5}-\frac {\sqrt {1-x^2}}{2 x^4}-\frac {3 \sqrt {1-x^2}}{5 x^3}-\frac {3 \sqrt {1-x^2}}{4 x^2}-\frac {6 \sqrt {1-x^2}}{5 x}+\frac {3}{4} \int \frac {1}{x \sqrt {1-x^2}} \, dx\\ &=-\frac {\sqrt {1-x^2}}{5 x^5}-\frac {\sqrt {1-x^2}}{2 x^4}-\frac {3 \sqrt {1-x^2}}{5 x^3}-\frac {3 \sqrt {1-x^2}}{4 x^2}-\frac {6 \sqrt {1-x^2}}{5 x}+\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1-x^2}}{5 x^5}-\frac {\sqrt {1-x^2}}{2 x^4}-\frac {3 \sqrt {1-x^2}}{5 x^3}-\frac {3 \sqrt {1-x^2}}{4 x^2}-\frac {6 \sqrt {1-x^2}}{5 x}-\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x^2}\right )\\ &=-\frac {\sqrt {1-x^2}}{5 x^5}-\frac {\sqrt {1-x^2}}{2 x^4}-\frac {3 \sqrt {1-x^2}}{5 x^3}-\frac {3 \sqrt {1-x^2}}{4 x^2}-\frac {6 \sqrt {1-x^2}}{5 x}-\frac {3}{4} \tanh ^{-1}\left (\sqrt {1-x^2}\right )\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 50, normalized size = 0.47 \begin {gather*} -\frac {\sqrt {1-x^2} \left (10 x^5 \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};1-x^2\right )+6 x^4+3 x^2+1\right )}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)^2/(x^6*Sqrt[1 - x^2]),x]

[Out]

-1/5*(Sqrt[1 - x^2]*(1 + 3*x^2 + 6*x^4 + 10*x^5*Hypergeometric2F1[1/2, 3, 3/2, 1 - x^2]))/x^5

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IntegrateAlgebraic [A]  time = 0.18, size = 63, normalized size = 0.59 \begin {gather*} \frac {3}{4} \log \left (\sqrt {1-x^2}-1\right )+\frac {\sqrt {1-x^2} \left (-24 x^4-15 x^3-12 x^2-10 x-4\right )}{20 x^5}-\frac {3 \log (x)}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x)^2/(x^6*Sqrt[1 - x^2]),x]

[Out]

(Sqrt[1 - x^2]*(-4 - 10*x - 12*x^2 - 15*x^3 - 24*x^4))/(20*x^5) - (3*Log[x])/4 + (3*Log[-1 + Sqrt[1 - x^2]])/4

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fricas [A]  time = 0.40, size = 58, normalized size = 0.54 \begin {gather*} \frac {15 \, x^{5} \log \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) - {\left (24 \, x^{4} + 15 \, x^{3} + 12 \, x^{2} + 10 \, x + 4\right )} \sqrt {-x^{2} + 1}}{20 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^2/x^6/(-x^2+1)^(1/2),x, algorithm="fricas")

[Out]

1/20*(15*x^5*log((sqrt(-x^2 + 1) - 1)/x) - (24*x^4 + 15*x^3 + 12*x^2 + 10*x + 4)*sqrt(-x^2 + 1))/x^5

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giac [B]  time = 0.20, size = 199, normalized size = 1.86 \begin {gather*} -\frac {x^{5} {\left (\frac {5 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}{x} - \frac {15 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + \frac {40 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{3}}{x^{3}} - \frac {110 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{4}}{x^{4}} - 1\right )}}{160 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{5}} - \frac {11 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}{16 \, x} + \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{4 \, x^{2}} - \frac {3 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}^{3}}{32 \, x^{3}} + \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{4}}{32 \, x^{4}} - \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{5}}{160 \, x^{5}} + \frac {3}{4} \, \log \left (-\frac {\sqrt {-x^{2} + 1} - 1}{{\left | x \right |}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^2/x^6/(-x^2+1)^(1/2),x, algorithm="giac")

[Out]

-1/160*x^5*(5*(sqrt(-x^2 + 1) - 1)/x - 15*(sqrt(-x^2 + 1) - 1)^2/x^2 + 40*(sqrt(-x^2 + 1) - 1)^3/x^3 - 110*(sq
rt(-x^2 + 1) - 1)^4/x^4 - 1)/(sqrt(-x^2 + 1) - 1)^5 - 11/16*(sqrt(-x^2 + 1) - 1)/x + 1/4*(sqrt(-x^2 + 1) - 1)^
2/x^2 - 3/32*(sqrt(-x^2 + 1) - 1)^3/x^3 + 1/32*(sqrt(-x^2 + 1) - 1)^4/x^4 - 1/160*(sqrt(-x^2 + 1) - 1)^5/x^5 +
3/4*log(-(sqrt(-x^2 + 1) - 1)/abs(x))

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maple [A]  time = 0.01, size = 84, normalized size = 0.79 \begin {gather*} -\frac {3 \arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )}{4}-\frac {6 \sqrt {-x^{2}+1}}{5 x}-\frac {3 \sqrt {-x^{2}+1}}{4 x^{2}}-\frac {3 \sqrt {-x^{2}+1}}{5 x^{3}}-\frac {\sqrt {-x^{2}+1}}{2 x^{4}}-\frac {\sqrt {-x^{2}+1}}{5 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)^2/x^6/(-x^2+1)^(1/2),x)

[Out]

-1/5*(-x^2+1)^(1/2)/x^5-3/5*(-x^2+1)^(1/2)/x^3-6/5*(-x^2+1)^(1/2)/x-1/2*(-x^2+1)^(1/2)/x^4-3/4*(-x^2+1)^(1/2)/
x^2-3/4*arctanh(1/(-x^2+1)^(1/2))

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maxima [A]  time = 0.97, size = 96, normalized size = 0.90 \begin {gather*} -\frac {6 \, \sqrt {-x^{2} + 1}}{5 \, x} - \frac {3 \, \sqrt {-x^{2} + 1}}{4 \, x^{2}} - \frac {3 \, \sqrt {-x^{2} + 1}}{5 \, x^{3}} - \frac {\sqrt {-x^{2} + 1}}{2 \, x^{4}} - \frac {\sqrt {-x^{2} + 1}}{5 \, x^{5}} - \frac {3}{4} \, \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)^2/x^6/(-x^2+1)^(1/2),x, algorithm="maxima")

[Out]

-6/5*sqrt(-x^2 + 1)/x - 3/4*sqrt(-x^2 + 1)/x^2 - 3/5*sqrt(-x^2 + 1)/x^3 - 1/2*sqrt(-x^2 + 1)/x^4 - 1/5*sqrt(-x
^2 + 1)/x^5 - 3/4*log(2*sqrt(-x^2 + 1)/abs(x) + 2/abs(x))

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mupad [B]  time = 0.04, size = 90, normalized size = 0.84 \begin {gather*} \frac {3\,\ln \left (\sqrt {\frac {1}{x^2}-1}-\sqrt {\frac {1}{x^2}}\right )}{4}-\sqrt {1-x^2}\,\left (\frac {2}{3\,x}+\frac {1}{3\,x^3}\right )-\sqrt {1-x^2}\,\left (\frac {3}{4\,x^2}+\frac {1}{2\,x^4}\right )-\sqrt {1-x^2}\,\left (\frac {8}{15\,x}+\frac {4}{15\,x^3}+\frac {1}{5\,x^5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)^2/(x^6*(1 - x^2)^(1/2)),x)

[Out]

(3*log((1/x^2 - 1)^(1/2) - (1/x^2)^(1/2)))/4 - (1 - x^2)^(1/2)*(2/(3*x) + 1/(3*x^3)) - (1 - x^2)^(1/2)*(3/(4*x
^2) + 1/(2*x^4)) - (1 - x^2)^(1/2)*(8/(15*x) + 4/(15*x^3) + 1/(5*x^5))

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sympy [C]  time = 12.69, size = 201, normalized size = 1.88 \begin {gather*} \begin {cases} - \frac {\sqrt {1 - x^{2}}}{x} - \frac {\left (1 - x^{2}\right )^{\frac {3}{2}}}{3 x^{3}} & \text {for}\: x > -1 \wedge x < 1 \end {cases} + \begin {cases} - \frac {\sqrt {1 - x^{2}}}{x} - \frac {2 \left (1 - x^{2}\right )^{\frac {3}{2}}}{3 x^{3}} - \frac {\left (1 - x^{2}\right )^{\frac {5}{2}}}{5 x^{5}} & \text {for}\: x > -1 \wedge x < 1 \end {cases} + 2 \left (\begin {cases} - \frac {3 \operatorname {acosh}{\left (\frac {1}{x} \right )}}{8} + \frac {3}{8 x \sqrt {-1 + \frac {1}{x^{2}}}} - \frac {1}{8 x^{3} \sqrt {-1 + \frac {1}{x^{2}}}} - \frac {1}{4 x^{5} \sqrt {-1 + \frac {1}{x^{2}}}} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > 1 \\\frac {3 i \operatorname {asin}{\left (\frac {1}{x} \right )}}{8} - \frac {3 i}{8 x \sqrt {1 - \frac {1}{x^{2}}}} + \frac {i}{8 x^{3} \sqrt {1 - \frac {1}{x^{2}}}} + \frac {i}{4 x^{5} \sqrt {1 - \frac {1}{x^{2}}}} & \text {otherwise} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)**2/x**6/(-x**2+1)**(1/2),x)

[Out]

Piecewise((-sqrt(1 - x**2)/x - (1 - x**2)**(3/2)/(3*x**3), (x > -1) & (x < 1))) + Piecewise((-sqrt(1 - x**2)/x
- 2*(1 - x**2)**(3/2)/(3*x**3) - (1 - x**2)**(5/2)/(5*x**5), (x > -1) & (x < 1))) + 2*Piecewise((-3*acosh(1/x
)/8 + 3/(8*x*sqrt(-1 + x**(-2))) - 1/(8*x**3*sqrt(-1 + x**(-2))) - 1/(4*x**5*sqrt(-1 + x**(-2))), 1/Abs(x**2)
> 1), (3*I*asin(1/x)/8 - 3*I/(8*x*sqrt(1 - 1/x**2)) + I/(8*x**3*sqrt(1 - 1/x**2)) + I/(4*x**5*sqrt(1 - 1/x**2)
), True))

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