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

Optimal. Leaf size=109 \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{x^2+1}+1}\right )}{2\ 2^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{x^2+1}\right )}{x}\right )}{2\ 2^{2/3} \sqrt{3}}-\frac{\tan ^{-1}(x)}{6\ 2^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{2\ 2^{2/3} \sqrt{3}} \]

[Out]

-ArcTan[x]/(6*2^(2/3)) + ArcTan[x/(1 + 2^(1/3)*(1 + x^2)^(1/3))]/(2*2^(2/3)) - ArcTanh[Sqrt[3]/x]/(2*2^(2/3)*S
qrt[3]) - ArcTanh[(Sqrt[3]*(1 - 2^(1/3)*(1 + x^2)^(1/3)))/x]/(2*2^(2/3)*Sqrt[3])

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Rubi [A]  time = 0.0121913, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {392} \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{x^2+1}+1}\right )}{2\ 2^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{x^2+1}\right )}{x}\right )}{2\ 2^{2/3} \sqrt{3}}-\frac{\tan ^{-1}(x)}{6\ 2^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{2\ 2^{2/3} \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[1/((3 - x^2)*(1 + x^2)^(1/3)),x]

[Out]

-ArcTan[x]/(6*2^(2/3)) + ArcTan[x/(1 + 2^(1/3)*(1 + x^2)^(1/3))]/(2*2^(2/3)) - ArcTanh[Sqrt[3]/x]/(2*2^(2/3)*S
qrt[3]) - ArcTanh[(Sqrt[3]*(1 - 2^(1/3)*(1 + x^2)^(1/3)))/x]/(2*2^(2/3)*Sqrt[3])

Rule 392

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[b/a, 2]}, Simp[(q*ArcTanh
[Sqrt[3]/(q*x)])/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d), x] + (-Simp[(q*ArcTan[(a^(1/3)*q*x)/(a^(1/3) + 2^(1/3)*(a + b*
x^2)^(1/3))])/(2*2^(2/3)*a^(1/3)*d), x] + Simp[(q*ArcTan[q*x])/(6*2^(2/3)*a^(1/3)*d), x] + Simp[(q*ArcTanh[(Sq
rt[3]*(a^(1/3) - 2^(1/3)*(a + b*x^2)^(1/3)))/(a^(1/3)*q*x)])/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d), x])] /; FreeQ[{a,
b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c + 3*a*d, 0] && PosQ[b/a]

Rubi steps

\begin{align*} \int \frac{1}{\left (3-x^2\right ) \sqrt [3]{1+x^2}} \, dx &=-\frac{\tan ^{-1}(x)}{6\ 2^{2/3}}+\frac{\tan ^{-1}\left (\frac{x}{1+\sqrt [3]{2} \sqrt [3]{1+x^2}}\right )}{2\ 2^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{2\ 2^{2/3} \sqrt{3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{1+x^2}\right )}{x}\right )}{2\ 2^{2/3} \sqrt{3}}\\ \end{align*}

Mathematica [C]  time = 0.0633664, size = 124, normalized size = 1.14 \[ -\frac{9 x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-x^2,\frac{x^2}{3}\right )}{\left (x^2-3\right ) \sqrt [3]{x^2+1} \left (2 x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};-x^2,\frac{x^2}{3}\right )-F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-x^2,\frac{x^2}{3}\right )\right )+9 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-x^2,\frac{x^2}{3}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-9*x*AppellF1[1/2, 1/3, 1, 3/2, -x^2, x^2/3])/((-3 + x^2)*(1 + x^2)^(1/3)*(9*AppellF1[1/2, 1/3, 1, 3/2, -x^2,
x^2/3] + 2*x^2*(AppellF1[3/2, 1/3, 2, 5/2, -x^2, x^2/3] - AppellF1[3/2, 4/3, 1, 5/2, -x^2, x^2/3])))

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-{x}^{2}+3}{\frac{1}{\sqrt [3]{{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 9.7741, size = 4852, normalized size = 44.51 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2592*432^(5/6)*sqrt(3)*arctan(-1/54*(2592*x^11 - 393984*x^9 - 699840*x^7 - 373248*x^5 - 69984*x^3 - sqrt(6)*
(18*sqrt(3)*2^(2/3)*(19*x^11 + 111*x^9 + 6030*x^7 + 7182*x^5 + 2511*x^3 + 243*x) + 3*432^(1/6)*sqrt(3)*(x^12 +
924*x^10 - 33363*x^8 - 60912*x^6 - 36693*x^4 - 8748*x^2 - 729) + (432^(5/6)*sqrt(3)*(x^10 - 78*x^8 - 720*x^6
- 594*x^4 - 81*x^2) + 432*sqrt(3)*2^(1/3)*(13*x^9 - 177*x^7 - 153*x^5 - 27*x^3))*(x^2 + 1)^(2/3) + 36*(96*x^10
- 4032*x^8 - 2592*x^6 + sqrt(3)*(x^11 + 369*x^9 - 3654*x^7 - 5454*x^5 - 2187*x^3 - 243*x))*(x^2 + 1)^(1/3))*s
qrt((2*2^(2/3)*(x^6 - 57*x^4 - 117*x^2 - 27) + (x^2 + 1)^(2/3)*(432^(5/6)*(x^3 + x) + 24*2^(1/3)*(x^4 + 9*x^2)
) - 8*(6*x^4 - 18*x^2 + sqrt(3)*(x^5 - 9*x))*(x^2 + 1)^(1/3) - 8*432^(1/6)*(x^5 + 18*x^3 + 9*x))/(x^6 - 9*x^4
+ 27*x^2 - 27)) + 216*(sqrt(3)*2^(2/3)*(x^10 + 276*x^8 + 1206*x^6 + 756*x^4 + 81*x^2) + 432^(1/6)*sqrt(3)*(31*
x^9 - 1620*x^7 - 2070*x^5 - 756*x^3 - 81*x))*(x^2 + 1)^(2/3) + 18*sqrt(3)*(x^12 + 1422*x^10 + 21447*x^8 + 2710
8*x^6 + 16767*x^4 + 6318*x^2 + 729) + (432^(5/6)*sqrt(3)*(x^11 - 681*x^9 + 4338*x^7 + 6102*x^5 + 2349*x^3 + 24
3*x) + 3888*sqrt(3)*2^(1/3)*(x^10 + 44*x^8 + 94*x^6 + 60*x^4 + 9*x^2))*(x^2 + 1)^(1/3))/(x^12 - 2178*x^10 + 46
791*x^8 + 83268*x^6 + 47871*x^4 + 10206*x^2 + 729)) + 1/2592*432^(5/6)*sqrt(3)*arctan(-1/54*(2592*x^11 - 39398
4*x^9 - 699840*x^7 - 373248*x^5 - 69984*x^3 + sqrt(6)*(18*sqrt(3)*2^(2/3)*(19*x^11 + 111*x^9 + 6030*x^7 + 7182
*x^5 + 2511*x^3 + 243*x) - 3*432^(1/6)*sqrt(3)*(x^12 + 924*x^10 - 33363*x^8 - 60912*x^6 - 36693*x^4 - 8748*x^2
- 729) - (432^(5/6)*sqrt(3)*(x^10 - 78*x^8 - 720*x^6 - 594*x^4 - 81*x^2) - 432*sqrt(3)*2^(1/3)*(13*x^9 - 177*
x^7 - 153*x^5 - 27*x^3))*(x^2 + 1)^(2/3) - 36*(96*x^10 - 4032*x^8 - 2592*x^6 - sqrt(3)*(x^11 + 369*x^9 - 3654*
x^7 - 5454*x^5 - 2187*x^3 - 243*x))*(x^2 + 1)^(1/3))*sqrt((2*2^(2/3)*(x^6 - 57*x^4 - 117*x^2 - 27) - (x^2 + 1)
^(2/3)*(432^(5/6)*(x^3 + x) - 24*2^(1/3)*(x^4 + 9*x^2)) - 8*(6*x^4 - 18*x^2 - sqrt(3)*(x^5 - 9*x))*(x^2 + 1)^(
1/3) + 8*432^(1/6)*(x^5 + 18*x^3 + 9*x))/(x^6 - 9*x^4 + 27*x^2 - 27)) - 216*(sqrt(3)*2^(2/3)*(x^10 + 276*x^8 +
1206*x^6 + 756*x^4 + 81*x^2) - 432^(1/6)*sqrt(3)*(31*x^9 - 1620*x^7 - 2070*x^5 - 756*x^3 - 81*x))*(x^2 + 1)^(
2/3) - 18*sqrt(3)*(x^12 + 1422*x^10 + 21447*x^8 + 27108*x^6 + 16767*x^4 + 6318*x^2 + 729) + (432^(5/6)*sqrt(3)
*(x^11 - 681*x^9 + 4338*x^7 + 6102*x^5 + 2349*x^3 + 243*x) - 3888*sqrt(3)*2^(1/3)*(x^10 + 44*x^8 + 94*x^6 + 60
*x^4 + 9*x^2))*(x^2 + 1)^(1/3))/(x^12 - 2178*x^10 + 46791*x^8 + 83268*x^6 + 47871*x^4 + 10206*x^2 + 729)) + 1/
5184*432^(5/6)*log(-(432^(5/6)*(x^6 + 69*x^4 + 63*x^2 + 27) + 864*(9*x^3 + sqrt(3)*(x^4 + 9*x^2) + 9*x)*(x^2 +
1)^(2/3) + 432*2^(1/3)*(5*x^5 + 30*x^3 + 9*x) + 432*(x^2 + 1)^(1/3)*(2^(2/3)*(x^5 + 18*x^3 + 9*x) + 4*432^(1/
6)*(x^4 + 3*x^2)))/(x^6 - 9*x^4 + 27*x^2 - 27)) - 1/5184*432^(5/6)*log((432^(5/6)*(x^6 + 69*x^4 + 63*x^2 + 27)
- 864*(9*x^3 - sqrt(3)*(x^4 + 9*x^2) + 9*x)*(x^2 + 1)^(2/3) - 432*2^(1/3)*(5*x^5 + 30*x^3 + 9*x) - 432*(x^2 +
1)^(1/3)*(2^(2/3)*(x^5 + 18*x^3 + 9*x) - 4*432^(1/6)*(x^4 + 3*x^2)))/(x^6 - 9*x^4 + 27*x^2 - 27)) - 1/10368*4
32^(5/6)*log(31104*(2*2^(2/3)*(x^6 - 57*x^4 - 117*x^2 - 27) + (x^2 + 1)^(2/3)*(432^(5/6)*(x^3 + x) + 24*2^(1/3
)*(x^4 + 9*x^2)) - 8*(6*x^4 - 18*x^2 + sqrt(3)*(x^5 - 9*x))*(x^2 + 1)^(1/3) - 8*432^(1/6)*(x^5 + 18*x^3 + 9*x)
)/(x^6 - 9*x^4 + 27*x^2 - 27)) + 1/10368*432^(5/6)*log(31104*(2*2^(2/3)*(x^6 - 57*x^4 - 117*x^2 - 27) - (x^2 +
1)^(2/3)*(432^(5/6)*(x^3 + x) - 24*2^(1/3)*(x^4 + 9*x^2)) - 8*(6*x^4 - 18*x^2 - sqrt(3)*(x^5 - 9*x))*(x^2 + 1
)^(1/3) + 8*432^(1/6)*(x^5 + 18*x^3 + 9*x))/(x^6 - 9*x^4 + 27*x^2 - 27))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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