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

Optimal. Leaf size=96 \[ -\frac{\log \left (x^2+3\right )}{2\ 2^{2/3}}+\frac{3 \log \left ((x+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{1-x}\right )}{2\ 2^{2/3}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2^{2/3} (x+1)^{2/3}}{\sqrt{3} \sqrt [3]{1-x}}\right )}{2^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.0183386, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {1008} \[ -\frac{\log \left (x^2+3\right )}{2\ 2^{2/3}}+\frac{3 \log \left ((x+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{1-x}\right )}{2\ 2^{2/3}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2^{2/3} (x+1)^{2/3}}{\sqrt{3} \sqrt [3]{1-x}}\right )}{2^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1008

Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)^(1/3)*((d_) + (f_.)*(x_)^2)), x_Symbol] :> Simp[(Sqrt[3]*h*ArcT
an[1/Sqrt[3] - (2^(2/3)*(1 - (3*h*x)/g)^(2/3))/(Sqrt[3]*(1 + (3*h*x)/g)^(1/3))])/(2^(2/3)*a^(1/3)*f), x] + (-S
imp[(3*h*Log[(1 - (3*h*x)/g)^(2/3) + 2^(1/3)*(1 + (3*h*x)/g)^(1/3)])/(2^(5/3)*a^(1/3)*f), x] + Simp[(h*Log[d +
f*x^2])/(2^(5/3)*a^(1/3)*f), x]) /; FreeQ[{a, c, d, f, g, h}, x] && EqQ[c*d + 3*a*f, 0] && EqQ[c*g^2 + 9*a*h^
2, 0] && GtQ[a, 0]

Rubi steps

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

Mathematica [C]  time = 0.144073, size = 143, normalized size = 1.49 \[ -\frac{1}{6} x^2 F_1\left (1;\frac{1}{3},1;2;x^2,-\frac{x^2}{3}\right )-\frac{27 x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right )}{\sqrt [3]{1-x^2} \left (x^2+3\right ) \left (2 x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};x^2,-\frac{x^2}{3}\right )-F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};x^2,-\frac{x^2}{3}\right )\right )-9 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};x^2,-\frac{x^2}{3}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(x^2*AppellF1[1, 1/3, 1, 2, x^2, -x^2/3])/6 - (27*x*AppellF1[1/2, 1/3, 1, 3/2, x^2, -x^2/3])/((1 - x^2)^(1/3)
*(3 + x^2)*(-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] - Appe
llF1[3/2, 4/3, 1, 5/2, x^2, -x^2/3])))

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Maple [F]  time = 0.035, size = 0, normalized size = 0. \begin{align*} \int{\frac{3-x}{{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((3-x)/(-x^2+1)^(1/3)/(x^2+3),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 54.8404, size = 775, normalized size = 8.07 \begin{align*} -\frac{1}{6} \cdot 4^{\frac{1}{6}} \sqrt{3} \arctan \left (\frac{4^{\frac{1}{6}} \sqrt{3}{\left (12 \cdot 4^{\frac{2}{3}}{\left (x^{4} + 3 \, x^{3} + 3 \, x^{2} + 9 \, x\right )}{\left (-x^{2} + 1\right )}^{\frac{2}{3}} + 4^{\frac{1}{3}}{\left (x^{6} - 18 \, x^{5} - 117 \, x^{4} - 36 \, x^{3} + 207 \, x^{2} + 54 \, x - 27\right )} + 12 \,{\left (x^{5} + 19 \, x^{4} + 42 \, x^{3} + 6 \, x^{2} - 27 \, x - 9\right )}{\left (-x^{2} + 1\right )}^{\frac{1}{3}}\right )}}{6 \,{\left (x^{6} + 54 \, x^{5} + 171 \, x^{4} + 108 \, x^{3} - 81 \, x^{2} - 162 \, x - 27\right )}}\right ) - \frac{1}{24} \cdot 4^{\frac{2}{3}} \log \left (\frac{6 \cdot 4^{\frac{2}{3}}{\left (x^{2} + 3 \, x\right )}{\left (-x^{2} + 1\right )}^{\frac{2}{3}} + 4^{\frac{1}{3}}{\left (x^{4} + 18 \, x^{3} + 24 \, x^{2} - 18 \, x - 9\right )} - 6 \,{\left (x^{3} + 7 \, x^{2} + 3 \, x - 3\right )}{\left (-x^{2} + 1\right )}^{\frac{1}{3}}}{x^{4} + 6 \, x^{2} + 9}\right ) + \frac{1}{12} \cdot 4^{\frac{2}{3}} \log \left (\frac{4^{\frac{2}{3}}{\left (x^{2} + 3\right )} + 6 \cdot 4^{\frac{1}{3}}{\left (-x^{2} + 1\right )}^{\frac{1}{3}}{\left (x + 1\right )} + 12 \,{\left (-x^{2} + 1\right )}^{\frac{2}{3}}}{x^{2} + 3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*4^(1/6)*sqrt(3)*arctan(1/6*4^(1/6)*sqrt(3)*(12*4^(2/3)*(x^4 + 3*x^3 + 3*x^2 + 9*x)*(-x^2 + 1)^(2/3) + 4^(
1/3)*(x^6 - 18*x^5 - 117*x^4 - 36*x^3 + 207*x^2 + 54*x - 27) + 12*(x^5 + 19*x^4 + 42*x^3 + 6*x^2 - 27*x - 9)*(
-x^2 + 1)^(1/3))/(x^6 + 54*x^5 + 171*x^4 + 108*x^3 - 81*x^2 - 162*x - 27)) - 1/24*4^(2/3)*log((6*4^(2/3)*(x^2
+ 3*x)*(-x^2 + 1)^(2/3) + 4^(1/3)*(x^4 + 18*x^3 + 24*x^2 - 18*x - 9) - 6*(x^3 + 7*x^2 + 3*x - 3)*(-x^2 + 1)^(1
/3))/(x^4 + 6*x^2 + 9)) + 1/12*4^(2/3)*log((4^(2/3)*(x^2 + 3) + 6*4^(1/3)*(-x^2 + 1)^(1/3)*(x + 1) + 12*(-x^2
+ 1)^(2/3))/(x^2 + 3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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