3.12.58 \(\int \frac {3+x^2}{\sqrt [3]{1+x^2} (-1-x^2+x^3)} \, dx\) [1158]

Optimal. Leaf size=86 \[ -\sqrt {3} \text {ArcTan}\left (\frac {\frac {2 x}{\sqrt {3}}+\frac {\sqrt [3]{1+x^2}}{\sqrt {3}}}{\sqrt [3]{1+x^2}}\right )+\log \left (-x+\sqrt [3]{1+x^2}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right ) \]

[Out]

-3^(1/2)*arctan((2/3*x*3^(1/2)+1/3*(x^2+1)^(1/3)*3^(1/2))/(x^2+1)^(1/3))+ln(-x+(x^2+1)^(1/3))-1/2*ln(x^2+x*(x^
2+1)^(1/3)+(x^2+1)^(2/3))

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Rubi [F]
time = 0.28, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {3+x^2}{\sqrt [3]{1+x^2} \left (-1-x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.10, size = 72, normalized size = 0.84 \begin {gather*} -\sqrt {3} \text {ArcTan}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )+\log \left (-x+\sqrt [3]{1+x^2}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.47, size = 260, normalized size = 3.02

method	result	size
trager	\(\ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {2}{3}} x +\left (x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+2 \left (x^{2}+1\right )^{\frac {2}{3}} x -\left (x^{2}+1\right )^{\frac {1}{3}} x^{2}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-x^{3}-x^{2}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{x^{3}-x^{2}-1}\right )+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {2}{3}} x -2 \left (x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-\left (x^{2}+1\right )^{\frac {2}{3}} x -\left (x^{2}+1\right )^{\frac {1}{3}} x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1}{x^{3}-x^{2}-1}\right )\)	\(260\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(-(-RootOf(_Z^2+_Z+1)^2*x^3+RootOf(_Z^2+_Z+1)*(x^2+1)^(2/3)*x+(x^2+1)^(1/3)*RootOf(_Z^2+_Z+1)*x^2-2*RootOf(_
Z^2+_Z+1)*x^3+2*(x^2+1)^(2/3)*x-(x^2+1)^(1/3)*x^2-RootOf(_Z^2+_Z+1)*x^2-x^3-x^2-RootOf(_Z^2+_Z+1)-1)/(x^3-x^2-
1))+RootOf(_Z^2+_Z+1)*ln((-RootOf(_Z^2+_Z+1)^2*x^3+RootOf(_Z^2+_Z+1)*(x^2+1)^(2/3)*x-2*(x^2+1)^(1/3)*RootOf(_Z
^2+_Z+1)*x^2-RootOf(_Z^2+_Z+1)*x^3-(x^2+1)^(2/3)*x-(x^2+1)^(1/3)*x^2+RootOf(_Z^2+_Z+1)*x^2+x^2+RootOf(_Z^2+_Z+
1)+1)/(x^3-x^2-1))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.90, size = 104, normalized size = 1.21 \begin {gather*} -\sqrt {3} \arctan \left (\frac {\sqrt {3} x^{3} - 2 \, \sqrt {3} {\left (x^{2} + 1\right )}^{\frac {1}{3}} x^{2} + 4 \, \sqrt {3} {\left (x^{2} + 1\right )}^{\frac {2}{3}} x}{x^{3} + 8 \, x^{2} + 8}\right ) + \frac {1}{2} \, \log \left (\frac {x^{3} - 3 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} x^{2} - x^{2} + 3 \, {\left (x^{2} + 1\right )}^{\frac {2}{3}} x - 1}{x^{3} - x^{2} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(3)*arctan((sqrt(3)*x^3 - 2*sqrt(3)*(x^2 + 1)^(1/3)*x^2 + 4*sqrt(3)*(x^2 + 1)^(2/3)*x)/(x^3 + 8*x^2 + 8))
+ 1/2*log((x^3 - 3*(x^2 + 1)^(1/3)*x^2 - x^2 + 3*(x^2 + 1)^(2/3)*x - 1)/(x^3 - x^2 - 1))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} + 3}{\sqrt [3]{x^{2} + 1} \left (x^{3} - x^{2} - 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {x^2+3}{{\left (x^2+1\right )}^{1/3}\,\left (-x^3+x^2+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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