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

Optimal. Leaf size=171 \[ \frac {\log \left (2 \sqrt [3]{1-x^3}+2^{2/3} x-2^{2/3}\right )}{3\ 2^{2/3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{1-x^3}}{\sqrt [3]{1-x^3}-2^{2/3} x+2^{2/3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (-2 \left (1-x^3\right )^{2/3}+\left (2^{2/3} x-2^{2/3}\right ) \sqrt [3]{1-x^3}-\sqrt [3]{2} x^2+2 \sqrt [3]{2} x-\sqrt [3]{2}\right )}{6\ 2^{2/3}} \]

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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 {1+x}{\left (1+4 x+x^2\right ) \sqrt [3]{1-x^3}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 + x)/((1 + 4*x + x^2)*(1 - x^3)^(1/3)),x]

[Out]

((3 - Sqrt[3])*Defer[Int][1/((4 - 2*Sqrt[3] + 2*x)*(1 - x^3)^(1/3)), x])/3 + ((3 + Sqrt[3])*Defer[Int][1/((4 +
2*Sqrt[3] + 2*x)*(1 - x^3)^(1/3)), x])/3

Rubi steps

\begin {align*} \int \frac {1+x}{\left (1+4 x+x^2\right ) \sqrt [3]{1-x^3}} \, dx &=\int \left (\frac {1-\frac {1}{\sqrt {3}}}{\left (4-2 \sqrt {3}+2 x\right ) \sqrt [3]{1-x^3}}+\frac {1+\frac {1}{\sqrt {3}}}{\left (4+2 \sqrt {3}+2 x\right ) \sqrt [3]{1-x^3}}\right ) \, dx\\ &=\frac {1}{3} \left (3-\sqrt {3}\right ) \int \frac {1}{\left (4-2 \sqrt {3}+2 x\right ) \sqrt [3]{1-x^3}} \, dx+\frac {1}{3} \left (3+\sqrt {3}\right ) \int \frac {1}{\left (4+2 \sqrt {3}+2 x\right ) \sqrt [3]{1-x^3}} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.19, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x}{\left (1+4 x+x^2\right ) \sqrt [3]{1-x^3}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(1 + x)/((1 + 4*x + x^2)*(1 - x^3)^(1/3)),x]

[Out]

Integrate[(1 + x)/((1 + 4*x + x^2)*(1 - x^3)^(1/3)), x]

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IntegrateAlgebraic [A]  time = 1.15, size = 171, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{1-x^3}}{2^{2/3}-2^{2/3} x+\sqrt [3]{1-x^3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (-2^{2/3}+2^{2/3} x+2 \sqrt [3]{1-x^3}\right )}{3\ 2^{2/3}}-\frac {\log \left (-\sqrt [3]{2}+2 \sqrt [3]{2} x-\sqrt [3]{2} x^2+\left (-2^{2/3}+2^{2/3} x\right ) \sqrt [3]{1-x^3}-2 \left (1-x^3\right )^{2/3}\right )}{6\ 2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x)/((1 + 4*x + x^2)*(1 - x^3)^(1/3)),x]

[Out]

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

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fricas [B]  time = 9.96, size = 316, normalized size = 1.85 \begin {gather*} \frac {1}{18} \cdot 4^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {4^{\frac {1}{6}} {\left (12 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (2 \, x^{4} + 7 \, x^{3} + 7 \, x + 2\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} \sqrt {3} {\left (91 \, x^{6} - 42 \, x^{5} + 105 \, x^{4} - 92 \, x^{3} + 105 \, x^{2} - 42 \, x + 91\right )} - 12 \, \sqrt {3} {\left (19 \, x^{5} - 29 \, x^{4} + 28 \, x^{3} - 28 \, x^{2} + 29 \, x - 19\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (53 \, x^{6} - 174 \, x^{5} + 111 \, x^{4} - 196 \, x^{3} + 111 \, x^{2} - 174 \, x + 53\right )}}\right ) - \frac {1}{72} \cdot 4^{\frac {2}{3}} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} {\left (2 \, x^{2} - x + 2\right )} + 4^{\frac {1}{3}} {\left (19 \, x^{4} - 10 \, x^{3} + 18 \, x^{2} - 10 \, x + 19\right )} - 6 \, {\left (5 \, x^{3} - 3 \, x^{2} + 3 \, x - 5\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{4} + 8 \, x^{3} + 18 \, x^{2} + 8 \, x + 1}\right ) + \frac {1}{36} \cdot 4^{\frac {2}{3}} \log \left (\frac {4^{\frac {2}{3}} {\left (x^{2} + 4 \, x + 1\right )} - 6 \cdot 4^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} - 12 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2} + 4 \, x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x^2+4*x+1)/(-x^3+1)^(1/3),x, algorithm="fricas")

[Out]

1/18*4^(1/6)*sqrt(3)*arctan(1/6*4^(1/6)*(12*4^(2/3)*sqrt(3)*(2*x^4 + 7*x^3 + 7*x + 2)*(-x^3 + 1)^(2/3) + 4^(1/
3)*sqrt(3)*(91*x^6 - 42*x^5 + 105*x^4 - 92*x^3 + 105*x^2 - 42*x + 91) - 12*sqrt(3)*(19*x^5 - 29*x^4 + 28*x^3 -
28*x^2 + 29*x - 19)*(-x^3 + 1)^(1/3))/(53*x^6 - 174*x^5 + 111*x^4 - 196*x^3 + 111*x^2 - 174*x + 53)) - 1/72*4
^(2/3)*log((6*4^(2/3)*(-x^3 + 1)^(2/3)*(2*x^2 - x + 2) + 4^(1/3)*(19*x^4 - 10*x^3 + 18*x^2 - 10*x + 19) - 6*(5
*x^3 - 3*x^2 + 3*x - 5)*(-x^3 + 1)^(1/3))/(x^4 + 8*x^3 + 18*x^2 + 8*x + 1)) + 1/36*4^(2/3)*log((4^(2/3)*(x^2 +
4*x + 1) - 6*4^(1/3)*(-x^3 + 1)^(1/3)*(x - 1) - 12*(-x^3 + 1)^(2/3))/(x^2 + 4*x + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 1}{{\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} + 4 \, x + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x^2+4*x+1)/(-x^3+1)^(1/3),x, algorithm="giac")

[Out]

integrate((x + 1)/((-x^3 + 1)^(1/3)*(x^2 + 4*x + 1)), x)

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maple [C]  time = 9.21, size = 1157, normalized size = 6.77

method	result	size
trager	\(\text {Expression too large to display}\)	\(1157\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)/(x^2+4*x+1)/(-x^3+1)^(1/3),x,method=_RETURNVERBOSE)

[Out]

-1/6*ln((90*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*RootOf(_Z^3-2)^3*x+756*RootOf(RootOf(_Z^3-2)^
2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)^2*RootOf(_Z^3-2)^2*x+108*(-x^3+1)^(2/3)*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^
3-2)+36*_Z^2)*RootOf(_Z^3-2)^2-33*(-x^3+1)^(1/3)*RootOf(_Z^3-2)^2*x-90*(-x^3+1)^(1/3)*RootOf(_Z^3-2)*RootOf(Ro
otOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*x+33*(-x^3+1)^(1/3)*RootOf(_Z^3-2)^2+90*(-x^3+1)^(1/3)*RootOf(_Z^3
-2)*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)-35*RootOf(_Z^3-2)*x^2-294*RootOf(RootOf(_Z^3-2)^2+6*_
Z*RootOf(_Z^3-2)+36*_Z^2)*x^2+10*RootOf(_Z^3-2)*x+84*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*x-30
*(-x^3+1)^(2/3)-35*RootOf(_Z^3-2)-294*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2))/(x^2+4*x+1))*RootO
f(_Z^3-2)-ln((90*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*RootOf(_Z^3-2)^3*x+756*RootOf(RootOf(_Z^
3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)^2*RootOf(_Z^3-2)^2*x+108*(-x^3+1)^(2/3)*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootO
f(_Z^3-2)+36*_Z^2)*RootOf(_Z^3-2)^2-33*(-x^3+1)^(1/3)*RootOf(_Z^3-2)^2*x-90*(-x^3+1)^(1/3)*RootOf(_Z^3-2)*Root
Of(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*x+33*(-x^3+1)^(1/3)*RootOf(_Z^3-2)^2+90*(-x^3+1)^(1/3)*RootOf
(_Z^3-2)*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)-35*RootOf(_Z^3-2)*x^2-294*RootOf(RootOf(_Z^3-2)^
2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*x^2+10*RootOf(_Z^3-2)*x+84*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)
*x-30*(-x^3+1)^(2/3)-35*RootOf(_Z^3-2)-294*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2))/(x^2+4*x+1))*
RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)+1/6*RootOf(_Z^3-2)*ln((90*RootOf(RootOf(_Z^3-2)^2+6*_Z*Ro
otOf(_Z^3-2)+36*_Z^2)*RootOf(_Z^3-2)^3*x-216*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)^2*RootOf(_Z^
3-2)^2*x-108*(-x^3+1)^(2/3)*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*RootOf(_Z^3-2)^2-15*(-x^3+1)^
(1/3)*RootOf(_Z^3-2)^2*x-198*(-x^3+1)^(1/3)*RootOf(_Z^3-2)*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2
)*x+15*(-x^3+1)^(1/3)*RootOf(_Z^3-2)^2+198*(-x^3+1)^(1/3)*RootOf(_Z^3-2)*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_
Z^3-2)+36*_Z^2)+35*RootOf(_Z^3-2)*x^2-84*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*x^2+20*RootOf(_Z
^3-2)*x-48*RootOf(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2)*x-66*(-x^3+1)^(2/3)+35*RootOf(_Z^3-2)-84*RootO
f(RootOf(_Z^3-2)^2+6*_Z*RootOf(_Z^3-2)+36*_Z^2))/(x^2+4*x+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 1}{{\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} + 4 \, x + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x^2+4*x+1)/(-x^3+1)^(1/3),x, algorithm="maxima")

[Out]

integrate((x + 1)/((-x^3 + 1)^(1/3)*(x^2 + 4*x + 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x+1}{{\left (1-x^3\right )}^{1/3}\,\left (x^2+4\,x+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)/((1 - x^3)^(1/3)*(4*x + x^2 + 1)),x)

[Out]

int((x + 1)/((1 - x^3)^(1/3)*(4*x + x^2 + 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 1}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} + 4 x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x**2+4*x+1)/(-x**3+1)**(1/3),x)

[Out]

Integral((x + 1)/((-(x - 1)*(x**2 + x + 1))**(1/3)*(x**2 + 4*x + 1)), x)

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