3.2.49 \(\int \frac {3+x}{\sqrt [3]{1-x^2} (3+x^2)} \, dx\) [149]

Optimal. Leaf size=95 \[ \frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2^{2/3} (1-x)^{2/3}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{2^{2/3}}+\frac {\log \left (3+x^2\right )}{2\ 2^{2/3}}-\frac {3 \log \left ((1-x)^{2/3}+\sqrt [3]{2} \sqrt [3]{1+x}\right )}{2\ 2^{2/3}} \]

[Out]

1/4*ln(x^2+3)*2^(1/3)-3/4*ln((1-x)^(2/3)+2^(1/3)*(1+x)^(1/3))*2^(1/3)-1/2*arctan(-1/3*3^(1/2)+1/3*2^(2/3)*(1-x
)^(2/3)/(1+x)^(1/3)*3^(1/2))*3^(1/2)*2^(1/3)

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Rubi [A]
time = 0.01, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1022} \begin {gather*} \frac {\sqrt {3} \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2^{2/3} (1-x)^{2/3}}{\sqrt {3} \sqrt [3]{x+1}}\right )}{2^{2/3}}+\frac {\log \left (x^2+3\right )}{2\ 2^{2/3}}-\frac {3 \log \left ((1-x)^{2/3}+\sqrt [3]{2} \sqrt [3]{x+1}\right )}{2\ 2^{2/3}} \end {gather*}

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

Rule 1022

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 ((1-x)^{2/3}+\sqrt [3]{2} \sqrt [3]{1+x}\right )}{2\ 2^{2/3}}\\ \end {align*}

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Mathematica [A]
time = 0.20, size = 151, normalized size = 1.59 \begin {gather*} \frac {-2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{1-x^2}}{-2^{2/3}+2^{2/3} x+\sqrt [3]{1-x^2}}\right )-2 \log \left (-2^{2/3}+2^{2/3} x-2 \sqrt [3]{1-x^2}\right )+\log \left (\sqrt [3]{2}-2 \sqrt [3]{2} x+\sqrt [3]{2} x^2+2^{2/3} (-1+x) \sqrt [3]{1-x^2}+2 \left (1-x^2\right )^{2/3}\right )}{2\ 2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 8.49, size = 1042, normalized size = 10.97

method	result	size
trager	\(\text {Expression too large to display}\)	\(1042\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*ln(-(4*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)
^2*RootOf(_Z^3+2)^2*x^2+6*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x^2-12*RootOf(R
ootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x-18*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2
)+4*_Z^2)*RootOf(_Z^3+2)^3*x+18*RootOf(_Z^3+2)^2*(-x^2+1)^(2/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*
_Z^2)+6*RootOf(_Z^3+2)*(-x^2+1)^(1/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x+9*RootOf(_Z^3+2)^2
*(-x^2+1)^(1/3)*x-6*RootOf(_Z^3+2)*(-x^2+1)^(1/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)-2*RootOf
(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^2-9*RootOf(_Z^3+2)^2*(-x^2+1)^(1/3)-3*RootOf(_Z^3+2)*x^2-12*(-
x^2+1)^(2/3)-6*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)-9*RootOf(_Z^3+2))/(2*RootOf(RootOf(_Z^3+2)^
2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^2*x-x+3)/(2*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*R
ootOf(_Z^3+2)^2*x-x-3))+1/2*RootOf(_Z^3+2)*ln((12*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf
(_Z^3+2)^2*x^2+2*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x^2-36*RootOf(RootOf(_Z^
3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x-6*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*
RootOf(_Z^3+2)^3*x+18*RootOf(_Z^3+2)^2*(-x^2+1)^(2/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)+12*R
ootOf(_Z^3+2)*(-x^2+1)^(1/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x+9*RootOf(_Z^3+2)^2*(-x^2+1)
^(1/3)*x-12*RootOf(_Z^3+2)*(-x^2+1)^(1/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)-6*RootOf(RootOf(
_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^2-9*RootOf(_Z^3+2)^2*(-x^2+1)^(1/3)-RootOf(_Z^3+2)*x^2+36*RootOf(RootO
f(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x+6*RootOf(_Z^3+2)*x-6*(-x^2+1)^(2/3)+18*RootOf(RootOf(_Z^3+2)^2+2*_Z*
RootOf(_Z^3+2)+4*_Z^2)+3*RootOf(_Z^3+2))/(2*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)
^2*x-x+3)/(2*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^2*x-x-3))

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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((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] Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (70) = 140\).
time = 9.40, size = 315, normalized size = 3.32 \begin {gather*} -\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (12 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} - 3 \, x^{3} + 3 \, x^{2} - 9 \, x\right )} {\left (-x^{2} + 1\right )}^{\frac {2}{3}} + 12 \, \left (-1\right )^{\frac {1}{3}} {\left (x^{5} - 19 \, x^{4} + 42 \, x^{3} - 6 \, x^{2} - 27 \, x + 9\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 4^{\frac {1}{3}} {\left (x^{6} + 18 \, x^{5} - 117 \, x^{4} + 36 \, x^{3} + 207 \, x^{2} - 54 \, x - 27\right )}\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}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {6 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{2} - 3 \, x\right )} {\left (-x^{2} + 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{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}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {6 \cdot 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{2} + 3\right )} - 12 \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{x^{2} + 3}\right ) \end {gather*}

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

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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