3.17.28 \(\int \frac {\sqrt [3]{-1+x}}{\sqrt [3]{1+x}} \, dx\) [1628]

Optimal. Leaf size=77 \[ \sqrt [3]{-1+x} (1+x)^{2/3}+\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{-1+x}}\right )}{\sqrt {3}}+\frac {1}{3} \log (-1+x)+\log \left (-1+\frac {\sqrt [3]{1+x}}{\sqrt [3]{-1+x}}\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {52, 61} \begin {gather*} \sqrt [3]{x-1} (x+1)^{2/3}+\frac {1}{3} \log (x-1)+\log \left (\frac {\sqrt [3]{x+1}}{\sqrt [3]{x-1}}-1\right )+\frac {2 \tan ^{-1}\left (\frac {2 \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + x)^(1/3)/(1 + x)^(1/3),x]

[Out]

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

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 61

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

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(158\) vs. \(2(77)=154\).
time = 0.18, size = 158, normalized size = 2.05 \begin {gather*} \frac {\sqrt [3]{\frac {-1+x}{1+x}} \left (3 \sqrt [3]{-1+x}+3 \sqrt [3]{-1+x} x+2 \sqrt {3} \sqrt [3]{1+x} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{1+x}}{2 \sqrt [3]{-1+x}+\sqrt [3]{1+x}}\right )+2 \sqrt [3]{1+x} \log \left (\sqrt [3]{-1+x}-\sqrt [3]{1+x}\right )-\sqrt [3]{1+x} \log \left ((-1+x)^{2/3}+\sqrt [3]{-1+x} \sqrt [3]{1+x}+(1+x)^{2/3}\right )\right )}{3 \sqrt [3]{-1+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 3.36, size = 25, normalized size = 0.32 \begin {gather*} \frac {3 2^{\frac {2}{3}} \left (-1+x\right )^{\frac {4}{3}} \text {hyper}\left [\left \{\frac {1}{3},\frac {4}{3}\right \},\left \{\frac {7}{3}\right \},\frac {\left (-1+x\right ) \text {exp\_polar}\left [I \text {Pi}\right ]}{2}\right ]}{8} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(-1 + x)^(1/3)/(1 + x)^(1/3),x]')

[Out]

3 2 ^ (2 / 3) (-1 + x) ^ (4 / 3) hyper[{1 / 3, 4 / 3}, {7 / 3}, (-1 + x) exp_polar[I Pi] / 2] / 8

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.44, size = 577, normalized size = 7.49

method	result	size
risch	\(\left (-1+x \right )^{\frac {1}{3}} \left (1+x \right )^{\frac {2}{3}}+\frac {\left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x +5 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}-4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +2 x^{2}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2}{-1+x}\right )}{3}-\frac {2 \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x -\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}-3 \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}-3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x -x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}+2 x -1}{-1+x}\right ) \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{3}-\frac {2 \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x -\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}-3 \left (x^{3}-x^{2}-x +1\right )^{\frac {2}{3}}-3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}} x -x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+3 \left (x^{3}-x^{2}-x +1\right )^{\frac {1}{3}}+2 x -1}{-1+x}\right )}{3}\right ) \left (\left (-1+x \right )^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{\left (-1+x \right )^{\frac {2}{3}} \left (1+x \right )^{\frac {1}{3}}}\)	\(577\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-1+x)^(1/3)*(1+x)^(2/3)+(2/3*RootOf(_Z^2+_Z+1)*ln(-(2*RootOf(_Z^2+_Z+1)^2*x^2-2*RootOf(_Z^2+_Z+1)^2*x+3*RootO
f(_Z^2+_Z+1)*(x^3-x^2-x+1)^(2/3)+3*RootOf(_Z^2+_Z+1)*(x^3-x^2-x+1)^(1/3)*x+5*RootOf(_Z^2+_Z+1)*x^2-3*RootOf(_Z
^2+_Z+1)*(x^3-x^2-x+1)^(1/3)-4*RootOf(_Z^2+_Z+1)*x+2*x^2-RootOf(_Z^2+_Z+1)-2)/(-1+x))-2/3*ln(-(2*RootOf(_Z^2+_
Z+1)^2*x^2-2*RootOf(_Z^2+_Z+1)^2*x-3*RootOf(_Z^2+_Z+1)*(x^3-x^2-x+1)^(2/3)-3*RootOf(_Z^2+_Z+1)*(x^3-x^2-x+1)^(
1/3)*x-RootOf(_Z^2+_Z+1)*x^2+3*RootOf(_Z^2+_Z+1)*(x^3-x^2-x+1)^(1/3)-3*(x^3-x^2-x+1)^(2/3)-3*(x^3-x^2-x+1)^(1/
3)*x-x^2+RootOf(_Z^2+_Z+1)+3*(x^3-x^2-x+1)^(1/3)+2*x-1)/(-1+x))*RootOf(_Z^2+_Z+1)-2/3*ln(-(2*RootOf(_Z^2+_Z+1)
^2*x^2-2*RootOf(_Z^2+_Z+1)^2*x-3*RootOf(_Z^2+_Z+1)*(x^3-x^2-x+1)^(2/3)-3*RootOf(_Z^2+_Z+1)*(x^3-x^2-x+1)^(1/3)
*x-RootOf(_Z^2+_Z+1)*x^2+3*RootOf(_Z^2+_Z+1)*(x^3-x^2-x+1)^(1/3)-3*(x^3-x^2-x+1)^(2/3)-3*(x^3-x^2-x+1)^(1/3)*x
-x^2+RootOf(_Z^2+_Z+1)+3*(x^3-x^2-x+1)^(1/3)+2*x-1)/(-1+x)))/(-1+x)^(2/3)*((-1+x)^2*(1+x))^(1/3)/(1+x)^(1/3)

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

[Out]

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

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Fricas [A]
time = 0.30, size = 107, normalized size = 1.39 \begin {gather*} -\frac {2}{3} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (x + 1\right )} + 2 \, \sqrt {3} {\left (x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )}^{\frac {1}{3}}}{3 \, {\left (x + 1\right )}}\right ) + {\left (x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \log \left (\frac {{\left (x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )}^{\frac {1}{3}} + {\left (x + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )}^{\frac {2}{3}} + x + 1}{x + 1}\right ) + \frac {2}{3} \, \log \left (\frac {{\left (x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )}^{\frac {1}{3}} - x - 1}{x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 1.72, size = 39, normalized size = 0.51 \begin {gather*} \frac {2^{\frac {2}{3}} \left (x - 1\right )^{\frac {4}{3}} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {\left (x - 1\right ) e^{i \pi }}{2}} \right )}}{2 \Gamma \left (\frac {7}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2**(2/3)*(x - 1)**(4/3)*gamma(4/3)*hyper((1/3, 4/3), (7/3,), (x - 1)*exp_polar(I*pi)/2)/(2*gamma(7/3))

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Giac [F] N/A
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((-1+x)^(1/3)/(1+x)^(1/3),x)

[Out]

Could not integrate

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 1)^(1/3)/(x + 1)^(1/3),x)

[Out]

int((x - 1)^(1/3)/(x + 1)^(1/3), x)

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