3.18 \(\int \frac {\cos (x)+\sin (x)}{\sqrt [3]{-\cos (x)+\sin (x)}} \, dx\)

Optimal. Leaf size=15 \[ \frac {3}{2} (\sin (x)-\cos (x))^{2/3} \]

[Out]

3/2*(-cos(x)+sin(x))^(2/3)

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Rubi [A]  time = 0.03, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {3145} \[ \frac {3}{2} (\sin (x)-\cos (x))^{2/3} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[x] + Sin[x])/(-Cos[x] + Sin[x])^(1/3),x]

[Out]

(3*(-Cos[x] + Sin[x])^(2/3))/2

Rule 3145

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(n_.)*(cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_
.)*sin[(d_.) + (e_.)*(x_)]), x_Symbol] :> Simp[((c*B - b*C)*(b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1))/(e*(n +
1)*(b^2 + c^2)), x] /; FreeQ[{b, c, d, e, B, C}, x] && NeQ[n, -1] && NeQ[b^2 + c^2, 0] && EqQ[b*B + c*C, 0]

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 15, normalized size = 1.00 \[ \frac {3}{2} (\sin (x)-\cos (x))^{2/3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(-Cos[x] + Sin[x])^(2/3))/2

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fricas [A]  time = 0.41, size = 11, normalized size = 0.73 \[ \frac {3}{2} \, {\left (-\cos \relax (x) + \sin \relax (x)\right )}^{\frac {2}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(-cos(x) + sin(x))^(2/3)

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giac [A]  time = 0.03, size = 11, normalized size = 0.73 \[ \frac {3}{2} \, {\left (-\cos \relax (x) + \sin \relax (x)\right )}^{\frac {2}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(-cos(x) + sin(x))^(2/3)

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maple [A]  time = 0.03, size = 12, normalized size = 0.80 \[ \frac {3 \left (-\cos \relax (x )+\sin \relax (x )\right )^{\frac {2}{3}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(x)+sin(x))/(-cos(x)+sin(x))^(1/3),x)

[Out]

3/2*(-cos(x)+sin(x))^(2/3)

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maxima [A]  time = 0.72, size = 11, normalized size = 0.73 \[ \frac {3}{2} \, {\left (-\cos \relax (x) + \sin \relax (x)\right )}^{\frac {2}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(-cos(x) + sin(x))^(2/3)

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mupad [B]  time = 0.24, size = 15, normalized size = 1.00 \[ \frac {3\,2^{1/3}\,{\left (-\cos \left (x+\frac {\pi }{4}\right )\right )}^{2/3}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(x) + sin(x))/(sin(x) - cos(x))^(1/3),x)

[Out]

(3*2^(1/3)*(-cos(x + pi/4))^(2/3))/2

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sympy [A]  time = 0.35, size = 12, normalized size = 0.80 \[ \frac {3 \left (\sin {\relax (x )} - \cos {\relax (x )}\right )^{\frac {2}{3}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*(sin(x) - cos(x))**(2/3)/2

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