3.1.18 \(\int \frac {\cos (x)+\sin (x)}{\sqrt [3]{-\cos (x)+\sin (x)}} \, dx\) [18]

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

[Out]

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

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Rubi [A]
time = 0.02, 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 = {3224} \begin {gather*} \frac {3}{2} (\sin (x)-\cos (x))^{2/3} \end {gather*}

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 3224

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 = 0.04, size = 15, normalized size = 1.00 \begin {gather*} \frac {3}{2} (-\cos (x)+\sin (x))^{2/3} \end {gather*}

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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Maple [A]
time = 0.08, size = 12, normalized size = 0.80

method	result	size
derivativedivides	\(\frac {3 \left (-\cos \left (x \right )+\sin \left (x \right )\right )^{\frac {2}{3}}}{2}\)	\(12\)
default	\(\frac {3 \left (-\cos \left (x \right )+\sin \left (x \right )\right )^{\frac {2}{3}}}{2}\)	\(12\)
risch	\(\frac {\left (-\frac {3}{2}-\frac {3 i}{2}\right ) \left (\left (1+i\right ) \left (-{\mathrm e}^{4 i x}+i {\mathrm e}^{2 i x}\right )\right )^{\frac {1}{3}} \left ({\mathrm e}^{i x}-i {\mathrm e}^{-i x}\right )}{\left (-8 \cos \left (x \right )+8 \sin \left (x \right )\right )^{\frac {1}{3}} \left (\left (-1-i\right ) \left ({\mathrm e}^{4 i x}-i {\mathrm e}^{2 i x}\right )\right )^{\frac {1}{3}}}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 4.39, size = 11, normalized size = 0.73 \begin {gather*} \frac {3}{2} \, {\left (-\cos \left (x\right ) + \sin \left (x\right )\right )}^{\frac {2}{3}} \end {gather*}

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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Fricas [A]
time = 3.36, size = 11, normalized size = 0.73 \begin {gather*} \frac {3}{2} \, {\left (-\cos \left (x\right ) + \sin \left (x\right )\right )}^{\frac {2}{3}} \end {gather*}

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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Sympy [A]
time = 0.10, size = 12, normalized size = 0.80 \begin {gather*} \frac {3 \left (\sin {\left (x \right )} - \cos {\left (x \right )}\right )^{\frac {2}{3}}}{2} \end {gather*}

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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Giac [A]
time = 0.75, size = 11, normalized size = 0.73 \begin {gather*} \frac {3}{2} \, {\left (-\cos \left (x\right ) + \sin \left (x\right )\right )}^{\frac {2}{3}} \end {gather*}

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

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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