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*(-Cos[x] + Sin[x])^(2/3))/2

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Rubi [A]  time = 0.0264653, antiderivative size = 15, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0494849, size = 15, normalized size = 1. \[ \frac{3}{2} (\sin (x)-\cos (x))^{2/3} \]

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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Maple [A]  time = 0.012, size = 12, normalized size = 0.8 \begin{align*}{\frac{3}{2} \left ( -\cos \left ( x \right ) +\sin \left ( x \right ) \right ) ^{{\frac{2}{3}}}} \end{align*}

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 = 0.973147, size = 15, normalized size = 1. \begin{align*} \frac{3}{2} \,{\left (-\cos \left (x\right ) + \sin \left (x\right )\right )}^{\frac{2}{3}} \end{align*}

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 = 0.464619, size = 41, normalized size = 2.73 \begin{align*} \frac{3}{2} \,{\left (-\cos \left (x\right ) + \sin \left (x\right )\right )}^{\frac{2}{3}} \end{align*}

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 = 0.344041, size = 12, normalized size = 0.8 \begin{align*} \frac{3 \left (\sin{\left (x \right )} - \cos{\left (x \right )}\right )^{\frac{2}{3}}}{2} \end{align*}

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 = 1.09935, size = 15, normalized size = 1. \begin{align*} \frac{3}{2} \,{\left (-\cos \left (x\right ) + \sin \left (x\right )\right )}^{\frac{2}{3}} \end{align*}

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)