3.885 \(\int (-\cos (x)+\sin (x)) (\cos (x)+\sin (x))^5 \, dx\)

Optimal. Leaf size=11 \[ -\frac{1}{6} (\sin (x)+\cos (x))^6 \]

[Out]

-(Cos[x] + Sin[x])^6/6

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Rubi [A]  time = 0.0207024, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {3145} \[ -\frac{1}{6} (\sin (x)+\cos (x))^6 \]

Antiderivative was successfully verified.

[In]

Int[(-Cos[x] + Sin[x])*(Cos[x] + Sin[x])^5,x]

[Out]

-(Cos[x] + Sin[x])^6/6

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 (-\cos (x)+\sin (x)) (\cos (x)+\sin (x))^5 \, dx &=-\frac{1}{6} (\cos (x)+\sin (x))^6\\ \end{align*}

Mathematica [B]  time = 0.0777261, size = 25, normalized size = 2.27 \[ -\frac{5}{8} \sin (2 x)+\frac{1}{24} \sin (6 x)+\frac{1}{4} \cos (4 x) \]

Antiderivative was successfully verified.

[In]

Integrate[(-Cos[x] + Sin[x])*(Cos[x] + Sin[x])^5,x]

[Out]

Cos[4*x]/4 - (5*Sin[2*x])/8 + Sin[6*x]/24

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Maple [B]  time = 0.032, size = 97, normalized size = 8.8 \begin{align*} -{\frac{\cos \left ( x \right ) }{6} \left ( \left ( \sin \left ( x \right ) \right ) ^{5}+{\frac{5\, \left ( \sin \left ( x \right ) \right ) ^{3}}{4}}+{\frac{15\,\sin \left ( x \right ) }{8}} \right ) }+{\frac{2\, \left ( \sin \left ( x \right ) \right ) ^{6}}{3}}-{\frac{5\, \left ( \cos \left ( x \right ) \right ) ^{3} \left ( \sin \left ( x \right ) \right ) ^{3}}{6}}-{\frac{5\, \left ( \cos \left ( x \right ) \right ) ^{3}\sin \left ( x \right ) }{8}}+{\frac{5\,\cos \left ( x \right ) \sin \left ( x \right ) }{16}}+{\frac{5\, \left ( \cos \left ( x \right ) \right ) ^{5}\sin \left ( x \right ) }{6}}-{\frac{5\,\sin \left ( x \right ) }{24} \left ( \left ( \cos \left ( x \right ) \right ) ^{3}+{\frac{3\,\cos \left ( x \right ) }{2}} \right ) }+{\frac{2\, \left ( \cos \left ( x \right ) \right ) ^{6}}{3}}-{\frac{\sin \left ( x \right ) }{6} \left ( \left ( \cos \left ( x \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( x \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( x \right ) }{8}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-cos(x)+sin(x))*(cos(x)+sin(x))^5,x)

[Out]

-1/6*(sin(x)^5+5/4*sin(x)^3+15/8*sin(x))*cos(x)+2/3*sin(x)^6-5/6*cos(x)^3*sin(x)^3-5/8*cos(x)^3*sin(x)+5/16*co
s(x)*sin(x)+5/6*cos(x)^5*sin(x)-5/24*(cos(x)^3+3/2*cos(x))*sin(x)+2/3*cos(x)^6-1/6*(cos(x)^5+5/4*cos(x)^3+15/8
*cos(x))*sin(x)

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Maxima [A]  time = 0.943586, size = 12, normalized size = 1.09 \begin{align*} -\frac{1}{6} \,{\left (\cos \left (x\right ) + \sin \left (x\right )\right )}^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cos(x)+sin(x))*(cos(x)+sin(x))^5,x, algorithm="maxima")

[Out]

-1/6*(cos(x) + sin(x))^6

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Fricas [B]  time = 2.39975, size = 101, normalized size = 9.18 \begin{align*} 2 \, \cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + \frac{1}{3} \,{\left (4 \, \cos \left (x\right )^{5} - 4 \, \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )\right )} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cos(x)+sin(x))*(cos(x)+sin(x))^5,x, algorithm="fricas")

[Out]

2*cos(x)^4 - 2*cos(x)^2 + 1/3*(4*cos(x)^5 - 4*cos(x)^3 - 3*cos(x))*sin(x)

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Sympy [B]  time = 2.08743, size = 54, normalized size = 4.91 \begin{align*} - \sin ^{5}{\left (x \right )} \cos{\left (x \right )} - 2 \sin ^{4}{\left (x \right )} \cos ^{2}{\left (x \right )} - \frac{10 \sin ^{3}{\left (x \right )} \cos ^{3}{\left (x \right )}}{3} - 2 \sin ^{2}{\left (x \right )} \cos ^{4}{\left (x \right )} - \sin{\left (x \right )} \cos ^{5}{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cos(x)+sin(x))*(cos(x)+sin(x))**5,x)

[Out]

-sin(x)**5*cos(x) - 2*sin(x)**4*cos(x)**2 - 10*sin(x)**3*cos(x)**3/3 - 2*sin(x)**2*cos(x)**4 - sin(x)*cos(x)**
5

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Giac [B]  time = 1.07106, size = 26, normalized size = 2.36 \begin{align*} \frac{1}{4} \, \cos \left (4 \, x\right ) + \frac{1}{24} \, \sin \left (6 \, x\right ) - \frac{5}{8} \, \sin \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-cos(x)+sin(x))*(cos(x)+sin(x))^5,x, algorithm="giac")

[Out]

1/4*cos(4*x) + 1/24*sin(6*x) - 5/8*sin(2*x)