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]

-1/6*(cos(x)+sin(x))^6

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Rubi [A]  time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, 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*}

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Mathematica [B]  time = 0.08, 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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fricas [B]  time = 0.63, size = 34, normalized size = 3.09 \[ 2 \, \cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + \frac {1}{3} \, {\left (4 \, \cos \relax (x)^{5} - 4 \, \cos \relax (x)^{3} - 3 \, \cos \relax (x)\right )} \sin \relax (x) \]

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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giac [B]  time = 0.13, size = 19, normalized size = 1.73 \[ \frac {1}{4} \, \cos \left (4 \, x\right ) + \frac {1}{24} \, \sin \left (6 \, x\right ) - \frac {5}{8} \, \sin \left (2 \, x\right ) \]

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)

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maple [B]  time = 0.08, size = 97, normalized size = 8.82 \[ -\frac {\left (\sin ^{5}\relax (x )+\frac {5 \left (\sin ^{3}\relax (x )\right )}{4}+\frac {15 \sin \relax (x )}{8}\right ) \cos \relax (x )}{6}+\frac {2 \left (\sin ^{6}\relax (x )\right )}{3}-\frac {5 \left (\cos ^{3}\relax (x )\right ) \left (\sin ^{3}\relax (x )\right )}{6}-\frac {5 \left (\cos ^{3}\relax (x )\right ) \sin \relax (x )}{8}+\frac {5 \cos \relax (x ) \sin \relax (x )}{16}+\frac {5 \left (\cos ^{5}\relax (x )\right ) \sin \relax (x )}{6}-\frac {5 \left (\cos ^{3}\relax (x )+\frac {3 \cos \relax (x )}{2}\right ) \sin \relax (x )}{24}+\frac {2 \left (\cos ^{6}\relax (x )\right )}{3}-\frac {\left (\cos ^{5}\relax (x )+\frac {5 \left (\cos ^{3}\relax (x )\right )}{4}+\frac {15 \cos \relax (x )}{8}\right ) \sin \relax (x )}{6} \]

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 = 0.33, size = 9, normalized size = 0.82 \[ -\frac {1}{6} \, {\left (\cos \relax (x) + \sin \relax (x)\right )}^{6} \]

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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mupad [B]  time = 3.19, size = 20, normalized size = 1.82 \[ -\frac {\sin \left (2\,x\right )\,\left ({\sin \left (2\,x\right )}^2+3\,\sin \left (2\,x\right )+3\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(sin(2*x)*(3*sin(2*x) + sin(2*x)^2 + 3))/6

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sympy [B]  time = 1.66, size = 46, normalized size = 4.18 \[ \frac {2 \sin ^{6}{\relax (x )}}{3} - \sin ^{5}{\relax (x )} \cos {\relax (x )} - \frac {10 \sin ^{3}{\relax (x )} \cos ^{3}{\relax (x )}}{3} - \sin {\relax (x )} \cos ^{5}{\relax (x )} + \frac {2 \cos ^{6}{\relax (x )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sin(x)**6/3 - sin(x)**5*cos(x) - 10*sin(x)**3*cos(x)**3/3 - sin(x)*cos(x)**5 + 2*cos(x)**6/3

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