3.345 \(\int \cos ^6(x) \sin ^7(x) \, dx\)

Optimal. Leaf size=33 \[ \frac{\cos ^{13}(x)}{13}-\frac{3 \cos ^{11}(x)}{11}+\frac{\cos ^9(x)}{3}-\frac{\cos ^7(x)}{7} \]

[Out]

-Cos[x]^7/7 + Cos[x]^9/3 - (3*Cos[x]^11)/11 + Cos[x]^13/13

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Rubi [A]  time = 0.0300866, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2565, 270} \[ \frac{\cos ^{13}(x)}{13}-\frac{3 \cos ^{11}(x)}{11}+\frac{\cos ^9(x)}{3}-\frac{\cos ^7(x)}{7} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^6*Sin[x]^7,x]

[Out]

-Cos[x]^7/7 + Cos[x]^9/3 - (3*Cos[x]^11)/11 + Cos[x]^13/13

Rule 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \cos ^6(x) \sin ^7(x) \, dx &=-\operatorname{Subst}\left (\int x^6 \left (1-x^2\right )^3 \, dx,x,\cos (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (x^6-3 x^8+3 x^{10}-x^{12}\right ) \, dx,x,\cos (x)\right )\\ &=-\frac{1}{7} \cos ^7(x)+\frac{\cos ^9(x)}{3}-\frac{3 \cos ^{11}(x)}{11}+\frac{\cos ^{13}(x)}{13}\\ \end{align*}

Mathematica [A]  time = 0.0272894, size = 55, normalized size = 1.67 \[ -\frac{5 \cos (x)}{1024}-\frac{5 \cos (3 x)}{4096}+\frac{3 \cos (5 x)}{4096}+\frac{3 \cos (7 x)}{14336}-\frac{\cos (9 x)}{6144}-\frac{\cos (11 x)}{45056}+\frac{\cos (13 x)}{53248} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^6*Sin[x]^7,x]

[Out]

(-5*Cos[x])/1024 - (5*Cos[3*x])/4096 + (3*Cos[5*x])/4096 + (3*Cos[7*x])/14336 - Cos[9*x]/6144 - Cos[11*x]/4505
6 + Cos[13*x]/53248

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Maple [A]  time = 0.008, size = 38, normalized size = 1.2 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{7} \left ( \sin \left ( x \right ) \right ) ^{6}}{13}}-{\frac{6\, \left ( \sin \left ( x \right ) \right ) ^{4} \left ( \cos \left ( x \right ) \right ) ^{7}}{143}}-{\frac{8\, \left ( \sin \left ( x \right ) \right ) ^{2} \left ( \cos \left ( x \right ) \right ) ^{7}}{429}}-{\frac{16\, \left ( \cos \left ( x \right ) \right ) ^{7}}{3003}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^6*sin(x)^7,x)

[Out]

-1/13*cos(x)^7*sin(x)^6-6/143*sin(x)^4*cos(x)^7-8/429*sin(x)^2*cos(x)^7-16/3003*cos(x)^7

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Maxima [A]  time = 0.934259, size = 34, normalized size = 1.03 \begin{align*} \frac{1}{13} \, \cos \left (x\right )^{13} - \frac{3}{11} \, \cos \left (x\right )^{11} + \frac{1}{3} \, \cos \left (x\right )^{9} - \frac{1}{7} \, \cos \left (x\right )^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^6*sin(x)^7,x, algorithm="maxima")

[Out]

1/13*cos(x)^13 - 3/11*cos(x)^11 + 1/3*cos(x)^9 - 1/7*cos(x)^7

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Fricas [A]  time = 2.05043, size = 85, normalized size = 2.58 \begin{align*} \frac{1}{13} \, \cos \left (x\right )^{13} - \frac{3}{11} \, \cos \left (x\right )^{11} + \frac{1}{3} \, \cos \left (x\right )^{9} - \frac{1}{7} \, \cos \left (x\right )^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^6*sin(x)^7,x, algorithm="fricas")

[Out]

1/13*cos(x)^13 - 3/11*cos(x)^11 + 1/3*cos(x)^9 - 1/7*cos(x)^7

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Sympy [A]  time = 0.064692, size = 27, normalized size = 0.82 \begin{align*} \frac{\cos ^{13}{\left (x \right )}}{13} - \frac{3 \cos ^{11}{\left (x \right )}}{11} + \frac{\cos ^{9}{\left (x \right )}}{3} - \frac{\cos ^{7}{\left (x \right )}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**6*sin(x)**7,x)

[Out]

cos(x)**13/13 - 3*cos(x)**11/11 + cos(x)**9/3 - cos(x)**7/7

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Giac [A]  time = 1.04935, size = 34, normalized size = 1.03 \begin{align*} \frac{1}{13} \, \cos \left (x\right )^{13} - \frac{3}{11} \, \cos \left (x\right )^{11} + \frac{1}{3} \, \cos \left (x\right )^{9} - \frac{1}{7} \, \cos \left (x\right )^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^6*sin(x)^7,x, algorithm="giac")

[Out]

1/13*cos(x)^13 - 3/11*cos(x)^11 + 1/3*cos(x)^9 - 1/7*cos(x)^7