\(\int \cos ^6(x) \sin ^7(x) \, dx\) [345]

Optimal result

Integrand size = 9, antiderivative size = 33 \[ \int \cos ^6(x) \sin ^7(x) \, dx=-\frac {1}{7} \cos ^7(x)+\frac {\cos ^9(x)}{3}-\frac {3 \cos ^{11}(x)}{11}+\frac {\cos ^{13}(x)}{13} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2645, 276} \[ \int \cos ^6(x) \sin ^7(x) \, dx=\frac {\cos ^{13}(x)}{13}-\frac {3 \cos ^{11}(x)}{11}+\frac {\cos ^9(x)}{3}-\frac {\cos ^7(x)}{7} \]

[In]

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

[Out]

-1/7*Cos[x]^7 + Cos[x]^9/3 - (3*Cos[x]^11)/11 + Cos[x]^13/13

Rule 276

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]

Rule 2645

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])

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int x^6 \left (1-x^2\right )^3 \, dx,x,\cos (x)\right ) \\ & = -\text {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] (verified)

Time = 0.10 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.67 \[ \int \cos ^6(x) \sin ^7(x) \, dx=-\frac {5 \cos (x)}{1024}-\frac {5 \cos (3 x)}{4096}+\frac {3 \cos (5 x)}{4096}+\frac {3 \cos (7 x)}{14336}-\frac {\cos (9 x)}{6144}-\frac {\cos (11 x)}{45056}+\frac {\cos (13 x)}{53248} \]

[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

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(-\frac {\left (\cos ^{7}\left (x \right )\right )}{7}+\frac {\left (\cos ^{9}\left (x \right )\right )}{3}-\frac {3 \left (\cos ^{11}\left (x \right )\right )}{11}+\frac {\left (\cos ^{13}\left (x \right )\right )}{13}\)	\(26\)
default	\(-\frac {\left (\cos ^{7}\left (x \right )\right )}{7}+\frac {\left (\cos ^{9}\left (x \right )\right )}{3}-\frac {3 \left (\cos ^{11}\left (x \right )\right )}{11}+\frac {\left (\cos ^{13}\left (x \right )\right )}{13}\)	\(26\)
risch	\(-\frac {5 \cos \left (x \right )}{1024}+\frac {\cos \left (13 x \right )}{53248}-\frac {\cos \left (11 x \right )}{45056}-\frac {\cos \left (9 x \right )}{6144}+\frac {3 \cos \left (7 x \right )}{14336}+\frac {3 \cos \left (5 x \right )}{4096}-\frac {5 \cos \left (3 x \right )}{4096}\)	\(42\)
parallelrisch	\(\frac {320}{3003}-\frac {5 \cos \left (x \right )}{1024}+\frac {\cos \left (13 x \right )}{53248}-\frac {\cos \left (11 x \right )}{45056}-\frac {\cos \left (9 x \right )}{6144}+\frac {3 \cos \left (7 x \right )}{14336}+\frac {3 \cos \left (5 x \right )}{4096}-\frac {5 \cos \left (3 x \right )}{4096}\)	\(43\)

[In]

int(cos(x)^6*sin(x)^7,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \cos ^6(x) \sin ^7(x) \, dx=\frac {1}{13} \, \cos \left (x\right )^{13} - \frac {3}{11} \, \cos \left (x\right )^{11} + \frac {1}{3} \, \cos \left (x\right )^{9} - \frac {1}{7} \, \cos \left (x\right )^{7} \]

[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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \cos ^6(x) \sin ^7(x) \, dx=\frac {\cos ^{13}{\left (x \right )}}{13} - \frac {3 \cos ^{11}{\left (x \right )}}{11} + \frac {\cos ^{9}{\left (x \right )}}{3} - \frac {\cos ^{7}{\left (x \right )}}{7} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \cos ^6(x) \sin ^7(x) \, dx=\frac {1}{13} \, \cos \left (x\right )^{13} - \frac {3}{11} \, \cos \left (x\right )^{11} + \frac {1}{3} \, \cos \left (x\right )^{9} - \frac {1}{7} \, \cos \left (x\right )^{7} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \cos ^6(x) \sin ^7(x) \, dx=\frac {1}{13} \, \cos \left (x\right )^{13} - \frac {3}{11} \, \cos \left (x\right )^{11} + \frac {1}{3} \, \cos \left (x\right )^{9} - \frac {1}{7} \, \cos \left (x\right )^{7} \]

[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

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \cos ^6(x) \sin ^7(x) \, dx=\frac {{\cos \left (x\right )}^{13}}{13}-\frac {3\,{\cos \left (x\right )}^{11}}{11}+\frac {{\cos \left (x\right )}^9}{3}-\frac {{\cos \left (x\right )}^7}{7} \]

[In]

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

[Out]

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