\(\int \cos (x) \sin ^3(6 x) \, dx\) [132]

Optimal result

Integrand size = 9, antiderivative size = 33 \[ \int \cos (x) \sin ^3(6 x) \, dx=-\frac {3}{40} \cos (5 x)-\frac {3}{56} \cos (7 x)+\frac {1}{136} \cos (17 x)+\frac {1}{152} \cos (19 x) \]

[Out]

-3/40*cos(5*x)-3/56*cos(7*x)+1/136*cos(17*x)+1/152*cos(19*x)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4439, 2718} \[ \int \cos (x) \sin ^3(6 x) \, dx=-\frac {3}{40} \cos (5 x)-\frac {3}{56} \cos (7 x)+\frac {1}{136} \cos (17 x)+\frac {1}{152} \cos (19 x) \]

[In]

Int[Cos[x]*Sin[6*x]^3,x]

[Out]

(-3*Cos[5*x])/40 - (3*Cos[7*x])/56 + Cos[17*x]/136 + Cos[19*x]/152

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 4439

Int[(F_)[(a_.) + (b_.)*(x_)]^(p_.)*(G_)[(c_.) + (d_.)*(x_)]^(q_.), x_Symbol] :> Int[ExpandTrigReduce[ActivateT
rig[F[a + b*x]^p*G[c + d*x]^q], x], x] /; FreeQ[{a, b, c, d}, x] && (EqQ[F, sin] || EqQ[F, cos]) && (EqQ[G, si
n] || EqQ[G, cos]) && IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3}{8} \sin (5 x)+\frac {3}{8} \sin (7 x)-\frac {1}{8} \sin (17 x)-\frac {1}{8} \sin (19 x)\right ) \, dx \\ & = -\left (\frac {1}{8} \int \sin (17 x) \, dx\right )-\frac {1}{8} \int \sin (19 x) \, dx+\frac {3}{8} \int \sin (5 x) \, dx+\frac {3}{8} \int \sin (7 x) \, dx \\ & = -\frac {3}{40} \cos (5 x)-\frac {3}{56} \cos (7 x)+\frac {1}{136} \cos (17 x)+\frac {1}{152} \cos (19 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \cos (x) \sin ^3(6 x) \, dx=-\frac {3}{40} \cos (5 x)-\frac {3}{56} \cos (7 x)+\frac {1}{136} \cos (17 x)+\frac {1}{152} \cos (19 x) \]

[In]

Integrate[Cos[x]*Sin[6*x]^3,x]

[Out]

(-3*Cos[5*x])/40 - (3*Cos[7*x])/56 + Cos[17*x]/136 + Cos[19*x]/152

Maple [A] (verified)

Time = 0.77 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79

method	result	size
default	\(-\frac {3 \cos \left (5 x \right )}{40}-\frac {3 \cos \left (7 x \right )}{56}+\frac {\cos \left (17 x \right )}{136}+\frac {\cos \left (19 x \right )}{152}\)	\(26\)
risch	\(-\frac {3 \cos \left (5 x \right )}{40}-\frac {3 \cos \left (7 x \right )}{56}+\frac {\cos \left (17 x \right )}{136}+\frac {\cos \left (19 x \right )}{152}\)	\(26\)
parallelrisch	\(\frac {1272}{11305}-\frac {3 \cos \left (5 x \right )}{40}-\frac {3 \cos \left (7 x \right )}{56}+\frac {\cos \left (17 x \right )}{136}+\frac {\cos \left (19 x \right )}{152}\)	\(27\)

[In]

int(cos(x)*sin(6*x)^3,x,method=_RETURNVERBOSE)

[Out]

-3/40*cos(5*x)-3/56*cos(7*x)+1/136*cos(17*x)+1/152*cos(19*x)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48 \[ \int \cos (x) \sin ^3(6 x) \, dx=\frac {32768}{19} \, \cos \left (x\right )^{19} - \frac {131072}{17} \, \cos \left (x\right )^{17} + 14336 \, \cos \left (x\right )^{15} - 14336 \, \cos \left (x\right )^{13} + 8320 \, \cos \left (x\right )^{11} - 2816 \, \cos \left (x\right )^{9} + \frac {3672}{7} \, \cos \left (x\right )^{7} - \frac {216}{5} \, \cos \left (x\right )^{5} \]

[In]

integrate(cos(x)*sin(6*x)^3,x, algorithm="fricas")

[Out]

32768/19*cos(x)^19 - 131072/17*cos(x)^17 + 14336*cos(x)^15 - 14336*cos(x)^13 + 8320*cos(x)^11 - 2816*cos(x)^9
+ 3672/7*cos(x)^7 - 216/5*cos(x)^5

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (29) = 58\).

Time = 0.61 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.97 \[ \int \cos (x) \sin ^3(6 x) \, dx=- \frac {251 \sin {\left (x \right )} \sin ^{3}{\left (6 x \right )}}{11305} - \frac {216 \sin {\left (x \right )} \sin {\left (6 x \right )} \cos ^{2}{\left (6 x \right )}}{11305} - \frac {1926 \sin ^{2}{\left (6 x \right )} \cos {\left (x \right )} \cos {\left (6 x \right )}}{11305} - \frac {1296 \cos {\left (x \right )} \cos ^{3}{\left (6 x \right )}}{11305} \]

[In]

integrate(cos(x)*sin(6*x)**3,x)

[Out]

-251*sin(x)*sin(6*x)**3/11305 - 216*sin(x)*sin(6*x)*cos(6*x)**2/11305 - 1926*sin(6*x)**2*cos(x)*cos(6*x)/11305
- 1296*cos(x)*cos(6*x)**3/11305

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \cos (x) \sin ^3(6 x) \, dx=\frac {1}{152} \, \cos \left (19 \, x\right ) + \frac {1}{136} \, \cos \left (17 \, x\right ) - \frac {3}{56} \, \cos \left (7 \, x\right ) - \frac {3}{40} \, \cos \left (5 \, x\right ) \]

[In]

integrate(cos(x)*sin(6*x)^3,x, algorithm="maxima")

[Out]

1/152*cos(19*x) + 1/136*cos(17*x) - 3/56*cos(7*x) - 3/40*cos(5*x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48 \[ \int \cos (x) \sin ^3(6 x) \, dx=\frac {32768}{19} \, \cos \left (x\right )^{19} - \frac {131072}{17} \, \cos \left (x\right )^{17} + 14336 \, \cos \left (x\right )^{15} - 14336 \, \cos \left (x\right )^{13} + 8320 \, \cos \left (x\right )^{11} - 2816 \, \cos \left (x\right )^{9} + \frac {3672}{7} \, \cos \left (x\right )^{7} - \frac {216}{5} \, \cos \left (x\right )^{5} \]

[In]

integrate(cos(x)*sin(6*x)^3,x, algorithm="giac")

[Out]

32768/19*cos(x)^19 - 131072/17*cos(x)^17 + 14336*cos(x)^15 - 14336*cos(x)^13 + 8320*cos(x)^11 - 2816*cos(x)^9
+ 3672/7*cos(x)^7 - 216/5*cos(x)^5

Mupad [B] (verification not implemented)

Time = 27.03 (sec) , antiderivative size = 150, normalized size of antiderivative = 4.55 \[ \int \cos (x) \sin ^3(6 x) \, dx=-\frac {32\,\left (305235\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{34}-9665775\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{32}[phone]\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{30}-1348695544\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{28}[phone]\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{26}-23578828164\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{24}[phone]\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{22}-75928491144\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{20}[phone]\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{18}-51607368282\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{16}[phone]\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{14}-7081614792\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{12}[phone]\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{10}-153524484\,{\mathrm {tan}\left (\frac {x}{2}\right )}^8+9744264\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6-291384\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+1539\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+81\right )}{11305\,{\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}^{19}} \]

[In]

int(sin(6*x)^3*cos(x),x)

[Out]

-(32*(1539*tan(x/2)^2 - 291384*tan(x/2)^4 + 9744264*tan(x/2)^6 - 153524484*tan(x/2)^8 + 1349637412*tan(x/2)^10
- 7081614792*tan(x/2)^12 + 23582909592*tan(x/2)^14 - 51607368282*tan(x/2)^16 + 75935973762*tan(x/2)^18 - 7592
8491144*tan(x/2)^20 + 51613490424*tan(x/2)^22 - 23578828164*tan(x/2)^24 + 7083812484*tan(x/2)^26 - 1348695544*
tan(x/2)^28 + 153838440*tan(x/2)^30 - 9665775*tan(x/2)^32 + 305235*tan(x/2)^34 + 81))/(11305*(tan(x/2)^2 + 1)^
19)