3.133 \(\int \cos (7 x) \sin ^3(6 x) \, dx\)

Optimal. Leaf size=31 \[ \frac {3 \cos (x)}{8}+\frac {1}{88} \cos (11 x)-\frac {3}{104} \cos (13 x)+\frac {1}{200} \cos (25 x) \]

[Out]

3/8*cos(x)+1/88*cos(11*x)-3/104*cos(13*x)+1/200*cos(25*x)

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4354, 2638} \[ \frac {3 \cos (x)}{8}+\frac {1}{88} \cos (11 x)-\frac {3}{104} \cos (13 x)+\frac {1}{200} \cos (25 x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[7*x]*Sin[6*x]^3,x]

[Out]

(3*Cos[x])/8 + Cos[11*x]/88 - (3*Cos[13*x])/104 + Cos[25*x]/200

Rule 2638

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

Rule 4354

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*} \int \cos (7 x) \sin ^3(6 x) \, dx &=\int \left (-\frac {3 \sin (x)}{8}-\frac {1}{8} \sin (11 x)+\frac {3}{8} \sin (13 x)-\frac {1}{8} \sin (25 x)\right ) \, dx\\ &=-\left (\frac {1}{8} \int \sin (11 x) \, dx\right )-\frac {1}{8} \int \sin (25 x) \, dx-\frac {3}{8} \int \sin (x) \, dx+\frac {3}{8} \int \sin (13 x) \, dx\\ &=\frac {3 \cos (x)}{8}+\frac {1}{88} \cos (11 x)-\frac {3}{104} \cos (13 x)+\frac {1}{200} \cos (25 x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 31, normalized size = 1.00 \[ \frac {3 \cos (x)}{8}+\frac {1}{88} \cos (11 x)-\frac {3}{104} \cos (13 x)+\frac {1}{200} \cos (25 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[7*x]*Sin[6*x]^3,x]

[Out]

(3*Cos[x])/8 + Cos[11*x]/88 - (3*Cos[13*x])/104 + Cos[25*x]/200

________________________________________________________________________________________

fricas [B]  time = 1.29, size = 67, normalized size = 2.16 \[ \frac {2097152}{25} \, \cos \relax (x)^{25} - 524288 \, \cos \relax (x)^{23} + 1441792 \, \cos \relax (x)^{21} - 2293760 \, \cos \relax (x)^{19} + 2334720 \, \cos \relax (x)^{17} - \frac {7938048}{5} \, \cos \relax (x)^{15} + \frac {9503232}{13} \, \cos \relax (x)^{13} - \frac {2484992}{11} \, \cos \relax (x)^{11} + 45248 \, \cos \relax (x)^{9} - 5400 \, \cos \relax (x)^{7} + \frac {1512}{5} \, \cos \relax (x)^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(7*x)*sin(6*x)^3,x, algorithm="fricas")

[Out]

2097152/25*cos(x)^25 - 524288*cos(x)^23 + 1441792*cos(x)^21 - 2293760*cos(x)^19 + 2334720*cos(x)^17 - 7938048/
5*cos(x)^15 + 9503232/13*cos(x)^13 - 2484992/11*cos(x)^11 + 45248*cos(x)^9 - 5400*cos(x)^7 + 1512/5*cos(x)^5

________________________________________________________________________________________

giac [A]  time = 0.14, size = 23, normalized size = 0.74 \[ \frac {1}{200} \, \cos \left (25 \, x\right ) - \frac {3}{104} \, \cos \left (13 \, x\right ) + \frac {1}{88} \, \cos \left (11 \, x\right ) + \frac {3}{8} \, \cos \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(7*x)*sin(6*x)^3,x, algorithm="giac")

[Out]

1/200*cos(25*x) - 3/104*cos(13*x) + 1/88*cos(11*x) + 3/8*cos(x)

________________________________________________________________________________________

maple [A]  time = 0.45, size = 24, normalized size = 0.77 \[ \frac {3 \cos \relax (x )}{8}+\frac {\cos \left (11 x \right )}{88}-\frac {3 \cos \left (13 x \right )}{104}+\frac {\cos \left (25 x \right )}{200} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(7*x)*sin(6*x)^3,x)

[Out]

3/8*cos(x)+1/88*cos(11*x)-3/104*cos(13*x)+1/200*cos(25*x)

________________________________________________________________________________________

maxima [A]  time = 0.73, size = 23, normalized size = 0.74 \[ \frac {1}{200} \, \cos \left (25 \, x\right ) - \frac {3}{104} \, \cos \left (13 \, x\right ) + \frac {1}{88} \, \cos \left (11 \, x\right ) + \frac {3}{8} \, \cos \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(7*x)*sin(6*x)^3,x, algorithm="maxima")

[Out]

1/200*cos(25*x) - 3/104*cos(13*x) + 1/88*cos(11*x) + 3/8*cos(x)

________________________________________________________________________________________

mupad [B]  time = 3.18, size = 198, normalized size = 6.39 \[ \frac {32\,\left (-96525\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{46}+8655075\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{44}-300482325\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{42}[phone]\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{40}-67792485475\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{38}[phone]\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{36}-2750448633075\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{34}[phone]\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{32}-26325778958050\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{30}[phone]\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{28}-67895787973650\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{26}[phone]\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{24}-49575456537350\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{22}[phone]\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{20}-10084340561350\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{18}[phone]\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{16}-523829476225\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{14}[phone]\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{12}-5739623945\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{10}[phone]\,{\mathrm {tan}\left (\frac {x}{2}\right )}^8-8468775\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+120825\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+2025\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+81\right )}{3575\,{\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}^{25}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(7*x)*sin(6*x)^3,x)

[Out]

(32*(2025*tan(x/2)^2 + 120825*tan(x/2)^4 - 8468775*tan(x/2)^6 + 301506975*tan(x/2)^8 - 5739623945*tan(x/2)^10
+ 67806830575*tan(x/2)^12 - 523829476225*tan(x/2)^14 + 2750536240650*tan(x/2)^16 - 10084340561350*tan(x/2)^18
+ 26326043727610*tan(x/2)^20 - 49575456537350*tan(x/2)^22 + 67896209197950*tan(x/2)^24 - 67895787973650*tan(x/
2)^26 + 49575817586750*tan(x/2)^28 - 26325778958050*tan(x/2)^30 + 10084506042325*tan(x/2)^32 - 2750448633075*t
an(x/2)^34 + 523868412925*tan(x/2)^36 - 67792485475*tan(x/2)^38 + 5743927475*tan(x/2)^40 - 300482325*tan(x/2)^
42 + 8655075*tan(x/2)^44 - 96525*tan(x/2)^46 + 81))/(3575*(tan(x/2)^2 + 1)^25)

________________________________________________________________________________________

sympy [B]  time = 5.27, size = 70, normalized size = 2.26 \[ \frac {1421 \sin ^{3}{\left (6 x \right )} \sin {\left (7 x \right )}}{3575} + \frac {1062 \sin ^{2}{\left (6 x \right )} \cos {\left (6 x \right )} \cos {\left (7 x \right )}}{3575} + \frac {1512 \sin {\left (6 x \right )} \sin {\left (7 x \right )} \cos ^{2}{\left (6 x \right )}}{3575} + \frac {1296 \cos ^{3}{\left (6 x \right )} \cos {\left (7 x \right )}}{3575} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(7*x)*sin(6*x)**3,x)

[Out]

1421*sin(6*x)**3*sin(7*x)/3575 + 1062*sin(6*x)**2*cos(6*x)*cos(7*x)/3575 + 1512*sin(6*x)*sin(7*x)*cos(6*x)**2/
3575 + 1296*cos(6*x)**3*cos(7*x)/3575

________________________________________________________________________________________