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

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Rubi [A]  time = 0.0299518, antiderivative size = 31, normalized size of antiderivative = 1., 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 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]

Rule 2638

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

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 = 0.0153507, size = 31, normalized size = 1. \[ \frac{3 \cos (x)}{8}+\frac{1}{88} \cos (11 x)-\frac{3}{104} \cos (13 x)+\frac{1}{200} \cos (25 x) \]

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

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Maple [A]  time = 0.18, size = 24, normalized size = 0.8 \begin{align*}{\frac{3\,\cos \left ( x \right ) }{8}}+{\frac{\cos \left ( 11\,x \right ) }{88}}-{\frac{3\,\cos \left ( 13\,x \right ) }{104}}+{\frac{\cos \left ( 25\,x \right ) }{200}} \end{align*}

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)

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Maxima [A]  time = 1.00676, size = 31, normalized size = 1. \begin{align*} \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 \left (x\right ) \end{align*}

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)

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Fricas [B]  time = 3.11891, size = 297, normalized size = 9.58 \begin{align*} \frac{2097152}{25} \, \cos \left (x\right )^{25} - 524288 \, \cos \left (x\right )^{23} + 1441792 \, \cos \left (x\right )^{21} - 2293760 \, \cos \left (x\right )^{19} + 2334720 \, \cos \left (x\right )^{17} - \frac{7938048}{5} \, \cos \left (x\right )^{15} + \frac{9503232}{13} \, \cos \left (x\right )^{13} - \frac{2484992}{11} \, \cos \left (x\right )^{11} + 45248 \, \cos \left (x\right )^{9} - 5400 \, \cos \left (x\right )^{7} + \frac{1512}{5} \, \cos \left (x\right )^{5} \end{align*}

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

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Sympy [B]  time = 11.8116, size = 70, normalized size = 2.26 \begin{align*} \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} \end{align*}

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

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Giac [A]  time = 1.1035, size = 31, normalized size = 1. \begin{align*} \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 \left (x\right ) \end{align*}

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)