3.3.92 \(\int (3 \cos (23 x) \sin (20 x)+20 \sin (43 x)) \, dx\) [292]

Optimal. Leaf size=9 \[ \sin (20 x) \sin (23 x) \]

[Out]

sin(20*x)*sin(23*x)

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Rubi [A]
time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.89, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {4369, 2718} \begin {gather*} \frac {1}{2} \cos (3 x)-\frac {1}{2} \cos (43 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[3*Cos[23*x]*Sin[20*x] + 20*Sin[43*x],x]

[Out]

Cos[3*x]/2 - Cos[43*x]/2

Rule 2718

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

Rule 4369

Int[cos[(c_.) + (d_.)*(x_)]*sin[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[-Cos[a - c + (b - d)*x]/(2*(b - d)), x]
- Simp[Cos[a + c + (b + d)*x]/(2*(b + d)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=3 \int \cos (23 x) \sin (20 x) \, dx+20 \int \sin (43 x) \, dx\\ &=\frac {1}{2} \cos (3 x)-\frac {1}{2} \cos (43 x)\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.02, size = 17, normalized size = 1.89 \begin {gather*} \frac {1}{2} \cos (3 x)-\frac {1}{2} \cos (43 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[3*Cos[23*x]*Sin[20*x] + 20*Sin[43*x],x]

[Out]

Cos[3*x]/2 - Cos[43*x]/2

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Maple [A]
time = 1.56, size = 14, normalized size = 1.56

method	result	size
default	\(\frac {\cos \left (3 x \right )}{2}-\frac {\cos \left (43 x \right )}{2}\)	\(14\)
risch	\(\frac {\cos \left (3 x \right )}{2}-\frac {\cos \left (43 x \right )}{2}\)	\(14\)
parts	\(\frac {\cos \left (3 x \right )}{2}-\frac {\cos \left (43 x \right )}{2}\)	\(14\)
parallelrisch	\(\frac {-\frac {40}{43}+\frac {40 \left (-\left (\tan ^{2}\left (10 x \right )\right )-\left (\tan ^{2}\left (\frac {43 x}{2}\right )\right )-2\right ) \left (\tan ^{2}\left (\frac {23 x}{2}\right )\right )}{43}+\frac {92 \tan \left (10 x \right ) \left (1+\tan ^{2}\left (\frac {43 x}{2}\right )\right ) \tan \left (\frac {23 x}{2}\right )}{43}-\frac {40 \left (\tan ^{2}\left (\frac {43 x}{2}\right )\right ) \left (\tan ^{2}\left (10 x \right )\right )}{43}-\frac {80 \left (\tan ^{2}\left (10 x \right )\right )}{43}}{\left (1+\tan ^{2}\left (\frac {23 x}{2}\right )\right ) \left (1+\tan ^{2}\left (10 x \right )\right ) \left (1+\tan ^{2}\left (\frac {43 x}{2}\right )\right )}\)	\(101\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(3*sin(20*x)*cos(23*x)+20*sin(43*x),x,method=_RETURNVERBOSE)

[Out]

1/2*cos(3*x)-1/2*cos(43*x)

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Maxima [A]
time = 0.33, size = 13, normalized size = 1.44 \begin {gather*} -\frac {1}{2} \, \cos \left (43 \, x\right ) + \frac {1}{2} \, \cos \left (3 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(3*sin(20*x)*cos(23*x)+20*sin(43*x),x, algorithm="maxima")

[Out]

-1/2*cos(43*x) + 1/2*cos(3*x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (9) = 18\).
time = 1.84, size = 131, normalized size = 14.56 \begin {gather*} -2199023255552 \, \cos \left (x\right )^{43} + 23639499997184 \, \cos \left (x\right )^{41} - 118197499985920 \, \cos \left (x\right )^{39} + 364934781206528 \, \cos \left (x\right )^{37} - 778995398344704 \, \cos \left (x\right )^{35} + 1219742794776576 \, \cos \left (x\right )^{33} - 1450504945139712 \, \cos \left (x\right )^{31} + 1338263491051520 \, \cos \left (x\right )^{29} - 970241031012352 \, \cos \left (x\right )^{27} + 556461767786496 \, \cos \left (x\right )^{25} - 252937167175680 \, \cos \left (x\right )^{23} + 90899294453760 \, \cos \left (x\right )^{21} - 25657058918400 \, \cos \left (x\right )^{19} + 5624816762880 \, \cos \left (x\right )^{17} - 942087536640 \, \cos \left (x\right )^{15} + 117760942080 \, \cos \left (x\right )^{13} - 10631196160 \, \cos \left (x\right )^{11} + 661443200 \, \cos \left (x\right )^{9} - 26457728 \, \cos \left (x\right )^{7} + 609224 \, \cos \left (x\right )^{5} - 6620 \, \cos \left (x\right )^{3} + 20 \, \cos \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(3*sin(20*x)*cos(23*x)+20*sin(43*x),x, algorithm="fricas")

[Out]

-2199023255552*cos(x)^43 + 23639499997184*cos(x)^41 - 118197499985920*cos(x)^39 + 364934781206528*cos(x)^37 -
778995398344704*cos(x)^35 + 1219742794776576*cos(x)^33 - 1450504945139712*cos(x)^31 + 1338263491051520*cos(x)^
29 - 970241031012352*cos(x)^27 + 556461767786496*cos(x)^25 - 252937167175680*cos(x)^23 + 90899294453760*cos(x)
^21 - 25657058918400*cos(x)^19 + 5624816762880*cos(x)^17 - 942087536640*cos(x)^15 + 117760942080*cos(x)^13 - 1
0631196160*cos(x)^11 + 661443200*cos(x)^9 - 26457728*cos(x)^7 + 609224*cos(x)^5 - 6620*cos(x)^3 + 20*cos(x)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (8) = 16\).
time = 0.13, size = 34, normalized size = 3.78 \begin {gather*} \frac {23 \sin {\left (20 x \right )} \sin {\left (23 x \right )}}{43} + \frac {20 \cos {\left (20 x \right )} \cos {\left (23 x \right )}}{43} - \frac {20 \cos {\left (43 x \right )}}{43} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(3*sin(20*x)*cos(23*x)+20*sin(43*x),x)

[Out]

23*sin(20*x)*sin(23*x)/43 + 20*cos(20*x)*cos(23*x)/43 - 20*cos(43*x)/43

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Giac [A]
time = 0.47, size = 13, normalized size = 1.44 \begin {gather*} -\frac {1}{2} \, \cos \left (43 \, x\right ) + \frac {1}{2} \, \cos \left (3 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(3*sin(20*x)*cos(23*x)+20*sin(43*x),x, algorithm="giac")

[Out]

-1/2*cos(43*x) + 1/2*cos(3*x)

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Mupad [B]
time = 0.20, size = 13, normalized size = 1.44 \begin {gather*} \frac {\cos \left (3\,x\right )}{2}-\frac {\cos \left (43\,x\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(20*sin(43*x) + 3*cos(23*x)*sin(20*x),x)

[Out]

cos(3*x)/2 - cos(43*x)/2

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Chatgpt [F] Failed to verify
time = 1.00, size = 25, normalized size = 2.78 \begin {gather*} \frac {3 \cos \left (20 x \right )}{23}+\frac {3 \cos \left (43 x \right )}{43}+\frac {10 \sin \left (20 x \right )}{23}-\frac {20 \sin \left (43 x \right )}{43} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(3*sin(20*x)*cos(23*x)+20*sin(43*x),x)

[Out]

3/23*cos(20*x)+3/43*cos(43*x)+10/23*sin(20*x)-20/43*sin(43*x)

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