3.215 \(\int a \cos (5+3 x) \sin ^2(5+3 x) \, dx\)

Optimal. Leaf size=13 \[ \frac{1}{9} a \sin ^3(3 x+5) \]

[Out]

(a*Sin[5 + 3*x]^3)/9

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Rubi [A]  time = 0.0274779, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{1}{9} a \sin ^3(3 x+5) \]

Antiderivative was successfully verified.

[In]  Int[a*Cos[5 + 3*x]*Sin[5 + 3*x]^2,x]

[Out]

(a*Sin[5 + 3*x]^3)/9

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Rubi in Sympy [A]  time = 1.41787, size = 10, normalized size = 0.77 \[ \frac{a \sin ^{3}{\left (3 x + 5 \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(a*cos(5+3*x)*sin(5+3*x)**2,x)

[Out]

a*sin(3*x + 5)**3/9

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Mathematica [A]  time = 0.00948142, size = 13, normalized size = 1. \[ \frac{1}{9} a \sin ^3(3 x+5) \]

Antiderivative was successfully verified.

[In]  Integrate[a*Cos[5 + 3*x]*Sin[5 + 3*x]^2,x]

[Out]

(a*Sin[5 + 3*x]^3)/9

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Maple [A]  time = 0.003, size = 12, normalized size = 0.9 \[{\frac{a \left ( \sin \left ( 5+3\,x \right ) \right ) ^{3}}{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(a*cos(5+3*x)*sin(5+3*x)^2,x)

[Out]

1/9*a*sin(5+3*x)^3

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Maxima [A]  time = 1.417, size = 15, normalized size = 1.15 \[ \frac{1}{9} \, a \sin \left (3 \, x + 5\right )^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(a*cos(3*x + 5)*sin(3*x + 5)^2,x, algorithm="maxima")

[Out]

1/9*a*sin(3*x + 5)^3

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Fricas [A]  time = 0.217906, size = 30, normalized size = 2.31 \[ -\frac{1}{9} \,{\left (a \cos \left (3 \, x + 5\right )^{2} - a\right )} \sin \left (3 \, x + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(a*cos(3*x + 5)*sin(3*x + 5)^2,x, algorithm="fricas")

[Out]

-1/9*(a*cos(3*x + 5)^2 - a)*sin(3*x + 5)

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Sympy [A]  time = 0.370279, size = 10, normalized size = 0.77 \[ \frac{a \sin ^{3}{\left (3 x + 5 \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(a*cos(5+3*x)*sin(5+3*x)**2,x)

[Out]

a*sin(3*x + 5)**3/9

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GIAC/XCAS [A]  time = 0.200689, size = 15, normalized size = 1.15 \[ \frac{1}{9} \, a \sin \left (3 \, x + 5\right )^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(a*cos(3*x + 5)*sin(3*x + 5)^2,x, algorithm="giac")

[Out]

1/9*a*sin(3*x + 5)^3