3.114 \(\int \sin ^3(a+b x) \, dx\)

Optimal. Leaf size=27 \[ \frac{\cos ^3(a+b x)}{3 b}-\frac{\cos (a+b x)}{b} \]

[Out]

-(Cos[a + b*x]/b) + Cos[a + b*x]^3/(3*b)

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Rubi [A]  time = 0.0212904, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\cos ^3(a+b x)}{3 b}-\frac{\cos (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]  Int[Sin[a + b*x]^3,x]

[Out]

-(Cos[a + b*x]/b) + Cos[a + b*x]^3/(3*b)

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Rubi in Sympy [A]  time = 1.17945, size = 19, normalized size = 0.7 \[ \frac{\cos ^{3}{\left (a + b x \right )}}{3 b} - \frac{\cos{\left (a + b x \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(sin(b*x+a)**3,x)

[Out]

cos(a + b*x)**3/(3*b) - cos(a + b*x)/b

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Mathematica [A]  time = 0.00971692, size = 29, normalized size = 1.07 \[ \frac{\cos (3 (a+b x))}{12 b}-\frac{3 \cos (a+b x)}{4 b} \]

Antiderivative was successfully verified.

[In]  Integrate[Sin[a + b*x]^3,x]

[Out]

(-3*Cos[a + b*x])/(4*b) + Cos[3*(a + b*x)]/(12*b)

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Maple [A]  time = 0.002, size = 22, normalized size = 0.8 \[ -{\frac{ \left ( 2+ \left ( \sin \left ( bx+a \right ) \right ) ^{2} \right ) \cos \left ( bx+a \right ) }{3\,b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sin(b*x+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.41262, size = 30, normalized size = 1.11 \[ \frac{\cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(cos(b*x + a)^3 - 3*cos(b*x + a))/b

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Fricas [A]  time = 0.227319, size = 30, normalized size = 1.11 \[ \frac{\cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(cos(b*x + a)^3 - 3*cos(b*x + a))/b

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Sympy [A]  time = 0.611178, size = 37, normalized size = 1.37 \[ \begin{cases} - \frac{\sin ^{2}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{b} - \frac{2 \cos ^{3}{\left (a + b x \right )}}{3 b} & \text{for}\: b \neq 0 \\x \sin ^{3}{\left (a \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(b*x+a)**3,x)

[Out]

Piecewise((-sin(a + b*x)**2*cos(a + b*x)/b - 2*cos(a + b*x)**3/(3*b), Ne(b, 0)),
(x*sin(a)**3, True))

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GIAC/XCAS [A]  time = 0.204047, size = 34, normalized size = 1.26 \[ \frac{\cos \left (b x + a\right )^{3}}{3 \, b} - \frac{\cos \left (b x + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*cos(b*x + a)^3/b - cos(b*x + a)/b