3.4 \(\int \sin ^3(e+f x) (4-5 \sin ^2(e+f x)) \, dx\)

Optimal. Leaf size=18 \[ \frac{\sin ^4(e+f x) \cos (e+f x)}{f} \]

[Out]

(Cos[e + f*x]*Sin[e + f*x]^4)/f

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Rubi [A]  time = 0.0212001, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {3011} \[ \frac{\sin ^4(e+f x) \cos (e+f x)}{f} \]

Antiderivative was successfully verified.

[In]

Int[Sin[e + f*x]^3*(4 - 5*Sin[e + f*x]^2),x]

[Out]

(Cos[e + f*x]*Sin[e + f*x]^4)/f

Rule 3011

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*Cos[e
+ f*x]*(b*Sin[e + f*x])^(m + 1))/(b*f*(m + 1)), x] /; FreeQ[{b, e, f, A, C, m}, x] && EqQ[A*(m + 2) + C*(m +
1), 0]

Rubi steps

\begin{align*} \int \sin ^3(e+f x) \left (4-5 \sin ^2(e+f x)\right ) \, dx &=\frac{\cos (e+f x) \sin ^4(e+f x)}{f}\\ \end{align*}

Mathematica [B]  time = 0.0266386, size = 44, normalized size = 2.44 \[ \frac{\cos (e+f x)}{8 f}-\frac{3 \cos (3 (e+f x))}{16 f}+\frac{\cos (5 (e+f x))}{16 f} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[e + f*x]^3*(4 - 5*Sin[e + f*x]^2),x]

[Out]

Cos[e + f*x]/(8*f) - (3*Cos[3*(e + f*x)])/(16*f) + Cos[5*(e + f*x)]/(16*f)

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Maple [B]  time = 0.04, size = 51, normalized size = 2.8 \begin{align*}{\frac{1}{f} \left ( \left ({\frac{8}{3}}+ \left ( \sin \left ( fx+e \right ) \right ) ^{4}+{\frac{4\, \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{3}} \right ) \cos \left ( fx+e \right ) -{\frac{ \left ( 8+4\, \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) }{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(f*x+e)^3*(4-5*sin(f*x+e)^2),x)

[Out]

1/f*((8/3+sin(f*x+e)^4+4/3*sin(f*x+e)^2)*cos(f*x+e)-4/3*(2+sin(f*x+e)^2)*cos(f*x+e))

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Maxima [A]  time = 0.957005, size = 39, normalized size = 2.17 \begin{align*} \frac{\cos \left (f x + e\right )^{5} - 2 \, \cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^3*(4-5*sin(f*x+e)^2),x, algorithm="maxima")

[Out]

(cos(f*x + e)^5 - 2*cos(f*x + e)^3 + cos(f*x + e))/f

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Fricas [A]  time = 1.75468, size = 73, normalized size = 4.06 \begin{align*} \frac{\cos \left (f x + e\right )^{5} - 2 \, \cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^3*(4-5*sin(f*x+e)^2),x, algorithm="fricas")

[Out]

(cos(f*x + e)^5 - 2*cos(f*x + e)^3 + cos(f*x + e))/f

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Sympy [A]  time = 4.06879, size = 100, normalized size = 5.56 \begin{align*} \begin{cases} \frac{5 \sin ^{4}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{f} + \frac{20 \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac{4 \sin ^{2}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{f} + \frac{8 \cos ^{5}{\left (e + f x \right )}}{3 f} - \frac{8 \cos ^{3}{\left (e + f x \right )}}{3 f} & \text{for}\: f \neq 0 \\x \left (4 - 5 \sin ^{2}{\left (e \right )}\right ) \sin ^{3}{\left (e \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)**3*(4-5*sin(f*x+e)**2),x)

[Out]

Piecewise((5*sin(e + f*x)**4*cos(e + f*x)/f + 20*sin(e + f*x)**2*cos(e + f*x)**3/(3*f) - 4*sin(e + f*x)**2*cos
(e + f*x)/f + 8*cos(e + f*x)**5/(3*f) - 8*cos(e + f*x)**3/(3*f), Ne(f, 0)), (x*(4 - 5*sin(e)**2)*sin(e)**3, Tr
ue))

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Giac [B]  time = 1.14129, size = 53, normalized size = 2.94 \begin{align*} \frac{\cos \left (f x + e\right )^{5}}{f} - \frac{2 \, \cos \left (f x + e\right )^{3}}{f} + \frac{\cos \left (f x + e\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^3*(4-5*sin(f*x+e)^2),x, algorithm="giac")

[Out]

cos(f*x + e)^5/f - 2*cos(f*x + e)^3/f + cos(f*x + e)/f