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(f*x+e)*sin(f*x+e)^4/f

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Rubi [A]  time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, 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*}

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Mathematica [B]  time = 0.03, 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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fricas [A]  time = 0.42, size = 29, normalized size = 1.61 \[ \frac {\cos \left (f x + e\right )^{5} - 2 \, \cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )}{f} \]

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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giac [B]  time = 0.16, size = 39, normalized size = 2.17 \[ \frac {\cos \left (f x + e\right )^{5}}{f} - \frac {2 \, \cos \left (f x + e\right )^{3}}{f} + \frac {\cos \left (f x + e\right )}{f} \]

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

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maple [B]  time = 0.42, size = 51, normalized size = 2.83 \[ \frac {\left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )-\frac {4 \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}}{f} \]

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 = 0.62, size = 29, normalized size = 1.61 \[ \frac {\cos \left (f x + e\right )^{5} - 2 \, \cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )}{f} \]

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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mupad [B]  time = 13.19, size = 22, normalized size = 1.22 \[ \frac {\cos \left (e+f\,x\right )\,{\left ({\cos \left (e+f\,x\right )}^2-1\right )}^2}{f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(cos(e + f*x)*(cos(e + f*x)^2 - 1)^2)/f

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sympy [A]  time = 2.30, size = 100, normalized size = 5.56 \[ \begin {cases} \frac {5 \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {20 \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {4 \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {8 \cos ^{5}{\left (e + f x \right )}}{3 f} - \frac {8 \cos ^{3}{\left (e + f x \right )}}{3 f} & \text {for}\: f \neq 0 \\x \left (4 - 5 \sin ^{2}{\relax (e )}\right ) \sin ^{3}{\relax (e )} & \text {otherwise} \end {cases} \]

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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