3.3 \(\int \sin ^4(e+f x) (5-6 \sin ^2(e+f x)) \, dx\)

Optimal. Leaf size=18 \[ \frac{\sin ^5(e+f x) \cos (e+f x)}{f} \]

[Out]

(Cos[e + f*x]*Sin[e + f*x]^5)/f

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Rubi [A]  time = 0.0217338, 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} \[ \frac{\sin ^5(e+f x) \cos (e+f x)}{f} \]

Antiderivative was successfully verified.

[In]

Int[Sin[e + f*x]^4*(5 - 6*Sin[e + f*x]^2),x]

[Out]

(Cos[e + f*x]*Sin[e + f*x]^5)/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 ^4(e+f x) \left (5-6 \sin ^2(e+f x)\right ) \, dx &=\frac{\cos (e+f x) \sin ^5(e+f x)}{f}\\ \end{align*}

Mathematica [B]  time = 0.11032, size = 39, normalized size = 2.17 \[ \frac{5 \sin (2 (e+f x))-4 \sin (4 (e+f x))+\sin (6 (e+f x))+24 e}{32 f} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[e + f*x]^4*(5 - 6*Sin[e + f*x]^2),x]

[Out]

(24*e + 5*Sin[2*(e + f*x)] - 4*Sin[4*(e + f*x)] + Sin[6*(e + f*x)])/(32*f)

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Maple [B]  time = 0.03, size = 65, normalized size = 3.6 \begin{align*}{\frac{1}{f} \left ( \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{5}+{\frac{5\, \left ( \sin \left ( fx+e \right ) \right ) ^{3}}{4}}+{\frac{15\,\sin \left ( fx+e \right ) }{8}} \right ) \cos \left ( fx+e \right ) -{\frac{5\,\cos \left ( fx+e \right ) }{4} \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{3}+{\frac{3\,\sin \left ( fx+e \right ) }{2}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(f*x+e)^4*(5-6*sin(f*x+e)^2),x)

[Out]

1/f*((sin(f*x+e)^5+5/4*sin(f*x+e)^3+15/8*sin(f*x+e))*cos(f*x+e)-5/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e))

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Maxima [B]  time = 0.955289, size = 59, normalized size = 3.28 \begin{align*} \frac{\tan \left (f x + e\right )^{5}}{{\left (\tan \left (f x + e\right )^{6} + 3 \, \tan \left (f x + e\right )^{4} + 3 \, \tan \left (f x + e\right )^{2} + 1\right )} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^4*(5-6*sin(f*x+e)^2),x, algorithm="maxima")

[Out]

tan(f*x + e)^5/((tan(f*x + e)^6 + 3*tan(f*x + e)^4 + 3*tan(f*x + e)^2 + 1)*f)

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Fricas [A]  time = 1.6056, size = 90, normalized size = 5. \begin{align*} \frac{{\left (\cos \left (f x + e\right )^{5} - 2 \, \cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^4*(5-6*sin(f*x+e)^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 8.13626, size = 236, normalized size = 13.11 \begin{align*} \begin{cases} - \frac{15 x \sin ^{6}{\left (e + f x \right )}}{8} - \frac{45 x \sin ^{4}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{8} + \frac{15 x \sin ^{4}{\left (e + f x \right )}}{8} - \frac{45 x \sin ^{2}{\left (e + f x \right )} \cos ^{4}{\left (e + f x \right )}}{8} + \frac{15 x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} - \frac{15 x \cos ^{6}{\left (e + f x \right )}}{8} + \frac{15 x \cos ^{4}{\left (e + f x \right )}}{8} + \frac{33 \sin ^{5}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{8 f} + \frac{5 \sin ^{3}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{f} - \frac{25 \sin ^{3}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{8 f} + \frac{15 \sin{\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{8 f} - \frac{15 \sin{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} & \text{for}\: f \neq 0 \\x \left (5 - 6 \sin ^{2}{\left (e \right )}\right ) \sin ^{4}{\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)**4*(5-6*sin(f*x+e)**2),x)

[Out]

Piecewise((-15*x*sin(e + f*x)**6/8 - 45*x*sin(e + f*x)**4*cos(e + f*x)**2/8 + 15*x*sin(e + f*x)**4/8 - 45*x*si
n(e + f*x)**2*cos(e + f*x)**4/8 + 15*x*sin(e + f*x)**2*cos(e + f*x)**2/4 - 15*x*cos(e + f*x)**6/8 + 15*x*cos(e
+ f*x)**4/8 + 33*sin(e + f*x)**5*cos(e + f*x)/(8*f) + 5*sin(e + f*x)**3*cos(e + f*x)**3/f - 25*sin(e + f*x)**
3*cos(e + f*x)/(8*f) + 15*sin(e + f*x)*cos(e + f*x)**5/(8*f) - 15*sin(e + f*x)*cos(e + f*x)**3/(8*f), Ne(f, 0)
), (x*(5 - 6*sin(e)**2)*sin(e)**4, True))

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Giac [B]  time = 1.15432, size = 62, normalized size = 3.44 \begin{align*} \frac{\sin \left (6 \, f x + 6 \, e\right )}{32 \, f} - \frac{\sin \left (4 \, f x + 4 \, e\right )}{8 \, f} + \frac{5 \, \sin \left (2 \, f x + 2 \, e\right )}{32 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^4*(5-6*sin(f*x+e)^2),x, algorithm="giac")

[Out]

1/32*sin(6*f*x + 6*e)/f - 1/8*sin(4*f*x + 4*e)/f + 5/32*sin(2*f*x + 2*e)/f