3.2 \(\int \sin ^5(e+f x) (6-7 \sin ^2(e+f x)) \, dx\)

Optimal. Leaf size=18 \[ \frac{\sin ^6(e+f x) \cos (e+f x)}{f} \]

[Out]

(Cos[e + f*x]*Sin[e + f*x]^6)/f

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Rubi [A]  time = 0.021364, 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} \[ \frac{\sin ^6(e+f x) \cos (e+f x)}{f} \]

Antiderivative was successfully verified.

[In]

Int[Sin[e + f*x]^5*(6 - 7*Sin[e + f*x]^2),x]

[Out]

(Cos[e + f*x]*Sin[e + f*x]^6)/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 ^5(e+f x) \left (6-7 \sin ^2(e+f x)\right ) \, dx &=\frac{\cos (e+f x) \sin ^6(e+f x)}{f}\\ \end{align*}

Mathematica [B]  time = 0.0329857, size = 59, normalized size = 3.28 \[ \frac{5 \cos (e+f x)}{64 f}-\frac{9 \cos (3 (e+f x))}{64 f}+\frac{5 \cos (5 (e+f x))}{64 f}-\frac{\cos (7 (e+f x))}{64 f} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[e + f*x]^5*(6 - 7*Sin[e + f*x]^2),x]

[Out]

(5*Cos[e + f*x])/(64*f) - (9*Cos[3*(e + f*x)])/(64*f) + (5*Cos[5*(e + f*x)])/(64*f) - Cos[7*(e + f*x)]/(64*f)

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Maple [B]  time = 0.027, size = 71, normalized size = 3.9 \begin{align*}{\frac{1}{f} \left ( \left ({\frac{16}{5}}+ \left ( \sin \left ( fx+e \right ) \right ) ^{6}+{\frac{6\, \left ( \sin \left ( fx+e \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{5}} \right ) \cos \left ( fx+e \right ) -{\frac{6\,\cos \left ( fx+e \right ) }{5} \left ({\frac{8}{3}}+ \left ( \sin \left ( fx+e \right ) \right ) ^{4}+{\frac{4\, \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{3}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(f*x+e)^5*(6-7*sin(f*x+e)^2),x)

[Out]

1/f*((16/5+sin(f*x+e)^6+6/5*sin(f*x+e)^4+8/5*sin(f*x+e)^2)*cos(f*x+e)-6/5*(8/3+sin(f*x+e)^4+4/3*sin(f*x+e)^2)*
cos(f*x+e))

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Maxima [B]  time = 0.942769, size = 57, normalized size = 3.17 \begin{align*} -\frac{\cos \left (f x + e\right )^{7} - 3 \, \cos \left (f x + e\right )^{5} + 3 \, \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)^5*(6-7*sin(f*x+e)^2),x, algorithm="maxima")

[Out]

-(cos(f*x + e)^7 - 3*cos(f*x + e)^5 + 3*cos(f*x + e)^3 - cos(f*x + e))/f

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Fricas [B]  time = 1.575, size = 100, normalized size = 5.56 \begin{align*} -\frac{\cos \left (f x + e\right )^{7} - 3 \, \cos \left (f x + e\right )^{5} + 3 \, \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)^5*(6-7*sin(f*x+e)^2),x, algorithm="fricas")

[Out]

-(cos(f*x + e)^7 - 3*cos(f*x + e)^5 + 3*cos(f*x + e)^3 - cos(f*x + e))/f

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Sympy [A]  time = 12.4766, size = 141, normalized size = 7.83 \begin{align*} \begin{cases} \frac{7 \sin ^{6}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{f} + \frac{14 \sin ^{4}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{f} - \frac{6 \sin ^{4}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{f} + \frac{56 \sin ^{2}{\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{5 f} - \frac{8 \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{f} + \frac{16 \cos ^{7}{\left (e + f x \right )}}{5 f} - \frac{16 \cos ^{5}{\left (e + f x \right )}}{5 f} & \text{for}\: f \neq 0 \\x \left (6 - 7 \sin ^{2}{\left (e \right )}\right ) \sin ^{5}{\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)**5*(6-7*sin(f*x+e)**2),x)

[Out]

Piecewise((7*sin(e + f*x)**6*cos(e + f*x)/f + 14*sin(e + f*x)**4*cos(e + f*x)**3/f - 6*sin(e + f*x)**4*cos(e +
f*x)/f + 56*sin(e + f*x)**2*cos(e + f*x)**5/(5*f) - 8*sin(e + f*x)**2*cos(e + f*x)**3/f + 16*cos(e + f*x)**7/
(5*f) - 16*cos(e + f*x)**5/(5*f), Ne(f, 0)), (x*(6 - 7*sin(e)**2)*sin(e)**5, True))

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Giac [B]  time = 1.12451, size = 78, normalized size = 4.33 \begin{align*} -\frac{\cos \left (7 \, f x + 7 \, e\right )}{64 \, f} + \frac{5 \, \cos \left (5 \, f x + 5 \, e\right )}{64 \, f} - \frac{9 \, \cos \left (3 \, f x + 3 \, e\right )}{64 \, f} + \frac{5 \, \cos \left (f x + e\right )}{64 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^5*(6-7*sin(f*x+e)^2),x, algorithm="giac")

[Out]

-1/64*cos(7*f*x + 7*e)/f + 5/64*cos(5*f*x + 5*e)/f - 9/64*cos(3*f*x + 3*e)/f + 5/64*cos(f*x + e)/f