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(f*x+e)*sin(f*x+e)^6/f

________________________________________________________________________________________

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} \[ \frac {\sin ^6(e+f x) \cos (e+f x)}{f} \]

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 = 0.03, 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)

________________________________________________________________________________________

fricas [B]  time = 0.44, size = 42, normalized size = 2.33 \[ -\frac {\cos \left (f x + e\right )^{7} - 3 \, \cos \left (f x + e\right )^{5} + 3 \, \cos \left (f x + e\right )^{3} - \cos \left (f x + e\right )}{f} \]

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

________________________________________________________________________________________

giac [B]  time = 0.17, size = 58, normalized size = 3.22 \[ -\frac {\cos \left (7 \, f x + 7 \, e\right )}{64 \, f} + \frac {5 \, \cos \left (5 \, f x + 5 \, e\right )}{64 \, f} - \frac {9 \, \cos \left (3 \, f x + 3 \, e\right )}{64 \, f} + \frac {5 \, \cos \left (f x + e\right )}{64 \, f} \]

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

________________________________________________________________________________________

maple [B]  time = 0.49, size = 71, normalized size = 3.94 \[ \frac {\left (\frac {16}{5}+\sin ^{6}\left (f x +e \right )+\frac {6 \left (\sin ^{4}\left (f x +e \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (f x +e \right )\right )}{5}\right ) \cos \left (f x +e \right )-\frac {6 \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}}{f} \]

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))

________________________________________________________________________________________

maxima [B]  time = 0.62, size = 42, normalized size = 2.33 \[ -\frac {\cos \left (f x + e\right )^{7} - 3 \, \cos \left (f x + e\right )^{5} + 3 \, \cos \left (f x + e\right )^{3} - \cos \left (f x + e\right )}{f} \]

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

________________________________________________________________________________________

mupad [B]  time = 13.47, size = 23, normalized size = 1.28 \[ -\frac {\cos \left (e+f\,x\right )\,{\left ({\cos \left (e+f\,x\right )}^2-1\right )}^3}{f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 7.74, size = 141, normalized size = 7.83 \[ \begin {cases} \frac {7 \sin ^{6}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {14 \sin ^{4}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{f} - \frac {6 \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {56 \sin ^{2}{\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{5 f} - \frac {8 \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{f} + \frac {16 \cos ^{7}{\left (e + f x \right )}}{5 f} - \frac {16 \cos ^{5}{\left (e + f x \right )}}{5 f} & \text {for}\: f \neq 0 \\x \left (6 - 7 \sin ^{2}{\relax (e )}\right ) \sin ^{5}{\relax (e )} & \text {otherwise} \end {cases} \]

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))

________________________________________________________________________________________