\(\int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x)) \, dx\) [437]

Optimal result

Integrand size = 23, antiderivative size = 83 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x)) \, dx=\frac {9}{2} (3 c+2 d) x-\frac {6 (3 c+2 d) \cos (e+f x)}{f}-\frac {3 (3 c+2 d) \cos (e+f x) \sin (e+f x)}{2 f}-\frac {d \cos (e+f x) (3+3 \sin (e+f x))^2}{3 f} \]

[Out]

1/2*a^2*(3*c+2*d)*x-2/3*a^2*(3*c+2*d)*cos(f*x+e)/f-1/6*a^2*(3*c+2*d)*cos(f*x+e)*sin(f*x+e)/f-1/3*d*cos(f*x+e)*
(a+a*sin(f*x+e))^2/f

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.13, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2830, 2723} \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x)) \, dx=-\frac {2 a^2 (3 c+2 d) \cos (e+f x)}{3 f}-\frac {a^2 (3 c+2 d) \sin (e+f x) \cos (e+f x)}{6 f}+\frac {1}{2} a^2 x (3 c+2 d)-\frac {d \cos (e+f x) (a \sin (e+f x)+a)^2}{3 f} \]

[In]

Int[(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x]),x]

[Out]

(a^2*(3*c + 2*d)*x)/2 - (2*a^2*(3*c + 2*d)*Cos[e + f*x])/(3*f) - (a^2*(3*c + 2*d)*Cos[e + f*x]*Sin[e + f*x])/(
6*f) - (d*Cos[e + f*x]*(a + a*Sin[e + f*x])^2)/(3*f)

Rule 2723

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^2, x_Symbol] :> Simp[(2*a^2 + b^2)*(x/2), x] + (-Simp[2*a*b*(Cos[c
+ d*x]/d), x] - Simp[b^2*Cos[c + d*x]*(Sin[c + d*x]/(2*d)), x]) /; FreeQ[{a, b, c, d}, x]

Rule 2830

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d
)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/(f*(m + 1))), x] + Dist[(a*d*m + b*c*(m + 1))/(b*(m + 1)), Int[(a + b*S
in[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&  !LtQ[m
, -2^(-1)]

Rubi steps \begin{align*} \text {integral}& = -\frac {d \cos (e+f x) (a+a \sin (e+f x))^2}{3 f}+\frac {1}{3} (3 c+2 d) \int (a+a \sin (e+f x))^2 \, dx \\ & = \frac {1}{2} a^2 (3 c+2 d) x-\frac {2 a^2 (3 c+2 d) \cos (e+f x)}{3 f}-\frac {a^2 (3 c+2 d) \cos (e+f x) \sin (e+f x)}{6 f}-\frac {d \cos (e+f x) (a+a \sin (e+f x))^2}{3 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.07 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x)) \, dx=\frac {3 \cos (e+f x) \left (-2 (6 c+5 d)-\frac {6 (3 c+2 d) \arcsin \left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )}{\sqrt {\cos ^2(e+f x)}}-3 (c+2 d) \sin (e+f x)-2 d \sin ^2(e+f x)\right )}{2 f} \]

[In]

Integrate[(3 + 3*Sin[e + f*x])^2*(c + d*Sin[e + f*x]),x]

[Out]

(3*Cos[e + f*x]*(-2*(6*c + 5*d) - (6*(3*c + 2*d)*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt[2]])/Sqrt[Cos[e + f*x]^2]
- 3*(c + 2*d)*Sin[e + f*x] - 2*d*Sin[e + f*x]^2))/(2*f)

Maple [A] (verified)

Time = 1.09 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.84

method	result	size
parallelrisch	\(\frac {a^{2} \left (3 \left (-c -2 d \right ) \sin \left (2 f x +2 e \right )+d \cos \left (3 f x +3 e \right )+3 \left (-8 c -7 d \right ) \cos \left (f x +e \right )+18 f x c +12 d x f -24 c -20 d \right )}{12 f}\)	\(70\)
parts	\(a^{2} c x +\frac {\left (a^{2} c +2 a^{2} d \right ) \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {\left (2 a^{2} c +a^{2} d \right ) \cos \left (f x +e \right )}{f}-\frac {a^{2} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}\)	\(94\)
risch	\(\frac {3 a^{2} c x}{2}+a^{2} x d -\frac {2 a^{2} \cos \left (f x +e \right ) c}{f}-\frac {7 a^{2} \cos \left (f x +e \right ) d}{4 f}+\frac {a^{2} d \cos \left (3 f x +3 e \right )}{12 f}-\frac {\sin \left (2 f x +2 e \right ) a^{2} c}{4 f}-\frac {\sin \left (2 f x +2 e \right ) a^{2} d}{2 f}\)	\(99\)
derivativedivides	\(\frac {a^{2} c \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {a^{2} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-2 a^{2} c \cos \left (f x +e \right )+2 a^{2} d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+a^{2} c \left (f x +e \right )-a^{2} d \cos \left (f x +e \right )}{f}\)	\(117\)
default	\(\frac {a^{2} c \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {a^{2} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-2 a^{2} c \cos \left (f x +e \right )+2 a^{2} d \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+a^{2} c \left (f x +e \right )-a^{2} d \cos \left (f x +e \right )}{f}\)	\(117\)
norman	\(\frac {\left (\frac {3}{2} a^{2} c +a^{2} d \right ) x +\left (\frac {3}{2} a^{2} c +a^{2} d \right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {9}{2} a^{2} c +3 a^{2} d \right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {9}{2} a^{2} c +3 a^{2} d \right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {a^{2} \left (c +2 d \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {12 a^{2} c +10 a^{2} d}{3 f}-\frac {\left (4 a^{2} c +2 a^{2} d \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {\left (8 a^{2} c +8 a^{2} d \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {a^{2} \left (c +2 d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3}}\)	\(230\)

[In]

int((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e)),x,method=_RETURNVERBOSE)

[Out]

1/12*a^2*(3*(-c-2*d)*sin(2*f*x+2*e)+d*cos(3*f*x+3*e)+3*(-8*c-7*d)*cos(f*x+e)+18*f*x*c+12*d*x*f-24*c-20*d)/f

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.99 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x)) \, dx=\frac {2 \, a^{2} d \cos \left (f x + e\right )^{3} + 3 \, {\left (3 \, a^{2} c + 2 \, a^{2} d\right )} f x - 3 \, {\left (a^{2} c + 2 \, a^{2} d\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 12 \, {\left (a^{2} c + a^{2} d\right )} \cos \left (f x + e\right )}{6 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e)),x, algorithm="fricas")

[Out]

1/6*(2*a^2*d*cos(f*x + e)^3 + 3*(3*a^2*c + 2*a^2*d)*f*x - 3*(a^2*c + 2*a^2*d)*cos(f*x + e)*sin(f*x + e) - 12*(
a^2*c + a^2*d)*cos(f*x + e))/f

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (85) = 170\).

Time = 0.15 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.40 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x)) \, dx=\begin {cases} \frac {a^{2} c x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {a^{2} c x \cos ^{2}{\left (e + f x \right )}}{2} + a^{2} c x - \frac {a^{2} c \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {2 a^{2} c \cos {\left (e + f x \right )}}{f} + a^{2} d x \sin ^{2}{\left (e + f x \right )} + a^{2} d x \cos ^{2}{\left (e + f x \right )} - \frac {a^{2} d \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {a^{2} d \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {2 a^{2} d \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {a^{2} d \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (c + d \sin {\left (e \right )}\right ) \left (a \sin {\left (e \right )} + a\right )^{2} & \text {otherwise} \end {cases} \]

[In]

integrate((a+a*sin(f*x+e))**2*(c+d*sin(f*x+e)),x)

[Out]

Piecewise((a**2*c*x*sin(e + f*x)**2/2 + a**2*c*x*cos(e + f*x)**2/2 + a**2*c*x - a**2*c*sin(e + f*x)*cos(e + f*
x)/(2*f) - 2*a**2*c*cos(e + f*x)/f + a**2*d*x*sin(e + f*x)**2 + a**2*d*x*cos(e + f*x)**2 - a**2*d*sin(e + f*x)
**2*cos(e + f*x)/f - a**2*d*sin(e + f*x)*cos(e + f*x)/f - 2*a**2*d*cos(e + f*x)**3/(3*f) - a**2*d*cos(e + f*x)
/f, Ne(f, 0)), (x*(c + d*sin(e))*(a*sin(e) + a)**2, True))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.37 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x)) \, dx=\frac {3 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c + 12 \, {\left (f x + e\right )} a^{2} c + 4 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} d + 6 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} d - 24 \, a^{2} c \cos \left (f x + e\right ) - 12 \, a^{2} d \cos \left (f x + e\right )}{12 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e)),x, algorithm="maxima")

[Out]

1/12*(3*(2*f*x + 2*e - sin(2*f*x + 2*e))*a^2*c + 12*(f*x + e)*a^2*c + 4*(cos(f*x + e)^3 - 3*cos(f*x + e))*a^2*
d + 6*(2*f*x + 2*e - sin(2*f*x + 2*e))*a^2*d - 24*a^2*c*cos(f*x + e) - 12*a^2*d*cos(f*x + e))/f

Giac [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.27 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x)) \, dx=a^{2} c x + \frac {a^{2} d \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} - \frac {a^{2} d \cos \left (f x + e\right )}{f} + \frac {1}{2} \, {\left (a^{2} c + 2 \, a^{2} d\right )} x - \frac {{\left (8 \, a^{2} c + 3 \, a^{2} d\right )} \cos \left (f x + e\right )}{4 \, f} - \frac {{\left (a^{2} c + 2 \, a^{2} d\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^2*(c+d*sin(f*x+e)),x, algorithm="giac")

[Out]

a^2*c*x + 1/12*a^2*d*cos(3*f*x + 3*e)/f - a^2*d*cos(f*x + e)/f + 1/2*(a^2*c + 2*a^2*d)*x - 1/4*(8*a^2*c + 3*a^
2*d)*cos(f*x + e)/f - 1/4*(a^2*c + 2*a^2*d)*sin(2*f*x + 2*e)/f

Mupad [B] (verification not implemented)

Time = 6.90 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.10 \[ \int (3+3 \sin (e+f x))^2 (c+d \sin (e+f x)) \, dx=-\frac {\frac {3\,a^2\,c\,\sin \left (2\,e+2\,f\,x\right )}{2}-\frac {a^2\,d\,\cos \left (3\,e+3\,f\,x\right )}{2}+3\,a^2\,d\,\sin \left (2\,e+2\,f\,x\right )+12\,a^2\,c\,\cos \left (e+f\,x\right )+\frac {21\,a^2\,d\,\cos \left (e+f\,x\right )}{2}-9\,a^2\,c\,f\,x-6\,a^2\,d\,f\,x}{6\,f} \]

[In]

int((a + a*sin(e + f*x))^2*(c + d*sin(e + f*x)),x)

[Out]

-((3*a^2*c*sin(2*e + 2*f*x))/2 - (a^2*d*cos(3*e + 3*f*x))/2 + 3*a^2*d*sin(2*e + 2*f*x) + 12*a^2*c*cos(e + f*x)
+ (21*a^2*d*cos(e + f*x))/2 - 9*a^2*c*f*x - 6*a^2*d*f*x)/(6*f)