3.679 \(\int (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2 \, dx\)

Optimal. Leaf size=217 \[ -\frac{\left (8 a^2 b c d+a^3 \left (-d^2\right )+4 a b^2 \left (3 c^2+2 d^2\right )+8 b^3 c d\right ) \cos (e+f x)}{6 b f}+\frac{1}{8} x \left (4 a^2 \left (2 c^2+d^2\right )+16 a b c d+b^2 \left (4 c^2+3 d^2\right )\right )-\frac{\left (2 a d (8 b c-a d)+3 b^2 \left (4 c^2+3 d^2\right )\right ) \sin (e+f x) \cos (e+f x)}{24 f}-\frac{d (8 b c-a d) \cos (e+f x) (a+b \sin (e+f x))^2}{12 b f}-\frac{d^2 \cos (e+f x) (a+b \sin (e+f x))^3}{4 b f} \]

[Out]

((16*a*b*c*d + 4*a^2*(2*c^2 + d^2) + b^2*(4*c^2 + 3*d^2))*x)/8 - ((8*a^2*b*c*d + 8*b^3*c*d - a^3*d^2 + 4*a*b^2
*(3*c^2 + 2*d^2))*Cos[e + f*x])/(6*b*f) - ((2*a*d*(8*b*c - a*d) + 3*b^2*(4*c^2 + 3*d^2))*Cos[e + f*x]*Sin[e +
f*x])/(24*f) - (d*(8*b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^2)/(12*b*f) - (d^2*Cos[e + f*x]*(a + b*Sin[e
+ f*x])^3)/(4*b*f)

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Rubi [A]  time = 0.281144, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2791, 2753, 2734} \[ -\frac{\left (8 a^2 b c d+a^3 \left (-d^2\right )+4 a b^2 \left (3 c^2+2 d^2\right )+8 b^3 c d\right ) \cos (e+f x)}{6 b f}+\frac{1}{8} x \left (4 a^2 \left (2 c^2+d^2\right )+16 a b c d+b^2 \left (4 c^2+3 d^2\right )\right )-\frac{\left (2 a d (8 b c-a d)+3 b^2 \left (4 c^2+3 d^2\right )\right ) \sin (e+f x) \cos (e+f x)}{24 f}-\frac{d (8 b c-a d) \cos (e+f x) (a+b \sin (e+f x))^2}{12 b f}-\frac{d^2 \cos (e+f x) (a+b \sin (e+f x))^3}{4 b f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((16*a*b*c*d + 4*a^2*(2*c^2 + d^2) + b^2*(4*c^2 + 3*d^2))*x)/8 - ((8*a^2*b*c*d + 8*b^3*c*d - a^3*d^2 + 4*a*b^2
*(3*c^2 + 2*d^2))*Cos[e + f*x])/(6*b*f) - ((2*a*d*(8*b*c - a*d) + 3*b^2*(4*c^2 + 3*d^2))*Cos[e + f*x]*Sin[e +
f*x])/(24*f) - (d*(8*b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^2)/(12*b*f) - (d^2*Cos[e + f*x]*(a + b*Sin[e
+ f*x])^3)/(4*b*f)

Rule 2791

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

Rule 2753

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[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[
b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*
c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]

Rule 2734

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

Rubi steps

\begin{align*} \int (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2 \, dx &=-\frac{d^2 \cos (e+f x) (a+b \sin (e+f x))^3}{4 b f}+\frac{\int (a+b \sin (e+f x))^2 \left (b \left (4 c^2+3 d^2\right )+d (8 b c-a d) \sin (e+f x)\right ) \, dx}{4 b}\\ &=-\frac{d (8 b c-a d) \cos (e+f x) (a+b \sin (e+f x))^2}{12 b f}-\frac{d^2 \cos (e+f x) (a+b \sin (e+f x))^3}{4 b f}+\frac{\int (a+b \sin (e+f x)) \left (b \left (12 a c^2+16 b c d+7 a d^2\right )+\left (2 a d (8 b c-a d)+3 b^2 \left (4 c^2+3 d^2\right )\right ) \sin (e+f x)\right ) \, dx}{12 b}\\ &=\frac{1}{8} \left (16 a b c d+4 a^2 \left (2 c^2+d^2\right )+b^2 \left (4 c^2+3 d^2\right )\right ) x-\frac{\left (8 a^2 b c d+8 b^3 c d-a^3 d^2+4 a b^2 \left (3 c^2+2 d^2\right )\right ) \cos (e+f x)}{6 b f}-\frac{\left (2 a d (8 b c-a d)+3 b^2 \left (4 c^2+3 d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{24 f}-\frac{d (8 b c-a d) \cos (e+f x) (a+b \sin (e+f x))^2}{12 b f}-\frac{d^2 \cos (e+f x) (a+b \sin (e+f x))^3}{4 b f}\\ \end{align*}

Mathematica [A]  time = 0.762033, size = 160, normalized size = 0.74 \[ \frac{3 \left (4 (e+f x) \left (4 a^2 \left (2 c^2+d^2\right )+16 a b c d+b^2 \left (4 c^2+3 d^2\right )\right )-8 \left (a^2 d^2+4 a b c d+b^2 \left (c^2+d^2\right )\right ) \sin (2 (e+f x))+b^2 d^2 \sin (4 (e+f x))\right )-48 \left (4 a^2 c d+a b \left (4 c^2+3 d^2\right )+3 b^2 c d\right ) \cos (e+f x)+16 b d (a d+b c) \cos (3 (e+f x))}{96 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-48*(4*a^2*c*d + 3*b^2*c*d + a*b*(4*c^2 + 3*d^2))*Cos[e + f*x] + 16*b*d*(b*c + a*d)*Cos[3*(e + f*x)] + 3*(4*(
16*a*b*c*d + 4*a^2*(2*c^2 + d^2) + b^2*(4*c^2 + 3*d^2))*(e + f*x) - 8*(4*a*b*c*d + a^2*d^2 + b^2*(c^2 + d^2))*
Sin[2*(e + f*x)] + b^2*d^2*Sin[4*(e + f*x)]))/(96*f)

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Maple [A]  time = 0.03, size = 216, normalized size = 1. \begin{align*}{\frac{1}{f} \left ({a}^{2}{c}^{2} \left ( fx+e \right ) -2\,{a}^{2}cd\cos \left ( fx+e \right ) +{a}^{2}{d}^{2} \left ( -{\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }{2}}+{\frac{fx}{2}}+{\frac{e}{2}} \right ) -2\,ab{c}^{2}\cos \left ( fx+e \right ) +4\,abcd \left ( -1/2\,\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) +1/2\,fx+e/2 \right ) -{\frac{2\,ab{d}^{2} \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) }{3}}+{c}^{2}{b}^{2} \left ( -{\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }{2}}+{\frac{fx}{2}}+{\frac{e}{2}} \right ) -{\frac{2\,{b}^{2}cd \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) }{3}}+{b}^{2}{d}^{2} \left ( -{\frac{\cos \left ( fx+e \right ) }{4} \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{3}+{\frac{3\,\sin \left ( fx+e \right ) }{2}} \right ) }+{\frac{3\,fx}{8}}+{\frac{3\,e}{8}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.17893, size = 281, normalized size = 1.29 \begin{align*} \frac{96 \,{\left (f x + e\right )} a^{2} c^{2} + 24 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} b^{2} c^{2} + 96 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a b c d + 64 \,{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} b^{2} c d + 24 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} d^{2} + 64 \,{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a b d^{2} + 3 \,{\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} b^{2} d^{2} - 192 \, a b c^{2} \cos \left (f x + e\right ) - 192 \, a^{2} c d \cos \left (f x + e\right )}{96 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(96*(f*x + e)*a^2*c^2 + 24*(2*f*x + 2*e - sin(2*f*x + 2*e))*b^2*c^2 + 96*(2*f*x + 2*e - sin(2*f*x + 2*e))
*a*b*c*d + 64*(cos(f*x + e)^3 - 3*cos(f*x + e))*b^2*c*d + 24*(2*f*x + 2*e - sin(2*f*x + 2*e))*a^2*d^2 + 64*(co
s(f*x + e)^3 - 3*cos(f*x + e))*a*b*d^2 + 3*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*b^2*d^2 - 1
92*a*b*c^2*cos(f*x + e) - 192*a^2*c*d*cos(f*x + e))/f

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Fricas [A]  time = 1.70664, size = 371, normalized size = 1.71 \begin{align*} \frac{16 \,{\left (b^{2} c d + a b d^{2}\right )} \cos \left (f x + e\right )^{3} + 3 \,{\left (16 \, a b c d + 4 \,{\left (2 \, a^{2} + b^{2}\right )} c^{2} +{\left (4 \, a^{2} + 3 \, b^{2}\right )} d^{2}\right )} f x - 48 \,{\left (a b c^{2} + a b d^{2} +{\left (a^{2} + b^{2}\right )} c d\right )} \cos \left (f x + e\right ) + 3 \,{\left (2 \, b^{2} d^{2} \cos \left (f x + e\right )^{3} -{\left (4 \, b^{2} c^{2} + 16 \, a b c d +{\left (4 \, a^{2} + 5 \, b^{2}\right )} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{24 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(16*(b^2*c*d + a*b*d^2)*cos(f*x + e)^3 + 3*(16*a*b*c*d + 4*(2*a^2 + b^2)*c^2 + (4*a^2 + 3*b^2)*d^2)*f*x -
48*(a*b*c^2 + a*b*d^2 + (a^2 + b^2)*c*d)*cos(f*x + e) + 3*(2*b^2*d^2*cos(f*x + e)^3 - (4*b^2*c^2 + 16*a*b*c*d
+ (4*a^2 + 5*b^2)*d^2)*cos(f*x + e))*sin(f*x + e))/f

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Sympy [A]  time = 1.80745, size = 459, normalized size = 2.12 \begin{align*} \begin{cases} a^{2} c^{2} x - \frac{2 a^{2} c d \cos{\left (e + f x \right )}}{f} + \frac{a^{2} d^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac{a^{2} d^{2} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac{a^{2} d^{2} \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{2 f} - \frac{2 a b c^{2} \cos{\left (e + f x \right )}}{f} + 2 a b c d x \sin ^{2}{\left (e + f x \right )} + 2 a b c d x \cos ^{2}{\left (e + f x \right )} - \frac{2 a b c d \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{f} - \frac{2 a b d^{2} \sin ^{2}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{f} - \frac{4 a b d^{2} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac{b^{2} c^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac{b^{2} c^{2} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac{b^{2} c^{2} \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{2 f} - \frac{2 b^{2} c d \sin ^{2}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{f} - \frac{4 b^{2} c d \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac{3 b^{2} d^{2} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac{3 b^{2} d^{2} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac{3 b^{2} d^{2} x \cos ^{4}{\left (e + f x \right )}}{8} - \frac{5 b^{2} d^{2} \sin ^{3}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{8 f} - \frac{3 b^{2} d^{2} \sin{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} & \text{for}\: f \neq 0 \\x \left (a + b \sin{\left (e \right )}\right )^{2} \left (c + d \sin{\left (e \right )}\right )^{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*c**2*x - 2*a**2*c*d*cos(e + f*x)/f + a**2*d**2*x*sin(e + f*x)**2/2 + a**2*d**2*x*cos(e + f*x)*
*2/2 - a**2*d**2*sin(e + f*x)*cos(e + f*x)/(2*f) - 2*a*b*c**2*cos(e + f*x)/f + 2*a*b*c*d*x*sin(e + f*x)**2 + 2
*a*b*c*d*x*cos(e + f*x)**2 - 2*a*b*c*d*sin(e + f*x)*cos(e + f*x)/f - 2*a*b*d**2*sin(e + f*x)**2*cos(e + f*x)/f
- 4*a*b*d**2*cos(e + f*x)**3/(3*f) + b**2*c**2*x*sin(e + f*x)**2/2 + b**2*c**2*x*cos(e + f*x)**2/2 - b**2*c**
2*sin(e + f*x)*cos(e + f*x)/(2*f) - 2*b**2*c*d*sin(e + f*x)**2*cos(e + f*x)/f - 4*b**2*c*d*cos(e + f*x)**3/(3*
f) + 3*b**2*d**2*x*sin(e + f*x)**4/8 + 3*b**2*d**2*x*sin(e + f*x)**2*cos(e + f*x)**2/4 + 3*b**2*d**2*x*cos(e +
f*x)**4/8 - 5*b**2*d**2*sin(e + f*x)**3*cos(e + f*x)/(8*f) - 3*b**2*d**2*sin(e + f*x)*cos(e + f*x)**3/(8*f),
Ne(f, 0)), (x*(a + b*sin(e))**2*(c + d*sin(e))**2, True))

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Giac [A]  time = 1.23986, size = 238, normalized size = 1.1 \begin{align*} \frac{b^{2} d^{2} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} + \frac{1}{8} \,{\left (8 \, a^{2} c^{2} + 4 \, b^{2} c^{2} + 16 \, a b c d + 4 \, a^{2} d^{2} + 3 \, b^{2} d^{2}\right )} x + \frac{{\left (b^{2} c d + a b d^{2}\right )} \cos \left (3 \, f x + 3 \, e\right )}{6 \, f} - \frac{{\left (4 \, a b c^{2} + 4 \, a^{2} c d + 3 \, b^{2} c d + 3 \, a b d^{2}\right )} \cos \left (f x + e\right )}{2 \, f} - \frac{{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2} + b^{2} d^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*b^2*d^2*sin(4*f*x + 4*e)/f + 1/8*(8*a^2*c^2 + 4*b^2*c^2 + 16*a*b*c*d + 4*a^2*d^2 + 3*b^2*d^2)*x + 1/6*(b^
2*c*d + a*b*d^2)*cos(3*f*x + 3*e)/f - 1/2*(4*a*b*c^2 + 4*a^2*c*d + 3*b^2*c*d + 3*a*b*d^2)*cos(f*x + e)/f - 1/4
*(b^2*c^2 + 4*a*b*c*d + a^2*d^2 + b^2*d^2)*sin(2*f*x + 2*e)/f