3.295 \(\int (a+b \sin ^2(e+f x))^2 \, dx\)

Optimal. Leaf size=72 \[ \frac{1}{8} x \left (8 a^2+8 a b+3 b^2\right )-\frac{b (8 a+3 b) \sin (e+f x) \cos (e+f x)}{8 f}-\frac{b^2 \sin ^3(e+f x) \cos (e+f x)}{4 f} \]

[Out]

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

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Rubi [A]  time = 0.020182, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {3179} \[ \frac{1}{8} x \left (8 a^2+8 a b+3 b^2\right )-\frac{b (8 a+3 b) \sin (e+f x) \cos (e+f x)}{8 f}-\frac{b^2 \sin ^3(e+f x) \cos (e+f x)}{4 f} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sin[e + f*x]^2)^2,x]

[Out]

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

Rule 3179

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^2, x_Symbol] :> Simp[((8*a^2 + 8*a*b + 3*b^2)*x)/8, x] + (-Simp[(
b^2*Cos[e + f*x]*Sin[e + f*x]^3)/(4*f), x] - Simp[(b*(8*a + 3*b)*Cos[e + f*x]*Sin[e + f*x])/(8*f), x]) /; Free
Q[{a, b, e, f}, x]

Rubi steps

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

Mathematica [A]  time = 0.116051, size = 58, normalized size = 0.81 \[ \frac{4 \left (8 a^2+8 a b+3 b^2\right ) (e+f x)-8 b (2 a+b) \sin (2 (e+f x))+b^2 \sin (4 (e+f x))}{32 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sin[e + f*x]^2)^2,x]

[Out]

(4*(8*a^2 + 8*a*b + 3*b^2)*(e + f*x) - 8*b*(2*a + b)*Sin[2*(e + f*x)] + b^2*Sin[4*(e + f*x)])/(32*f)

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Maple [A]  time = 0.03, size = 78, normalized size = 1.1 \begin{align*}{\frac{1}{f} \left ({b}^{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 ) +2\,ab \left ( -1/2\,\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) +1/2\,fx+e/2 \right ) +{a}^{2} \left ( fx+e \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/f*(b^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)+2*a*b*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f
*x+1/2*e)+a^2*(f*x+e))

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Maxima [A]  time = 0.972697, size = 92, normalized size = 1.28 \begin{align*} a^{2} x + \frac{{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a b}{2 \, f} + \frac{{\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}}{32 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*x + 1/2*(2*f*x + 2*e - sin(2*f*x + 2*e))*a*b/f + 1/32*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*
e))*b^2/f

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Fricas [A]  time = 1.95607, size = 143, normalized size = 1.99 \begin{align*} \frac{{\left (8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} f x +{\left (2 \, b^{2} \cos \left (f x + e\right )^{3} -{\left (8 \, a b + 5 \, b^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{8 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.69381, size = 168, normalized size = 2.33 \begin{align*} \begin{cases} a^{2} x + a b x \sin ^{2}{\left (e + f x \right )} + a b x \cos ^{2}{\left (e + f x \right )} - \frac{a b \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{f} + \frac{3 b^{2} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac{3 b^{2} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac{3 b^{2} x \cos ^{4}{\left (e + f x \right )}}{8} - \frac{5 b^{2} \sin ^{3}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{8 f} - \frac{3 b^{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 ^{2}{\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)**2,x)

[Out]

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

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Giac [A]  time = 1.10804, size = 81, normalized size = 1.12 \begin{align*} \frac{1}{8} \,{\left (8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} x + \frac{b^{2} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} - \frac{{\left (2 \, a b + b^{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)^2,x, algorithm="giac")

[Out]

1/8*(8*a^2 + 8*a*b + 3*b^2)*x + 1/32*b^2*sin(4*f*x + 4*e)/f - 1/4*(2*a*b + b^2)*sin(2*f*x + 2*e)/f