3.75 \(\int (a+b \sin ^2(x))^2 \, dx\)

Optimal. Leaf size=50 \[ \frac{1}{8} x \left (8 a^2+8 a b+3 b^2\right )-\frac{1}{8} b (8 a+3 b) \sin (x) \cos (x)-\frac{1}{4} b^2 \sin ^3(x) \cos (x) \]

[Out]

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

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Rubi [A]  time = 0.0157437, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3179} \[ \frac{1}{8} x \left (8 a^2+8 a b+3 b^2\right )-\frac{1}{8} b (8 a+3 b) \sin (x) \cos (x)-\frac{1}{4} b^2 \sin ^3(x) \cos (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sin[x]^2)^2,x]

[Out]

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

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

Mathematica [A]  time = 0.0576988, size = 43, normalized size = 0.86 \[ \frac{1}{32} \left (4 x \left (8 a^2+8 a b+3 b^2\right )-8 b (2 a+b) \sin (2 x)+b^2 \sin (4 x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sin[x]^2)^2,x]

[Out]

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

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Maple [A]  time = 0.025, size = 42, normalized size = 0.8 \begin{align*}{b}^{2} \left ( -{\frac{\cos \left ( x \right ) }{4} \left ( \left ( \sin \left ( x \right ) \right ) ^{3}+{\frac{3\,\sin \left ( x \right ) }{2}} \right ) }+{\frac{3\,x}{8}} \right ) +2\,ab \left ( -1/2\,\sin \left ( x \right ) \cos \left ( x \right ) +x/2 \right ) +{a}^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.938432, size = 53, normalized size = 1.06 \begin{align*} \frac{1}{32} \, b^{2}{\left (12 \, x + \sin \left (4 \, x\right ) - 8 \, \sin \left (2 \, x\right )\right )} + \frac{1}{2} \, a b{\left (2 \, x - \sin \left (2 \, x\right )\right )} + a^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.62652, size = 116, normalized size = 2.32 \begin{align*} \frac{1}{8} \,{\left (8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} x + \frac{1}{8} \,{\left (2 \, b^{2} \cos \left (x\right )^{3} -{\left (8 \, a b + 5 \, b^{2}\right )} \cos \left (x\right )\right )} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 1.28142, size = 110, normalized size = 2.2 \begin{align*} a^{2} x + a b x \sin ^{2}{\left (x \right )} + a b x \cos ^{2}{\left (x \right )} - a b \sin{\left (x \right )} \cos{\left (x \right )} + \frac{3 b^{2} x \sin ^{4}{\left (x \right )}}{8} + \frac{3 b^{2} x \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )}}{4} + \frac{3 b^{2} x \cos ^{4}{\left (x \right )}}{8} - \frac{5 b^{2} \sin ^{3}{\left (x \right )} \cos{\left (x \right )}}{8} - \frac{3 b^{2} \sin{\left (x \right )} \cos ^{3}{\left (x \right )}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(x)**2)**2,x)

[Out]

a**2*x + a*b*x*sin(x)**2 + a*b*x*cos(x)**2 - a*b*sin(x)*cos(x) + 3*b**2*x*sin(x)**4/8 + 3*b**2*x*sin(x)**2*cos
(x)**2/4 + 3*b**2*x*cos(x)**4/8 - 5*b**2*sin(x)**3*cos(x)/8 - 3*b**2*sin(x)*cos(x)**3/8

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Giac [A]  time = 1.13907, size = 57, normalized size = 1.14 \begin{align*} \frac{1}{32} \, b^{2} \sin \left (4 \, x\right ) + \frac{1}{8} \,{\left (8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} x - \frac{1}{4} \,{\left (2 \, a b + b^{2}\right )} \sin \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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