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]

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

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Rubi [A]  time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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*}

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Mathematica [A]  time = 0.06, 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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fricas [A]  time = 0.41, size = 47, normalized size = 0.94 \[ \frac {1}{8} \, {\left (8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} x + \frac {1}{8} \, {\left (2 \, b^{2} \cos \relax (x)^{3} - {\left (8 \, a b + 5 \, b^{2}\right )} \cos \relax (x)\right )} \sin \relax (x) \]

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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giac [A]  time = 0.13, size = 42, normalized size = 0.84 \[ \frac {1}{32} \, b^{2} \sin \left (4 \, x\right ) + \frac {1}{8} \, {\left (8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} x - \frac {1}{4} \, {\left (2 \, a b + b^{2}\right )} \sin \left (2 \, x\right ) \]

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)

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maple [A]  time = 0.34, size = 42, normalized size = 0.84 \[ b^{2} \left (-\frac {\left (\sin ^{3}\relax (x )+\frac {3 \sin \relax (x )}{2}\right ) \cos \relax (x )}{4}+\frac {3 x}{8}\right )+2 a b \left (-\frac {\sin \relax (x ) \cos \relax (x )}{2}+\frac {x}{2}\right )+a^{2} x \]

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 = 0.33, size = 39, normalized size = 0.78 \[ \frac {1}{32} \, b^{2} {\left (12 \, x + \sin \left (4 \, x\right ) - 8 \, \sin \left (2 \, x\right )\right )} + \frac {1}{2} \, a b {\left (2 \, x - \sin \left (2 \, x\right )\right )} + a^{2} x \]

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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mupad [B]  time = 13.52, size = 44, normalized size = 0.88 \[ x\,a^2-\sin \relax (x)\,a\,b\,\cos \relax (x)+x\,a\,b+\frac {\sin \relax (x)\,b^2\,{\cos \relax (x)}^3}{4}-\frac {5\,\sin \relax (x)\,b^2\,\cos \relax (x)}{8}+\frac {3\,x\,b^2}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sin(x)^2)^2,x)

[Out]

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

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sympy [B]  time = 0.76, size = 110, normalized size = 2.20 \[ a^{2} x + a b x \sin ^{2}{\relax (x )} + a b x \cos ^{2}{\relax (x )} - a b \sin {\relax (x )} \cos {\relax (x )} + \frac {3 b^{2} x \sin ^{4}{\relax (x )}}{8} + \frac {3 b^{2} x \sin ^{2}{\relax (x )} \cos ^{2}{\relax (x )}}{4} + \frac {3 b^{2} x \cos ^{4}{\relax (x )}}{8} - \frac {5 b^{2} \sin ^{3}{\relax (x )} \cos {\relax (x )}}{8} - \frac {3 b^{2} \sin {\relax (x )} \cos ^{3}{\relax (x )}}{8} \]

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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