Optimal. Leaf size=50 \[ \frac {1}{8} x \left (8 a^2+8 a b+3 b^2\right )-\frac {1}{8} b (8 a+3 b) \sin (x) \cos (x)-\frac {1}{4} b^2 \sin ^3(x) \cos (x) \]
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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.
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Rule 3179
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.
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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.
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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.
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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.
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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.
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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.
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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.
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