Optimal. Leaf size=76 \[ \frac{\sin ^5(2 a+2 b x)}{20 b}-\frac{\sin ^3(2 a+2 b x) \cos (2 a+2 b x)}{16 b}-\frac{3 \sin (2 a+2 b x) \cos (2 a+2 b x)}{32 b}+\frac{3 x}{16} \]
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Rubi [A] time = 0.0644401, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4285, 2635, 8, 2564, 30} \[ \frac{\sin ^5(2 a+2 b x)}{20 b}-\frac{\sin ^3(2 a+2 b x) \cos (2 a+2 b x)}{16 b}-\frac{3 \sin (2 a+2 b x) \cos (2 a+2 b x)}{32 b}+\frac{3 x}{16} \]
Antiderivative was successfully verified.
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Rule 4285
Rule 2635
Rule 8
Rule 2564
Rule 30
Rubi steps
\begin{align*} \int \cos ^2(a+b x) \sin ^4(2 a+2 b x) \, dx &=\frac{1}{2} \int \sin ^4(2 a+2 b x) \, dx+\frac{1}{2} \int \cos (2 a+2 b x) \sin ^4(2 a+2 b x) \, dx\\ &=-\frac{\cos (2 a+2 b x) \sin ^3(2 a+2 b x)}{16 b}+\frac{3}{8} \int \sin ^2(2 a+2 b x) \, dx+\frac{\operatorname{Subst}\left (\int x^4 \, dx,x,\sin (2 a+2 b x)\right )}{4 b}\\ &=-\frac{3 \cos (2 a+2 b x) \sin (2 a+2 b x)}{32 b}-\frac{\cos (2 a+2 b x) \sin ^3(2 a+2 b x)}{16 b}+\frac{\sin ^5(2 a+2 b x)}{20 b}+\frac{3 \int 1 \, dx}{16}\\ &=\frac{3 x}{16}-\frac{3 \cos (2 a+2 b x) \sin (2 a+2 b x)}{32 b}-\frac{\cos (2 a+2 b x) \sin ^3(2 a+2 b x)}{16 b}+\frac{\sin ^5(2 a+2 b x)}{20 b}\\ \end{align*}
Mathematica [A] time = 0.188991, size = 62, normalized size = 0.82 \[ \frac{20 \sin (2 (a+b x))-40 \sin (4 (a+b x))-10 \sin (6 (a+b x))+5 \sin (8 (a+b x))+2 \sin (10 (a+b x))+120 b x}{640 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 75, normalized size = 1. \begin{align*}{\frac{3\,x}{16}}+{\frac{\sin \left ( 2\,bx+2\,a \right ) }{32\,b}}-{\frac{\sin \left ( 4\,bx+4\,a \right ) }{16\,b}}-{\frac{\sin \left ( 6\,bx+6\,a \right ) }{64\,b}}+{\frac{\sin \left ( 8\,bx+8\,a \right ) }{128\,b}}+{\frac{\sin \left ( 10\,bx+10\,a \right ) }{320\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15575, size = 88, normalized size = 1.16 \begin{align*} \frac{120 \, b x + 2 \, \sin \left (10 \, b x + 10 \, a\right ) + 5 \, \sin \left (8 \, b x + 8 \, a\right ) - 10 \, \sin \left (6 \, b x + 6 \, a\right ) - 40 \, \sin \left (4 \, b x + 4 \, a\right ) + 20 \, \sin \left (2 \, b x + 2 \, a\right )}{640 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.510664, size = 177, normalized size = 2.33 \begin{align*} \frac{15 \, b x +{\left (128 \, \cos \left (b x + a\right )^{9} - 176 \, \cos \left (b x + a\right )^{7} + 8 \, \cos \left (b x + a\right )^{5} + 10 \, \cos \left (b x + a\right )^{3} + 15 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{80 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34078, size = 100, normalized size = 1.32 \begin{align*} \frac{3}{16} \, x + \frac{\sin \left (10 \, b x + 10 \, a\right )}{320 \, b} + \frac{\sin \left (8 \, b x + 8 \, a\right )}{128 \, b} - \frac{\sin \left (6 \, b x + 6 \, a\right )}{64 \, b} - \frac{\sin \left (4 \, b x + 4 \, a\right )}{16 \, b} + \frac{\sin \left (2 \, b x + 2 \, a\right )}{32 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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