Optimal. Leaf size=172 \[ \frac{3 x^2 \cos ^4(a+b x)}{16 b^2}+\frac{9 x^2 \cos ^2(a+b x)}{16 b^2}-\frac{3 \cos ^4(a+b x)}{128 b^4}-\frac{45 \cos ^2(a+b x)}{128 b^4}-\frac{3 x \sin (a+b x) \cos ^3(a+b x)}{32 b^3}-\frac{45 x \sin (a+b x) \cos (a+b x)}{64 b^3}+\frac{x^3 \sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac{3 x^3 \sin (a+b x) \cos (a+b x)}{8 b}-\frac{45 x^2}{128 b^2}+\frac{3 x^4}{32} \]
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Rubi [A] time = 0.154006, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3311, 30, 3310} \[ \frac{3 x^2 \cos ^4(a+b x)}{16 b^2}+\frac{9 x^2 \cos ^2(a+b x)}{16 b^2}-\frac{3 \cos ^4(a+b x)}{128 b^4}-\frac{45 \cos ^2(a+b x)}{128 b^4}-\frac{3 x \sin (a+b x) \cos ^3(a+b x)}{32 b^3}-\frac{45 x \sin (a+b x) \cos (a+b x)}{64 b^3}+\frac{x^3 \sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac{3 x^3 \sin (a+b x) \cos (a+b x)}{8 b}-\frac{45 x^2}{128 b^2}+\frac{3 x^4}{32} \]
Antiderivative was successfully verified.
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Rule 3311
Rule 30
Rule 3310
Rubi steps
\begin{align*} \int x^3 \cos ^4(a+b x) \, dx &=\frac{3 x^2 \cos ^4(a+b x)}{16 b^2}+\frac{x^3 \cos ^3(a+b x) \sin (a+b x)}{4 b}+\frac{3}{4} \int x^3 \cos ^2(a+b x) \, dx-\frac{3 \int x \cos ^4(a+b x) \, dx}{8 b^2}\\ &=\frac{9 x^2 \cos ^2(a+b x)}{16 b^2}-\frac{3 \cos ^4(a+b x)}{128 b^4}+\frac{3 x^2 \cos ^4(a+b x)}{16 b^2}+\frac{3 x^3 \cos (a+b x) \sin (a+b x)}{8 b}-\frac{3 x \cos ^3(a+b x) \sin (a+b x)}{32 b^3}+\frac{x^3 \cos ^3(a+b x) \sin (a+b x)}{4 b}+\frac{3 \int x^3 \, dx}{8}-\frac{9 \int x \cos ^2(a+b x) \, dx}{32 b^2}-\frac{9 \int x \cos ^2(a+b x) \, dx}{8 b^2}\\ &=\frac{3 x^4}{32}-\frac{45 \cos ^2(a+b x)}{128 b^4}+\frac{9 x^2 \cos ^2(a+b x)}{16 b^2}-\frac{3 \cos ^4(a+b x)}{128 b^4}+\frac{3 x^2 \cos ^4(a+b x)}{16 b^2}-\frac{45 x \cos (a+b x) \sin (a+b x)}{64 b^3}+\frac{3 x^3 \cos (a+b x) \sin (a+b x)}{8 b}-\frac{3 x \cos ^3(a+b x) \sin (a+b x)}{32 b^3}+\frac{x^3 \cos ^3(a+b x) \sin (a+b x)}{4 b}-\frac{9 \int x \, dx}{64 b^2}-\frac{9 \int x \, dx}{16 b^2}\\ &=-\frac{45 x^2}{128 b^2}+\frac{3 x^4}{32}-\frac{45 \cos ^2(a+b x)}{128 b^4}+\frac{9 x^2 \cos ^2(a+b x)}{16 b^2}-\frac{3 \cos ^4(a+b x)}{128 b^4}+\frac{3 x^2 \cos ^4(a+b x)}{16 b^2}-\frac{45 x \cos (a+b x) \sin (a+b x)}{64 b^3}+\frac{3 x^3 \cos (a+b x) \sin (a+b x)}{8 b}-\frac{3 x \cos ^3(a+b x) \sin (a+b x)}{32 b^3}+\frac{x^3 \cos ^3(a+b x) \sin (a+b x)}{4 b}\\ \end{align*}
Mathematica [A] time = 0.414309, size = 100, normalized size = 0.58 \[ \frac{4 b x \left (32 \left (2 b^2 x^2-3\right ) \sin (2 (a+b x))+\left (8 b^2 x^2-3\right ) \sin (4 (a+b x))+24 b^3 x^3\right )+192 \left (2 b^2 x^2-1\right ) \cos (2 (a+b x))+3 \left (8 b^2 x^2-1\right ) \cos (4 (a+b x))}{1024 b^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.056, size = 440, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05618, size = 409, normalized size = 2.38 \begin{align*} \frac{96 \,{\left (b x + a\right )}^{4} - 32 \,{\left (12 \, b x + 12 \, a + \sin \left (4 \, b x + 4 \, a\right ) + 8 \, \sin \left (2 \, b x + 2 \, a\right )\right )} a^{3} + 24 \,{\left (24 \,{\left (b x + a\right )}^{2} + 4 \,{\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right ) + 32 \,{\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right ) + \cos \left (4 \, b x + 4 \, a\right ) + 16 \, \cos \left (2 \, b x + 2 \, a\right )\right )} a^{2} - 12 \,{\left (32 \,{\left (b x + a\right )}^{3} + 4 \,{\left (b x + a\right )} \cos \left (4 \, b x + 4 \, a\right ) + 64 \,{\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) +{\left (8 \,{\left (b x + a\right )}^{2} - 1\right )} \sin \left (4 \, b x + 4 \, a\right ) + 32 \,{\left (2 \,{\left (b x + a\right )}^{2} - 1\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} a + 3 \,{\left (8 \,{\left (b x + a\right )}^{2} - 1\right )} \cos \left (4 \, b x + 4 \, a\right ) + 192 \,{\left (2 \,{\left (b x + a\right )}^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + 4 \,{\left (8 \,{\left (b x + a\right )}^{3} - 3 \, b x - 3 \, a\right )} \sin \left (4 \, b x + 4 \, a\right ) + 128 \,{\left (2 \,{\left (b x + a\right )}^{3} - 3 \, b x - 3 \, a\right )} \sin \left (2 \, b x + 2 \, a\right )}{1024 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41015, size = 271, normalized size = 1.58 \begin{align*} \frac{12 \, b^{4} x^{4} + 3 \,{\left (8 \, b^{2} x^{2} - 1\right )} \cos \left (b x + a\right )^{4} - 45 \, b^{2} x^{2} + 9 \,{\left (8 \, b^{2} x^{2} - 5\right )} \cos \left (b x + a\right )^{2} + 2 \,{\left (2 \,{\left (8 \, b^{3} x^{3} - 3 \, b x\right )} \cos \left (b x + a\right )^{3} + 3 \,{\left (8 \, b^{3} x^{3} - 15 \, b x\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{128 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.4835, size = 253, normalized size = 1.47 \begin{align*} \begin{cases} \frac{3 x^{4} \sin ^{4}{\left (a + b x \right )}}{32} + \frac{3 x^{4} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{16} + \frac{3 x^{4} \cos ^{4}{\left (a + b x \right )}}{32} + \frac{3 x^{3} \sin ^{3}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{8 b} + \frac{5 x^{3} \sin{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b} - \frac{45 x^{2} \sin ^{4}{\left (a + b x \right )}}{128 b^{2}} - \frac{9 x^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{64 b^{2}} + \frac{51 x^{2} \cos ^{4}{\left (a + b x \right )}}{128 b^{2}} - \frac{45 x \sin ^{3}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{64 b^{3}} - \frac{51 x \sin{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{64 b^{3}} + \frac{45 \sin ^{4}{\left (a + b x \right )}}{256 b^{4}} - \frac{51 \cos ^{4}{\left (a + b x \right )}}{256 b^{4}} & \text{for}\: b \neq 0 \\\frac{x^{4} \cos ^{4}{\left (a \right )}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11344, size = 146, normalized size = 0.85 \begin{align*} \frac{3}{32} \, x^{4} + \frac{3 \,{\left (8 \, b^{2} x^{2} - 1\right )} \cos \left (4 \, b x + 4 \, a\right )}{1024 \, b^{4}} + \frac{3 \,{\left (2 \, b^{2} x^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right )}{16 \, b^{4}} + \frac{{\left (8 \, b^{3} x^{3} - 3 \, b x\right )} \sin \left (4 \, b x + 4 \, a\right )}{256 \, b^{4}} + \frac{{\left (2 \, b^{3} x^{3} - 3 \, b x\right )} \sin \left (2 \, b x + 2 \, a\right )}{8 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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