Optimal. Leaf size=46 \[ -\frac {\sin ^3(a+b x) \cos (a+b x)}{4 b}-\frac {3 \sin (a+b x) \cos (a+b x)}{8 b}+\frac {3 x}{8} \]
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Rubi [A] time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2635, 8} \[ -\frac {\sin ^3(a+b x) \cos (a+b x)}{4 b}-\frac {3 \sin (a+b x) \cos (a+b x)}{8 b}+\frac {3 x}{8} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rubi steps
\begin {align*} \int \sin ^4(a+b x) \, dx &=-\frac {\cos (a+b x) \sin ^3(a+b x)}{4 b}+\frac {3}{4} \int \sin ^2(a+b x) \, dx\\ &=-\frac {3 \cos (a+b x) \sin (a+b x)}{8 b}-\frac {\cos (a+b x) \sin ^3(a+b x)}{4 b}+\frac {3 \int 1 \, dx}{8}\\ &=\frac {3 x}{8}-\frac {3 \cos (a+b x) \sin (a+b x)}{8 b}-\frac {\cos (a+b x) \sin ^3(a+b x)}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 0.72 \[ \frac {12 (a+b x)-8 \sin (2 (a+b x))+\sin (4 (a+b x))}{32 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 36, normalized size = 0.78 \[ \frac {3 \, b x + {\left (2 \, \cos \left (b x + a\right )^{3} - 5 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 32, normalized size = 0.70 \[ \frac {3}{8} \, x + \frac {\sin \left (4 \, b x + 4 \, a\right )}{32 \, b} - \frac {\sin \left (2 \, b x + 2 \, a\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 38, normalized size = 0.83 \[ \frac {-\frac {\left (\sin ^{3}\left (b x +a \right )+\frac {3 \sin \left (b x +a \right )}{2}\right ) \cos \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 33, normalized size = 0.72 \[ \frac {12 \, b x + 12 \, a + \sin \left (4 \, b x + 4 \, a\right ) - 8 \, \sin \left (2 \, b x + 2 \, a\right )}{32 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 50, normalized size = 1.09 \[ \frac {3\,x}{8}-\frac {\frac {5\,{\mathrm {tan}\left (a+b\,x\right )}^3}{8}+\frac {3\,\mathrm {tan}\left (a+b\,x\right )}{8}}{b\,\left ({\mathrm {tan}\left (a+b\,x\right )}^4+2\,{\mathrm {tan}\left (a+b\,x\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.24, size = 95, normalized size = 2.07 \[ \begin {cases} \frac {3 x \sin ^{4}{\left (a + b x \right )}}{8} + \frac {3 x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{4} + \frac {3 x \cos ^{4}{\left (a + b x \right )}}{8} - \frac {5 \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b} - \frac {3 \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b} & \text {for}\: b \neq 0 \\x \sin ^{4}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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