Integrand size = 8, antiderivative size = 25 \[ \int \sin ^2(a+b x) \, dx=\frac {x}{2}-\frac {\cos (a+b x) \sin (a+b x)}{2 b} \]
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Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2715, 8} \[ \int \sin ^2(a+b x) \, dx=\frac {x}{2}-\frac {\sin (a+b x) \cos (a+b x)}{2 b} \]
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Rule 8
Rule 2715
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (a+b x) \sin (a+b x)}{2 b}+\frac {\int 1 \, dx}{2} \\ & = \frac {x}{2}-\frac {\cos (a+b x) \sin (a+b x)}{2 b} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \sin ^2(a+b x) \, dx=-\frac {-2 (a+b x)+\sin (2 (a+b x))}{4 b} \]
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Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76
method | result | size |
risch | \(\frac {x}{2}-\frac {\sin \left (2 b x +2 a \right )}{4 b}\) | \(19\) |
parallelrisch | \(\frac {2 b x -\sin \left (2 b x +2 a \right )}{4 b}\) | \(22\) |
derivativedivides | \(\frac {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}}{b}\) | \(27\) |
default | \(\frac {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}}{b}\) | \(27\) |
norman | \(\frac {\frac {\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )}{b}+x \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\frac {x}{2}-\frac {\tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b}+\frac {x \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{2}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{2}}\) | \(77\) |
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Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \sin ^2(a+b x) \, dx=\frac {b x - \cos \left (b x + a\right ) \sin \left (b x + a\right )}{2 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (19) = 38\).
Time = 0.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.84 \[ \int \sin ^2(a+b x) \, dx=\begin {cases} \frac {x \sin ^{2}{\left (a + b x \right )}}{2} + \frac {x \cos ^{2}{\left (a + b x \right )}}{2} - \frac {\sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{2 b} & \text {for}\: b \neq 0 \\x \sin ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \sin ^2(a+b x) \, dx=\frac {2 \, b x + 2 \, a - \sin \left (2 \, b x + 2 \, a\right )}{4 \, b} \]
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Time = 0.30 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int \sin ^2(a+b x) \, dx=\frac {1}{2} \, x - \frac {\sin \left (2 \, b x + 2 \, a\right )}{4 \, b} \]
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Time = 0.15 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int \sin ^2(a+b x) \, dx=\frac {x}{2}-\frac {\sin \left (2\,a+2\,b\,x\right )}{4\,b} \]
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