Integrand size = 21, antiderivative size = 44 \[ \int \frac {\sin ^2(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {x}{2 a}+\frac {\sin (c+d x)}{a d}-\frac {\cos (c+d x) \sin (c+d x)}{2 a d} \]
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Time = 0.15 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3957, 2918, 2717, 2715, 8} \[ \int \frac {\sin ^2(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\sin (c+d x)}{a d}-\frac {\sin (c+d x) \cos (c+d x)}{2 a d}-\frac {x}{2 a} \]
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Rule 8
Rule 2715
Rule 2717
Rule 2918
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cos (c+d x) \sin ^2(c+d x)}{-a-a \cos (c+d x)} \, dx \\ & = \frac {\int \cos (c+d x) \, dx}{a}-\frac {\int \cos ^2(c+d x) \, dx}{a} \\ & = \frac {\sin (c+d x)}{a d}-\frac {\cos (c+d x) \sin (c+d x)}{2 a d}-\frac {\int 1 \, dx}{2 a} \\ & = -\frac {x}{2 a}+\frac {\sin (c+d x)}{a d}-\frac {\cos (c+d x) \sin (c+d x)}{2 a d} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.55 \[ \int \frac {\sin ^2(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \left (-c+2 d x-4 \sin (c+d x)+\sin (2 (c+d x))+\tan \left (\frac {c}{2}\right )\right )}{2 a d (1+\sec (c+d x))} \]
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Time = 0.61 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.75
method | result | size |
parallelrisch | \(\frac {-2 d x +4 \sin \left (d x +c \right )-\sin \left (2 d x +2 c \right )}{4 d a}\) | \(33\) |
risch | \(-\frac {x}{2 a}+\frac {\sin \left (d x +c \right )}{a d}-\frac {\sin \left (2 d x +2 c \right )}{4 d a}\) | \(38\) |
derivativedivides | \(\frac {-\frac {4 \left (-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}-\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}\) | \(64\) |
default | \(\frac {-\frac {4 \left (-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}-\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}\) | \(64\) |
norman | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {x}{2 a}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{a d}-\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a}-\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2 a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}\) | \(93\) |
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Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.61 \[ \int \frac {\sin ^2(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {d x + {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right )}{2 \, a d} \]
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\[ \int \frac {\sin ^2(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\int \frac {\sin ^{2}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (40) = 80\).
Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.55 \[ \int \frac {\sin ^2(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\frac {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a + \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac {\arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{d} \]
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Time = 0.31 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.32 \[ \int \frac {\sin ^2(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {d x + c}{a} - \frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a}}{2 \, d} \]
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Time = 13.79 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.68 \[ \int \frac {\sin ^2(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\sin \left (2\,c+2\,d\,x\right )-4\,\sin \left (c+d\,x\right )+2\,d\,x}{4\,a\,d} \]
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