Integrand size = 23, antiderivative size = 26 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {2 a \sec (c+d x) \sqrt {a+a \sin (c+d x)}}{d} \]
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Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2752} \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {2 a \sec (c+d x) \sqrt {a \sin (c+d x)+a}}{d} \]
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Rule 2752
Rubi steps \begin{align*} \text {integral}& = \frac {2 a \sec (c+d x) \sqrt {a+a \sin (c+d x)}}{d} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {2 a \sec (c+d x) \sqrt {a (1+\sin (c+d x))}}{d} \]
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Time = 0.30 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42
method | result | size |
default | \(\frac {2 a^{2} \left (1+\sin \left (d x +c \right )\right )}{\cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(37\) |
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none
Time = 0.33 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {2 \, \sqrt {a \sin \left (d x + c\right ) + a} a}{d \cos \left (d x + c\right )} \]
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Timed out. \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (24) = 48\).
Time = 0.33 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.77 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {2 \, {\left (a^{\frac {3}{2}} + \frac {2 \, a^{\frac {3}{2}} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{\frac {3}{2}} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )}}{d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{\frac {3}{2}}} \]
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none
Time = 0.33 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.46 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {\sqrt {2} a^{\frac {3}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )} \]
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Time = 4.79 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {4\,a\,\cos \left (c+d\,x\right )\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}}{d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \]
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