Integrand size = 24, antiderivative size = 27 \[ \int \frac {\sin ^3(c+d x)}{a-a \sin ^2(c+d x)} \, dx=\frac {\cos (c+d x)}{a d}+\frac {\sec (c+d x)}{a d} \]
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Time = 0.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3254, 2670, 14} \[ \int \frac {\sin ^3(c+d x)}{a-a \sin ^2(c+d x)} \, dx=\frac {\cos (c+d x)}{a d}+\frac {\sec (c+d x)}{a d} \]
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Rule 14
Rule 2670
Rule 3254
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sin (c+d x) \tan ^2(c+d x) \, dx}{a} \\ & = -\frac {\text {Subst}\left (\int \frac {1-x^2}{x^2} \, dx,x,\cos (c+d x)\right )}{a d} \\ & = -\frac {\text {Subst}\left (\int \left (-1+\frac {1}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{a d} \\ & = \frac {\cos (c+d x)}{a d}+\frac {\sec (c+d x)}{a d} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {\sin ^3(c+d x)}{a-a \sin ^2(c+d x)} \, dx=\frac {\frac {\cos (c+d x)}{d}+\frac {\sec (c+d x)}{d}}{a} \]
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Time = 0.40 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(\frac {\cos \left (d x +c \right )+\frac {1}{\cos \left (d x +c \right )}}{d a}\) | \(23\) |
default | \(\frac {\cos \left (d x +c \right )+\frac {1}{\cos \left (d x +c \right )}}{d a}\) | \(23\) |
parallelrisch | \(\frac {3+\cos \left (2 d x +2 c \right )+4 \cos \left (d x +c \right )}{2 a d \cos \left (d x +c \right )}\) | \(36\) |
risch | \(\frac {{\mathrm e}^{3 i \left (d x +c \right )}+7 \cos \left (d x +c \right )+5 i \sin \left (d x +c \right )}{2 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}\) | \(49\) |
norman | \(\frac {-\frac {4 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {4}{a d}-\frac {8 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\) | \(79\) |
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Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {\sin ^3(c+d x)}{a-a \sin ^2(c+d x)} \, dx=\frac {\cos \left (d x + c\right )^{2} + 1}{a d \cos \left (d x + c\right )} \]
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Time = 1.52 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {\sin ^3(c+d x)}{a-a \sin ^2(c+d x)} \, dx=\begin {cases} - \frac {4}{a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - a d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{3}{\left (c \right )}}{- a \sin ^{2}{\left (c \right )} + a} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\sin ^3(c+d x)}{a-a \sin ^2(c+d x)} \, dx=\frac {\frac {\cos \left (d x + c\right )}{a} + \frac {1}{a \cos \left (d x + c\right )}}{d} \]
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Time = 0.31 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {\sin ^3(c+d x)}{a-a \sin ^2(c+d x)} \, dx=\frac {\cos \left (d x + c\right )}{a d} + \frac {1}{a d \cos \left (d x + c\right )} \]
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Time = 13.95 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {\sin ^3(c+d x)}{a-a \sin ^2(c+d x)} \, dx=\frac {{\cos \left (c+d\,x\right )}^2+1}{a\,d\,\cos \left (c+d\,x\right )} \]
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