Integrand size = 30, antiderivative size = 17 \[ \int \csc ^2(c+d x) (a-a \csc (c+d x)) (A+A \csc (c+d x)) \, dx=\frac {a A \cot ^3(c+d x)}{3 d} \]
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Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4047, 2687, 30} \[ \int \csc ^2(c+d x) (a-a \csc (c+d x)) (A+A \csc (c+d x)) \, dx=\frac {a A \cot ^3(c+d x)}{3 d} \]
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Rule 30
Rule 2687
Rule 4047
Rubi steps \begin{align*} \text {integral}& = -\left ((a A) \int \cot ^2(c+d x) \csc ^2(c+d x) \, dx\right ) \\ & = -\frac {(a A) \text {Subst}\left (\int x^2 \, dx,x,-\cot (c+d x)\right )}{d} \\ & = \frac {a A \cot ^3(c+d x)}{3 d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \csc ^2(c+d x) (a-a \csc (c+d x)) (A+A \csc (c+d x)) \, dx=\frac {a A \cot ^3(c+d x)}{3 d} \]
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Result contains complex when optimal does not.
Time = 0.93 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.06
method | result | size |
risch | \(-\frac {2 i A a \left (3 \,{\mathrm e}^{4 i \left (d x +c \right )}+1\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}\) | \(35\) |
derivativedivides | \(\frac {-A a \left (-\frac {2}{3}-\frac {\csc \left (d x +c \right )^{2}}{3}\right ) \cot \left (d x +c \right )-A a \cot \left (d x +c \right )}{d}\) | \(38\) |
default | \(\frac {-A a \left (-\frac {2}{3}-\frac {\csc \left (d x +c \right )^{2}}{3}\right ) \cot \left (d x +c \right )-A a \cot \left (d x +c \right )}{d}\) | \(38\) |
parts | \(-\frac {a A \cot \left (d x +c \right )}{d}-\frac {A a \left (-\frac {2}{3}-\frac {\csc \left (d x +c \right )^{2}}{3}\right ) \cot \left (d x +c \right )}{d}\) | \(40\) |
parallelrisch | \(\frac {A a \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}{96 d}\) | \(48\) |
norman | \(\frac {\frac {A a}{24 d}-\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 d}+\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8 d}-\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{24 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}\) | \(75\) |
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Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (15) = 30\).
Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.12 \[ \int \csc ^2(c+d x) (a-a \csc (c+d x)) (A+A \csc (c+d x)) \, dx=-\frac {A a \cos \left (d x + c\right )^{3}}{3 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (14) = 28\).
Time = 0.85 (sec) , antiderivative size = 54, normalized size of antiderivative = 3.18 \[ \int \csc ^2(c+d x) (a-a \csc (c+d x)) (A+A \csc (c+d x)) \, dx=\begin {cases} \frac {- A a \left (- \frac {\cot ^{3}{\left (c + d x \right )}}{3} - \cot {\left (c + d x \right )}\right ) - A a \cot {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A \csc {\left (c \right )} + A\right ) \left (- a \csc {\left (c \right )} + a\right ) \csc ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (15) = 30\).
Time = 0.23 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.47 \[ \int \csc ^2(c+d x) (a-a \csc (c+d x)) (A+A \csc (c+d x)) \, dx=-\frac {\frac {3 \, A a}{\tan \left (d x + c\right )} - \frac {{\left (3 \, \tan \left (d x + c\right )^{2} + 1\right )} A a}{\tan \left (d x + c\right )^{3}}}{3 \, d} \]
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none
Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \csc ^2(c+d x) (a-a \csc (c+d x)) (A+A \csc (c+d x)) \, dx=\frac {A a}{3 \, d \tan \left (d x + c\right )^{3}} \]
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Time = 19.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \csc ^2(c+d x) (a-a \csc (c+d x)) (A+A \csc (c+d x)) \, dx=\frac {A\,a\,{\mathrm {cot}\left (c+d\,x\right )}^3}{3\,d} \]
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