Integrand size = 14, antiderivative size = 15 \[ \int \csc ^3(x) (a \cos (x)+b \sin (x)) \, dx=-b \cot (x)-\frac {1}{2} a \csc ^2(x) \]
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Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3168, 3852, 8, 2686, 30} \[ \int \csc ^3(x) (a \cos (x)+b \sin (x)) \, dx=-\frac {1}{2} a \csc ^2(x)-b \cot (x) \]
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Rule 8
Rule 30
Rule 2686
Rule 3168
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \int \left (b \csc ^2(x)+a \cot (x) \csc ^2(x)\right ) \, dx \\ & = a \int \cot (x) \csc ^2(x) \, dx+b \int \csc ^2(x) \, dx \\ & = -(a \text {Subst}(\int x \, dx,x,\csc (x)))-b \text {Subst}(\int 1 \, dx,x,\cot (x)) \\ & = -b \cot (x)-\frac {1}{2} a \csc ^2(x) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \csc ^3(x) (a \cos (x)+b \sin (x)) \, dx=-b \cot (x)-\frac {1}{2} a \csc ^2(x) \]
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Time = 0.57 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93
method | result | size |
default | \(-\frac {a}{2 \sin \left (x \right )^{2}}-b \cot \left (x \right )\) | \(14\) |
parts | \(-b \cot \left (x \right )-\frac {a \csc \left (x \right )^{2}}{2}\) | \(14\) |
risch | \(-\frac {2 i \left (i a \,{\mathrm e}^{2 i x}+b \,{\mathrm e}^{2 i x}-b \right )}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}\) | \(33\) |
parallelrisch | \(\frac {-a \tan \left (\frac {x}{2}\right )^{4}+4 b \tan \left (\frac {x}{2}\right )^{3}-4 b \tan \left (\frac {x}{2}\right )-a}{8 \tan \left (\frac {x}{2}\right )^{2}}\) | \(38\) |
norman | \(\frac {-\frac {a}{8}-\frac {a \tan \left (\frac {x}{2}\right )^{6}}{8}-\frac {b \tan \left (\frac {x}{2}\right )}{2}+\frac {b \tan \left (\frac {x}{2}\right )^{5}}{2}}{\tan \left (\frac {x}{2}\right )^{2} \left (1+\tan \left (\frac {x}{2}\right )^{2}\right )}\) | \(47\) |
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Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.27 \[ \int \csc ^3(x) (a \cos (x)+b \sin (x)) \, dx=\frac {2 \, b \cos \left (x\right ) \sin \left (x\right ) + a}{2 \, {\left (\cos \left (x\right )^{2} - 1\right )}} \]
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Time = 1.84 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \csc ^3(x) (a \cos (x)+b \sin (x)) \, dx=- \frac {a}{2 \sin ^{2}{\left (x \right )}} - \frac {b \cos {\left (x \right )}}{\sin {\left (x \right )}} \]
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Time = 0.22 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \csc ^3(x) (a \cos (x)+b \sin (x)) \, dx=-\frac {b}{\tan \left (x\right )} - \frac {a}{2 \, \sin \left (x\right )^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \csc ^3(x) (a \cos (x)+b \sin (x)) \, dx=-\frac {2 \, b \tan \left (x\right ) + a}{2 \, \tan \left (x\right )^{2}} \]
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Time = 20.82 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \csc ^3(x) (a \cos (x)+b \sin (x)) \, dx=-\frac {a+b\,\sin \left (2\,x\right )}{2\,{\sin \left (x\right )}^2} \]
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