Integrand size = 13, antiderivative size = 26 \[ \int \csc (c+b x) \sin (a+b x) \, dx=x \cos (a-c)+\frac {\log (\sin (c+b x)) \sin (a-c)}{b} \]
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Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {4678, 3556, 8} \[ \int \csc (c+b x) \sin (a+b x) \, dx=\frac {\sin (a-c) \log (\sin (b x+c))}{b}+x \cos (a-c) \]
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Rule 8
Rule 3556
Rule 4678
Rubi steps \begin{align*} \text {integral}& = \cos (a-c) \int 1 \, dx+\sin (a-c) \int \cot (c+b x) \, dx \\ & = x \cos (a-c)+\frac {\log (\sin (c+b x)) \sin (a-c)}{b} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \csc (c+b x) \sin (a+b x) \, dx=x \cos (a-c)+\frac {\log (\sin (c+b x)) \sin (a-c)}{b} \]
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Result contains complex when optimal does not.
Time = 1.06 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.62
method | result | size |
risch | \(-2 i \sin \left (a -c \right ) x -\frac {2 i \sin \left (a -c \right ) a}{b}+x \,{\mathrm e}^{i \left (a -c \right )}+\frac {\ln \left ({\mathrm e}^{2 i \left (x b +a \right )}-{\mathrm e}^{2 i \left (a -c \right )}\right ) \sin \left (a -c \right )}{b}\) | \(68\) |
default | \(\frac {\frac {\frac {\left (\cos \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )\right ) \ln \left (1+\tan \left (x b +a \right )^{2}\right )}{2}+\left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right ) \arctan \left (\tan \left (x b +a \right )\right )}{\left (\cos \left (c \right )^{2}+\sin \left (c \right )^{2}\right ) \left (\cos \left (a \right )^{2}+\sin \left (a \right )^{2}\right )}+\frac {\left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right ) \ln \left (\tan \left (x b +a \right ) \cos \left (a \right ) \cos \left (c \right )+\tan \left (x b +a \right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )}{\cos \left (a \right )^{2} \cos \left (c \right )^{2}+\cos \left (c \right )^{2} \sin \left (a \right )^{2}+\cos \left (a \right )^{2} \sin \left (c \right )^{2}+\sin \left (a \right )^{2} \sin \left (c \right )^{2}}}{b}\) | \(161\) |
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none
Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \csc (c+b x) \sin (a+b x) \, dx=\frac {b x \cos \left (-a + c\right ) - \log \left (\frac {1}{2} \, \sin \left (b x + c\right )\right ) \sin \left (-a + c\right )}{b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (20) = 40\).
Time = 4.45 (sec) , antiderivative size = 333, normalized size of antiderivative = 12.81 \[ \int \csc (c+b x) \sin (a+b x) \, dx=\left (\begin {cases} 0 & \text {for}\: b = 0 \wedge c = 0 \\x & \text {for}\: c = 0 \\0 & \text {for}\: b = 0 \\- \frac {b x \tan ^{2}{\left (\frac {c}{2} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} + \frac {b x}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} - \frac {2 \log {\left (\tan {\left (\frac {c}{2} \right )} + \tan {\left (\frac {b x}{2} \right )} \right )} \tan {\left (\frac {c}{2} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} - \frac {2 \log {\left (\tan {\left (\frac {b x}{2} \right )} - \frac {1}{\tan {\left (\frac {c}{2} \right )}} \right )} \tan {\left (\frac {c}{2} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} + \frac {2 \log {\left (\tan ^{2}{\left (\frac {b x}{2} \right )} + 1 \right )} \tan {\left (\frac {c}{2} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} & \text {otherwise} \end {cases}\right ) \cos {\left (a \right )} + \left (\begin {cases} \tilde {\infty } x & \text {for}\: b = 0 \wedge c = 0 \\\frac {\log {\left (\sin {\left (b x \right )} \right )}}{b} & \text {for}\: c = 0 \\\frac {x}{\sin {\left (c \right )}} & \text {for}\: b = 0 \\\frac {2 b x \tan {\left (\frac {c}{2} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} - \frac {\log {\left (\tan {\left (\frac {c}{2} \right )} + \tan {\left (\frac {b x}{2} \right )} \right )} \tan ^{2}{\left (\frac {c}{2} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} + \frac {\log {\left (\tan {\left (\frac {c}{2} \right )} + \tan {\left (\frac {b x}{2} \right )} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} - \frac {\log {\left (\tan {\left (\frac {b x}{2} \right )} - \frac {1}{\tan {\left (\frac {c}{2} \right )}} \right )} \tan ^{2}{\left (\frac {c}{2} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} + \frac {\log {\left (\tan {\left (\frac {b x}{2} \right )} - \frac {1}{\tan {\left (\frac {c}{2} \right )}} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} + \frac {\log {\left (\tan ^{2}{\left (\frac {b x}{2} \right )} + 1 \right )} \tan ^{2}{\left (\frac {c}{2} \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} - \frac {\log {\left (\tan ^{2}{\left (\frac {b x}{2} \right )} + 1 \right )}}{b \tan ^{2}{\left (\frac {c}{2} \right )} + b} & \text {otherwise} \end {cases}\right ) \sin {\left (a \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (26) = 52\).
Time = 0.29 (sec) , antiderivative size = 108, normalized size of antiderivative = 4.15 \[ \int \csc (c+b x) \sin (a+b x) \, dx=\frac {2 \, b x \cos \left (-a + c\right ) - \log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right ) \sin \left (-a + c\right ) - \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right ) \sin \left (-a + c\right )}{2 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (26) = 52\).
Time = 0.30 (sec) , antiderivative size = 236, normalized size of antiderivative = 9.08 \[ \int \csc (c+b x) \sin (a+b x) \, dx=\frac {\frac {{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right )^{2} + 4 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, c\right )^{2} + 1\right )} {\left (b x + c\right )}}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1} - \frac {2 \, {\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )\right )} \log \left (\tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )^{2} + 1\right )}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1} + \frac {2 \, {\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \right |}\right )}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1}}{b} \]
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Time = 20.65 (sec) , antiderivative size = 111, normalized size of antiderivative = 4.27 \[ \int \csc (c+b x) \sin (a+b x) \, dx=x\,\left (\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}+c\,1{}\mathrm {i}}}{2}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}-c\,1{}\mathrm {i}}}{2}\right )+x\,\left (\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}+c\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}-c\,1{}\mathrm {i}}}{2}\right )+\frac {\ln \left (-{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\right )\,\left (\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}+c\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}-c\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}{b} \]
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