Integrand size = 18, antiderivative size = 28 \[ \int \csc ^2(2 a+2 b x) \sin (a+b x) \, dx=-\frac {\text {arctanh}(\cos (a+b x))}{4 b}+\frac {\sec (a+b x)}{4 b} \]
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Time = 0.05 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4373, 2702, 327, 213} \[ \int \csc ^2(2 a+2 b x) \sin (a+b x) \, dx=\frac {\sec (a+b x)}{4 b}-\frac {\text {arctanh}(\cos (a+b x))}{4 b} \]
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Rule 213
Rule 327
Rule 2702
Rule 4373
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \int \csc (a+b x) \sec ^2(a+b x) \, dx \\ & = \frac {\text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{4 b} \\ & = \frac {\sec (a+b x)}{4 b}+\frac {\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{4 b} \\ & = -\frac {\text {arctanh}(\cos (a+b x))}{4 b}+\frac {\sec (a+b x)}{4 b} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79 \[ \int \csc ^2(2 a+2 b x) \sin (a+b x) \, dx=-\frac {\log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )}{4 b}+\frac {\log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )}{4 b}+\frac {\sec (a+b x)}{4 b} \]
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Time = 0.77 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11
method | result | size |
default | \(\frac {\frac {1}{\cos \left (x b +a \right )}+\ln \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}{4 b}\) | \(31\) |
risch | \(\frac {{\mathrm e}^{i \left (x b +a \right )}}{2 b \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )}+\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{4 b}-\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right )}{4 b}\) | \(63\) |
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Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (24) = 48\).
Time = 0.25 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86 \[ \int \csc ^2(2 a+2 b x) \sin (a+b x) \, dx=-\frac {\cos \left (b x + a\right ) \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - \cos \left (b x + a\right ) \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - 2}{8 \, b \cos \left (b x + a\right )} \]
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Timed out. \[ \int \csc ^2(2 a+2 b x) \sin (a+b x) \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (24) = 48\).
Time = 0.24 (sec) , antiderivative size = 236, normalized size of antiderivative = 8.43 \[ \int \csc ^2(2 a+2 b x) \sin (a+b x) \, dx=\frac {4 \, \cos \left (2 \, b x + 2 \, a\right ) \cos \left (b x + a\right ) - {\left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right ) + {\left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right ) + 4 \, \sin \left (2 \, b x + 2 \, a\right ) \sin \left (b x + a\right ) + 4 \, \cos \left (b x + a\right )}{8 \, {\left (b \cos \left (2 \, b x + 2 \, a\right )^{2} + b \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, b \cos \left (2 \, b x + 2 \, a\right ) + b\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (24) = 48\).
Time = 0.30 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86 \[ \int \csc ^2(2 a+2 b x) \sin (a+b x) \, dx=\frac {\frac {4}{\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1} + \log \left (-\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1}\right )}{8 \, b} \]
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Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \csc ^2(2 a+2 b x) \sin (a+b x) \, dx=\frac {1}{4\,b\,\cos \left (a+b\,x\right )}-\frac {\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{4\,b} \]
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