Integrand size = 20, antiderivative size = 43 \[ \int \cos ^3(a+b x) \csc ^4(2 a+2 b x) \, dx=\frac {\text {arctanh}(\sin (a+b x))}{16 b}-\frac {\csc (a+b x)}{16 b}-\frac {\csc ^3(a+b x)}{48 b} \]
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Time = 0.07 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4372, 2701, 308, 213} \[ \int \cos ^3(a+b x) \csc ^4(2 a+2 b x) \, dx=\frac {\text {arctanh}(\sin (a+b x))}{16 b}-\frac {\csc ^3(a+b x)}{48 b}-\frac {\csc (a+b x)}{16 b} \]
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Rule 213
Rule 308
Rule 2701
Rule 4372
Rubi steps \begin{align*} \text {integral}& = \frac {1}{16} \int \csc ^4(a+b x) \sec (a+b x) \, dx \\ & = -\frac {\text {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{16 b} \\ & = -\frac {\text {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\csc (a+b x)\right )}{16 b} \\ & = -\frac {\csc (a+b x)}{16 b}-\frac {\csc ^3(a+b x)}{48 b}-\frac {\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{16 b} \\ & = \frac {\text {arctanh}(\sin (a+b x))}{16 b}-\frac {\csc (a+b x)}{16 b}-\frac {\csc ^3(a+b x)}{48 b} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.72 \[ \int \cos ^3(a+b x) \csc ^4(2 a+2 b x) \, dx=-\frac {\csc ^3(a+b x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\sin ^2(a+b x)\right )}{48 b} \]
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Time = 10.38 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.95
method | result | size |
default | \(\frac {-\frac {1}{3 \sin \left (x b +a \right )^{3}}-\frac {1}{\sin \left (x b +a \right )}+\ln \left (\sec \left (x b +a \right )+\tan \left (x b +a \right )\right )}{16 b}\) | \(41\) |
risch | \(-\frac {i \left (3 \,{\mathrm e}^{5 i \left (x b +a \right )}-10 \,{\mathrm e}^{3 i \left (x b +a \right )}+3 \,{\mathrm e}^{i \left (x b +a \right )}\right )}{24 b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{3}}-\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}-i\right )}{16 b}+\frac {\ln \left (i+{\mathrm e}^{i \left (x b +a \right )}\right )}{16 b}\) | \(91\) |
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Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (37) = 74\).
Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.19 \[ \int \cos ^3(a+b x) \csc ^4(2 a+2 b x) \, dx=\frac {3 \, {\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 3 \, {\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (-\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 6 \, \cos \left (b x + a\right )^{2} + 8}{96 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \]
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Timed out. \[ \int \cos ^3(a+b x) \csc ^4(2 a+2 b x) \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 834 vs. \(2 (37) = 74\).
Time = 0.32 (sec) , antiderivative size = 834, normalized size of antiderivative = 19.40 \[ \int \cos ^3(a+b x) \csc ^4(2 a+2 b x) \, dx=\text {Too large to display} \]
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none
Time = 0.33 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.21 \[ \int \cos ^3(a+b x) \csc ^4(2 a+2 b x) \, dx=-\frac {\frac {2 \, {\left (3 \, \sin \left (b x + a\right )^{2} + 1\right )}}{\sin \left (b x + a\right )^{3}} - 3 \, \log \left (\sin \left (b x + a\right ) + 1\right ) + 3 \, \log \left (-\sin \left (b x + a\right ) + 1\right )}{96 \, b} \]
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Time = 19.31 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.88 \[ \int \cos ^3(a+b x) \csc ^4(2 a+2 b x) \, dx=\frac {\mathrm {atanh}\left (\sin \left (a+b\,x\right )\right )}{16\,b}-\frac {\frac {{\sin \left (a+b\,x\right )}^2}{16}+\frac {1}{48}}{b\,{\sin \left (a+b\,x\right )}^3} \]
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