Integrand size = 21, antiderivative size = 96 \[ \int (d \cos (a+b x))^{9/2} \csc ^2(a+b x) \, dx=-\frac {d (d \cos (a+b x))^{7/2} \csc (a+b x)}{b}-\frac {21 d^4 \sqrt {d \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b \sqrt {\cos (a+b x)}}-\frac {7 d^3 (d \cos (a+b x))^{3/2} \sin (a+b x)}{5 b} \]
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Time = 0.05 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2647, 2715, 2721, 2719} \[ \int (d \cos (a+b x))^{9/2} \csc ^2(a+b x) \, dx=-\frac {21 d^4 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {d \cos (a+b x)}}{5 b \sqrt {\cos (a+b x)}}-\frac {7 d^3 \sin (a+b x) (d \cos (a+b x))^{3/2}}{5 b}-\frac {d \csc (a+b x) (d \cos (a+b x))^{7/2}}{b} \]
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Rule 2647
Rule 2715
Rule 2719
Rule 2721
Rubi steps \begin{align*} \text {integral}& = -\frac {d (d \cos (a+b x))^{7/2} \csc (a+b x)}{b}-\frac {1}{2} \left (7 d^2\right ) \int (d \cos (a+b x))^{5/2} \, dx \\ & = -\frac {d (d \cos (a+b x))^{7/2} \csc (a+b x)}{b}-\frac {7 d^3 (d \cos (a+b x))^{3/2} \sin (a+b x)}{5 b}-\frac {1}{10} \left (21 d^4\right ) \int \sqrt {d \cos (a+b x)} \, dx \\ & = -\frac {d (d \cos (a+b x))^{7/2} \csc (a+b x)}{b}-\frac {7 d^3 (d \cos (a+b x))^{3/2} \sin (a+b x)}{5 b}-\frac {\left (21 d^4 \sqrt {d \cos (a+b x)}\right ) \int \sqrt {\cos (a+b x)} \, dx}{10 \sqrt {\cos (a+b x)}} \\ & = -\frac {d (d \cos (a+b x))^{7/2} \csc (a+b x)}{b}-\frac {21 d^4 \sqrt {d \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b \sqrt {\cos (a+b x)}}-\frac {7 d^3 (d \cos (a+b x))^{3/2} \sin (a+b x)}{5 b} \\ \end{align*}
Time = 0.48 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.77 \[ \int (d \cos (a+b x))^{9/2} \csc ^2(a+b x) \, dx=-\frac {d^4 \sqrt {d \cos (a+b x)} \left (21 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )+\sqrt {\cos (a+b x)} (5 \cot (a+b x)+\sin (2 (a+b x)))\right )}{5 b \sqrt {\cos (a+b x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(228\) vs. \(2(110)=220\).
Time = 2.85 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.39
method | result | size |
default | \(\frac {\sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, d^{6} \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (-64 \left (\sin ^{10}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+160 \left (\sin ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+42 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) {\left (2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}^{\frac {3}{2}} E\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}-104 \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-4 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+22 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-5\right )}{10 {\left (-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d +d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right )}^{\frac {3}{2}} \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b}\) | \(229\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.26 \[ \int (d \cos (a+b x))^{9/2} \csc ^2(a+b x) \, dx=\frac {-21 i \, \sqrt {2} d^{\frac {9}{2}} \sin \left (b x + a\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + 21 i \, \sqrt {2} d^{\frac {9}{2}} \sin \left (b x + a\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) + 2 \, {\left (2 \, d^{4} \cos \left (b x + a\right )^{3} - 7 \, d^{4} \cos \left (b x + a\right )\right )} \sqrt {d \cos \left (b x + a\right )}}{10 \, b \sin \left (b x + a\right )} \]
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Timed out. \[ \int (d \cos (a+b x))^{9/2} \csc ^2(a+b x) \, dx=\text {Timed out} \]
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\[ \int (d \cos (a+b x))^{9/2} \csc ^2(a+b x) \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {9}{2}} \csc \left (b x + a\right )^{2} \,d x } \]
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\[ \int (d \cos (a+b x))^{9/2} \csc ^2(a+b x) \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {9}{2}} \csc \left (b x + a\right )^{2} \,d x } \]
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Timed out. \[ \int (d \cos (a+b x))^{9/2} \csc ^2(a+b x) \, dx=\int \frac {{\left (d\,\cos \left (a+b\,x\right )\right )}^{9/2}}{{\sin \left (a+b\,x\right )}^2} \,d x \]
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