Integrand size = 25, antiderivative size = 166 \[ \int (d \cos (a+b x))^{9/2} (c \sin (a+b x))^{5/2} \, dx=\frac {c d^3 (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2}}{20 b}+\frac {3 c d (d \cos (a+b x))^{7/2} (c \sin (a+b x))^{3/2}}{70 b}-\frac {c (d \cos (a+b x))^{11/2} (c \sin (a+b x))^{3/2}}{7 b d}+\frac {3 c^2 d^4 \sqrt {d \cos (a+b x)} E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {c \sin (a+b x)}}{40 b \sqrt {\sin (2 a+2 b x)}} \]
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Time = 0.15 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2648, 2649, 2652, 2719} \[ \int (d \cos (a+b x))^{9/2} (c \sin (a+b x))^{5/2} \, dx=\frac {3 c^2 d^4 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}{40 b \sqrt {\sin (2 a+2 b x)}}+\frac {c d^3 (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{3/2}}{20 b}-\frac {c (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{11/2}}{7 b d}+\frac {3 c d (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{7/2}}{70 b} \]
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Rule 2648
Rule 2649
Rule 2652
Rule 2719
Rubi steps \begin{align*} \text {integral}& = -\frac {c (d \cos (a+b x))^{11/2} (c \sin (a+b x))^{3/2}}{7 b d}+\frac {1}{14} \left (3 c^2\right ) \int (d \cos (a+b x))^{9/2} \sqrt {c \sin (a+b x)} \, dx \\ & = \frac {3 c d (d \cos (a+b x))^{7/2} (c \sin (a+b x))^{3/2}}{70 b}-\frac {c (d \cos (a+b x))^{11/2} (c \sin (a+b x))^{3/2}}{7 b d}+\frac {1}{20} \left (3 c^2 d^2\right ) \int (d \cos (a+b x))^{5/2} \sqrt {c \sin (a+b x)} \, dx \\ & = \frac {c d^3 (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2}}{20 b}+\frac {3 c d (d \cos (a+b x))^{7/2} (c \sin (a+b x))^{3/2}}{70 b}-\frac {c (d \cos (a+b x))^{11/2} (c \sin (a+b x))^{3/2}}{7 b d}+\frac {1}{40} \left (3 c^2 d^4\right ) \int \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)} \, dx \\ & = \frac {c d^3 (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2}}{20 b}+\frac {3 c d (d \cos (a+b x))^{7/2} (c \sin (a+b x))^{3/2}}{70 b}-\frac {c (d \cos (a+b x))^{11/2} (c \sin (a+b x))^{3/2}}{7 b d}+\frac {\left (3 c^2 d^4 \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}\right ) \int \sqrt {\sin (2 a+2 b x)} \, dx}{40 \sqrt {\sin (2 a+2 b x)}} \\ & = \frac {c d^3 (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2}}{20 b}+\frac {3 c d (d \cos (a+b x))^{7/2} (c \sin (a+b x))^{3/2}}{70 b}-\frac {c (d \cos (a+b x))^{11/2} (c \sin (a+b x))^{3/2}}{7 b d}+\frac {3 c^2 d^4 \sqrt {d \cos (a+b x)} E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {c \sin (a+b x)}}{40 b \sqrt {\sin (2 a+2 b x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.12 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.43 \[ \int (d \cos (a+b x))^{9/2} (c \sin (a+b x))^{5/2} \, dx=\frac {2 (d \cos (a+b x))^{9/2} \sqrt [4]{\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},\frac {7}{4},\frac {11}{4},\sin ^2(a+b x)\right ) \sec ^5(a+b x) (c \sin (a+b x))^{7/2}}{7 b c} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(438\) vs. \(2(165)=330\).
Time = 1.70 (sec) , antiderivative size = 439, normalized size of antiderivative = 2.64
method | result | size |
default | \(-\frac {\sqrt {2}\, \sqrt {d \cos \left (b x +a \right )}\, \sqrt {c \sin \left (b x +a \right )}\, \left (-40 \sqrt {2}\, \left (\cos ^{8}\left (b x +a \right )\right )+52 \sqrt {2}\, \left (\cos ^{6}\left (b x +a \right )\right )+42 \sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )-21 \sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, F\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )+2 \sqrt {2}\, \left (\cos ^{4}\left (b x +a \right )\right )+42 \sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )-21 \sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, F\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )+7 \sqrt {2}\, \left (\cos ^{2}\left (b x +a \right )\right )-21 \sqrt {2}\, \cos \left (b x +a \right )\right ) d^{4} c^{2} \sec \left (b x +a \right ) \csc \left (b x +a \right )}{560 b}\) | \(439\) |
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\[ \int (d \cos (a+b x))^{9/2} (c \sin (a+b x))^{5/2} \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {9}{2}} \left (c \sin \left (b x + a\right )\right )^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int (d \cos (a+b x))^{9/2} (c \sin (a+b x))^{5/2} \, dx=\text {Timed out} \]
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\[ \int (d \cos (a+b x))^{9/2} (c \sin (a+b x))^{5/2} \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {9}{2}} \left (c \sin \left (b x + a\right )\right )^{\frac {5}{2}} \,d x } \]
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\[ \int (d \cos (a+b x))^{9/2} (c \sin (a+b x))^{5/2} \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {9}{2}} \left (c \sin \left (b x + a\right )\right )^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int (d \cos (a+b x))^{9/2} (c \sin (a+b x))^{5/2} \, dx=\int {\left (d\,\cos \left (a+b\,x\right )\right )}^{9/2}\,{\left (c\,\sin \left (a+b\,x\right )\right )}^{5/2} \,d x \]
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