Integrand size = 25, antiderivative size = 131 \[ \int (d \cos (a+b x))^{5/2} (c \sin (a+b x))^{5/2} \, dx=\frac {c d (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2}}{10 b}-\frac {c (d \cos (a+b x))^{7/2} (c \sin (a+b x))^{3/2}}{5 b d}+\frac {3 c^2 d^2 \sqrt {d \cos (a+b x)} E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {c \sin (a+b x)}}{20 b \sqrt {\sin (2 a+2 b x)}} \]
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Time = 0.11 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2648, 2649, 2652, 2719} \[ \int (d \cos (a+b x))^{5/2} (c \sin (a+b x))^{5/2} \, dx=\frac {3 c^2 d^2 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {c \sin (a+b x)} \sqrt {d \cos (a+b x)}}{20 b \sqrt {\sin (2 a+2 b x)}}-\frac {c (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{7/2}}{5 b d}+\frac {c d (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{3/2}}{10 b} \]
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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))^{7/2} (c \sin (a+b x))^{3/2}}{5 b d}+\frac {1}{10} \left (3 c^2\right ) \int (d \cos (a+b x))^{5/2} \sqrt {c \sin (a+b x)} \, dx \\ & = \frac {c d (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2}}{10 b}-\frac {c (d \cos (a+b x))^{7/2} (c \sin (a+b x))^{3/2}}{5 b d}+\frac {1}{20} \left (3 c^2 d^2\right ) \int \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)} \, dx \\ & = \frac {c d (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2}}{10 b}-\frac {c (d \cos (a+b x))^{7/2} (c \sin (a+b x))^{3/2}}{5 b d}+\frac {\left (3 c^2 d^2 \sqrt {d \cos (a+b x)} \sqrt {c \sin (a+b x)}\right ) \int \sqrt {\sin (2 a+2 b x)} \, dx}{20 \sqrt {\sin (2 a+2 b x)}} \\ & = \frac {c d (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2}}{10 b}-\frac {c (d \cos (a+b x))^{7/2} (c \sin (a+b x))^{3/2}}{5 b d}+\frac {3 c^2 d^2 \sqrt {d \cos (a+b x)} E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {c \sin (a+b x)}}{20 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 = 70, normalized size of antiderivative = 0.53 \[ \int (d \cos (a+b x))^{5/2} (c \sin (a+b x))^{5/2} \, dx=\frac {2 d^2 \sqrt {d \cos (a+b x)} \sqrt [4]{\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {7}{4},\frac {11}{4},\sin ^2(a+b x)\right ) (c \sin (a+b x))^{5/2} \tan (a+b x)}{7 b} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(425\) vs. \(2(136)=272\).
Time = 1.42 (sec) , antiderivative size = 426, normalized size of antiderivative = 3.25
method | result | size |
default | \(\frac {\sqrt {2}\, d^{2} c^{2} \left (4 \sqrt {2}\, \left (\cos ^{6}\left (b x +a \right )\right )-6 \sqrt {2}\, \left (\cos ^{4}\left (b x +a \right )\right )-6 \sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )+3 \sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, F\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right ) \cos \left (b x +a \right )-6 \sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, E\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )+3 \sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )+1}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, F\left (\sqrt {-\cot \left (b x +a \right )+\csc \left (b x +a \right )+1}, \frac {\sqrt {2}}{2}\right )-\sqrt {2}\, \left (\cos ^{2}\left (b x +a \right )\right )+3 \sqrt {2}\, \cos \left (b x +a \right )\right ) \sqrt {c \sin \left (b x +a \right )}\, \sqrt {d \cos \left (b x +a \right )}\, \sec \left (b x +a \right ) \csc \left (b x +a \right )}{40 b}\) | \(426\) |
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\[ \int (d \cos (a+b x))^{5/2} (c \sin (a+b x))^{5/2} \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {5}{2}} \left (c \sin \left (b x + a\right )\right )^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int (d \cos (a+b x))^{5/2} (c \sin (a+b x))^{5/2} \, dx=\text {Timed out} \]
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\[ \int (d \cos (a+b x))^{5/2} (c \sin (a+b x))^{5/2} \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {5}{2}} \left (c \sin \left (b x + a\right )\right )^{\frac {5}{2}} \,d x } \]
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\[ \int (d \cos (a+b x))^{5/2} (c \sin (a+b x))^{5/2} \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {5}{2}} \left (c \sin \left (b x + a\right )\right )^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int (d \cos (a+b x))^{5/2} (c \sin (a+b x))^{5/2} \, dx=\int {\left (d\,\cos \left (a+b\,x\right )\right )}^{5/2}\,{\left (c\,\sin \left (a+b\,x\right )\right )}^{5/2} \,d x \]
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