Integrand size = 21, antiderivative size = 156 \[ \int (d \cos (a+b x))^{9/2} \sin ^4(a+b x) \, dx=\frac {56 d^4 \sqrt {d \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{1105 b \sqrt {\cos (a+b x)}}+\frac {56 d^3 (d \cos (a+b x))^{3/2} \sin (a+b x)}{3315 b}+\frac {8 d (d \cos (a+b x))^{7/2} \sin (a+b x)}{663 b}-\frac {12 (d \cos (a+b x))^{11/2} \sin (a+b x)}{221 b d}-\frac {2 (d \cos (a+b x))^{11/2} \sin ^3(a+b x)}{17 b d} \]
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Time = 0.11 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2648, 2715, 2721, 2719} \[ \int (d \cos (a+b x))^{9/2} \sin ^4(a+b x) \, dx=\frac {56 d^4 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {d \cos (a+b x)}}{1105 b \sqrt {\cos (a+b x)}}+\frac {56 d^3 \sin (a+b x) (d \cos (a+b x))^{3/2}}{3315 b}-\frac {2 \sin ^3(a+b x) (d \cos (a+b x))^{11/2}}{17 b d}-\frac {12 \sin (a+b x) (d \cos (a+b x))^{11/2}}{221 b d}+\frac {8 d \sin (a+b x) (d \cos (a+b x))^{7/2}}{663 b} \]
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Rule 2648
Rule 2715
Rule 2719
Rule 2721
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (d \cos (a+b x))^{11/2} \sin ^3(a+b x)}{17 b d}+\frac {6}{17} \int (d \cos (a+b x))^{9/2} \sin ^2(a+b x) \, dx \\ & = -\frac {12 (d \cos (a+b x))^{11/2} \sin (a+b x)}{221 b d}-\frac {2 (d \cos (a+b x))^{11/2} \sin ^3(a+b x)}{17 b d}+\frac {12}{221} \int (d \cos (a+b x))^{9/2} \, dx \\ & = \frac {8 d (d \cos (a+b x))^{7/2} \sin (a+b x)}{663 b}-\frac {12 (d \cos (a+b x))^{11/2} \sin (a+b x)}{221 b d}-\frac {2 (d \cos (a+b x))^{11/2} \sin ^3(a+b x)}{17 b d}+\frac {1}{663} \left (28 d^2\right ) \int (d \cos (a+b x))^{5/2} \, dx \\ & = \frac {56 d^3 (d \cos (a+b x))^{3/2} \sin (a+b x)}{3315 b}+\frac {8 d (d \cos (a+b x))^{7/2} \sin (a+b x)}{663 b}-\frac {12 (d \cos (a+b x))^{11/2} \sin (a+b x)}{221 b d}-\frac {2 (d \cos (a+b x))^{11/2} \sin ^3(a+b x)}{17 b d}+\frac {\left (28 d^4\right ) \int \sqrt {d \cos (a+b x)} \, dx}{1105} \\ & = \frac {56 d^3 (d \cos (a+b x))^{3/2} \sin (a+b x)}{3315 b}+\frac {8 d (d \cos (a+b x))^{7/2} \sin (a+b x)}{663 b}-\frac {12 (d \cos (a+b x))^{11/2} \sin (a+b x)}{221 b d}-\frac {2 (d \cos (a+b x))^{11/2} \sin ^3(a+b x)}{17 b d}+\frac {\left (28 d^4 \sqrt {d \cos (a+b x)}\right ) \int \sqrt {\cos (a+b x)} \, dx}{1105 \sqrt {\cos (a+b x)}} \\ & = \frac {56 d^4 \sqrt {d \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{1105 b \sqrt {\cos (a+b x)}}+\frac {56 d^3 (d \cos (a+b x))^{3/2} \sin (a+b x)}{3315 b}+\frac {8 d (d \cos (a+b x))^{7/2} \sin (a+b x)}{663 b}-\frac {12 (d \cos (a+b x))^{11/2} \sin (a+b x)}{221 b d}-\frac {2 (d \cos (a+b x))^{11/2} \sin ^3(a+b x)}{17 b d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.10 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.37 \[ \int (d \cos (a+b x))^{9/2} \sin ^4(a+b x) \, dx=\frac {(d \cos (a+b x))^{9/2} \sqrt [4]{\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},\frac {5}{2},\frac {7}{2},\sin ^2(a+b x)\right ) \tan ^5(a+b x)}{5 b} \]
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Time = 13.67 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.76
method | result | size |
default | \(-\frac {8 \sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, d^{5} \left (24960 \left (\cos ^{19}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-124800 \left (\cos ^{17}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+265440 \left (\cos ^{15}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-312960 \left (\cos ^{13}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+222520 \left (\cos ^{11}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-96360 \left (\cos ^{9}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+23866 \left (\cos ^{7}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-2652 \left (\cos ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-35 \left (\cos ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-21 \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {1-2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, E\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )+21 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3315 \sqrt {-d \left (2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right )}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b}\) | \(275\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.85 \[ \int (d \cos (a+b x))^{9/2} \sin ^4(a+b x) \, dx=-\frac {2 \, {\left (-42 i \, \sqrt {2} d^{\frac {9}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + 42 i \, \sqrt {2} d^{\frac {9}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) - {\left (195 \, d^{4} \cos \left (b x + a\right )^{7} - 285 \, d^{4} \cos \left (b x + a\right )^{5} + 20 \, d^{4} \cos \left (b x + a\right )^{3} + 28 \, d^{4} \cos \left (b x + a\right )\right )} \sqrt {d \cos \left (b x + a\right )} \sin \left (b x + a\right )\right )}}{3315 \, b} \]
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Timed out. \[ \int (d \cos (a+b x))^{9/2} \sin ^4(a+b x) \, dx=\text {Timed out} \]
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\[ \int (d \cos (a+b x))^{9/2} \sin ^4(a+b x) \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {9}{2}} \sin \left (b x + a\right )^{4} \,d x } \]
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\[ \int (d \cos (a+b x))^{9/2} \sin ^4(a+b x) \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{\frac {9}{2}} \sin \left (b x + a\right )^{4} \,d x } \]
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Timed out. \[ \int (d \cos (a+b x))^{9/2} \sin ^4(a+b x) \, dx=\int {\sin \left (a+b\,x\right )}^4\,{\left (d\,\cos \left (a+b\,x\right )\right )}^{9/2} \,d x \]
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