Optimal. Leaf size=156 \[ \frac {56 d^4 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {d \cos (a+b x)}}{1105 b \sqrt {\cos (a+b x)}}+\frac {56 d^3 \sin (a+b x) (d \cos (a+b x))^{3/2}}{3315 b}-\frac {2 \sin ^3(a+b x) (d \cos (a+b x))^{11/2}}{17 b d}-\frac {12 \sin (a+b x) (d \cos (a+b x))^{11/2}}{221 b d}+\frac {8 d \sin (a+b x) (d \cos (a+b x))^{7/2}}{663 b} \]
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Rubi [A] time = 0.15, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2568, 2635, 2640, 2639} \[ \frac {56 d^3 \sin (a+b x) (d \cos (a+b x))^{3/2}}{3315 b}+\frac {56 d^4 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {d \cos (a+b x)}}{1105 b \sqrt {\cos (a+b x)}}-\frac {2 \sin ^3(a+b x) (d \cos (a+b x))^{11/2}}{17 b d}-\frac {12 \sin (a+b x) (d \cos (a+b x))^{11/2}}{221 b d}+\frac {8 d \sin (a+b x) (d \cos (a+b x))^{7/2}}{663 b} \]
Antiderivative was successfully verified.
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Rule 2568
Rule 2635
Rule 2639
Rule 2640
Rubi steps
\begin {align*} \int (d \cos (a+b x))^{9/2} \sin ^4(a+b x) \, dx &=-\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*}
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Mathematica [C] time = 0.13, size = 57, normalized size = 0.37 \[ \frac {\sqrt [4]{\cos ^2(a+b x)} \tan ^5(a+b x) (d \cos (a+b x))^{9/2} \, _2F_1\left (-\frac {7}{4},\frac {5}{2};\frac {7}{2};\sin ^2(a+b x)\right )}{5 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d^{4} \cos \left (b x + a\right )^{8} - 2 \, d^{4} \cos \left (b x + a\right )^{6} + d^{4} \cos \left (b x + a\right )^{4}\right )} \sqrt {d \cos \left (b x + a\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \cos \left (b x + a\right )\right )^{\frac {9}{2}} \sin \left (b x + a\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 275, normalized size = 1.76 \[ -\frac {8 \sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, d^{5} \left (24960 \left (\cos ^{19}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-124800 \left (\cos ^{17}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+265440 \left (\cos ^{15}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-312960 \left (\cos ^{13}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+222520 \left (\cos ^{11}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-96360 \left (\cos ^{9}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+23866 \left (\cos ^{7}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-2652 \left (\cos ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-35 \left (\cos ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-21 \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )+21 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3315 \sqrt {-d \left (2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right )}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \cos \left (b x + a\right )\right )^{\frac {9}{2}} \sin \left (b x + a\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\sin \left (a+b\,x\right )}^4\,{\left (d\,\cos \left (a+b\,x\right )\right )}^{9/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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