Optimal. Leaf size=100 \[ \frac {2 \sin (c+d x) (b \cos (c+d x))^{7/2}}{9 b^4 d}+\frac {14 \sin (c+d x) (b \cos (c+d x))^{3/2}}{45 b^2 d}+\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \cos (c+d x)}}{15 b d \sqrt {\cos (c+d x)}} \]
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Rubi [A] time = 0.06, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {16, 2635, 2640, 2639} \[ \frac {2 \sin (c+d x) (b \cos (c+d x))^{7/2}}{9 b^4 d}+\frac {14 \sin (c+d x) (b \cos (c+d x))^{3/2}}{45 b^2 d}+\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \cos (c+d x)}}{15 b d \sqrt {\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2635
Rule 2639
Rule 2640
Rubi steps
\begin {align*} \int \frac {\cos ^5(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx &=\frac {\int (b \cos (c+d x))^{9/2} \, dx}{b^5}\\ &=\frac {2 (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^4 d}+\frac {7 \int (b \cos (c+d x))^{5/2} \, dx}{9 b^3}\\ &=\frac {14 (b \cos (c+d x))^{3/2} \sin (c+d x)}{45 b^2 d}+\frac {2 (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^4 d}+\frac {7 \int \sqrt {b \cos (c+d x)} \, dx}{15 b}\\ &=\frac {14 (b \cos (c+d x))^{3/2} \sin (c+d x)}{45 b^2 d}+\frac {2 (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^4 d}+\frac {\left (7 \sqrt {b \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{15 b \sqrt {\cos (c+d x)}}\\ &=\frac {14 \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 b d \sqrt {\cos (c+d x)}}+\frac {14 (b \cos (c+d x))^{3/2} \sin (c+d x)}{45 b^2 d}+\frac {2 (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^4 d}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 71, normalized size = 0.71 \[ \frac {(38 \sin (2 (c+d x))+5 \sin (4 (c+d x))) \cos (c+d x)+168 \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{180 d \sqrt {b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \cos \left (d x + c\right )} \cos \left (d x + c\right )^{4}}{b}, x\right ) \]
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 \[ \int \frac {\cos \left (d x + c\right )^{5}}{\sqrt {b \cos \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 220, normalized size = 2.20 \[ -\frac {2 \sqrt {b \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (160 \left (\cos ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-480 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+616 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-432 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+160 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-21 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-24 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{45 \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d} \]
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 \[ \int \frac {\cos \left (d x + c\right )^{5}}{\sqrt {b \cos \left (d x + c\right )}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^5}{\sqrt {b\,\cos \left (c+d\,x\right )}} \,d x \]
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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