Optimal. Leaf size=128 \[ \frac {2 \sin (c+d x) (b \cos (c+d x))^{9/2}}{11 b^7 d}+\frac {18 \sin (c+d x) (b \cos (c+d x))^{5/2}}{77 b^5 d}+\frac {30 \sin (c+d x) \sqrt {b \cos (c+d x)}}{77 b^3 d}+\frac {30 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{77 b^2 d \sqrt {b \cos (c+d x)}} \]
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Rubi [A] time = 0.08, antiderivative size = 128, 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 = {16, 2635, 2642, 2641} \[ \frac {2 \sin (c+d x) (b \cos (c+d x))^{9/2}}{11 b^7 d}+\frac {18 \sin (c+d x) (b \cos (c+d x))^{5/2}}{77 b^5 d}+\frac {30 \sin (c+d x) \sqrt {b \cos (c+d x)}}{77 b^3 d}+\frac {30 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{77 b^2 d \sqrt {b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2635
Rule 2641
Rule 2642
Rubi steps
\begin {align*} \int \frac {\cos ^8(c+d x)}{(b \cos (c+d x))^{5/2}} \, dx &=\frac {\int (b \cos (c+d x))^{11/2} \, dx}{b^8}\\ &=\frac {2 (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^7 d}+\frac {9 \int (b \cos (c+d x))^{7/2} \, dx}{11 b^6}\\ &=\frac {18 (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 b^5 d}+\frac {2 (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^7 d}+\frac {45 \int (b \cos (c+d x))^{3/2} \, dx}{77 b^4}\\ &=\frac {30 \sqrt {b \cos (c+d x)} \sin (c+d x)}{77 b^3 d}+\frac {18 (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 b^5 d}+\frac {2 (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^7 d}+\frac {15 \int \frac {1}{\sqrt {b \cos (c+d x)}} \, dx}{77 b^2}\\ &=\frac {30 \sqrt {b \cos (c+d x)} \sin (c+d x)}{77 b^3 d}+\frac {18 (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 b^5 d}+\frac {2 (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^7 d}+\frac {\left (15 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{77 b^2 \sqrt {b \cos (c+d x)}}\\ &=\frac {30 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{77 b^2 d \sqrt {b \cos (c+d x)}}+\frac {30 \sqrt {b \cos (c+d x)} \sin (c+d x)}{77 b^3 d}+\frac {18 (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 b^5 d}+\frac {2 (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^7 d}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 76, normalized size = 0.59 \[ \frac {347 \sin (2 (c+d x))+64 \sin (4 (c+d x))+7 \sin (6 (c+d x))+480 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{1232 b^2 d \sqrt {b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \cos \left (d x + c\right )} \cos \left (d x + c\right )^{5}}{b^{3}}, x\right ) \]
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 \[ \int \frac {\cos \left (d x + c\right )^{8}}{\left (b \cos \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 236, normalized size = 1.84 \[ -\frac {2 \sqrt {b \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (448 \left (\cos ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1568 \left (\cos ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2384 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2040 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1084 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-370 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+62 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{77 b^{2} \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d} \]
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 \[ \int \frac {\cos \left (d x + c\right )^{8}}{\left (b \cos \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^8}{{\left (b\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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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