Optimal. Leaf size=147 \[ \frac {2 (11 A+9 C) \sin (c+d x) (b \cos (c+d x))^{5/2}}{77 b^3 d}+\frac {10 (11 A+9 C) \sin (c+d x) \sqrt {b \cos (c+d x)}}{231 b d}+\frac {10 (11 A+9 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d \sqrt {b \cos (c+d x)}}+\frac {2 C \sin (c+d x) (b \cos (c+d x))^{9/2}}{11 b^5 d} \]
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Rubi [A] time = 0.13, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {16, 3014, 2635, 2642, 2641} \[ \frac {2 (11 A+9 C) \sin (c+d x) (b \cos (c+d x))^{5/2}}{77 b^3 d}+\frac {10 (11 A+9 C) \sin (c+d x) \sqrt {b \cos (c+d x)}}{231 b d}+\frac {10 (11 A+9 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d \sqrt {b \cos (c+d x)}}+\frac {2 C \sin (c+d x) (b \cos (c+d x))^{9/2}}{11 b^5 d} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2635
Rule 2641
Rule 2642
Rule 3014
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x) \left (A+C \cos ^2(c+d x)\right )}{\sqrt {b \cos (c+d x)}} \, dx &=\frac {\int (b \cos (c+d x))^{7/2} \left (A+C \cos ^2(c+d x)\right ) \, dx}{b^4}\\ &=\frac {2 C (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^5 d}+\frac {(11 A+9 C) \int (b \cos (c+d x))^{7/2} \, dx}{11 b^4}\\ &=\frac {2 (11 A+9 C) (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 b^3 d}+\frac {2 C (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^5 d}+\frac {(5 (11 A+9 C)) \int (b \cos (c+d x))^{3/2} \, dx}{77 b^2}\\ &=\frac {10 (11 A+9 C) \sqrt {b \cos (c+d x)} \sin (c+d x)}{231 b d}+\frac {2 (11 A+9 C) (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 b^3 d}+\frac {2 C (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^5 d}+\frac {1}{231} (5 (11 A+9 C)) \int \frac {1}{\sqrt {b \cos (c+d x)}} \, dx\\ &=\frac {10 (11 A+9 C) \sqrt {b \cos (c+d x)} \sin (c+d x)}{231 b d}+\frac {2 (11 A+9 C) (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 b^3 d}+\frac {2 C (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^5 d}+\frac {\left (5 (11 A+9 C) \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{231 \sqrt {b \cos (c+d x)}}\\ &=\frac {10 (11 A+9 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d \sqrt {b \cos (c+d x)}}+\frac {10 (11 A+9 C) \sqrt {b \cos (c+d x)} \sin (c+d x)}{231 b d}+\frac {2 (11 A+9 C) (b \cos (c+d x))^{5/2} \sin (c+d x)}{77 b^3 d}+\frac {2 C (b \cos (c+d x))^{9/2} \sin (c+d x)}{11 b^5 d}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 94, normalized size = 0.64 \[ \frac {\sin (2 (c+d x)) (12 (11 A+16 C) \cos (2 (c+d x))+572 A+21 C \cos (4 (c+d x))+531 C)+80 (11 A+9 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{1848 d \sqrt {b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \cos \left (d x + c\right )^{5} + A \cos \left (d x + c\right )^{3}\right )} \sqrt {b \cos \left (d x + c\right )}}{b}, x\right ) \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{4}}{\sqrt {b \cos \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.44, size = 349, normalized size = 2.37 \[ -\frac {2 \sqrt {b \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (1344 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3360 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (528 A +3792 C \right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-792 A -2328 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (616 A +924 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-176 A -186 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+55 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+45 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{231 \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d} \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{4}}{\sqrt {b \cos \left (d x + c\right )}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^4\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )}{\sqrt {b\,\cos \left (c+d\,x\right )}} \,d x \]
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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