Optimal. Leaf size=173 \[ \frac {2 A \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^2 d}+\frac {6 A E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \cos (c+d x)}}{5 b d \sqrt {\cos (c+d x)}}+\frac {2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^3 d}+\frac {10 B \sin (c+d x) \sqrt {b \cos (c+d x)}}{21 b d}+\frac {10 B \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {b \cos (c+d x)}} \]
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Rubi [A] time = 0.13, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {16, 2748, 2635, 2640, 2639, 2642, 2641} \[ \frac {2 A \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^2 d}+\frac {6 A E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \cos (c+d x)}}{5 b d \sqrt {\cos (c+d x)}}+\frac {2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^3 d}+\frac {10 B \sin (c+d x) \sqrt {b \cos (c+d x)}}{21 b d}+\frac {10 B \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2635
Rule 2639
Rule 2640
Rule 2641
Rule 2642
Rule 2748
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{\sqrt {b \cos (c+d x)}} \, dx &=\frac {\int (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx}{b^3}\\ &=\frac {A \int (b \cos (c+d x))^{5/2} \, dx}{b^3}+\frac {B \int (b \cos (c+d x))^{7/2} \, dx}{b^4}\\ &=\frac {2 A (b \cos (c+d x))^{3/2} \sin (c+d x)}{5 b^2 d}+\frac {2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b^3 d}+\frac {(3 A) \int \sqrt {b \cos (c+d x)} \, dx}{5 b}+\frac {(5 B) \int (b \cos (c+d x))^{3/2} \, dx}{7 b^2}\\ &=\frac {10 B \sqrt {b \cos (c+d x)} \sin (c+d x)}{21 b d}+\frac {2 A (b \cos (c+d x))^{3/2} \sin (c+d x)}{5 b^2 d}+\frac {2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b^3 d}+\frac {1}{21} (5 B) \int \frac {1}{\sqrt {b \cos (c+d x)}} \, dx+\frac {\left (3 A \sqrt {b \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 b \sqrt {\cos (c+d x)}}\\ &=\frac {6 A \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b d \sqrt {\cos (c+d x)}}+\frac {10 B \sqrt {b \cos (c+d x)} \sin (c+d x)}{21 b d}+\frac {2 A (b \cos (c+d x))^{3/2} \sin (c+d x)}{5 b^2 d}+\frac {2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b^3 d}+\frac {\left (5 B \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 \sqrt {b \cos (c+d x)}}\\ &=\frac {6 A \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b d \sqrt {\cos (c+d x)}}+\frac {10 B \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {10 B \sqrt {b \cos (c+d x)} \sin (c+d x)}{21 b d}+\frac {2 A (b \cos (c+d x))^{3/2} \sin (c+d x)}{5 b^2 d}+\frac {2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b^3 d}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 101, normalized size = 0.58 \[ \frac {\sin (2 (c+d x)) (42 A \cos (c+d x)+15 B \cos (2 (c+d x))+65 B)+252 A \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+100 B \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{210 d \sqrt {b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.23, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B \cos \left (d x + c\right )^{3} + A \cos \left (d x + c\right )^{2}\right )} \sqrt {b \cos \left (d x + c\right )}}{b}, x\right ) \]
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 \[ \int \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{\sqrt {b \cos \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.96, size = 298, normalized size = 1.72 \[ -\frac {2 \sqrt {b \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (240 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-168 A -360 B \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (168 A +280 B \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-42 A -80 B \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-63 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+25 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{105 \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d} \]
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 \[ \int \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{\sqrt {b \cos \left (d x + c\right )}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^3\,\left (A+B\,\cos \left (c+d\,x\right )\right )}{\sqrt {b\,\cos \left (c+d\,x\right )}} \,d x \]
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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