Optimal. Leaf size=111 \[ \frac {6 A E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}+\frac {10 B F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}+\frac {10 B \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d} \]
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Rubi [A] time = 0.07, antiderivative size = 111, 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 = {2748, 2635, 2639, 2641} \[ \frac {6 A E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}+\frac {10 B F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}+\frac {10 B \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2641
Rule 2748
Rubi steps
\begin {align*} \int \cos ^{\frac {5}{2}}(c+d x) (A+B \cos (c+d x)) \, dx &=A \int \cos ^{\frac {5}{2}}(c+d x) \, dx+B \int \cos ^{\frac {7}{2}}(c+d x) \, dx\\ &=\frac {2 A \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 B \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {1}{5} (3 A) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{7} (5 B) \int \cos ^{\frac {3}{2}}(c+d x) \, dx\\ &=\frac {6 A E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {10 B \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 B \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {1}{21} (5 B) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {6 A E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {10 B F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {10 B \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 B \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.53, size = 77, normalized size = 0.69 \[ \frac {\sin (c+d x) \sqrt {\cos (c+d x)} (42 A \cos (c+d x)+15 B \cos (2 (c+d x))+65 B)+126 A E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+50 B F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{105 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 2.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B \cos \left (d x + c\right )^{3} + A \cos \left (d x + c\right )^{2}\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]
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 \[ \int {\left (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.71, size = 290, normalized size = 2.61 \[ -\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (240 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-168 A -360 B \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (168 A +280 B \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-42 A -80 B \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-63 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+25 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{105 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]
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 \[ \int {\left (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.96, size = 87, normalized size = 0.78 \[ -\frac {2\,A\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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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