Optimal. Leaf size=140 \[ \frac {2 (7 a A+5 b B) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {6 (a B+A b) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 (a B+A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}+\frac {2 (7 a A+5 b B) \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d}+\frac {2 b B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d} \]
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Rubi [A] time = 0.18, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2968, 3023, 2748, 2635, 2641, 2639} \[ \frac {2 (7 a A+5 b B) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {6 (a B+A b) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 (a B+A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}+\frac {2 (7 a A+5 b B) \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d}+\frac {2 b B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2641
Rule 2748
Rule 2968
Rule 3023
Rubi steps
\begin {align*} \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x)) (A+B \cos (c+d x)) \, dx &=\int \cos ^{\frac {3}{2}}(c+d x) \left (a A+(A b+a B) \cos (c+d x)+b B \cos ^2(c+d x)\right ) \, dx\\ &=\frac {2 b B \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {2}{7} \int \cos ^{\frac {3}{2}}(c+d x) \left (\frac {1}{2} (7 a A+5 b B)+\frac {7}{2} (A b+a B) \cos (c+d x)\right ) \, dx\\ &=\frac {2 b B \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d}+(A b+a B) \int \cos ^{\frac {5}{2}}(c+d x) \, dx+\frac {1}{7} (7 a A+5 b B) \int \cos ^{\frac {3}{2}}(c+d x) \, dx\\ &=\frac {2 (7 a A+5 b B) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 (A b+a B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 b B \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {1}{5} (3 (A b+a B)) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{21} (7 a A+5 b B) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {6 (A b+a B) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 (7 a A+5 b B) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 (7 a A+5 b B) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 (A b+a B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 b B \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.86, size = 103, normalized size = 0.74 \[ \frac {10 (7 a A+5 b B) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+126 (a B+A b) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\sin (c+d x) \sqrt {\cos (c+d x)} (42 (a B+A b) \cos (c+d x)+70 a A+15 b B \cos (2 (c+d x))+65 b B)}{105 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B b \cos \left (d x + c\right )^{3} + A a \cos \left (d x + c\right ) + {\left (B a + A b\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-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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maple [B] time = 1.39, size = 413, normalized size = 2.95 \[ -\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 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 b -168 a B -360 B 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 (140 a A +168 A b +168 a B +280 B 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 (-70 a A -42 A b -42 a B -80 B 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 )+35 a 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 )-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 ) b +25 B 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 )-63 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}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a \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 \[ \int {\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.16, size = 166, normalized size = 1.19 \[ \frac {2\,A\,a\,\left (\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )+\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )\right )}{3\,d}-\frac {2\,A\,b\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,a\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,b\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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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