Optimal. Leaf size=134 \[ \frac {2 a (5 A+3 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}+\frac {2 b (7 A+5 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 b (7 A+5 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d}+\frac {2 b C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d} \]
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Rubi [A] time = 0.19, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3034, 3023, 2748, 2639, 2635, 2641} \[ \frac {2 a (5 A+3 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}+\frac {2 b (7 A+5 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 b (7 A+5 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d}+\frac {2 b C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2641
Rule 2748
Rule 3023
Rule 3034
Rubi steps
\begin {align*} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {2 b C \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {2}{7} \int \sqrt {\cos (c+d x)} \left (\frac {7 a A}{2}+\frac {1}{2} b (7 A+5 C) \cos (c+d x)+\frac {7}{2} a C \cos ^2(c+d x)\right ) \, dx\\ &=\frac {2 a C \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 b C \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {4}{35} \int \sqrt {\cos (c+d x)} \left (\frac {7}{4} a (5 A+3 C)+\frac {5}{4} b (7 A+5 C) \cos (c+d x)\right ) \, dx\\ &=\frac {2 a C \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 b C \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {1}{5} (a (5 A+3 C)) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{7} (b (7 A+5 C)) \int \cos ^{\frac {3}{2}}(c+d x) \, dx\\ &=\frac {2 a (5 A+3 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 b (7 A+5 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a C \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 b C \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {1}{21} (b (7 A+5 C)) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 a (5 A+3 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 b (7 A+5 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 b (7 A+5 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a C \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 b C \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.69, size = 98, normalized size = 0.73 \[ \frac {\sin (c+d x) \sqrt {\cos (c+d x)} (42 a C \cos (c+d x)+70 A b+15 b C \cos (2 (c+d x))+65 b C)+42 a (5 A+3 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+10 b (7 A+5 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{105 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b \cos \left (d x + c\right )^{3} + C a \cos \left (d x + c\right )^{2} + A b \cos \left (d x + c\right ) + A a\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )} \sqrt {\cos \left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.98, size = 401, normalized size = 2.99 \[ -\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (240 C b \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-168 a C -360 C b \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (140 A b +168 a C +280 C b \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-70 A b -42 a C -80 C b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+35 A b \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-105 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a +25 C b \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-63 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a \right )}{105 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )} \sqrt {\cos \left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.31, size = 139, normalized size = 1.04 \[ \frac {2\,A\,b\,\left (\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )+\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )\right )}{3\,d}+\frac {2\,A\,a\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}-\frac {2\,C\,a\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,b\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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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