Optimal. Leaf size=223 \[ \frac {2 \left (7 a^2 B+10 a b C+5 b^2 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 \left (7 a^2 B+10 a b C+5 b^2 B\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d}+\frac {2 \left (9 a (a C+2 b B)+7 b^2 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (9 a (a C+2 b B)+7 b^2 C\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{45 d}+\frac {2 b (11 a C+9 b B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{63 d}+\frac {2 b C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d} \]
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Rubi [A] time = 0.42, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3029, 2990, 3023, 2748, 2635, 2641, 2639} \[ \frac {2 \left (7 a^2 B+10 a b C+5 b^2 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 \left (7 a^2 B+10 a b C+5 b^2 B\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d}+\frac {2 \left (9 a (a C+2 b B)+7 b^2 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (9 a (a C+2 b B)+7 b^2 C\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{45 d}+\frac {2 b (11 a C+9 b B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{63 d}+\frac {2 b C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2641
Rule 2748
Rule 2990
Rule 3023
Rule 3029
Rubi steps
\begin {align*} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 (B+C \cos (c+d x)) \, dx\\ &=\frac {2 b C \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{9 d}+\frac {2}{9} \int \cos ^{\frac {3}{2}}(c+d x) \left (\frac {1}{2} a (9 a B+5 b C)+\frac {1}{2} \left (7 b^2 C+9 a (2 b B+a C)\right ) \cos (c+d x)+\frac {1}{2} b (9 b B+11 a C) \cos ^2(c+d x)\right ) \, dx\\ &=\frac {2 b (9 b B+11 a C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b C \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{9 d}+\frac {4}{63} \int \cos ^{\frac {3}{2}}(c+d x) \left (\frac {9}{4} \left (7 a^2 B+5 b^2 B+10 a b C\right )+\frac {7}{4} \left (7 b^2 C+9 a (2 b B+a C)\right ) \cos (c+d x)\right ) \, dx\\ &=\frac {2 b (9 b B+11 a C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b C \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{9 d}+\frac {1}{7} \left (7 a^2 B+5 b^2 B+10 a b C\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{9} \left (7 b^2 C+9 a (2 b B+a C)\right ) \int \cos ^{\frac {5}{2}}(c+d x) \, dx\\ &=\frac {2 \left (7 a^2 B+5 b^2 B+10 a b C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 \left (7 b^2 C+9 a (2 b B+a C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 b (9 b B+11 a C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b C \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{9 d}+\frac {1}{21} \left (7 a^2 B+5 b^2 B+10 a b C\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+\frac {1}{15} \left (7 b^2 C+9 a (2 b B+a C)\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {2 \left (7 b^2 C+9 a (2 b B+a C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (7 a^2 B+5 b^2 B+10 a b C\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 \left (7 a^2 B+5 b^2 B+10 a b C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 \left (7 b^2 C+9 a (2 b B+a C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 b (9 b B+11 a C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b C \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{9 d}\\ \end {align*}
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Mathematica [A] time = 1.46, size = 167, normalized size = 0.75 \[ \frac {60 \left (7 a^2 B+10 a b C+5 b^2 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+84 \left (9 a^2 C+18 a b B+7 b^2 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\sin (c+d x) \sqrt {\cos (c+d x)} \left (7 \left (36 a^2 C+72 a b B+43 b^2 C\right ) \cos (c+d x)+5 \left (84 a^2 B+18 b (2 a C+b B) \cos (2 (c+d x))+156 a b C+78 b^2 B+7 b^2 C \cos (3 (c+d x))\right )\right )}{630 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b^{2} \cos \left (d x + c\right )^{4} + B a^{2} \cos \left (d x + c\right ) + {\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{2} + 2 \, B a b\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2} \sqrt {\cos \left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.43, size = 610, normalized size = 2.74 \[ -\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 (-1120 C \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (720 b^{2} B +1440 C a b +2240 b^{2} 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 (-1008 B a b -1080 b^{2} B -504 a^{2} C -2160 C a b -2072 b^{2} 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 (420 a^{2} B +1008 B a b +840 b^{2} B +504 a^{2} C +1680 C a b +952 b^{2} 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 (-210 a^{2} B -252 B a b -240 b^{2} B -126 a^{2} C -480 C a b -168 b^{2} 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 )-378 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 b +105 a^{2} 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 )+75 b^{2} 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 )-189 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^{2}-147 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 ) b^{2}+150 C 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 )\right )}{315 \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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2} \sqrt {\cos \left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.61, size = 264, normalized size = 1.18 \[ \frac {2\,B\,a^2\,\left (\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )+\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )\right )}{3\,d}-\frac {2\,C\,a^2\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,b^2\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,b^2\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {4\,B\,a\,b\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {4\,C\,a\,b\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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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