Optimal. Leaf size=182 \[ \frac {2 \left (5 a^2 B+6 a b C+3 b^2 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \left (7 a (a C+2 b B)+5 b^2 C\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 \left (7 a (a C+2 b B)+5 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d}+\frac {2 b (9 a C+7 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{35 d}+\frac {2 b C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}{7 d} \]
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Rubi [A] time = 0.39, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3029, 2990, 3023, 2748, 2639, 2635, 2641} \[ \frac {2 \left (5 a^2 B+6 a b C+3 b^2 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \left (7 a (a C+2 b B)+5 b^2 C\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 \left (7 a (a C+2 b B)+5 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d}+\frac {2 b (9 a C+7 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{35 d}+\frac {2 b C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))}{7 d} \]
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 \frac {(a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx &=\int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 (B+C \cos (c+d x)) \, dx\\ &=\frac {2 b C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{7 d}+\frac {2}{7} \int \sqrt {\cos (c+d x)} \left (\frac {1}{2} a (7 a B+3 b C)+\frac {1}{2} \left (5 b^2 C+7 a (2 b B+a C)\right ) \cos (c+d x)+\frac {1}{2} b (7 b B+9 a C) \cos ^2(c+d x)\right ) \, dx\\ &=\frac {2 b (7 b B+9 a C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 b C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{7 d}+\frac {4}{35} \int \sqrt {\cos (c+d x)} \left (\frac {7}{4} \left (5 a^2 B+3 b^2 B+6 a b C\right )+\frac {5}{4} \left (5 b^2 C+7 a (2 b B+a C)\right ) \cos (c+d x)\right ) \, dx\\ &=\frac {2 b (7 b B+9 a C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 b C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{7 d}+\frac {1}{5} \left (5 a^2 B+3 b^2 B+6 a b C\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{7} \left (5 b^2 C+7 a (2 b B+a C)\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx\\ &=\frac {2 \left (5 a^2 B+3 b^2 B+6 a b C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \left (5 b^2 C+7 a (2 b B+a C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 b (7 b B+9 a C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 b C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{7 d}+\frac {1}{21} \left (5 b^2 C+7 a (2 b B+a C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 \left (5 a^2 B+3 b^2 B+6 a b C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \left (5 b^2 C+7 a (2 b B+a C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 \left (5 b^2 C+7 a (2 b B+a C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 b (7 b B+9 a C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 b C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 1.07, size = 139, normalized size = 0.76 \[ \frac {10 \left (7 a^2 C+14 a b B+5 b^2 C\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+42 \left (5 a^2 B+6 a b C+3 b^2 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\sin (c+d x) \sqrt {\cos (c+d x)} \left (5 \left (14 a^2 C+28 a b B+3 b^2 C \cos (2 (c+d x))+13 b^2 C\right )+42 b (2 a C+b B) \cos (c+d x)\right )}{105 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b^{2} \cos \left (d x + c\right )^{3} + B a^{2} + {\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (C a^{2} + 2 \, B a b\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2}}{\sqrt {\cos \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.32, size = 548, normalized size = 3.01 \[ -\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^{2} \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 b^{2} B -336 C a b -360 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 (280 B a b +168 b^{2} B +140 a^{2} C +336 C a b +280 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 (-140 B a b -42 b^{2} B -70 a^{2} C -84 C a b -80 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 )+70 B 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 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^{2}-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 ) b^{2}+35 a^{2} 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}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+25 b^{2} 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}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-126 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 b \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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2}}{\sqrt {\cos \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.64, size = 229, normalized size = 1.26 \[ \frac {2\,C\,a^2\,\left (\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )+\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )\right )}{3\,d}+\frac {2\,B\,a^2\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {2\,B\,a\,b\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}-\frac {2\,B\,b^2\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,b^2\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {4\,C\,a\,b\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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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