Optimal. Leaf size=404 \[ \frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{231 d}+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{3465 d}+\frac {2 F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (77 a^4 (3 A+C)+308 a^3 b B+66 a^2 b^2 (7 A+5 C)+220 a b^3 B+5 b^4 (11 A+9 C)\right )}{231 d}+\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right )}{15 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (64 a^4 C+682 a^3 b B+9 a^2 b^2 (143 A+101 C)+660 a b^3 B+15 b^4 (11 A+9 C)\right )}{693 d}+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{99 d}+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d} \]
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Rubi [A] time = 1.26, antiderivative size = 404, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {3049, 3033, 3023, 2748, 2641, 2639} \[ \frac {2 F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (66 a^2 b^2 (7 A+5 C)+77 a^4 (3 A+C)+308 a^3 b B+220 a b^3 B+5 b^4 (11 A+9 C)\right )}{231 d}+\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (12 a^3 b (5 A+3 C)+54 a^2 b^2 B+15 a^4 B+4 a b^3 (9 A+7 C)+7 b^4 B\right )}{15 d}+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (1353 a^2 b B+192 a^3 C+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{3465 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{231 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (9 a^2 b^2 (143 A+101 C)+682 a^3 b B+64 a^4 C+660 a b^3 B+15 b^4 (11 A+9 C)\right )}{693 d}+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{99 d}+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2748
Rule 3023
Rule 3033
Rule 3049
Rubi steps
\begin {align*} \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx &=\frac {2 C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d}+\frac {2}{11} \int \frac {(a+b \cos (c+d x))^3 \left (\frac {1}{2} a (11 A+C)+\frac {1}{2} (11 A b+11 a B+9 b C) \cos (c+d x)+\frac {1}{2} (11 b B+8 a C) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 (11 b B+8 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d}+\frac {2 C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d}+\frac {4}{99} \int \frac {(a+b \cos (c+d x))^2 \left (\frac {1}{4} a (99 a A+11 b B+17 a C)+\frac {1}{4} \left (198 a A b+99 a^2 B+77 b^2 B+146 a b C\right ) \cos (c+d x)+\frac {3}{4} \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d}+\frac {2 (11 b B+8 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d}+\frac {2 C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d}+\frac {8}{693} \int \frac {(a+b \cos (c+d x)) \left (\frac {1}{8} a \left (242 a b B+9 b^2 (11 A+9 C)+a^2 (693 A+167 C)\right )+\frac {1}{8} \left (693 a^3 B+1441 a b^2 B+45 b^3 (11 A+9 C)+a^2 b (2079 A+1381 C)\right ) \cos (c+d x)+\frac {1}{8} \left (1353 a^2 b B+539 b^3 B+192 a^3 C+2 a b^2 (891 A+673 C)\right ) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 b \left (1353 a^2 b B+539 b^3 B+192 a^3 C+2 a b^2 (891 A+673 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac {2 \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d}+\frac {2 (11 b B+8 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d}+\frac {2 C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d}+\frac {16 \int \frac {\frac {5}{16} a^2 \left (242 a b B+9 b^2 (11 A+9 C)+a^2 (693 A+167 C)\right )+\frac {231}{16} \left (15 a^4 B+54 a^2 b^2 B+7 b^4 B+12 a^3 b (5 A+3 C)+4 a b^3 (9 A+7 C)\right ) \cos (c+d x)+\frac {15}{16} \left (682 a^3 b B+660 a b^3 B+64 a^4 C+15 b^4 (11 A+9 C)+9 a^2 b^2 (143 A+101 C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx}{3465}\\ &=\frac {2 \left (682 a^3 b B+660 a b^3 B+64 a^4 C+15 b^4 (11 A+9 C)+9 a^2 b^2 (143 A+101 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{693 d}+\frac {2 b \left (1353 a^2 b B+539 b^3 B+192 a^3 C+2 a b^2 (891 A+673 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac {2 \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d}+\frac {2 (11 b B+8 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d}+\frac {2 C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d}+\frac {32 \int \frac {\frac {45}{32} \left (308 a^3 b B+220 a b^3 B+77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 C)\right )+\frac {693}{32} \left (15 a^4 B+54 a^2 b^2 B+7 b^4 B+12 a^3 b (5 A+3 C)+4 a b^3 (9 A+7 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx}{10395}\\ &=\frac {2 \left (682 a^3 b B+660 a b^3 B+64 a^4 C+15 b^4 (11 A+9 C)+9 a^2 b^2 (143 A+101 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{693 d}+\frac {2 b \left (1353 a^2 b B+539 b^3 B+192 a^3 C+2 a b^2 (891 A+673 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac {2 \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d}+\frac {2 (11 b B+8 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d}+\frac {2 C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d}+\frac {1}{15} \left (15 a^4 B+54 a^2 b^2 B+7 b^4 B+12 a^3 b (5 A+3 C)+4 a b^3 (9 A+7 C)\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{231} \left (308 a^3 b B+220 a b^3 B+77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 \left (15 a^4 B+54 a^2 b^2 B+7 b^4 B+12 a^3 b (5 A+3 C)+4 a b^3 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (308 a^3 b B+220 a b^3 B+77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {2 \left (682 a^3 b B+660 a b^3 B+64 a^4 C+15 b^4 (11 A+9 C)+9 a^2 b^2 (143 A+101 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{693 d}+\frac {2 b \left (1353 a^2 b B+539 b^3 B+192 a^3 C+2 a b^2 (891 A+673 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac {2 \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d}+\frac {2 (11 b B+8 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d}+\frac {2 C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d}\\ \end {align*}
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Mathematica [A] time = 2.45, size = 319, normalized size = 0.79 \[ \frac {10 F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (77 a^4 (3 A+C)+308 a^3 b B+66 a^2 b^2 (7 A+5 C)+220 a b^3 B+5 b^4 (11 A+9 C)\right )+154 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right )+\frac {1}{12} \sin (c+d x) \sqrt {\cos (c+d x)} \left (154 b \cos (c+d x) \left (144 a^3 C+216 a^2 b B+4 a b^2 (36 A+43 C)+43 b^3 B\right )+5 \left (1848 a^4 C+7392 a^3 b B+36 b^2 \cos (2 (c+d x)) \left (66 a^2 C+44 a b B+11 A b^2+16 b^2 C\right )+792 a^2 b^2 (14 A+13 C)+154 b^3 (4 a C+b B) \cos (3 (c+d x))+6864 a b^3 B+3 b^4 (572 A+531 C)+63 b^4 C \cos (4 (c+d x))\right )\right )}{1155 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C b^{4} \cos \left (d x + c\right )^{6} + {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{5} + A a^{4} + {\left (6 \, C a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (2 \, C a^{3} b + 3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{4} + 4 \, B a^{3} b + 6 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}, x\right ) \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\sqrt {\cos \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.95, size = 1273, normalized size = 3.15 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\sqrt {\cos \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.96, size = 600, normalized size = 1.49 \[ \frac {2\,\left (A\,a^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+4\,A\,a^3\,b\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+2\,A\,a^2\,b^2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+2\,A\,a^2\,b^2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{d}+\frac {C\,a^4\,\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\,a^4\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {4\,B\,a^3\,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\,A\,b^4\,{\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\,B\,b^4\,{\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 {2\,C\,b^4\,{\cos \left (c+d\,x\right )}^{13/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {13}{4};\ \frac {17}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{13\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {8\,A\,a\,b^3\,{\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 {8\,B\,a\,b^3\,{\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 {8\,C\,a^3\,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 {8\,C\,a\,b^3\,{\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 {12\,B\,a^2\,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 {4\,C\,a^2\,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 )}{3\,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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