Optimal. Leaf size=329 \[ \frac {8 a b \left (3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {4 a b \left (96 a^2 C+891 A b^2+673 b^2 C\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{3465 d}+\frac {2 \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{231 d}+\frac {2 \left (77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {2 \left (64 a^4 C+9 a^2 b^2 (143 A+101 C)+15 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{693 d}+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}+\frac {16 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{99 d} \]
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Rubi [A] time = 1.09, antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3050, 3049, 3033, 3023, 2748, 2641, 2639} \[ \frac {2 \left (66 a^2 b^2 (7 A+5 C)+77 a^4 (3 A+C)+5 b^4 (11 A+9 C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {8 a b \left (3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {4 a b \left (96 a^2 C+891 A b^2+673 b^2 C\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{3465 d}+\frac {2 \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{231 d}+\frac {2 \left (9 a^2 b^2 (143 A+101 C)+64 a^4 C+15 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{693 d}+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}+\frac {16 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{99 d} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2748
Rule 3023
Rule 3033
Rule 3049
Rule 3050
Rubi steps
\begin {align*} \int \frac {(a+b \cos (c+d x))^4 \left (A+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} b (11 A+9 C) \cos (c+d x)+4 a C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {16 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^2 (99 A+17 C)+\frac {1}{2} a b (99 A+73 C) \cos (c+d x)+\frac {3}{4} \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d}+\frac {16 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 (9 b^2 (11 A+9 C)+a^2 (693 A+167 C)\right )+\frac {1}{8} b \left (45 b^2 (11 A+9 C)+a^2 (2079 A+1381 C)\right ) \cos (c+d x)+\frac {1}{4} a \left (891 A b^2+96 a^2 C+673 b^2 C\right ) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {4 a b \left (891 A b^2+96 a^2 C+673 b^2 C\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac {2 \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d}+\frac {16 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 (9 b^2 (11 A+9 C)+a^2 (693 A+167 C)\right )+\frac {231}{4} a b \left (3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \cos (c+d x)+\frac {15}{16} \left (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 (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 {4 a b \left (891 A b^2+96 a^2 C+673 b^2 C\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac {2 \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d}+\frac {16 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 (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}{8} a b \left (3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx}{10395}\\ &=\frac {2 \left (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 {4 a b \left (891 A b^2+96 a^2 C+673 b^2 C\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac {2 \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d}+\frac {16 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 (4 a b \left (3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right )\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{231} \left (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 {8 a b \left (3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (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 (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 {4 a b \left (891 A b^2+96 a^2 C+673 b^2 C\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac {2 \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d}+\frac {16 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 = 2.12, size = 243, normalized size = 0.74 \[ \frac {14784 \left (3 a^3 b (5 A+3 C)+a b^3 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+240 \left (77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (616 a b \left (36 a^2 C+36 A b^2+43 b^2 C\right ) \cos (c+d x)+5 \left (1848 a^4 C+792 a^2 b^2 (14 A+13 C)+36 \left (66 a^2 b^2 C+11 A b^4+16 b^4 C\right ) \cos (2 (c+d x))+616 a b^3 C \cos (3 (c+d x))+3 b^4 (572 A+531 C)+63 b^4 C \cos (4 (c+d x))\right )\right )}{27720 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C b^{4} \cos \left (d x + c\right )^{6} + 4 \, C a b^{3} \cos \left (d x + c\right )^{5} + 4 \, A a^{3} b \cos \left (d x + c\right ) + A a^{4} + {\left (6 \, C a^{2} b^{2} + A b^{4}\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (C a^{3} b + A a b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{4} + 6 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}}{\sqrt {\cos \left (d x + c\right )}}, x\right ) \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\sqrt {\cos \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.16, size = 924, normalized size = 2.81 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\sqrt {\cos \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.19, size = 400, normalized size = 1.22 \[ \frac {2\,\left (A\,a^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+4\,A\,a^3\,b\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+2\,A\,a^2\,b^2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+2\,A\,a^2\,b^2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{d}+\frac {C\,a^4\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}-\frac {2\,A\,b^4\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,b^4\,{\cos \left (c+d\,x\right )}^{13/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {13}{4};\ \frac {17}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{13\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {8\,A\,a\,b^3\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {8\,C\,a^3\,b\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {8\,C\,a\,b^3\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {4\,C\,a^2\,b^2\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{3\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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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