Optimal. Leaf size=382 \[ \frac {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{1287 d}+\frac {4 a b \left (96 a^2 C+1573 A b^2+1259 b^2 C\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{9009 d}+\frac {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{231 d}+\frac {2 \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {2 \left (192 a^4 C+11 a^2 b^2 (637 A+491 C)+77 b^4 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{6435 d}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}+\frac {16 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{143 d} \]
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Rubi [A] time = 1.15, antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {3050, 3049, 3033, 3023, 2748, 2639, 2635, 2641} \[ \frac {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {2 \left (78 a^2 b^2 (9 A+7 C)+39 a^4 (5 A+3 C)+7 b^4 (13 A+11 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{1287 d}+\frac {4 a b \left (96 a^2 C+1573 A b^2+1259 b^2 C\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{9009 d}+\frac {2 \left (11 a^2 b^2 (637 A+491 C)+192 a^4 C+77 b^4 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{6435 d}+\frac {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{231 d}+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}+\frac {16 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{143 d} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2641
Rule 2748
Rule 3023
Rule 3033
Rule 3049
Rule 3050
Rubi steps
\begin {align*} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {2 C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {2}{13} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \left (\frac {1}{2} a (13 A+3 C)+\frac {1}{2} b (13 A+11 C) \cos (c+d x)+4 a C \cos ^2(c+d x)\right ) \, dx\\ &=\frac {16 a C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {4}{143} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \left (\frac {1}{4} a^2 (143 A+57 C)+\frac {1}{2} a b (143 A+113 C) \cos (c+d x)+\frac {1}{4} \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac {16 a C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {8 \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x)) \left (\frac {3}{8} a \left (11 b^2 (13 A+11 C)+3 a^2 (143 A+73 C)\right )+\frac {1}{8} b \left (77 b^2 (13 A+11 C)+3 a^2 (1287 A+961 C)\right ) \cos (c+d x)+\frac {1}{4} a \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx}{1287}\\ &=\frac {4 a b \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac {16 a C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {16 \int \sqrt {\cos (c+d x)} \left (\frac {21}{16} a^2 \left (11 b^2 (13 A+11 C)+3 a^2 (143 A+73 C)\right )+\frac {117}{4} a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \cos (c+d x)+\frac {7}{16} \left (192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \cos ^2(c+d x)\right ) \, dx}{9009}\\ &=\frac {2 \left (192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6435 d}+\frac {4 a b \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac {16 a C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {32 \int \sqrt {\cos (c+d x)} \left (\frac {231}{32} \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right )+\frac {585}{8} a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \cos (c+d x)\right ) \, dx}{45045}\\ &=\frac {2 \left (192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6435 d}+\frac {4 a b \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac {16 a C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {1}{77} \left (4 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right )\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{195} \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {2 \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6435 d}+\frac {4 a b \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac {16 a C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}+\frac {1}{231} \left (4 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6435 d}+\frac {4 a b \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac {16 a C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d}\\ \end {align*}
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Mathematica [A] time = 2.95, size = 281, normalized size = 0.74 \[ \frac {24960 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+7392 \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (5 b \left (77 \left (312 a^2 b C+52 A b^3+89 b^3 C\right ) \cos (3 (c+d x))+3744 a \left (11 a^2 C+11 A b^2+16 b^2 C\right ) \cos (2 (c+d x))+312 a \left (44 a^2 (14 A+13 C)+b^2 (572 A+531 C)\right )+6552 a b^2 C \cos (4 (c+d x))+693 b^3 C \cos (5 (c+d x))\right )+154 \left (936 a^4 C+156 a^2 b^2 (36 A+43 C)+b^4 (1118 A+1171 C)\right ) \cos (c+d x)\right )}{720720 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b^{4} \cos \left (d x + c\right )^{6} + 4 \, C a b^{3} \cos \left (d x + c\right )^{5} + 4 \, A a^{3} b \cos \left (d x + c\right ) + A a^{4} + {\left (6 \, C a^{2} b^{2} + A b^{4}\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (C a^{3} b + A a b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{4} + 6 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sqrt {\cos \left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.31, size = 1017, normalized size = 2.66 \[ \text {result too large to display} \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sqrt {\cos \left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.57, size = 677, normalized size = 1.77 \[ \frac {2\,A\,a^4\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}-\frac {136\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {15}{4};\ \frac {23}{4};\ {\cos \left (c+d\,x\right )}^2\right )\,\left (\frac {11\,C\,a^4\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {9\,C\,a^4\,{\cos \left (c+d\,x\right )}^{15/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {42\,C\,a^2\,b^2\,{\cos \left (c+d\,x\right )}^{15/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}\right )}{21945\,d}-\frac {2\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {15}{4};\ \frac {19}{4};\ {\cos \left (c+d\,x\right )}^2\right )\,\left (\frac {165\,C\,a^4\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {52\,C\,a^4\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {36\,C\,a^4\,{\cos \left (c+d\,x\right )}^{15/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {77\,C\,b^4\,{\cos \left (c+d\,x\right )}^{15/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {630\,C\,a^2\,b^2\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {168\,C\,a^2\,b^2\,{\cos \left (c+d\,x\right )}^{15/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}\right )}{1155\,d}-\frac {8\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {13}{4};\ \frac {17}{4};\ {\cos \left (c+d\,x\right )}^2\right )\,\left (\frac {13\,C\,a^3\,b\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {9\,C\,a\,b^3\,{\cos \left (c+d\,x\right )}^{13/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {4\,C\,a^3\,b\,{\cos \left (c+d\,x\right )}^{13/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}\right )}{117\,d}+\frac {4\,A\,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 )}^{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 {8\,A\,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 {160\,C\,a^3\,b\,{\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 {21}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{663\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {12\,A\,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}} \]
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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