Optimal. Leaf size=305 \[ \frac {2 F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (11 a^2 (7 A+5 C)+110 a b B+5 b^2 (11 A+9 C)\right )}{231 d}+\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (9 a^2 B+18 a A b+14 a b C+7 b^2 B\right )}{15 d}+\frac {2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (4 a^2 C+22 a b B+11 A b^2+9 b^2 C\right )}{77 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (9 a^2 B+18 a A b+14 a b C+7 b^2 B\right )}{45 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (11 a^2 (7 A+5 C)+110 a b B+5 b^2 (11 A+9 C)\right )}{231 d}+\frac {2 b (4 a C+11 b B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{99 d}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d} \]
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Rubi [A] time = 0.61, antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {3049, 3033, 3023, 2748, 2635, 2641, 2639} \[ \frac {2 F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (11 a^2 (7 A+5 C)+110 a b B+5 b^2 (11 A+9 C)\right )}{231 d}+\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (9 a^2 B+18 a A b+14 a b C+7 b^2 B\right )}{15 d}+\frac {2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (4 a^2 C+22 a b B+11 A b^2+9 b^2 C\right )}{77 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (9 a^2 B+18 a A b+14 a b C+7 b^2 B\right )}{45 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (11 a^2 (7 A+5 C)+110 a b B+5 b^2 (11 A+9 C)\right )}{231 d}+\frac {2 b (4 a C+11 b B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{99 d}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2641
Rule 2748
Rule 3023
Rule 3033
Rule 3049
Rubi steps
\begin {align*} \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac {2 C \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac {2}{11} \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x)) \left (\frac {1}{2} a (11 A+5 C)+\frac {1}{2} (11 A b+11 a B+9 b C) \cos (c+d x)+\frac {1}{2} (11 b B+4 a C) \cos ^2(c+d x)\right ) \, dx\\ &=\frac {2 b (11 b B+4 a C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 C \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac {4}{99} \int \cos ^{\frac {3}{2}}(c+d x) \left (\frac {9}{4} a^2 (11 A+5 C)+\frac {11}{4} \left (18 a A b+9 a^2 B+7 b^2 B+14 a b C\right ) \cos (c+d x)+\frac {9}{4} \left (11 A b^2+22 a b B+4 a^2 C+9 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac {2 \left (11 A b^2+22 a b B+4 a^2 C+9 b^2 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac {2 b (11 b B+4 a C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 C \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac {8}{693} \int \cos ^{\frac {3}{2}}(c+d x) \left (\frac {9}{8} \left (110 a b B+11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right )+\frac {77}{8} \left (18 a A b+9 a^2 B+7 b^2 B+14 a b C\right ) \cos (c+d x)\right ) \, dx\\ &=\frac {2 \left (11 A b^2+22 a b B+4 a^2 C+9 b^2 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac {2 b (11 b B+4 a C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 C \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac {1}{9} \left (18 a A b+9 a^2 B+7 b^2 B+14 a b C\right ) \int \cos ^{\frac {5}{2}}(c+d x) \, dx+\frac {1}{77} \left (110 a b B+11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx\\ &=\frac {2 \left (110 a b B+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 (18 a A b+9 a^2 B+7 b^2 B+14 a b C\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 \left (11 A b^2+22 a b B+4 a^2 C+9 b^2 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac {2 b (11 b B+4 a C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 C \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac {1}{15} \left (18 a A b+9 a^2 B+7 b^2 B+14 a b C\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{231} \left (110 a b B+11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 \left (18 a A b+9 a^2 B+7 b^2 B+14 a b C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (110 a b B+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 (110 a b B+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 (18 a A b+9 a^2 B+7 b^2 B+14 a b C\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 \left (11 A b^2+22 a b B+4 a^2 C+9 b^2 C\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac {2 b (11 b B+4 a C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 C \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d}\\ \end {align*}
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Mathematica [A] time = 1.67, size = 239, normalized size = 0.78 \[ \frac {10 F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (11 a^2 (7 A+5 C)+110 a b B+5 b^2 (11 A+9 C)\right )+154 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (9 a^2 B+2 a b (9 A+7 C)+7 b^2 B\right )+\frac {1}{12} \sin (c+d x) \sqrt {\cos (c+d x)} \left (154 \cos (c+d x) \left (36 a^2 B+72 a A b+86 a b C+43 b^2 B\right )+5 \left (36 \cos (2 (c+d x)) \left (11 a^2 C+22 a b B+11 A b^2+16 b^2 C\right )+132 a^2 (14 A+13 C)+154 b (2 a C+b B) \cos (3 (c+d x))+3432 a b B+3 b^2 (572 A+531 C)+63 b^2 C \cos (4 (c+d x))\right )\right )}{1155 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b^{2} \cos \left (d x + c\right )^{5} + {\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{4} + A a^{2} \cos \left (d x + c\right ) + {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{3} + {\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2} \cos \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.61, size = 863, normalized size = 2.83 \[ \text {result too large to display} \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2} \cos \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.41, size = 401, normalized size = 1.31 \[ \frac {2\,A\,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\,{\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\,A\,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\,a^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\,B\,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 {2\,C\,b^2\,{\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 {4\,A\,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\,B\,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}}-\frac {4\,C\,a\,b\,{\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}} \]
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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