Optimal. Leaf size=255 \[ \frac {2 b \left (22 a^2 C+27 a b B+7 b^2 C\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{45 d}+\frac {2 \left (7 a^3 C+21 a^2 b B+15 a b^2 C+5 b^3 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 \left (15 a^3 B+27 a^2 b C+27 a b^2 B+7 b^3 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (7 a^3 C+21 a^2 b B+15 a b^2 C+5 b^3 B\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d}+\frac {2 b^2 (13 a C+9 b B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{63 d}+\frac {2 b C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d} \]
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Rubi [A] time = 0.58, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3029, 2990, 3033, 3023, 2748, 2639, 2635, 2641} \[ \frac {2 \left (21 a^2 b B+7 a^3 C+15 a b^2 C+5 b^3 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 \left (27 a^2 b C+15 a^3 B+27 a b^2 B+7 b^3 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 b \left (22 a^2 C+27 a b B+7 b^2 C\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{45 d}+\frac {2 \left (21 a^2 b B+7 a^3 C+15 a b^2 C+5 b^3 B\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d}+\frac {2 b^2 (13 a C+9 b B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{63 d}+\frac {2 b C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2641
Rule 2748
Rule 2990
Rule 3023
Rule 3029
Rule 3033
Rubi steps
\begin {align*} \int \frac {(a+b \cos (c+d x))^3 \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))^3 (B+C \cos (c+d x)) \, dx\\ &=\frac {2 b C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac {2}{9} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x)) \left (\frac {3}{2} a (3 a B+b C)+\frac {1}{2} \left (7 b^2 C+9 a (2 b B+a C)\right ) \cos (c+d x)+\frac {1}{2} b (9 b B+13 a C) \cos ^2(c+d x)\right ) \, dx\\ &=\frac {2 b^2 (9 b B+13 a C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac {4}{63} \int \sqrt {\cos (c+d x)} \left (\frac {21}{4} a^2 (3 a B+b C)+\frac {9}{4} \left (21 a^2 b B+5 b^3 B+7 a^3 C+15 a b^2 C\right ) \cos (c+d x)+\frac {7}{4} b \left (27 a b B+22 a^2 C+7 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac {2 b \left (27 a b B+22 a^2 C+7 b^2 C\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 b^2 (9 b B+13 a C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac {8}{315} \int \sqrt {\cos (c+d x)} \left (\frac {21}{8} \left (15 a^3 B+27 a b^2 B+27 a^2 b C+7 b^3 C\right )+\frac {45}{8} \left (21 a^2 b B+5 b^3 B+7 a^3 C+15 a b^2 C\right ) \cos (c+d x)\right ) \, dx\\ &=\frac {2 b \left (27 a b B+22 a^2 C+7 b^2 C\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 b^2 (9 b B+13 a C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac {1}{7} \left (21 a^2 b B+5 b^3 B+7 a^3 C+15 a b^2 C\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{15} \left (15 a^3 B+27 a b^2 B+27 a^2 b C+7 b^3 C\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {2 \left (15 a^3 B+27 a b^2 B+27 a^2 b C+7 b^3 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (21 a^2 b B+5 b^3 B+7 a^3 C+15 a b^2 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 b \left (27 a b B+22 a^2 C+7 b^2 C\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 b^2 (9 b B+13 a C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac {1}{21} \left (21 a^2 b B+5 b^3 B+7 a^3 C+15 a b^2 C\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 \left (15 a^3 B+27 a b^2 B+27 a^2 b C+7 b^3 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (21 a^2 b B+5 b^3 B+7 a^3 C+15 a b^2 C\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 \left (21 a^2 b B+5 b^3 B+7 a^3 C+15 a b^2 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 b \left (27 a b B+22 a^2 C+7 b^2 C\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 b^2 (9 b B+13 a C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{9 d}\\ \end {align*}
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Mathematica [A] time = 1.22, size = 197, normalized size = 0.77 \[ \frac {60 \left (7 a^3 C+21 a^2 b B+15 a b^2 C+5 b^3 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+84 \left (15 a^3 B+27 a^2 b C+27 a b^2 B+7 b^3 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\sin (c+d x) \sqrt {\cos (c+d x)} \left (7 b \left (108 a^2 C+108 a b B+43 b^2 C\right ) \cos (c+d x)+5 \left (84 a^3 C+252 a^2 b B+18 b^2 (3 a C+b B) \cos (2 (c+d x))+234 a b^2 C+78 b^3 B+7 b^3 C \cos (3 (c+d x))\right )\right )}{630 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b^{3} \cos \left (d x + c\right )^{4} + B a^{3} + {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (C a^{2} b + B a b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (C a^{3} + 3 \, B a^{2} b\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\sqrt {\cos \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.35, size = 745, normalized size = 2.92 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\sqrt {\cos \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.88, size = 328, normalized size = 1.29 \[ \frac {2\,\left (B\,a^3\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+B\,a^2\,b\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+B\,a^2\,b\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{d}+\frac {C\,a^3\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}-\frac {2\,B\,b^3\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,b^3\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {6\,B\,a\,b^2\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {6\,C\,a^2\,b\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,a\,b^2\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{3\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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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