Optimal. Leaf size=264 \[ \frac {2 \left (9 a^2 B+14 a b C+7 b^2 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (9 a^2 B+14 a b C+7 b^2 B\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{45 d}+\frac {10 \left (11 a (a C+2 b B)+9 b^2 C\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {2 \left (11 a (a C+2 b B)+9 b^2 C\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{77 d}+\frac {10 \left (11 a (a C+2 b B)+9 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{231 d}+\frac {2 b (13 a C+11 b B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{99 d}+\frac {2 b C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))}{11 d} \]
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Rubi [A] time = 0.46, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3029, 2990, 3023, 2748, 2635, 2639, 2641} \[ \frac {2 \left (9 a^2 B+14 a b C+7 b^2 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (9 a^2 B+14 a b C+7 b^2 B\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{45 d}+\frac {10 \left (11 a (a C+2 b B)+9 b^2 C\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {2 \left (11 a (a C+2 b B)+9 b^2 C\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{77 d}+\frac {10 \left (11 a (a C+2 b B)+9 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{231 d}+\frac {2 b (13 a C+11 b B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{99 d}+\frac {2 b C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))}{11 d} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2641
Rule 2748
Rule 2990
Rule 3023
Rule 3029
Rubi steps
\begin {align*} \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\int \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2 (B+C \cos (c+d x)) \, dx\\ &=\frac {2 b C \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{11 d}+\frac {2}{11} \int \cos ^{\frac {5}{2}}(c+d x) \left (\frac {1}{2} a (11 a B+7 b C)+\frac {1}{2} \left (9 b^2 C+11 a (2 b B+a C)\right ) \cos (c+d x)+\frac {1}{2} b (11 b B+13 a C) \cos ^2(c+d x)\right ) \, dx\\ &=\frac {2 b (11 b B+13 a C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 b C \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{11 d}+\frac {4}{99} \int \cos ^{\frac {5}{2}}(c+d x) \left (\frac {11}{4} \left (9 a^2 B+7 b^2 B+14 a b C\right )+\frac {9}{4} \left (9 b^2 C+11 a (2 b B+a C)\right ) \cos (c+d x)\right ) \, dx\\ &=\frac {2 b (11 b B+13 a C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 b C \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{11 d}+\frac {1}{9} \left (9 a^2 B+7 b^2 B+14 a b C\right ) \int \cos ^{\frac {5}{2}}(c+d x) \, dx+\frac {1}{11} \left (9 b^2 C+11 a (2 b B+a C)\right ) \int \cos ^{\frac {7}{2}}(c+d x) \, dx\\ &=\frac {2 \left (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 (9 b^2 C+11 a (2 b B+a C)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac {2 b (11 b B+13 a C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 b C \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{11 d}+\frac {1}{15} \left (9 a^2 B+7 b^2 B+14 a b C\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{77} \left (5 \left (9 b^2 C+11 a (2 b B+a C)\right )\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx\\ &=\frac {2 \left (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 {10 \left (9 b^2 C+11 a (2 b B+a C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (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 (9 b^2 C+11 a (2 b B+a C)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac {2 b (11 b B+13 a C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 b C \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{11 d}+\frac {1}{231} \left (5 \left (9 b^2 C+11 a (2 b B+a C)\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 \left (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 {10 \left (9 b^2 C+11 a (2 b B+a C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {10 \left (9 b^2 C+11 a (2 b B+a C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (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 (9 b^2 C+11 a (2 b B+a C)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac {2 b (11 b B+13 a C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{99 d}+\frac {2 b C \cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{11 d}\\ \end {align*}
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Mathematica [A] time = 1.74, size = 196, normalized size = 0.74 \[ \frac {1200 \left (11 a^2 C+22 a b B+9 b^2 C\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+3696 \left (9 a^2 B+14 a b C+7 b^2 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (154 \left (36 a^2 B+86 a b C+43 b^2 B\right ) \cos (c+d x)+180 \left (11 a^2 C+22 a b B+16 b^2 C\right ) \cos (2 (c+d x))+15 \left (572 a^2 C+1144 a b B+21 b^2 C \cos (4 (c+d x))+531 b^2 C\right )+770 b (2 a C+b B) \cos (3 (c+d x))\right )}{27720 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b^{2} \cos \left (d x + c\right )^{5} + B a^{2} \cos \left (d x + c\right )^{2} + {\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{4} + {\left (C a^{2} + 2 \, B a b\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2} \cos \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.14, size = 666, normalized size = 2.52 \[ -\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (20160 C \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-12320 b^{2} B -24640 C a b -50400 b^{2} C \right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (15840 B a b +24640 b^{2} B +7920 a^{2} C +49280 C a b +56880 b^{2} C \right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-5544 a^{2} B -23760 B a b -22792 b^{2} B -11880 a^{2} C -45584 C a b -34920 b^{2} C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (5544 a^{2} B +18480 B a b +10472 b^{2} B +9240 a^{2} C +20944 C a b +13860 b^{2} C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-1386 a^{2} B -5280 B a b -1848 b^{2} B -2640 a^{2} C -3696 C a b -2790 b^{2} C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-2079 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2}-1617 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{2}+1650 B a b \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-3234 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a b +825 a^{2} C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+675 b^{2} C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{3465 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2} \cos \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.78, size = 275, normalized size = 1.04 \[ -\frac {2\,B\,a^2\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,a^2\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,b^2\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,b^2\,{\cos \left (c+d\,x\right )}^{13/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {13}{4};\ \frac {17}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{13\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {4\,B\,a\,b\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {4\,C\,a\,b\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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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