Optimal. Leaf size=140 \[ \frac {2 \left (3 a^2 B+2 a b C+b^2 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}+\frac {2 \left (5 a (a C+2 b B)+3 b^2 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 b (7 a C+5 b B) \sin (c+d x) \sqrt {\cos (c+d x)}}{15 d}+\frac {2 b C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}{5 d} \]
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Rubi [A] time = 0.36, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3029, 2990, 3023, 2748, 2641, 2639} \[ \frac {2 \left (3 a^2 B+2 a b C+b^2 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}+\frac {2 \left (5 a (a C+2 b B)+3 b^2 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 b (7 a C+5 b B) \sin (c+d x) \sqrt {\cos (c+d x)}}{15 d}+\frac {2 b C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}{5 d} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2748
Rule 2990
Rule 3023
Rule 3029
Rubi steps
\begin {align*} \int \frac {(a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx &=\int \frac {(a+b \cos (c+d x))^2 (B+C \cos (c+d x))}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 b C \sqrt {\cos (c+d x)} (a+b \cos (c+d x)) \sin (c+d x)}{5 d}+\frac {2}{5} \int \frac {\frac {1}{2} a (5 a B+b C)+\frac {1}{2} \left (3 b^2 C+5 a (2 b B+a C)\right ) \cos (c+d x)+\frac {1}{2} b (5 b B+7 a C) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 b (5 b B+7 a C) \sqrt {\cos (c+d x)} \sin (c+d x)}{15 d}+\frac {2 b C \sqrt {\cos (c+d x)} (a+b \cos (c+d x)) \sin (c+d x)}{5 d}+\frac {4}{15} \int \frac {\frac {5}{4} \left (3 a^2 B+b^2 B+2 a b C\right )+\frac {3}{4} \left (3 b^2 C+5 a (2 b B+a C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 b (5 b B+7 a C) \sqrt {\cos (c+d x)} \sin (c+d x)}{15 d}+\frac {2 b C \sqrt {\cos (c+d x)} (a+b \cos (c+d x)) \sin (c+d x)}{5 d}+\frac {1}{3} \left (3 a^2 B+b^2 B+2 a b C\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+\frac {1}{5} \left (3 b^2 C+5 a (2 b B+a C)\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {2 \left (3 b^2 C+5 a (2 b B+a C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \left (3 a^2 B+b^2 B+2 a b C\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}+\frac {2 b (5 b B+7 a C) \sqrt {\cos (c+d x)} \sin (c+d x)}{15 d}+\frac {2 b C \sqrt {\cos (c+d x)} (a+b \cos (c+d x)) \sin (c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.61, size = 106, normalized size = 0.76 \[ \frac {2 \left (5 \left (3 a^2 B+2 a b C+b^2 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+3 \left (5 a^2 C+10 a b B+3 b^2 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+b \sin (c+d x) \sqrt {\cos (c+d x)} (10 a C+5 b B+3 b C \cos (c+d x))\right )}{15 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C b^{2} \cos \left (d x + c\right )^{3} + B a^{2} + {\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (C a^{2} + 2 \, B a b\right )} \cos \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}, x\right ) \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2}}{\cos \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.99, size = 487, normalized size = 3.48 \[ -\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 (-24 C \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (20 b^{2} B +40 C a b +24 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 (-10 b^{2} B -20 C a b -6 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 )+15 a^{2} 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 )+5 b^{2} 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 )-30 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 b +10 C 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 )-15 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^{2}-9 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 ) b^{2}\right )}{15 \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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2}}{\cos \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.65, size = 177, normalized size = 1.26 \[ \frac {B\,b^2\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}+\frac {2\,B\,a^2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {2\,C\,a^2\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {2\,C\,a\,b\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}+\frac {4\,B\,a\,b\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}-\frac {2\,C\,b^2\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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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