Optimal. Leaf size=71 \[ \frac {2 (a C+b B) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 (a B-b C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {2 a B \sin (c+d x)}{d \sqrt {\cos (c+d x)}} \]
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Rubi [A] time = 0.21, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3029, 2968, 3021, 2748, 2641, 2639} \[ \frac {2 (a C+b B) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 (a B-b C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {2 a B \sin (c+d x)}{d \sqrt {\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2748
Rule 2968
Rule 3021
Rule 3029
Rubi steps
\begin {align*} \int \frac {(a+b \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx &=\int \frac {(a+b \cos (c+d x)) (B+C \cos (c+d x))}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\int \frac {a B+(b B+a C) \cos (c+d x)+b C \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 a B \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+2 \int \frac {\frac {1}{2} (b B+a C)-\frac {1}{2} (a B-b C) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 a B \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+(b B+a C) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+(-a B+b C) \int \sqrt {\cos (c+d x)} \, dx\\ &=-\frac {2 (a B-b C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {2 (b B+a C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {2 a B \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 64, normalized size = 0.90 \[ \frac {2 \left ((a C+b B) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+(b C-a B) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\frac {a B \sin (c+d x)}{\sqrt {\cos (c+d x)}}\right )}{d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C b \cos \left (d x + c\right )^{2} + B a + {\left (C a + B b\right )} \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{\frac {3}{2}}}, x\right ) \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}}{\cos \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.96, size = 244, normalized size = 3.44 \[ -\frac {2 \left (B b \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a -2 B a \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b \right )}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}}{\cos \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.69, size = 96, normalized size = 1.35 \[ \frac {2\,B\,b\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {2\,C\,a\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {2\,C\,b\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {2\,B\,a\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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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