Optimal. Leaf size=135 \[ \frac {14 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 a \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}+\frac {14 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{45 d}+\frac {10 b F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}+\frac {10 b \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d} \]
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Rubi [A] time = 0.10, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {4225, 2748, 2635, 2641, 2639} \[ \frac {14 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 a \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}+\frac {14 a \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{45 d}+\frac {10 b F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}+\frac {10 b \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2641
Rule 2748
Rule 4225
Rubi steps
\begin {align*} \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x)) \, dx &=\int \cos ^{\frac {7}{2}}(c+d x) (b+a \cos (c+d x)) \, dx\\ &=a \int \cos ^{\frac {9}{2}}(c+d x) \, dx+b \int \cos ^{\frac {7}{2}}(c+d x) \, dx\\ &=\frac {2 b \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {2 a \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {1}{9} (7 a) \int \cos ^{\frac {5}{2}}(c+d x) \, dx+\frac {1}{7} (5 b) \int \cos ^{\frac {3}{2}}(c+d x) \, dx\\ &=\frac {10 b \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {14 a \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 b \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {2 a \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {1}{15} (7 a) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{21} (5 b) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {14 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {10 b F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {10 b \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {14 a \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 b \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {2 a \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 d}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 90, normalized size = 0.67 \[ \frac {\sqrt {\cos (c+d x)} (266 a \sin (2 (c+d x))+35 a \sin (4 (c+d x))+690 b \sin (c+d x)+90 b \sin (3 (c+d x)))+1176 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+600 b F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{1260 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \cos \left (d x + c\right )^{4} \sec \left (d x + c\right ) + a \cos \left (d x + c\right )^{4}\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]
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 \[ \int {\left (b \sec \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {9}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 4.02, size = 318, normalized size = 2.36 \[ -\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-1120 a \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2240 a +720 b \right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-2072 a -1080 b \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (952 a +840 b \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-168 a -240 b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+75 b \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-147 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a \right )}{315 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]
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 \[ \int {\left (b \sec \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {9}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.31, size = 87, normalized size = 0.64 \[ -\frac {2\,a\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,b\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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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