Optimal. Leaf size=151 \[ \frac {2 A \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 A \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 B \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {10 B \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {10 B \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d} \]
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Rubi [A] time = 0.11, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3238, 3787, 3769, 3771, 2641, 2639} \[ \frac {2 A \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 A \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 B \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {10 B \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {10 B \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 3238
Rule 3769
Rule 3771
Rule 3787
Rubi steps
\begin {align*} \int \frac {A+B \cos (c+d x)}{\sec ^{\frac {5}{2}}(c+d x)} \, dx &=\int \frac {B+A \sec (c+d x)}{\sec ^{\frac {7}{2}}(c+d x)} \, dx\\ &=A \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x)} \, dx+B \int \frac {1}{\sec ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {2 B \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{5} (3 A) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{7} (5 B) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 B \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {10 B \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {1}{21} (5 B) \int \sqrt {\sec (c+d x)} \, dx+\frac {1}{5} \left (3 A \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {6 A \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 B \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {10 B \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {1}{21} \left (5 B \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {6 A \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {10 B \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 B \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {10 B \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.56, size = 99, normalized size = 0.66 \[ \frac {\sqrt {\sec (c+d x)} \left (\sin (2 (c+d x)) (42 A \cos (c+d x)+15 B \cos (2 (c+d x))+65 B)+252 A \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+100 B \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{210 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {B \cos \left (d x + c\right ) + A}{\sec \left (d x + c\right )^{\frac {5}{2}}}, x\right ) \]
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 \[ \int \frac {B \cos \left (d x + c\right ) + A}{\sec \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.74, size = 290, normalized size = 1.92 \[ -\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (240 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-168 A -360 B \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (168 A +280 B \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-42 A -80 B \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-63 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+25 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{105 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]
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 \[ \int \frac {B \cos \left (d x + c\right ) + A}{\sec \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {A+B\,\cos \left (c+d\,x\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
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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