Optimal. Leaf size=155 \[ \frac {(3 A+2 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b^2 d \sqrt {b \cos (c+d x)}}+\frac {B x \sqrt {\cos (c+d x)}}{2 b^2 \sqrt {b \cos (c+d x)}}+\frac {B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 b^2 d \sqrt {b \cos (c+d x)}}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 b^2 d \sqrt {b \cos (c+d x)}} \]
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Rubi [A] time = 0.06, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {17, 3023, 2734} \[ \frac {(3 A+2 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b^2 d \sqrt {b \cos (c+d x)}}+\frac {B x \sqrt {\cos (c+d x)}}{2 b^2 \sqrt {b \cos (c+d x)}}+\frac {B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 b^2 d \sqrt {b \cos (c+d x)}}+\frac {C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{3 b^2 d \sqrt {b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 2734
Rule 3023
Rubi steps
\begin {align*} \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{5/2}} \, dx &=\frac {\sqrt {\cos (c+d x)} \int \cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx}{b^2 \sqrt {b \cos (c+d x)}}\\ &=\frac {C \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3 b^2 d \sqrt {b \cos (c+d x)}}+\frac {\sqrt {\cos (c+d x)} \int \cos (c+d x) (3 A+2 C+3 B \cos (c+d x)) \, dx}{3 b^2 \sqrt {b \cos (c+d x)}}\\ &=\frac {B x \sqrt {\cos (c+d x)}}{2 b^2 \sqrt {b \cos (c+d x)}}+\frac {(3 A+2 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{3 b^2 d \sqrt {b \cos (c+d x)}}+\frac {B \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 b^2 d \sqrt {b \cos (c+d x)}}+\frac {C \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{3 b^2 d \sqrt {b \cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 78, normalized size = 0.50 \[ \frac {\sqrt {\cos (c+d x)} (3 (4 A+3 C) \sin (c+d x)+3 B \sin (2 (c+d x))+6 B c+6 B d x+C \sin (3 (c+d x)))}{12 b^2 d \sqrt {b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 242, normalized size = 1.56 \[ \left [-\frac {3 \, B \sqrt {-b} \cos \left (d x + c\right ) \log \left (2 \, b \cos \left (d x + c\right )^{2} + 2 \, \sqrt {b \cos \left (d x + c\right )} \sqrt {-b} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - b\right ) - 2 \, {\left (2 \, C \cos \left (d x + c\right )^{2} + 3 \, B \cos \left (d x + c\right ) + 6 \, A + 4 \, C\right )} \sqrt {b \cos \left (d x + c\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{12 \, b^{3} d \cos \left (d x + c\right )}, \frac {3 \, B \sqrt {b} \arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{\sqrt {b} \cos \left (d x + c\right )^{\frac {3}{2}}}\right ) \cos \left (d x + c\right ) + {\left (2 \, C \cos \left (d x + c\right )^{2} + 3 \, B \cos \left (d x + c\right ) + 6 \, A + 4 \, C\right )} \sqrt {b \cos \left (d x + c\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{6 \, b^{3} d \cos \left (d x + c\right )}\right ] \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {7}{2}}}{\left (b \cos \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 83, normalized size = 0.54 \[ \frac {\left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \left (2 C \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )+3 B \cos \left (d x +c \right ) \sin \left (d x +c \right )+6 A \sin \left (d x +c \right )+3 B \left (d x +c \right )+4 C \sin \left (d x +c \right )\right )}{6 d \left (b \cos \left (d x +c \right )\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 80, normalized size = 0.52 \[ \frac {\frac {3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B}{b^{\frac {5}{2}}} + \frac {C {\left (\sin \left (3 \, d x + 3 \, c\right ) + 9 \, \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (3 \, d x + 3 \, c\right ), \cos \left (3 \, d x + 3 \, c\right )\right )\right )\right )}}{b^{\frac {5}{2}}} + \frac {12 \, A \sin \left (d x + c\right )}{b^{\frac {5}{2}}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.16, size = 107, normalized size = 0.69 \[ \frac {\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (3\,B\,\sin \left (c+d\,x\right )+12\,A\,\sin \left (2\,c+2\,d\,x\right )+3\,B\,\sin \left (3\,c+3\,d\,x\right )+10\,C\,\sin \left (2\,c+2\,d\,x\right )+C\,\sin \left (4\,c+4\,d\,x\right )+12\,B\,d\,x\,\cos \left (c+d\,x\right )\right )}{12\,b^3\,d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \]
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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