Optimal. Leaf size=122 \[ \frac {x (4 A+3 C) \sqrt {\cos (c+d x)}}{8 b \sqrt {b \cos (c+d x)}}+\frac {(4 A+3 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{8 b d \sqrt {b \cos (c+d x)}}+\frac {C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 b d \sqrt {b \cos (c+d x)}} \]
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Rubi [A] time = 0.06, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {17, 3014, 2635, 8} \[ \frac {x (4 A+3 C) \sqrt {\cos (c+d x)}}{8 b \sqrt {b \cos (c+d x)}}+\frac {(4 A+3 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{8 b d \sqrt {b \cos (c+d x)}}+\frac {C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 b d \sqrt {b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 17
Rule 2635
Rule 3014
Rubi steps
\begin {align*} \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{3/2}} \, dx &=\frac {\sqrt {\cos (c+d x)} \int \cos ^2(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx}{b \sqrt {b \cos (c+d x)}}\\ &=\frac {C \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{4 b d \sqrt {b \cos (c+d x)}}+\frac {\left ((4 A+3 C) \sqrt {\cos (c+d x)}\right ) \int \cos ^2(c+d x) \, dx}{4 b \sqrt {b \cos (c+d x)}}\\ &=\frac {(4 A+3 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{8 b d \sqrt {b \cos (c+d x)}}+\frac {C \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{4 b d \sqrt {b \cos (c+d x)}}+\frac {\left ((4 A+3 C) \sqrt {\cos (c+d x)}\right ) \int 1 \, dx}{8 b \sqrt {b \cos (c+d x)}}\\ &=\frac {(4 A+3 C) x \sqrt {\cos (c+d x)}}{8 b \sqrt {b \cos (c+d x)}}+\frac {(4 A+3 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{8 b d \sqrt {b \cos (c+d x)}}+\frac {C \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{4 b d \sqrt {b \cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 67, normalized size = 0.55 \[ \frac {\cos ^{\frac {3}{2}}(c+d x) (4 (4 A+3 C) (c+d x)+8 (A+C) \sin (2 (c+d x))+C \sin (4 (c+d x)))}{32 d (b \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 207, normalized size = 1.70 \[ \left [\frac {2 \, {\left (2 \, C \cos \left (d x + c\right )^{2} + 4 \, A + 3 \, C\right )} \sqrt {b \cos \left (d x + c\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - {\left (4 \, A + 3 \, C\right )} \sqrt {-b} \log \left (2 \, b \cos \left (d x + c\right )^{2} + 2 \, \sqrt {b \cos \left (d x + c\right )} \sqrt {-b} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - b\right )}{16 \, b^{2} d}, \frac {{\left (2 \, C \cos \left (d x + c\right )^{2} + 4 \, A + 3 \, C\right )} \sqrt {b \cos \left (d x + c\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + {\left (4 \, A + 3 \, C\right )} \sqrt {b} \arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{\sqrt {b} \cos \left (d x + c\right )^{\frac {3}{2}}}\right )}{8 \, b^{2} d}\right ] \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{\frac {7}{2}}}{\left (b \cos \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 88, normalized size = 0.72 \[ \frac {\left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \left (2 C \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )+4 A \cos \left (d x +c \right ) \sin \left (d x +c \right )+3 C \sin \left (d x +c \right ) \cos \left (d x +c \right )+4 A \left (d x +c \right )+3 C \left (d x +c \right )\right )}{8 d \left (b \cos \left (d x +c \right )\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.14, size = 75, normalized size = 0.61 \[ \frac {\frac {8 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A}{b^{\frac {3}{2}}} + \frac {{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, d x + 4 \, c\right ), \cos \left (4 \, d x + 4 \, c\right )\right )\right )\right )} C}{b^{\frac {3}{2}}}}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.96, size = 115, normalized size = 0.94 \[ \frac {\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (8\,A\,\sin \left (c+d\,x\right )+8\,C\,\sin \left (c+d\,x\right )+8\,A\,\sin \left (3\,c+3\,d\,x\right )+9\,C\,\sin \left (3\,c+3\,d\,x\right )+C\,\sin \left (5\,c+5\,d\,x\right )+32\,A\,d\,x\,\cos \left (c+d\,x\right )+24\,C\,d\,x\,\cos \left (c+d\,x\right )\right )}{32\,b^2\,d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \]
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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