Optimal. Leaf size=80 \[ \frac {(A+C) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d \sqrt {b \cos (c+d x)}}-\frac {C \sin ^3(c+d x) \sqrt {\cos (c+d x)}}{3 b d \sqrt {b \cos (c+d x)}} \]
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Rubi [A] time = 0.03, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {17, 3013} \[ \frac {(A+C) \sin (c+d x) \sqrt {\cos (c+d x)}}{b d \sqrt {b \cos (c+d x)}}-\frac {C \sin ^3(c+d x) \sqrt {\cos (c+d x)}}{3 b d \sqrt {b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 3013
Rubi steps
\begin {align*} \int \frac {\cos ^{\frac {5}{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 (c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx}{b \sqrt {b \cos (c+d x)}}\\ &=-\frac {\sqrt {\cos (c+d x)} \operatorname {Subst}\left (\int \left (A+C-C x^2\right ) \, dx,x,-\sin (c+d x)\right )}{b d \sqrt {b \cos (c+d x)}}\\ &=\frac {(A+C) \sqrt {\cos (c+d x)} \sin (c+d x)}{b d \sqrt {b \cos (c+d x)}}-\frac {C \sqrt {\cos (c+d x)} \sin ^3(c+d x)}{3 b d \sqrt {b \cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 52, normalized size = 0.65 \[ \frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (6 A+C \cos (2 (c+d x))+5 C)}{6 d (b \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 49, normalized size = 0.61 \[ \frac {{\left (C \cos \left (d x + c\right )^{2} + 3 \, A + 2 \, C\right )} \sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{3 \, b^{2} d \sqrt {\cos \left (d x + c\right )}} \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{\frac {5}{2}}}{\left (b \cos \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 47, normalized size = 0.59 \[ \frac {\left (C \left (\cos ^{2}\left (d x +c \right )\right )+3 A +2 C \right ) \sin \left (d x +c \right ) \left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right )}{3 d \left (b \cos \left (d x +c \right )\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 57, normalized size = 0.71 \[ \frac {\frac {C {\left (\sin \left (3 \, d x + 3 \, c\right ) + 9 \, \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (3 \, d x + 3 \, c\right ), \cos \left (3 \, d x + 3 \, c\right )\right )\right )\right )}}{b^{\frac {3}{2}}} + \frac {12 \, A \sin \left (d x + c\right )}{b^{\frac {3}{2}}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.84, size = 75, normalized size = 0.94 \[ \frac {\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (12\,A\,\sin \left (2\,c+2\,d\,x\right )+10\,C\,\sin \left (2\,c+2\,d\,x\right )+C\,\sin \left (4\,c+4\,d\,x\right )\right )}{12\,b^2\,d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \]
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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