Optimal. Leaf size=80 \[ \frac {(A+C) \sqrt {\cos (c+d x)} \sin (c+d x)}{b^2 d \sqrt {b \cos (c+d x)}}-\frac {C \sqrt {\cos (c+d x)} \sin ^3(c+d x)}{3 b^2 d \sqrt {b \cos (c+d x)}} \]
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Rubi [A]
time = 0.02, 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, 3092}
\begin {gather*} \frac {(A+C) \sin (c+d x) \sqrt {\cos (c+d x)}}{b^2 d \sqrt {b \cos (c+d x)}}-\frac {C \sin ^3(c+d x) \sqrt {\cos (c+d x)}}{3 b^2 d \sqrt {b \cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 17
Rule 3092
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))^{5/2}} \, dx &=\frac {\sqrt {\cos (c+d x)} \int \cos (c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx}{b^2 \sqrt {b \cos (c+d x)}}\\ &=-\frac {\sqrt {\cos (c+d x)} \text {Subst}\left (\int \left (A+C-C x^2\right ) \, dx,x,-\sin (c+d x)\right )}{b^2 d \sqrt {b \cos (c+d x)}}\\ &=\frac {(A+C) \sqrt {\cos (c+d x)} \sin (c+d x)}{b^2 d \sqrt {b \cos (c+d x)}}-\frac {C \sqrt {\cos (c+d x)} \sin ^3(c+d x)}{3 b^2 d \sqrt {b \cos (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 55, normalized size = 0.69 \begin {gather*} \frac {\sqrt {\cos (c+d x)} (6 A+5 C+C \cos (2 (c+d x))) \sin (c+d x)}{6 b^2 d \sqrt {b \cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 47, normalized size = 0.59
method | result | size |
default | \(\frac {\left (C \left (\cos ^{2}\left (d x +c \right )\right )+3 A +2 C \right ) \sin \left (d x +c \right ) \left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right )}{3 d \left (b \cos \left (d x +c \right )\right )^{\frac {5}{2}}}\) | \(47\) |
risch | \(\frac {\left (\sqrt {\cos }\left (d x +c \right )\right ) \left (4 A +3 C \right ) \sin \left (d x +c \right )}{4 b^{2} \sqrt {b \cos \left (d x +c \right )}\, d}+\frac {\left (\sqrt {\cos }\left (d x +c \right )\right ) C \sin \left (3 d x +3 c \right )}{12 b^{2} \sqrt {b \cos \left (d x +c \right )}\, d}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.62, size = 57, normalized size = 0.71 \begin {gather*} \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 {5}{2}}} + \frac {12 \, A \sin \left (d x + c\right )}{b^{\frac {5}{2}}}}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 49, normalized size = 0.61 \begin {gather*} \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^{3} d \sqrt {\cos \left (d x + c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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^3\,d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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