Optimal. Leaf size=69 \[ \frac {x \sqrt {\cos (c+d x)}}{2 b \sqrt {b \cos (c+d x)}}+\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 b d \sqrt {b \cos (c+d x)}} \]
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Rubi [A]
time = 0.01, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {17, 2715, 8}
\begin {gather*} \frac {x \sqrt {\cos (c+d x)}}{2 b \sqrt {b \cos (c+d x)}}+\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 b d \sqrt {b \cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 17
Rule 2715
Rubi steps
\begin {align*} \int \frac {\cos ^{\frac {7}{2}}(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx &=\frac {\sqrt {\cos (c+d x)} \int \cos ^2(c+d x) \, dx}{b \sqrt {b \cos (c+d x)}}\\ &=\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 b d \sqrt {b \cos (c+d x)}}+\frac {\sqrt {\cos (c+d x)} \int 1 \, dx}{2 b \sqrt {b \cos (c+d x)}}\\ &=\frac {x \sqrt {\cos (c+d x)}}{2 b \sqrt {b \cos (c+d x)}}+\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 b d \sqrt {b \cos (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 45, normalized size = 0.65 \begin {gather*} \frac {\cos ^{\frac {3}{2}}(c+d x) (2 (c+d x)+\sin (2 (c+d x)))}{4 d (b \cos (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 42, normalized size = 0.61
method | result | size |
default | \(\frac {\left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \left (\sin \left (d x +c \right ) \cos \left (d x +c \right )+d x +c \right )}{2 d \left (b \cos \left (d x +c \right )\right )^{\frac {3}{2}}}\) | \(42\) |
risch | \(\frac {x \left (\sqrt {\cos }\left (d x +c \right )\right )}{2 b \sqrt {b \cos \left (d x +c \right )}}+\frac {\left (\sqrt {\cos }\left (d x +c \right )\right ) \sin \left (2 d x +2 c \right )}{4 b \sqrt {b \cos \left (d x +c \right )}\, d}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.58, size = 25, normalized size = 0.36 \begin {gather*} \frac {2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )}{4 \, b^{\frac {3}{2}} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 157, normalized size = 2.28 \begin {gather*} \left [\frac {2 \, \sqrt {b \cos \left (d x + c\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + 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 )}{4 \, b^{2} d}, \frac {\sqrt {b \cos \left (d x + c\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + 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 )}{2 \, b^{2} d}\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 = 0.59, size = 65, normalized size = 0.94 \begin {gather*} \frac {\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (\sin \left (c+d\,x\right )+\sin \left (3\,c+3\,d\,x\right )+4\,d\,x\,\cos \left (c+d\,x\right )\right )}{4\,b^2\,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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