Optimal. Leaf size=107 \[ \frac {3 x \sqrt {\cos (c+d x)}}{8 b^2 \sqrt {b \cos (c+d x)}}+\frac {3 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{8 b^2 d \sqrt {b \cos (c+d x)}}+\frac {\cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{4 b^2 d \sqrt {b \cos (c+d x)}} \]
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Rubi [A]
time = 0.02, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps
used = 4, 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 {3 x \sqrt {\cos (c+d x)}}{8 b^2 \sqrt {b \cos (c+d x)}}+\frac {\sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{4 b^2 d \sqrt {b \cos (c+d x)}}+\frac {3 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{8 b^2 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 {13}{2}}(c+d x)}{(b \cos (c+d x))^{5/2}} \, dx &=\frac {\sqrt {\cos (c+d x)} \int \cos ^4(c+d x) \, dx}{b^2 \sqrt {b \cos (c+d x)}}\\ &=\frac {\cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{4 b^2 d \sqrt {b \cos (c+d x)}}+\frac {\left (3 \sqrt {\cos (c+d x)}\right ) \int \cos ^2(c+d x) \, dx}{4 b^2 \sqrt {b \cos (c+d x)}}\\ &=\frac {3 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{8 b^2 d \sqrt {b \cos (c+d x)}}+\frac {\cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{4 b^2 d \sqrt {b \cos (c+d x)}}+\frac {\left (3 \sqrt {\cos (c+d x)}\right ) \int 1 \, dx}{8 b^2 \sqrt {b \cos (c+d x)}}\\ &=\frac {3 x \sqrt {\cos (c+d x)}}{8 b^2 \sqrt {b \cos (c+d x)}}+\frac {3 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{8 b^2 d \sqrt {b \cos (c+d x)}}+\frac {\cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{4 b^2 d \sqrt {b \cos (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 58, normalized size = 0.54 \begin {gather*} \frac {\sqrt {\cos (c+d x)} (12 (c+d x)+8 \sin (2 (c+d x))+\sin (4 (c+d x)))}{32 b^2 d \sqrt {b \cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 62, normalized size = 0.58
method | result | size |
default | \(\frac {\left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \left (2 \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )+3 \sin \left (d x +c \right ) \cos \left (d x +c \right )+3 d x +3 c \right )}{8 d \left (b \cos \left (d x +c \right )\right )^{\frac {5}{2}}}\) | \(62\) |
risch | \(\frac {3 x \left (\sqrt {\cos }\left (d x +c \right )\right )}{8 b^{2} \sqrt {b \cos \left (d x +c \right )}}+\frac {\left (\sqrt {\cos }\left (d x +c \right )\right ) \sin \left (4 d x +4 c \right )}{32 b^{2} \sqrt {b \cos \left (d x +c \right )}\, d}+\frac {\left (\sqrt {\cos }\left (d x +c \right )\right ) \sin \left (2 d x +2 c \right )}{4 b^{2} \sqrt {b \cos \left (d x +c \right )}\, d}\) | \(96\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.58, size = 49, normalized size = 0.46 \begin {gather*} \frac {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 )}{32 \, b^{\frac {5}{2}} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 182, normalized size = 1.70 \begin {gather*} \left [\frac {2 \, \sqrt {b \cos \left (d x + c\right )} {\left (2 \, \cos \left (d x + c\right )^{2} + 3\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 3 \, \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^{3} d}, \frac {\sqrt {b \cos \left (d x + c\right )} {\left (2 \, \cos \left (d x + c\right )^{2} + 3\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 3 \, \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^{3} 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 = 1.00, size = 78, normalized size = 0.73 \begin {gather*} \frac {\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (8\,\sin \left (c+d\,x\right )+9\,\sin \left (3\,c+3\,d\,x\right )+\sin \left (5\,c+5\,d\,x\right )+24\,d\,x\,\cos \left (c+d\,x\right )\right )}{32\,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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