Optimal. Leaf size=27 \[ \frac {x \sqrt {\cos (c+d x)}}{b^2 \sqrt {b \cos (c+d x)}} \]
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Rubi [A] time = 0.00, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {17, 8} \[ \frac {x \sqrt {\cos (c+d x)}}{b^2 \sqrt {b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 17
Rubi steps
\begin {align*} \int \frac {\cos ^{\frac {5}{2}}(c+d x)}{(b \cos (c+d x))^{5/2}} \, dx &=\frac {\sqrt {\cos (c+d x)} \int 1 \, dx}{b^2 \sqrt {b \cos (c+d x)}}\\ &=\frac {x \sqrt {\cos (c+d x)}}{b^2 \sqrt {b \cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 0.89 \[ \frac {x \cos ^{\frac {5}{2}}(c+d x)}{(b \cos (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 97, normalized size = 3.59 \[ \left [-\frac {\sqrt {-b} \log \left (2 \, b \cos \left (d x + c\right )^{2} + 2 \, \sqrt {b \cos \left (d x + c\right )} \sqrt {-b} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - b\right )}{2 \, b^{3} d}, \frac {\arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{\sqrt {b} \cos \left (d x + c\right )^{\frac {3}{2}}}\right )}{b^{\frac {5}{2}} d}\right ] \]
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 \[ \int \frac {\cos \left (d x + c\right )^{\frac {5}{2}}}{\left (b \cos \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 28, normalized size = 1.04 \[ \frac {\left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \left (d x +c \right )}{d \left (b \cos \left (d x +c \right )\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 26, normalized size = 0.96 \[ \frac {2 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{b^{\frac {5}{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 37, normalized size = 1.37 \[ \frac {2\,x\,{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {b\,\cos \left (c+d\,x\right )}}{b^3\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \]
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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