Optimal. Leaf size=27 \[ \frac {x \sqrt {\sec (c+d x)}}{b^2 \sqrt {b \sec (c+d x)}} \]
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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} \[ \frac {x \sqrt {\sec (c+d x)}}{b^2 \sqrt {b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 17
Rubi steps
\begin {align*} \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(b \sec (c+d x))^{5/2}} \, dx &=\frac {\sqrt {\sec (c+d x)} \int 1 \, dx}{b^2 \sqrt {b \sec (c+d x)}}\\ &=\frac {x \sqrt {\sec (c+d x)}}{b^2 \sqrt {b \sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 24, normalized size = 0.89 \[ \frac {x \sec ^{\frac {5}{2}}(c+d x)}{(b \sec (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 101, normalized size = 3.74 \[ \left [-\frac {\sqrt {-b} \log \left (2 \, \sqrt {-b} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )^{\frac {3}{2}} \sin \left (d x + c\right ) + 2 \, b \cos \left (d x + c\right )^{2} - b\right )}{2 \, b^{3} d}, \frac {\arctan \left (\frac {\sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {b} \sqrt {\cos \left (d x + c\right )}}\right )}{b^{\frac {5}{2}} d}\right ] \]
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 \[ \int \frac {\sec \left (d x + c\right )^{\frac {5}{2}}}{\left (b \sec \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.81, size = 32, normalized size = 1.19 \[ \frac {\left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {5}{2}} \left (d x +c \right )}{d \left (\frac {b}{\cos \left (d x +c \right )}\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.82, 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 = 0.32, size = 27, normalized size = 1.00 \[ \frac {x\,\sqrt {\frac {b}{\cos \left (c+d\,x\right )}}}{b^3\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \]
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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