Optimal. Leaf size=32 \[ \frac {\sin (c+d x) \sqrt {b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}} \]
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Rubi [A] time = 0.01, antiderivative size = 32, 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, 2637} \[ \frac {\sin (c+d x) \sqrt {b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 2637
Rubi steps
\begin {align*} \int \frac {\sqrt {b \sec (c+d x)}}{\sec ^{\frac {3}{2}}(c+d x)} \, dx &=\frac {\sqrt {b \sec (c+d x)} \int \cos (c+d x) \, dx}{\sqrt {\sec (c+d x)}}\\ &=\frac {\sqrt {b \sec (c+d x)} \sin (c+d x)}{d \sqrt {\sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 32, normalized size = 1.00 \[ \frac {\sin (c+d x) \sqrt {b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 30, normalized size = 0.94 \[ \frac {\sqrt {\frac {b}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{d} \]
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 \[ \int \frac {\sqrt {b \sec \left (d x + c\right )}}{\sec \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.16, size = 41, normalized size = 1.28 \[ \frac {\sqrt {\frac {b}{\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )}{d \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \cos \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 13, normalized size = 0.41 \[ \frac {\sqrt {b} \sin \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 32, normalized size = 1.00 \[ \frac {\sin \left (c+d\,x\right )\,\sqrt {\frac {b}{\cos \left (c+d\,x\right )}}}{d\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 13.39, size = 36, normalized size = 1.12 \[ \begin {cases} \frac {\sqrt {b} \tan {\left (c + d x \right )}}{d \sec {\left (c + d x \right )}} & \text {for}\: d \neq 0 \\\frac {x \sqrt {b \sec {\relax (c )}}}{\sec ^{\frac {3}{2}}{\relax (c )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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