Optimal. Leaf size=35 \[ \frac {\sin (c+d x) \sqrt {\sec (c+d x)}}{b d \sqrt {b \sec (c+d x)}} \]
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Rubi [A] time = 0.01, antiderivative size = 35, 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} \[ \frac {\sin (c+d x) \sqrt {\sec (c+d x)}}{b d \sqrt {b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 2637
Rubi steps
\begin {align*} \int \frac {\sqrt {\sec (c+d x)}}{(b \sec (c+d x))^{3/2}} \, dx &=\frac {\sqrt {\sec (c+d x)} \int \cos (c+d x) \, dx}{b \sqrt {b \sec (c+d x)}}\\ &=\frac {\sqrt {\sec (c+d x)} \sin (c+d x)}{b d \sqrt {b \sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 32, normalized size = 0.91 \[ \frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d (b \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 33, normalized size = 0.94 \[ \frac {\sqrt {\frac {b}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{b^{2} d} \]
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 \[ \int \frac {\sqrt {\sec \left (d x + c\right )}}{\left (b \sec \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.86, size = 41, normalized size = 1.17 \[ \frac {\sin \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )}}}{d \left (\frac {b}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \cos \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 13, normalized size = 0.37 \[ \frac {\sin \left (d x + c\right )}{b^{\frac {3}{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 39, normalized size = 1.11 \[ \frac {\sin \left (2\,c+2\,d\,x\right )\,\sqrt {\frac {b}{\cos \left (c+d\,x\right )}}\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}}{2\,b^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 15.08, size = 36, normalized size = 1.03 \[ \begin {cases} \frac {\tan {\left (c + d x \right )}}{b^{\frac {3}{2}} d \sec {\left (c + d x \right )}} & \text {for}\: d \neq 0 \\\frac {x \sqrt {\sec {\relax (c )}}}{\left (b \sec {\relax (c )}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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