Optimal. Leaf size=69 \[ \frac {x \sqrt {\sec (c+d x)}}{2 b^2 \sqrt {b \sec (c+d x)}}+\frac {\sin (c+d x)}{2 b^2 d \sqrt {\sec (c+d x)} \sqrt {b \sec (c+d x)}} \]
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Rubi [A] time = 0.01, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {17, 2635, 8} \[ \frac {x \sqrt {\sec (c+d x)}}{2 b^2 \sqrt {b \sec (c+d x)}}+\frac {\sin (c+d x)}{2 b^2 d \sqrt {\sec (c+d x)} \sqrt {b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 17
Rule 2635
Rubi steps
\begin {align*} \int \frac {\sqrt {\sec (c+d x)}}{(b \sec (c+d x))^{5/2}} \, dx &=\frac {\sqrt {\sec (c+d x)} \int \cos ^2(c+d x) \, dx}{b^2 \sqrt {b \sec (c+d x)}}\\ &=\frac {\sin (c+d x)}{2 b^2 d \sqrt {\sec (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {\sqrt {\sec (c+d x)} \int 1 \, dx}{2 b^2 \sqrt {b \sec (c+d x)}}\\ &=\frac {x \sqrt {\sec (c+d x)}}{2 b^2 \sqrt {b \sec (c+d x)}}+\frac {\sin (c+d x)}{2 b^2 d \sqrt {\sec (c+d x)} \sqrt {b \sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 48, normalized size = 0.70 \[ \frac {(2 (c+d x)+\sin (2 (c+d x))) \sqrt {\sec (c+d x)}}{4 b^2 d \sqrt {b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 165, normalized size = 2.39 \[ \left [\frac {2 \, \sqrt {\frac {b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )^{\frac {3}{2}} \sin \left (d x + c\right ) - \sqrt {-b} \log \left (2 \, \sqrt {-b} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )^{\frac {3}{2}} \sin \left (d x + c\right ) + 2 \, b \cos \left (d x + c\right )^{2} - b\right )}{4 \, b^{3} d}, \frac {\sqrt {\frac {b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )^{\frac {3}{2}} \sin \left (d x + c\right ) + \sqrt {b} \arctan \left (\frac {\sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {b} \sqrt {\cos \left (d x + c\right )}}\right )}{2 \, b^{3} d}\right ] \]
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 \[ \int \frac {\sqrt {\sec \left (d x + c\right )}}{\left (b \sec \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.91, size = 54, normalized size = 0.78 \[ \frac {\left (\cos \left (d x +c \right ) \sin \left (d x +c \right )+d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )}}}{2 d \cos \left (d x +c \right )^{2} \left (\frac {b}{\cos \left (d x +c \right )}\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 25, normalized size = 0.36 \[ \frac {2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )}{4 \, b^{\frac {5}{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.63, size = 64, normalized size = 0.93 \[ \frac {\sqrt {\frac {b}{\cos \left (c+d\,x\right )}}\,\left (\sin \left (c+d\,x\right )+\sin \left (3\,c+3\,d\,x\right )+4\,d\,x\,\cos \left (c+d\,x\right )\right )}{8\,b^3\,d\,\cos \left (c+d\,x\right )\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 110.56, size = 82, normalized size = 1.19 \[ \begin {cases} \frac {x \tan ^{2}{\left (c + d x \right )}}{2 b^{\frac {5}{2}} \sec ^{2}{\left (c + d x \right )}} + \frac {x}{2 b^{\frac {5}{2}} \sec ^{2}{\left (c + d x \right )}} + \frac {\tan {\left (c + d x \right )}}{2 b^{\frac {5}{2}} d \sec ^{2}{\left (c + d x \right )}} & \text {for}\: d \neq 0 \\\frac {x \sqrt {\sec {\relax (c )}}}{\left (b \sec {\relax (c )}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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