Optimal. Leaf size=69 \[ \frac {b^2 x \sqrt {b \sec (c+d x)}}{2 \sqrt {\sec (c+d x)}}+\frac {b^2 \sin (c+d x) \sqrt {b \sec (c+d x)}}{2 d \sec ^{\frac {3}{2}}(c+d x)} \]
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Rubi [A] time = 0.02, 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} \[ \frac {b^2 x \sqrt {b \sec (c+d x)}}{2 \sqrt {\sec (c+d x)}}+\frac {b^2 \sin (c+d x) \sqrt {b \sec (c+d x)}}{2 d \sec ^{\frac {3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 8
Rule 17
Rule 2635
Rubi steps
\begin {align*} \int \frac {(b \sec (c+d x))^{5/2}}{\sec ^{\frac {9}{2}}(c+d x)} \, dx &=\frac {\left (b^2 \sqrt {b \sec (c+d x)}\right ) \int \cos ^2(c+d x) \, dx}{\sqrt {\sec (c+d x)}}\\ &=\frac {b^2 \sqrt {b \sec (c+d x)} \sin (c+d x)}{2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (b^2 \sqrt {b \sec (c+d x)}\right ) \int 1 \, dx}{2 \sqrt {\sec (c+d x)}}\\ &=\frac {b^2 x \sqrt {b \sec (c+d x)}}{2 \sqrt {\sec (c+d x)}}+\frac {b^2 \sqrt {b \sec (c+d x)} \sin (c+d x)}{2 d \sec ^{\frac {3}{2}}(c+d x)}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 45, normalized size = 0.65 \[ \frac {(2 (c+d x)+\sin (2 (c+d x))) (b \sec (c+d x))^{5/2}}{4 d \sec ^{\frac {5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 167, normalized size = 2.42 \[ \left [\frac {2 \, b^{2} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )^{\frac {3}{2}} \sin \left (d x + c\right ) + \sqrt {-b} b^{2} \log \left (-2 \, \sqrt {-b} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )^{\frac {3}{2}} \sin \left (d x + c\right ) + 2 \, b \cos \left (d x + c\right )^{2} - b\right )}{4 \, d}, \frac {b^{2} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )^{\frac {3}{2}} \sin \left (d x + c\right ) + b^{\frac {5}{2}} \arctan \left (\frac {\sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {b} \sqrt {\cos \left (d x + c\right )}}\right )}{2 \, d}\right ] \]
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 \[ \int \frac {\left (b \sec \left (d x + c\right )\right )^{\frac {5}{2}}}{\sec \left (d x + c\right )^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.98, size = 54, normalized size = 0.78 \[ \frac {\left (\cos \left (d x +c \right ) \sin \left (d x +c \right )+d x +c \right ) \left (\frac {b}{\cos \left (d x +c \right )}\right )^{\frac {5}{2}}}{2 d \cos \left (d x +c \right )^{2} \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 32, normalized size = 0.46 \[ \frac {{\left (2 \, {\left (d x + c\right )} b^{2} + b^{2} \sin \left (2 \, d x + 2 \, c\right )\right )} \sqrt {b}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.37, size = 44, normalized size = 0.64 \[ \frac {b^2\,\left (\sin \left (2\,c+2\,d\,x\right )+2\,d\,x\right )\,\sqrt {\frac {b}{\cos \left (c+d\,x\right )}}}{4\,d\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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