Optimal. Leaf size=35 \[ \frac {b^2 \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {b \sec (c+d x)}}{d} \]
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Rubi [A] time = 0.01, antiderivative size = 35, 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, 3767, 8} \[ \frac {b^2 \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {b \sec (c+d x)}}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 17
Rule 3767
Rubi steps
\begin {align*} \int \frac {(b \sec (c+d x))^{5/2}}{\sqrt {\sec (c+d x)}} \, dx &=\frac {\left (b^2 \sqrt {b \sec (c+d x)}\right ) \int \sec ^2(c+d x) \, dx}{\sqrt {\sec (c+d x)}}\\ &=-\frac {\left (b^2 \sqrt {b \sec (c+d x)}\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d \sqrt {\sec (c+d x)}}\\ &=\frac {b^2 \sqrt {\sec (c+d x)} \sqrt {b \sec (c+d x)} \sin (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 32, normalized size = 0.91 \[ \frac {\sin (c+d x) (b \sec (c+d x))^{5/2}}{d \sec ^{\frac {3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 33, normalized size = 0.94 \[ \frac {b^{2} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{d \sqrt {\cos \left (d x + c\right )}} \]
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 \[ \int \frac {\left (b \sec \left (d x + c\right )\right )^{\frac {5}{2}}}{\sqrt {\sec \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.09, size = 39, normalized size = 1.11 \[ \frac {\left (\frac {b}{\cos \left (d x +c \right )}\right )^{\frac {5}{2}} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{d \sqrt {\frac {1}{\cos \left (d x +c \right )}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.20, size = 54, normalized size = 1.54 \[ \frac {2 \, b^{\frac {5}{2}} \sin \left (2 \, d x + 2 \, c\right )}{{\left (\cos \left (2 \, d x + 2 \, c\right )^{2} + \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.72, size = 66, normalized size = 1.89 \[ \frac {b^2\,\sqrt {\frac {b}{\cos \left (c+d\,x\right )}}\,\left (\cos \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}+\sin \left (2\,c+2\,d\,x\right )+1{}\mathrm {i}\right )}{d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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