Optimal. Leaf size=70 \[ \frac {\sin (c+d x) \sqrt {\sec (c+d x)}}{d \sqrt {b \sec (c+d x)}}-\frac {\sin ^3(c+d x) \sqrt {\sec (c+d x)}}{3 d \sqrt {b \sec (c+d x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {18, 2633} \[ \frac {\sin (c+d x) \sqrt {\sec (c+d x)}}{d \sqrt {b \sec (c+d x)}}-\frac {\sin ^3(c+d x) \sqrt {\sec (c+d x)}}{3 d \sqrt {b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 18
Rule 2633
Rubi steps
\begin {align*} \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {b \sec (c+d x)}} \, dx &=\frac {\sqrt {\sec (c+d x)} \int \cos ^3(c+d x) \, dx}{\sqrt {b \sec (c+d x)}}\\ &=-\frac {\sqrt {\sec (c+d x)} \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d \sqrt {b \sec (c+d x)}}\\ &=\frac {\sqrt {\sec (c+d x)} \sin (c+d x)}{d \sqrt {b \sec (c+d x)}}-\frac {\sqrt {\sec (c+d x)} \sin ^3(c+d x)}{3 d \sqrt {b \sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 45, normalized size = 0.64 \[ \frac {\sin (c+d x) (\cos (2 (c+d x))+5) \sqrt {\sec (c+d x)}}{6 d \sqrt {b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 51, normalized size = 0.73 \[ \frac {{\left (\cos \left (d x + c\right )^{3} + 2 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{3 \, b d \sqrt {\cos \left (d x + c\right )}} \]
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 \[ \int \frac {1}{\sqrt {b \sec \left (d x + c\right )} \sec \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.05, size = 52, normalized size = 0.74 \[ \frac {\sin \left (d x +c \right ) \left (2+\cos ^{2}\left (d x +c \right )\right )}{3 d \cos \left (d x +c \right )^{3} \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {5}{2}} \sqrt {\frac {b}{\cos \left (d x +c \right )}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 42, normalized size = 0.60 \[ \frac {\sin \left (3 \, d x + 3 \, c\right ) + 9 \, \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (3 \, d x + 3 \, c\right ), \cos \left (3 \, d x + 3 \, c\right )\right )\right )}{12 \, \sqrt {b} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.52, size = 48, normalized size = 0.69 \[ \frac {\left (9\,\sin \left (c+d\,x\right )+\sin \left (3\,c+3\,d\,x\right )\right )\,\sqrt {\frac {b}{\cos \left (c+d\,x\right )}}}{12\,b\,d\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \]
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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