Optimal. Leaf size=98 \[ \frac{3 x \sqrt{b \sec (c+d x)}}{8 \sqrt{\sec (c+d x)}}+\frac{3 \sin (c+d x) \sqrt{b \sec (c+d x)}}{8 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\sin (c+d x) \sqrt{b \sec (c+d x)}}{4 d \sec ^{\frac{7}{2}}(c+d x)} \]
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Rubi [A] time = 0.0251071, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {17, 2635, 8} \[ \frac{3 x \sqrt{b \sec (c+d x)}}{8 \sqrt{\sec (c+d x)}}+\frac{3 \sin (c+d x) \sqrt{b \sec (c+d x)}}{8 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\sin (c+d x) \sqrt{b \sec (c+d x)}}{4 d \sec ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 17
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\sqrt{b \sec (c+d x)}}{\sec ^{\frac{9}{2}}(c+d x)} \, dx &=\frac{\sqrt{b \sec (c+d x)} \int \cos ^4(c+d x) \, dx}{\sqrt{\sec (c+d x)}}\\ &=\frac{\sqrt{b \sec (c+d x)} \sin (c+d x)}{4 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{\left (3 \sqrt{b \sec (c+d x)}\right ) \int \cos ^2(c+d x) \, dx}{4 \sqrt{\sec (c+d x)}}\\ &=\frac{\sqrt{b \sec (c+d x)} \sin (c+d x)}{4 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{3 \sqrt{b \sec (c+d x)} \sin (c+d x)}{8 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\left (3 \sqrt{b \sec (c+d x)}\right ) \int 1 \, dx}{8 \sqrt{\sec (c+d x)}}\\ &=\frac{3 x \sqrt{b \sec (c+d x)}}{8 \sqrt{\sec (c+d x)}}+\frac{\sqrt{b \sec (c+d x)} \sin (c+d x)}{4 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{3 \sqrt{b \sec (c+d x)} \sin (c+d x)}{8 d \sec ^{\frac{3}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 0.116334, size = 55, normalized size = 0.56 \[ \frac{(12 (c+d x)+8 \sin (2 (c+d x))+\sin (4 (c+d x))) \sqrt{b \sec (c+d x)}}{32 d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.157, size = 74, normalized size = 0.8 \begin{align*}{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) +3\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +3\,dx+3\,c}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}\sqrt{{\frac{b}{\cos \left ( dx+c \right ) }}} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{-{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.13244, size = 66, normalized size = 0.67 \begin{align*} \frac{{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (\frac{1}{2} \, \arctan \left (\sin \left (4 \, d x + 4 \, c\right ), \cos \left (4 \, d x + 4 \, c\right )\right )\right )\right )} \sqrt{b}}{32 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78951, size = 537, normalized size = 5.48 \begin{align*} \left [\frac{\frac{2 \,{\left (2 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2}\right )} \sqrt{\frac{b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}} + 3 \, \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 )}{16 \, d}, \frac{\frac{{\left (2 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2}\right )} \sqrt{\frac{b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}} + 3 \, \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 )}{8 \, d}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b \sec \left (d x + c\right )}}{\sec \left (d x + c\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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