Optimal. Leaf size=107 \[ \frac{3 x \sqrt{\sec (c+d x)}}{8 b^2 \sqrt{b \sec (c+d x)}}+\frac{\sin (c+d x)}{4 b^2 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{b \sec (c+d x)}}+\frac{3 \sin (c+d x)}{8 b^2 d \sqrt{\sec (c+d x)} \sqrt{b \sec (c+d x)}} \]
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Rubi [A] time = 0.0284181, antiderivative size = 107, 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 = {18, 2635, 8} \[ \frac{3 x \sqrt{\sec (c+d x)}}{8 b^2 \sqrt{b \sec (c+d x)}}+\frac{\sin (c+d x)}{4 b^2 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{b \sec (c+d x)}}+\frac{3 \sin (c+d x)}{8 b^2 d \sqrt{\sec (c+d x)} \sqrt{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 18
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) (b \sec (c+d x))^{5/2}} \, dx &=\frac{\sqrt{\sec (c+d x)} \int \cos ^4(c+d x) \, dx}{b^2 \sqrt{b \sec (c+d x)}}\\ &=\frac{\sin (c+d x)}{4 b^2 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{b \sec (c+d x)}}+\frac{\left (3 \sqrt{\sec (c+d x)}\right ) \int \cos ^2(c+d x) \, dx}{4 b^2 \sqrt{b \sec (c+d x)}}\\ &=\frac{\sin (c+d x)}{4 b^2 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{b \sec (c+d x)}}+\frac{3 \sin (c+d x)}{8 b^2 d \sqrt{\sec (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{\left (3 \sqrt{\sec (c+d x)}\right ) \int 1 \, dx}{8 b^2 \sqrt{b \sec (c+d x)}}\\ &=\frac{3 x \sqrt{\sec (c+d x)}}{8 b^2 \sqrt{b \sec (c+d x)}}+\frac{\sin (c+d x)}{4 b^2 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{b \sec (c+d x)}}+\frac{3 \sin (c+d x)}{8 b^2 d \sqrt{\sec (c+d x)} \sqrt{b \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0575848, size = 58, normalized size = 0.54 \[ \frac{(12 (c+d x)+8 \sin (2 (c+d x))+\sin (4 (c+d x))) \sqrt{\sec (c+d x)}}{32 b^2 d \sqrt{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.127, size = 74, normalized size = 0.7 \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}} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{-{\frac{3}{2}}} \left ({\frac{b}{\cos \left ( dx+c \right ) }} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.97597, size = 66, normalized size = 0.62 \begin{align*} \frac{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 )}{32 \, b^{\frac{5}{2}} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.46266, size = 552, normalized size = 5.16 \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 \, b^{3} 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 \, b^{3} 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{1}{\left (b \sec \left (d x + c\right )\right )^{\frac{5}{2}} \sec \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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