Optimal. Leaf size=72 \[ \frac{b \sin ^3(c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{b \sec (c+d x)}}{3 d}+\frac{b \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{b \sec (c+d x)}}{d} \]
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Rubi [A] time = 0.0169584, antiderivative size = 72, normalized size of antiderivative = 1., 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 = {17, 3767} \[ \frac{b \sin ^3(c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{b \sec (c+d x)}}{3 d}+\frac{b \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{b \sec (c+d x)}}{d} \]
Antiderivative was successfully verified.
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Rule 17
Rule 3767
Rubi steps
\begin{align*} \int \sec ^{\frac{5}{2}}(c+d x) (b \sec (c+d x))^{3/2} \, dx &=\frac{\left (b \sqrt{b \sec (c+d x)}\right ) \int \sec ^4(c+d x) \, dx}{\sqrt{\sec (c+d x)}}\\ &=-\frac{\left (b \sqrt{b \sec (c+d x)}\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d \sqrt{\sec (c+d x)}}\\ &=\frac{b \sqrt{\sec (c+d x)} \sqrt{b \sec (c+d x)} \sin (c+d x)}{d}+\frac{b \sec ^{\frac{5}{2}}(c+d x) \sqrt{b \sec (c+d x)} \sin ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.088353, size = 45, normalized size = 0.62 \[ \frac{\left (\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right ) (b \sec (c+d x))^{3/2}}{d \sec ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.118, size = 52, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+1 \right ) \cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{3\,d} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{{\frac{5}{2}}} \left ({\frac{b}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.10283, size = 404, normalized size = 5.61 \begin{align*} -\frac{4 \,{\left (3 \, b \cos \left (6 \, d x + 6 \, c\right ) \sin \left (2 \, d x + 2 \, c\right ) + 9 \, b \cos \left (4 \, d x + 4 \, c\right ) \sin \left (2 \, d x + 2 \, c\right ) -{\left (3 \, b \cos \left (2 \, d x + 2 \, c\right ) + b\right )} \sin \left (6 \, d x + 6 \, c\right ) - 3 \,{\left (3 \, b \cos \left (2 \, d x + 2 \, c\right ) + b\right )} \sin \left (4 \, d x + 4 \, c\right )\right )} \sqrt{b}}{3 \,{\left (2 \,{\left (3 \, \cos \left (4 \, d x + 4 \, c\right ) + 3 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \cos \left (6 \, d x + 6 \, c\right ) + \cos \left (6 \, d x + 6 \, c\right )^{2} + 6 \,{\left (3 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \cos \left (4 \, d x + 4 \, c\right ) + 9 \, \cos \left (4 \, d x + 4 \, c\right )^{2} + 9 \, \cos \left (2 \, d x + 2 \, c\right )^{2} + 6 \,{\left (\sin \left (4 \, d x + 4 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right )\right )} \sin \left (6 \, d x + 6 \, c\right ) + \sin \left (6 \, d x + 6 \, c\right )^{2} + 9 \, \sin \left (4 \, d x + 4 \, c\right )^{2} + 18 \, \sin \left (4 \, d x + 4 \, c\right ) \sin \left (2 \, d x + 2 \, c\right ) + 9 \, \sin \left (2 \, d x + 2 \, c\right )^{2} + 6 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35406, size = 117, normalized size = 1.62 \begin{align*} \frac{{\left (2 \, b \cos \left (d x + c\right )^{2} + b\right )} \sqrt{\frac{b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{3 \, d \cos \left (d x + c\right )^{\frac{5}{2}}} \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 \left (b \sec \left (d x + c\right )\right )^{\frac{3}{2}} \sec \left (d x + c\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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