Optimal. Leaf size=116 \[ \frac{b^2 \sin ^5(c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{b \sec (c+d x)}}{5 d}+\frac{2 b^2 \sin ^3(c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{b \sec (c+d x)}}{3 d}+\frac{b^2 \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{b \sec (c+d x)}}{d} \]
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Rubi [A] time = 0.0239371, antiderivative size = 116, 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} \[ \frac{b^2 \sin ^5(c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{b \sec (c+d x)}}{5 d}+\frac{2 b^2 \sin ^3(c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{b \sec (c+d x)}}{3 d}+\frac{b^2 \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{b \sec (c+d x)}}{d} \]
Antiderivative was successfully verified.
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Rule 17
Rule 3767
Rubi steps
\begin{align*} \int \sec ^{\frac{7}{2}}(c+d x) (b \sec (c+d x))^{5/2} \, dx &=\frac{\left (b^2 \sqrt{b \sec (c+d x)}\right ) \int \sec ^6(c+d x) \, dx}{\sqrt{\sec (c+d x)}}\\ &=-\frac{\left (b^2 \sqrt{b \sec (c+d x)}\right ) \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (c+d x)\right )}{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}+\frac{2 b^2 \sec ^{\frac{5}{2}}(c+d x) \sqrt{b \sec (c+d x)} \sin ^3(c+d x)}{3 d}+\frac{b^2 \sec ^{\frac{9}{2}}(c+d x) \sqrt{b \sec (c+d x)} \sin ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.216542, size = 57, normalized size = 0.49 \[ \frac{\left (\frac{1}{5} \tan ^5(c+d x)+\frac{2}{3} \tan ^3(c+d x)+\tan (c+d x)\right ) (b \sec (c+d x))^{5/2}}{d \sec ^{\frac{5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.119, size = 62, normalized size = 0.5 \begin{align*}{\frac{ \left ( 8\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}+4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+3 \right ) \cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{15\,d} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{{\frac{7}{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 [B] time = 2.53733, size = 952, normalized size = 8.21 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72835, size = 158, normalized size = 1.36 \begin{align*} \frac{{\left (8 \, b^{2} \cos \left (d x + c\right )^{4} + 4 \, b^{2} \cos \left (d x + c\right )^{2} + 3 \, b^{2}\right )} \sqrt{\frac{b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{15 \, d \cos \left (d x + c\right )^{\frac{9}{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{5}{2}} \sec \left (d x + c\right )^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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