Optimal. Leaf size=100 \[ \frac{10 \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \sqrt{b \sec (c+d x)}}{21 b d}+\frac{2 \sin (c+d x) (b \sec (c+d x))^{7/2}}{7 b^4 d}+\frac{10 \sin (c+d x) (b \sec (c+d x))^{3/2}}{21 b^2 d} \]
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Rubi [A] time = 0.0574047, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {16, 3768, 3771, 2641} \[ \frac{2 \sin (c+d x) (b \sec (c+d x))^{7/2}}{7 b^4 d}+\frac{10 \sin (c+d x) (b \sec (c+d x))^{3/2}}{21 b^2 d}+\frac{10 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{21 b d} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3768
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{\sec ^5(c+d x)}{\sqrt{b \sec (c+d x)}} \, dx &=\frac{\int (b \sec (c+d x))^{9/2} \, dx}{b^5}\\ &=\frac{2 (b \sec (c+d x))^{7/2} \sin (c+d x)}{7 b^4 d}+\frac{5 \int (b \sec (c+d x))^{5/2} \, dx}{7 b^3}\\ &=\frac{10 (b \sec (c+d x))^{3/2} \sin (c+d x)}{21 b^2 d}+\frac{2 (b \sec (c+d x))^{7/2} \sin (c+d x)}{7 b^4 d}+\frac{5 \int \sqrt{b \sec (c+d x)} \, dx}{21 b}\\ &=\frac{10 (b \sec (c+d x))^{3/2} \sin (c+d x)}{21 b^2 d}+\frac{2 (b \sec (c+d x))^{7/2} \sin (c+d x)}{7 b^4 d}+\frac{\left (5 \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{21 b}\\ &=\frac{10 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{21 b d}+\frac{10 (b \sec (c+d x))^{3/2} \sin (c+d x)}{21 b^2 d}+\frac{2 (b \sec (c+d x))^{7/2} \sin (c+d x)}{7 b^4 d}\\ \end{align*}
Mathematica [A] time = 0.0827893, size = 69, normalized size = 0.69 \[ \frac{\sec ^3(c+d x) \left (10 \cos ^{\frac{5}{2}}(c+d x) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+5 \sin (2 (c+d x))+6 \tan (c+d x)\right )}{21 d \sqrt{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.23, size = 152, normalized size = 1.5 \begin{align*} -{\frac{ \left ( -2+2\,\cos \left ( dx+c \right ) \right ) \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}{21\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{4}} \left ( 5\,i \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i \right ) -5\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+5\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-3\,\cos \left ( dx+c \right ) +3 \right ){\frac{1}{\sqrt{{\frac{b}{\cos \left ( dx+c \right ) }}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (d x + c\right )^{5}}{\sqrt{b \sec \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b \sec \left (d x + c\right )} \sec \left (d x + c\right )^{4}}{b}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec ^{5}{\left (c + d x \right )}}{\sqrt{b \sec{\left (c + d x \right )}}}\, dx \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{\sec \left (d x + c\right )^{5}}{\sqrt{b \sec \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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