Optimal. Leaf size=98 \[ \frac{10 b \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \sqrt{b \sec (c+d x)}}{21 d}+\frac{2 b^4 \sin (c+d x)}{7 d (b \sec (c+d x))^{5/2}}+\frac{10 b^2 \sin (c+d x)}{21 d \sqrt{b \sec (c+d x)}} \]
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Rubi [A] time = 0.0730224, antiderivative size = 98, 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, 3769, 3771, 2641} \[ \frac{2 b^4 \sin (c+d x)}{7 d (b \sec (c+d x))^{5/2}}+\frac{10 b^2 \sin (c+d x)}{21 d \sqrt{b \sec (c+d x)}}+\frac{10 b \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{21 d} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3769
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \cos ^5(c+d x) (b \sec (c+d x))^{3/2} \, dx &=b^5 \int \frac{1}{(b \sec (c+d x))^{7/2}} \, dx\\ &=\frac{2 b^4 \sin (c+d x)}{7 d (b \sec (c+d x))^{5/2}}+\frac{1}{7} \left (5 b^3\right ) \int \frac{1}{(b \sec (c+d x))^{3/2}} \, dx\\ &=\frac{2 b^4 \sin (c+d x)}{7 d (b \sec (c+d x))^{5/2}}+\frac{10 b^2 \sin (c+d x)}{21 d \sqrt{b \sec (c+d x)}}+\frac{1}{21} (5 b) \int \sqrt{b \sec (c+d x)} \, dx\\ &=\frac{2 b^4 \sin (c+d x)}{7 d (b \sec (c+d x))^{5/2}}+\frac{10 b^2 \sin (c+d x)}{21 d \sqrt{b \sec (c+d x)}}+\frac{1}{21} \left (5 b \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{10 b \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{21 d}+\frac{2 b^4 \sin (c+d x)}{7 d (b \sec (c+d x))^{5/2}}+\frac{10 b^2 \sin (c+d x)}{21 d \sqrt{b \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0779546, size = 64, normalized size = 0.65 \[ \frac{b \sqrt{b \sec (c+d x)} \left (40 \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+26 \sin (2 (c+d x))+3 \sin (4 (c+d x))\right )}{84 d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.184, size = 151, normalized size = 1.5 \begin{align*} -{\frac{2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2} \left ( -1+\cos \left ( dx+c \right ) \right ) \cos \left ( dx+c \right ) }{21\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}} \left ({\frac{b}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{3}{2}}} \left ( 5\,i{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sin \left ( dx+c \right ) -3\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}+3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}-5\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+5\,\cos \left ( dx+c \right ) \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 \left (b \sec \left (d x + c\right )\right )^{\frac{3}{2}} \cos \left (d x + c\right )^{5}\,{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 (\sqrt{b \sec \left (d x + c\right )} b \cos \left (d x + c\right )^{5} \sec \left (d x + c\right ), x\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(-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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