Optimal. Leaf size=95 \[ \frac{10 \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \sqrt{b \sec (c+d x)}}{21 d}+\frac{2 b^3 \sin (c+d x)}{7 d (b \sec (c+d x))^{5/2}}+\frac{10 b \sin (c+d x)}{21 d \sqrt{b \sec (c+d x)}} \]
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Rubi [A] time = 0.0701063, antiderivative size = 95, 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} \[ \frac{2 b^3 \sin (c+d x)}{7 d (b \sec (c+d x))^{5/2}}+\frac{10 b \sin (c+d x)}{21 d \sqrt{b \sec (c+d x)}}+\frac{10 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \sec (c+d x)}}{21 d} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3769
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \cos ^4(c+d x) \sqrt{b \sec (c+d x)} \, dx &=b^4 \int \frac{1}{(b \sec (c+d x))^{7/2}} \, dx\\ &=\frac{2 b^3 \sin (c+d x)}{7 d (b \sec (c+d x))^{5/2}}+\frac{1}{7} \left (5 b^2\right ) \int \frac{1}{(b \sec (c+d x))^{3/2}} \, dx\\ &=\frac{2 b^3 \sin (c+d x)}{7 d (b \sec (c+d x))^{5/2}}+\frac{10 b \sin (c+d x)}{21 d \sqrt{b \sec (c+d x)}}+\frac{5}{21} \int \sqrt{b \sec (c+d x)} \, dx\\ &=\frac{2 b^3 \sin (c+d x)}{7 d (b \sec (c+d x))^{5/2}}+\frac{10 b \sin (c+d x)}{21 d \sqrt{b \sec (c+d x)}}+\frac{1}{21} \left (5 \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\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 d}+\frac{2 b^3 \sin (c+d x)}{7 d (b \sec (c+d x))^{5/2}}+\frac{10 b \sin (c+d x)}{21 d \sqrt{b \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.089581, size = 63, normalized size = 0.66 \[ \frac{\sqrt{b \sec (c+d x)} \left (40 \sqrt{\cos (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )+26 \sin (2 (c+d x))+3 \sin (4 (c+d x))\right )}{84 d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.188, size = 145, 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 ( 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 ) \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 \sqrt{b \sec \left (d x + c\right )} \cos \left (d x + c\right )^{4}\,{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 )} \cos \left (d x + c\right )^{4}, 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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