Optimal. Leaf size=95 \[ \frac{2 b^3 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac{14 b \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}+\frac{14 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}} \]
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Rubi [A] time = 0.069369, 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, 2639} \[ \frac{2 b^3 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac{14 b \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}+\frac{14 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3769
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{\cos ^4(c+d x)}{\sqrt{b \sec (c+d x)}} \, dx &=b^4 \int \frac{1}{(b \sec (c+d x))^{9/2}} \, dx\\ &=\frac{2 b^3 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac{1}{9} \left (7 b^2\right ) \int \frac{1}{(b \sec (c+d x))^{5/2}} \, dx\\ &=\frac{2 b^3 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac{14 b \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}+\frac{7}{15} \int \frac{1}{\sqrt{b \sec (c+d x)}} \, dx\\ &=\frac{2 b^3 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac{14 b \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}+\frac{7 \int \sqrt{\cos (c+d x)} \, dx}{15 \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}\\ &=\frac{14 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 b^3 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac{14 b \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.189403, size = 70, normalized size = 0.74 \[ \frac{4 (33 \sin (c+d x)+5 \sin (3 (c+d x))) \cos (c+d x)+\frac{336 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{\sqrt{\cos (c+d x)}}}{360 d \sqrt{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.202, size = 328, normalized size = 3.5 \begin{align*}{\frac{2}{45\,d\sin \left ( dx+c \right ) b} \left ( 21\,i\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 ) \sin \left ( dx+c \right ) \cos \left ( dx+c \right ) -21\,i\cos \left ( dx+c \right ) \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 EllipticE} \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 ) ^{6}+21\,i\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 ) \sin \left ( dx+c \right ) -21\,i\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 EllipticE} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i \right ) -2\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}-14\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+21\,\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 \frac{\cos \left (d x + c\right )^{4}}{\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 )} \cos \left (d x + c\right )^{4}}{b \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right )^{4}}{\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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