Optimal. Leaf size=97 \[ \frac{2 \sin (c+d x) (b \cos (c+d x))^{7/2}}{9 b^3 d}+\frac{14 \sin (c+d x) (b \cos (c+d x))^{3/2}}{45 b d}+\frac{14 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}} \]
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Rubi [A] time = 0.069299, antiderivative size = 97, 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, 2635, 2640, 2639} \[ \frac{2 \sin (c+d x) (b \cos (c+d x))^{7/2}}{9 b^3 d}+\frac{14 \sin (c+d x) (b \cos (c+d x))^{3/2}}{45 b d}+\frac{14 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{b \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2635
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \cos ^4(c+d x) \sqrt{b \cos (c+d x)} \, dx &=\frac{\int (b \cos (c+d x))^{9/2} \, dx}{b^4}\\ &=\frac{2 (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^3 d}+\frac{7 \int (b \cos (c+d x))^{5/2} \, dx}{9 b^2}\\ &=\frac{14 (b \cos (c+d x))^{3/2} \sin (c+d x)}{45 b d}+\frac{2 (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^3 d}+\frac{7}{15} \int \sqrt{b \cos (c+d x)} \, dx\\ &=\frac{14 (b \cos (c+d x))^{3/2} \sin (c+d x)}{45 b d}+\frac{2 (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^3 d}+\frac{\left (7 \sqrt{b \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{15 \sqrt{\cos (c+d x)}}\\ &=\frac{14 \sqrt{b \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d \sqrt{\cos (c+d x)}}+\frac{14 (b \cos (c+d x))^{3/2} \sin (c+d x)}{45 b d}+\frac{2 (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^3 d}\\ \end{align*}
Mathematica [A] time = 0.153787, size = 75, normalized size = 0.77 \[ \frac{\sqrt{b \cos (c+d x)} \left (168 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+(38 \sin (2 (c+d x))+5 \sin (4 (c+d x))) \sqrt{\cos (c+d x)}\right )}{180 d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.934, size = 221, normalized size = 2.3 \begin{align*} -{\frac{2\,b}{45\,d}\sqrt{b \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 160\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{11}-480\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{9}+616\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{7}-432\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}+160\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}-21\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -24\,\cos \left ( 1/2\,dx+c/2 \right ) \right ){\frac{1}{\sqrt{-b \left ( 2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}- \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) }}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{b \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \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 \cos \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 \cos \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \cos \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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