Optimal. Leaf size=70 \[ \frac{6 c^2 E\left (\left .\frac{1}{2} (a+b x)\right |2\right ) \sqrt{c \cos (a+b x)}}{5 b \sqrt{\cos (a+b x)}}+\frac{2 c \sin (a+b x) (c \cos (a+b x))^{3/2}}{5 b} \]
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Rubi [A] time = 0.0364114, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2635, 2640, 2639} \[ \frac{6 c^2 E\left (\left .\frac{1}{2} (a+b x)\right |2\right ) \sqrt{c \cos (a+b x)}}{5 b \sqrt{\cos (a+b x)}}+\frac{2 c \sin (a+b x) (c \cos (a+b x))^{3/2}}{5 b} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int (c \cos (a+b x))^{5/2} \, dx &=\frac{2 c (c \cos (a+b x))^{3/2} \sin (a+b x)}{5 b}+\frac{1}{5} \left (3 c^2\right ) \int \sqrt{c \cos (a+b x)} \, dx\\ &=\frac{2 c (c \cos (a+b x))^{3/2} \sin (a+b x)}{5 b}+\frac{\left (3 c^2 \sqrt{c \cos (a+b x)}\right ) \int \sqrt{\cos (a+b x)} \, dx}{5 \sqrt{\cos (a+b x)}}\\ &=\frac{6 c^2 \sqrt{c \cos (a+b x)} E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{5 b \sqrt{\cos (a+b x)}}+\frac{2 c (c \cos (a+b x))^{3/2} \sin (a+b x)}{5 b}\\ \end{align*}
Mathematica [A] time = 0.0844326, size = 62, normalized size = 0.89 \[ \frac{(c \cos (a+b x))^{5/2} \left (6 E\left (\left .\frac{1}{2} (a+b x)\right |2\right )+\sin (2 (a+b x)) \sqrt{\cos (a+b x)}\right )}{5 b \cos ^{\frac{5}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.855, size = 213, normalized size = 3. \begin{align*} -{\frac{2\,{c}^{3}}{5\,b}\sqrt{c \left ( 2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}} \left ( -8\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{6}\cos \left ( 1/2\,bx+a/2 \right ) +8\,\cos \left ( 1/2\,bx+a/2 \right ) \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}-3\,\sqrt{2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}{\it EllipticE} \left ( \cos \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) -2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}\cos \left ( 1/2\,bx+a/2 \right ) \right ){\frac{1}{\sqrt{-c \left ( 2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}- \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2} \right ) }}} \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{c \left ( 2\, \left ( \cos \left ( 1/2\,bx+a/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 \left (c \cos \left (b x + a\right )\right )^{\frac{5}{2}}\,{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{c \cos \left (b x + a\right )} c^{2} \cos \left (b x + a\right )^{2}, 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 \left (c \cos \left (b x + a\right )\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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