Optimal. Leaf size=70 \[ \frac{2 c^2 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{3 b \sqrt{c \cos (a+b x)}}+\frac{2 c \sin (a+b x) \sqrt{c \cos (a+b x)}}{3 b} \]
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Rubi [A] time = 0.0365162, 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, 2642, 2641} \[ \frac{2 c^2 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{3 b \sqrt{c \cos (a+b x)}}+\frac{2 c \sin (a+b x) \sqrt{c \cos (a+b x)}}{3 b} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int (c \cos (a+b x))^{3/2} \, dx &=\frac{2 c \sqrt{c \cos (a+b x)} \sin (a+b x)}{3 b}+\frac{1}{3} c^2 \int \frac{1}{\sqrt{c \cos (a+b x)}} \, dx\\ &=\frac{2 c \sqrt{c \cos (a+b x)} \sin (a+b x)}{3 b}+\frac{\left (c^2 \sqrt{\cos (a+b x)}\right ) \int \frac{1}{\sqrt{\cos (a+b x)}} \, dx}{3 \sqrt{c \cos (a+b x)}}\\ &=\frac{2 c^2 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{3 b \sqrt{c \cos (a+b x)}}+\frac{2 c \sqrt{c \cos (a+b x)} \sin (a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0462831, size = 58, normalized size = 0.83 \[ \frac{2 (c \cos (a+b x))^{3/2} \left (F\left (\left .\frac{1}{2} (a+b x)\right |2\right )+\sin (a+b x) \sqrt{\cos (a+b x)}\right )}{3 b \cos ^{\frac{3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.884, size = 190, normalized size = 2.7 \begin{align*} -{\frac{2\,{c}^{2}}{3\,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 ( 4\,\cos \left ( 1/2\,bx+a/2 \right ) \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+\sqrt{ \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ({\frac{bx}{2}}+{\frac{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{3}{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 \cos \left (b x + a\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 \left (c \cos \left (b x + a\right )\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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