Optimal. Leaf size=68 \[ \frac{2 \sin (a+b x)}{b c \sqrt{c \cos (a+b x)}}-\frac{2 E\left (\left .\frac{1}{2} (a+b x)\right |2\right ) \sqrt{c \cos (a+b x)}}{b c^2 \sqrt{\cos (a+b x)}} \]
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Rubi [A] time = 0.0360195, antiderivative size = 68, 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 = {2636, 2640, 2639} \[ \frac{2 \sin (a+b x)}{b c \sqrt{c \cos (a+b x)}}-\frac{2 E\left (\left .\frac{1}{2} (a+b x)\right |2\right ) \sqrt{c \cos (a+b x)}}{b c^2 \sqrt{\cos (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{(c \cos (a+b x))^{3/2}} \, dx &=\frac{2 \sin (a+b x)}{b c \sqrt{c \cos (a+b x)}}-\frac{\int \sqrt{c \cos (a+b x)} \, dx}{c^2}\\ &=\frac{2 \sin (a+b x)}{b c \sqrt{c \cos (a+b x)}}-\frac{\sqrt{c \cos (a+b x)} \int \sqrt{\cos (a+b x)} \, dx}{c^2 \sqrt{\cos (a+b x)}}\\ &=-\frac{2 \sqrt{c \cos (a+b x)} E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b c^2 \sqrt{\cos (a+b x)}}+\frac{2 \sin (a+b x)}{b c \sqrt{c \cos (a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.0303218, size = 50, normalized size = 0.74 \[ \frac{2 \left (\sin (a+b x)-\sqrt{\cos (a+b x)} E\left (\left .\frac{1}{2} (a+b x)\right |2\right )\right )}{b c \sqrt{c \cos (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 2.045, size = 168, normalized size = 2.5 \begin{align*} -2\,{\frac{\sqrt{-2\,c \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+c \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}} \left ( \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 ) }{c\sqrt{-c \left ( 2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}- \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2} \right ) }\sin \left ( 1/2\,bx+a/2 \right ) \sqrt{c \left ( 2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) }b}} \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{1}{\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 (\frac{\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 \frac{1}{\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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