Optimal. Leaf size=72 \[ \frac{2 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{3 b c^2 \sqrt{c \cos (a+b x)}}+\frac{2 \sin (a+b x)}{3 b c (c \cos (a+b x))^{3/2}} \]
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Rubi [A] time = 0.0353468, antiderivative size = 72, 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, 2642, 2641} \[ \frac{2 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{3 b c^2 \sqrt{c \cos (a+b x)}}+\frac{2 \sin (a+b x)}{3 b c (c \cos (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{(c \cos (a+b x))^{5/2}} \, dx &=\frac{2 \sin (a+b x)}{3 b c (c \cos (a+b x))^{3/2}}+\frac{\int \frac{1}{\sqrt{c \cos (a+b x)}} \, dx}{3 c^2}\\ &=\frac{2 \sin (a+b x)}{3 b c (c \cos (a+b x))^{3/2}}+\frac{\sqrt{\cos (a+b x)} \int \frac{1}{\sqrt{\cos (a+b x)}} \, dx}{3 c^2 \sqrt{c \cos (a+b x)}}\\ &=\frac{2 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{3 b c^2 \sqrt{c \cos (a+b x)}}+\frac{2 \sin (a+b x)}{3 b c (c \cos (a+b x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0591189, size = 51, normalized size = 0.71 \[ \frac{2 \left (\tan (a+b x)+\sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )\right )}{3 b c^2 \sqrt{c \cos (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.763, size = 241, normalized size = 3.4 \begin{align*} -{\frac{2}{3\,{c}^{2}b} \left ( -2\,\sqrt{ \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}+\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 ) \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}}{\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 ( 2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) ^{-1} \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 \frac{1}{\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 (\frac{\sqrt{c \cos \left (b x + a\right )}}{c^{3} \cos \left (b x + a\right )^{3}}, 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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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