Optimal. Leaf size=42 \[ \frac{2 F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{3 b}+\frac{2 \sin (a+b x)}{3 b \cos ^{\frac{3}{2}}(a+b x)} \]
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Rubi [A] time = 0.0183912, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2636, 2641} \[ \frac{2 F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{3 b}+\frac{2 \sin (a+b x)}{3 b \cos ^{\frac{3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{\cos ^{\frac{5}{2}}(a+b x)} \, dx &=\frac{2 \sin (a+b x)}{3 b \cos ^{\frac{3}{2}}(a+b x)}+\frac{1}{3} \int \frac{1}{\sqrt{\cos (a+b x)}} \, dx\\ &=\frac{2 F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{3 b}+\frac{2 \sin (a+b x)}{3 b \cos ^{\frac{3}{2}}(a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0596565, size = 36, normalized size = 0.86 \[ \frac{2 \left (F\left (\left .\frac{1}{2} (a+b x)\right |2\right )+\frac{\sin (a+b x)}{\cos ^{\frac{3}{2}}(a+b x)}\right )}{3 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 2.024, size = 213, normalized size = 5.1 \begin{align*} -{\frac{2}{3\,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{ \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{-2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+ \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}}}} \left ( 2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) ^{-{\frac{3}{2}}} \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-1}} \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}{\cos \left (b x + a\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{1}{\cos \left (b x + a\right )^{\frac{5}{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}{\cos \left (b x + a\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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