Optimal. Leaf size=80 \[ -\frac{2 \sin (a+b x) \cos (a+b x) \, _2F_1\left (\frac{1}{2},\frac{2-m}{4};\frac{6-m}{4};\cos ^2(a+b x)\right )}{b (2-m) \sqrt{\sin ^2(a+b x)} \sqrt{c \cos ^m(a+b x)}} \]
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Rubi [A] time = 0.0466379, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3208, 2643} \[ -\frac{2 \sin (a+b x) \cos (a+b x) \, _2F_1\left (\frac{1}{2},\frac{2-m}{4};\frac{6-m}{4};\cos ^2(a+b x)\right )}{b (2-m) \sqrt{\sin ^2(a+b x)} \sqrt{c \cos ^m(a+b x)}} \]
Antiderivative was successfully verified.
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Rule 3208
Rule 2643
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{c \cos ^m(a+b x)}} \, dx &=\frac{\cos ^{\frac{m}{2}}(a+b x) \int \cos ^{-\frac{m}{2}}(a+b x) \, dx}{\sqrt{c \cos ^m(a+b x)}}\\ &=-\frac{2 \cos (a+b x) \, _2F_1\left (\frac{1}{2},\frac{2-m}{4};\frac{6-m}{4};\cos ^2(a+b x)\right ) \sin (a+b x)}{b (2-m) \sqrt{c \cos ^m(a+b x)} \sqrt{\sin ^2(a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.0749802, size = 72, normalized size = 0.9 \[ \frac{2 \sqrt{\sin ^2(a+b x)} \cot (a+b x) \, _2F_1\left (\frac{1}{2},\frac{2-m}{4};\frac{6-m}{4};\cos ^2(a+b x)\right )}{b (m-2) \sqrt{c \cos ^m(a+b x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.2, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt{c \left ( \cos \left ( bx+a \right ) \right ) ^{m}}}}\, dx \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}{\sqrt{c \cos \left (b x + a\right )^{m}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c \cos ^{m}{\left (a + b x \right )}}}\, dx \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}{\sqrt{c \cos \left (b x + a\right )^{m}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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