Optimal. Leaf size=24 \[ \frac{\tan (a+b x) \left (c \cos ^m(a+b x)\right )^{\frac{1}{m}}}{b} \]
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Rubi [A] time = 0.0198843, antiderivative size = 24, 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, 2637} \[ \frac{\tan (a+b x) \left (c \cos ^m(a+b x)\right )^{\frac{1}{m}}}{b} \]
Antiderivative was successfully verified.
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Rule 3208
Rule 2637
Rubi steps
\begin{align*} \int \left (c \cos ^m(a+b x)\right )^{\frac{1}{m}} \, dx &=\left (\left (c \cos ^m(a+b x)\right )^{\frac{1}{m}} \sec (a+b x)\right ) \int \cos (a+b x) \, dx\\ &=\frac{\left (c \cos ^m(a+b x)\right )^{\frac{1}{m}} \tan (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.029611, size = 24, normalized size = 1. \[ \frac{\tan (a+b x) \left (c \cos ^m(a+b x)\right )^{\frac{1}{m}}}{b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.26, size = 0, normalized size = 0. \begin{align*} \int \sqrt [m]{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 \left (c \cos \left (b x + a\right )^{m}\right )^{\left (\frac{1}{m}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.0493, size = 32, normalized size = 1.33 \begin{align*} \frac{c^{\left (\frac{1}{m}\right )} \sin \left (b x + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.54197, size = 65, normalized size = 2.71 \begin{align*} \begin{cases} x \left (c \cos ^{m}{\left (a \right )}\right )^{\frac{1}{m}} & \text{for}\: b = 0 \\x \left (0^{m} c\right )^{\frac{1}{m}} & \text{for}\: a = - b x + \frac{\pi }{2} \vee a = - b x + \frac{3 \pi }{2} \\\frac{c^{\frac{1}{m}} \left (\cos ^{m}{\left (a + b x \right )}\right )^{\frac{1}{m}} \sin{\left (a + b x \right )}}{b \cos{\left (a + b x \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 8.07252, size = 405, normalized size = 16.88 \begin{align*} \frac{2 \,{\left ({\left | c \right |}^{\left (\frac{1}{m}\right )} \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a + \frac{\pi \mathrm{sgn}\left (c\right )}{4 \, m} - \frac{\pi }{4 \, m}\right )^{2} \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{3} -{\left | c \right |}^{\left (\frac{1}{m}\right )} \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a + \frac{\pi \mathrm{sgn}\left (c\right )}{4 \, m} - \frac{\pi }{4 \, m}\right )^{2} \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right ) + 4 \,{\left | c \right |}^{\left (\frac{1}{m}\right )} \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a + \frac{\pi \mathrm{sgn}\left (c\right )}{4 \, m} - \frac{\pi }{4 \, m}\right ) \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{2} -{\left | c \right |}^{\left (\frac{1}{m}\right )} \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{3} +{\left | c \right |}^{\left (\frac{1}{m}\right )} \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )\right )}}{b \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a + \frac{\pi \mathrm{sgn}\left (c\right )}{4 \, m} - \frac{\pi }{4 \, m}\right )^{2} \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{4} + 2 \, b \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a + \frac{\pi \mathrm{sgn}\left (c\right )}{4 \, m} - \frac{\pi }{4 \, m}\right )^{2} \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{2} + b \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{4} + b \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a + \frac{\pi \mathrm{sgn}\left (c\right )}{4 \, m} - \frac{\pi }{4 \, m}\right )^{2} + 2 \, b \tan \left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )^{2} + b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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