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 = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, 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 2637
Rule 3208
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*}
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Mathematica [A] time = 0.03, size = 24, normalized size = 1.00 \[ \frac {\tan (a+b x) \left (c \cos ^m(a+b x)\right )^{\frac {1}{m}}}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 15, normalized size = 0.62 \[ \frac {c^{\left (\frac {1}{m}\right )} \sin \left (b x + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 5.14, size = 300, normalized size = 12.50 \[ \frac {2 \, {\left ({\left | c \right |}^{\left (\frac {1}{m}\right )} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a + \frac {\pi \mathrm {sgn}\relax (c)}{4 \, m} - \frac {\pi }{4 \, m}\right )^{2} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{3} - {\left | c \right |}^{\left (\frac {1}{m}\right )} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a + \frac {\pi \mathrm {sgn}\relax (c)}{4 \, m} - \frac {\pi }{4 \, m}\right )^{2} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) + 4 \, {\left | c \right |}^{\left (\frac {1}{m}\right )} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a + \frac {\pi \mathrm {sgn}\relax (c)}{4 \, m} - \frac {\pi }{4 \, m}\right ) \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - {\left | c \right |}^{\left (\frac {1}{m}\right )} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{3} + {\left | c \right |}^{\left (\frac {1}{m}\right )} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )\right )}}{b \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a + \frac {\pi \mathrm {sgn}\relax (c)}{4 \, m} - \frac {\pi }{4 \, m}\right )^{2} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + 2 \, b \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a + \frac {\pi \mathrm {sgn}\relax (c)}{4 \, m} - \frac {\pi }{4 \, m}\right )^{2} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} + b \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + b \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a + \frac {\pi \mathrm {sgn}\relax (c)}{4 \, m} - \frac {\pi }{4 \, m}\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} + b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \left (c \left (\cos ^{m}\left (b x +a \right )\right )\right )^{\frac {1}{m}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \cos \left (b x + a\right )^{m}\right )^{\left (\frac {1}{m}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.40, size = 40, normalized size = 1.67 \[ \frac {\sin \left (2\,a+2\,b\,x\right )\,{\left (c\,{\cos \left (a+b\,x\right )}^m\right )}^{1/m}}{b\,\left (\cos \left (2\,a+2\,b\,x\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.58, size = 65, normalized size = 2.71 \[ \begin {cases} x \left (c \cos ^{m}{\relax (a )}\right )^{\frac {1}{m}} & \text {for}\: b = 0 \\x \left (0^{m} c\right )^{\frac {1}{m}} & \text {for}\: a = - b x + \frac {\pi }{2} \vee a = - b x + \frac {3 \pi }{2} \\\frac {c^{\frac {1}{m}} \left (\cos ^{m}{\left (a + b x \right )}\right )^{\frac {1}{m}} \sin {\left (a + b x \right )}}{b \cos {\left (a + b x \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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