Optimal. Leaf size=25 \[ -\frac {\cot (a+b x) \left (c \sin ^n(a+b x)\right )^{\frac {1}{n}}}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 25, 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, 2638} \[ -\frac {\cot (a+b x) \left (c \sin ^n(a+b x)\right )^{\frac {1}{n}}}{b} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3208
Rubi steps
\begin {align*} \int \left (c \sin ^n(a+b x)\right )^{\frac {1}{n}} \, dx &=\left (\csc (a+b x) \left (c \sin ^n(a+b x)\right )^{\frac {1}{n}}\right ) \int \sin (a+b x) \, dx\\ &=-\frac {\cot (a+b x) \left (c \sin ^n(a+b x)\right )^{\frac {1}{n}}}{b}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 25, normalized size = 1.00 \[ -\frac {\cot (a+b x) \left (c \sin ^n(a+b x)\right )^{\frac {1}{n}}}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 16, normalized size = 0.64 \[ -\frac {c^{\left (\frac {1}{n}\right )} \cos \left (b x + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.43, size = 384, normalized size = 15.36 \[ \frac {{\left | c \right |}^{\left (\frac {1}{n}\right )} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a + \frac {\pi \mathrm {sgn}\relax (c)}{4 \, n} - \frac {\pi }{4 \, n}\right )^{2} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} - 2 \, {\left | c \right |}^{\left (\frac {1}{n}\right )} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a + \frac {\pi \mathrm {sgn}\relax (c)}{4 \, n} - \frac {\pi }{4 \, n}\right )^{2} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} + 4 \, {\left | c \right |}^{\left (\frac {1}{n}\right )} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a + \frac {\pi \mathrm {sgn}\relax (c)}{4 \, n} - \frac {\pi }{4 \, n}\right ) \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{3} - {\left | c \right |}^{\left (\frac {1}{n}\right )} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + {\left | c \right |}^{\left (\frac {1}{n}\right )} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a + \frac {\pi \mathrm {sgn}\relax (c)}{4 \, n} - \frac {\pi }{4 \, n}\right )^{2} - 4 \, {\left | c \right |}^{\left (\frac {1}{n}\right )} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a + \frac {\pi \mathrm {sgn}\relax (c)}{4 \, n} - \frac {\pi }{4 \, n}\right ) \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) + 2 \, {\left | c \right |}^{\left (\frac {1}{n}\right )} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - {\left | c \right |}^{\left (\frac {1}{n}\right )}}{b \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a + \frac {\pi \mathrm {sgn}\relax (c)}{4 \, n} - \frac {\pi }{4 \, n}\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 \, n} - \frac {\pi }{4 \, n}\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 \, n} - \frac {\pi }{4 \, n}\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 = 0.85, size = 0, normalized size = 0.00 \[ \int \left (c \left (\sin ^{n}\left (b x +a \right )\right )\right )^{\frac {1}{n}}\, dx \]
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 \[ \int \left (c \sin \left (b x + a\right )^{n}\right )^{\left (\frac {1}{n}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.75, size = 36, normalized size = 1.44 \[ -\frac {\sin \left (2\,a+2\,b\,x\right )\,{\left (c\,{\sin \left (a+b\,x\right )}^n\right )}^{1/n}}{2\,b\,{\sin \left (a+b\,x\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.11, size = 61, normalized size = 2.44 \[ \begin {cases} x \left (c \sin ^{n}{\relax (a )}\right )^{\frac {1}{n}} & \text {for}\: b = 0 \\x \left (0^{n} c\right )^{\frac {1}{n}} & \text {for}\: a = - b x \vee a = - b x + \pi \\- \frac {c^{\frac {1}{n}} \left (\sin ^{n}{\left (a + b x \right )}\right )^{\frac {1}{n}} \cos {\left (a + b x \right )}}{b \sin {\left (a + b x \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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