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 = 0.0189031, antiderivative size = 25, 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, 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 3208
Rule 2638
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*}
Mathematica [A] time = 0.0337482, size = 25, normalized size = 1. \[ -\frac{\cot (a+b x) \left (c \sin ^n(a+b x)\right )^{\frac{1}{n}}}{b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.292, size = 0, normalized size = 0. \begin{align*} \int \sqrt [n]{c \left ( \sin \left ( bx+a \right ) \right ) ^{n}}\, 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 \sin \left (b x + a\right )^{n}\right )^{\left (\frac{1}{n}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.643, size = 34, normalized size = 1.36 \begin{align*} -\frac{c^{\left (\frac{1}{n}\right )} \cos \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 = 6.30427, size = 61, normalized size = 2.44 \begin{align*} \begin{cases} x \left (c \sin ^{n}{\left (a \right )}\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 8.92025, size = 518, normalized size = 20.72 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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