Optimal. Leaf size=25 \[ -\frac {\cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b} \]
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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 = {3207, 2638} \[ -\frac {\cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3207
Rubi steps
\begin {align*} \int \sqrt [3]{c \sin ^3(a+b x)} \, dx &=\left (\csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int \sin (a+b x) \, dx\\ &=-\frac {\cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 25, normalized size = 1.00 \[ -\frac {\cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 43, normalized size = 1.72 \[ -\frac {\left (-{\left (c \cos \left (b x + a\right )^{2} - c\right )} \sin \left (b x + a\right )\right )^{\frac {1}{3}} \cos \left (b x + a\right )}{b \sin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \sin \left (b x + a\right )^{3}\right )^{\frac {1}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.26, size = 105, normalized size = 4.20 \[ -\frac {i \left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {1}{3}} {\mathrm e}^{2 i \left (b x +a \right )}}{2 b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}-\frac {i \left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {1}{3}}}{2 b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 31, normalized size = 1.24 \[ -\frac {2 \, c^{\frac {1}{3}}}{b {\left (\frac {\sin \left (b x + a\right )^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.85, size = 49, normalized size = 1.96 \[ -\frac {\sin \left (2\,a+2\,b\,x\right )\,{\left (2\,c\,\left (3\,\sin \left (a+b\,x\right )-\sin \left (3\,a+3\,b\,x\right )\right )\right )}^{1/3}}{4\,b\,{\sin \left (a+b\,x\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.82, size = 53, normalized size = 2.12 \[ \begin {cases} 0 & \text {for}\: a = - b x \vee a = - b x + \pi \\x \sqrt [3]{c \sin ^{3}{\relax (a )}} & \text {for}\: b = 0 \\- \frac {\sqrt [3]{c} \sqrt [3]{\sin ^{3}{\left (a + b x \right )}} \cos {\left (a + b x \right )}}{b \sin {\left (a + b x \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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