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 = 0.0175031, 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 = {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 3207
Rule 2638
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*}
Mathematica [A] time = 0.0627671, size = 25, normalized size = 1. \[ -\frac{\cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.111, size = 105, normalized size = 4.2 \begin{align*}{\frac{-{\frac{i}{2}}{{\rm e}^{2\,i \left ( bx+a \right ) }}}{ \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) b}\sqrt [3]{ic \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( bx+a \right ) }}}}-{\frac{{\frac{i}{2}}}{ \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) b}\sqrt [3]{ic \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( bx+a \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45972, size = 42, normalized size = 1.68 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69497, size = 104, normalized size = 4.16 \begin{align*} -\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.94314, size = 53, normalized size = 2.12 \begin{align*} \begin{cases} x \sqrt [3]{c \sin ^{3}{\left (a \right )}} & \text{for}\: b = 0 \\0 & \text{for}\: a = - b x \vee a = - b x + \pi \\- \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \sin \left (b x + a\right )^{3}\right )^{\frac{1}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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