Optimal. Leaf size=55 \[ \frac{1}{2} x \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}-\frac{\cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{2 b} \]
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Rubi [A] time = 0.023013, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3207, 2635, 8} \[ \frac{1}{2} x \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}-\frac{\cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{2 b} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \left (c \sin ^3(a+b x)\right )^{2/3} \, dx &=\left (\csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int \sin ^2(a+b x) \, dx\\ &=-\frac{\cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{2 b}+\frac{1}{2} \left (\csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int 1 \, dx\\ &=-\frac{\cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{2 b}+\frac{1}{2} x \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\\ \end{align*}
Mathematica [A] time = 0.0939423, size = 47, normalized size = 0.85 \[ \frac{(2 (a+b x)-\sin (2 (a+b x))) \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{4 b} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.113, size = 158, normalized size = 2.9 \begin{align*} -{\frac{x{{\rm e}^{2\,i \left ( bx+a \right ) }}}{2\, \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) ^{2}} \left ( ic \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( bx+a \right ) }} \right ) ^{{\frac{2}{3}}}}-{\frac{{\frac{i}{8}}{{\rm e}^{4\,i \left ( bx+a \right ) }}}{ \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) ^{2}b} \left ( ic \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( bx+a \right ) }} \right ) ^{{\frac{2}{3}}}}+{\frac{{\frac{i}{8}}}{ \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) ^{2}b} \left ( ic \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( bx+a \right ) }} \right ) ^{{\frac{2}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.47115, size = 157, normalized size = 2.85 \begin{align*} \frac{c^{\frac{2}{3}} \arctan \left (\frac{\sin \left (b x + a\right )}{\cos \left (b x + a\right ) + 1}\right ) - \frac{\frac{c^{\frac{2}{3}} \sin \left (b x + a\right )}{\cos \left (b x + a\right ) + 1} - \frac{c^{\frac{2}{3}} \sin \left (b x + a\right )^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}}}{\frac{2 \, \sin \left (b x + a\right )^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac{\sin \left (b x + a\right )^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + 1}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62901, size = 146, normalized size = 2.65 \begin{align*} -\frac{{\left (b x - \cos \left (b x + a\right ) \sin \left (b x + a\right )\right )} \left (-{\left (c \cos \left (b x + a\right )^{2} - c\right )} \sin \left (b x + a\right )\right )^{\frac{2}{3}}}{2 \,{\left (b \cos \left (b x + a\right )^{2} - b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \sin ^{3}{\left (a + b x \right )}\right )^{\frac{2}{3}}\, dx \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{2}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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