Optimal. Leaf size=44 \[ -\frac{3 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d \sqrt [3]{e (c+d x)}} \]
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Rubi [A] time = 0.0628817, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {3435, 3381, 3379, 2638} \[ -\frac{3 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d \sqrt [3]{e (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3435
Rule 3381
Rule 3379
Rule 2638
Rubi steps
\begin{align*} \int \frac{\sin \left (a+b (c+d x)^{2/3}\right )}{\sqrt [3]{c e+d e x}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sin \left (a+b x^{2/3}\right )}{\sqrt [3]{e x}} \, dx,x,c+d x\right )}{d}\\ &=\frac{\sqrt [3]{c+d x} \operatorname{Subst}\left (\int \frac{\sin \left (a+b x^{2/3}\right )}{\sqrt [3]{x}} \, dx,x,c+d x\right )}{d \sqrt [3]{e (c+d x)}}\\ &=\frac{\left (3 \sqrt [3]{c+d x}\right ) \operatorname{Subst}\left (\int \sin (a+b x) \, dx,x,(c+d x)^{2/3}\right )}{2 d \sqrt [3]{e (c+d x)}}\\ &=-\frac{3 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d \sqrt [3]{e (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.074525, size = 44, normalized size = 1. \[ -\frac{3 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d \sqrt [3]{e (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{\sin \left ( a+b \left ( dx+c \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{dex+ce}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04933, size = 31, normalized size = 0.7 \begin{align*} -\frac{3 \, \cos \left ({\left (d x + c\right )}^{\frac{2}{3}} b + a\right )}{2 \, b d e^{\frac{1}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71172, size = 123, normalized size = 2.8 \begin{align*} -\frac{3 \,{\left (d e x + c e\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}} \cos \left ({\left (d x + c\right )}^{\frac{2}{3}} b + a\right )}{2 \,{\left (b d^{2} e x + b c d e\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 \frac{\sin{\left (a + b \left (c + d x\right )^{\frac{2}{3}} \right )}}{\sqrt [3]{e \left (c + d x\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22065, size = 70, normalized size = 1.59 \begin{align*} -\frac{3 \,{\left (\cos \left ({\left (d x e + c e\right )}^{\frac{2}{3}} b e^{\left (-\frac{2}{3}\right )} + a\right ) + \cos \left (-{\left (d x e + c e\right )}^{\frac{2}{3}} b e^{\left (-\frac{2}{3}\right )} - a\right )\right )} e^{\left (-\frac{1}{3}\right )}}{4 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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