Optimal. Leaf size=47 \[ \frac{3 (c+d x)^{2/3} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 b d e (e (c+d x))^{2/3}} \]
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Rubi [A] time = 0.066342, antiderivative size = 47, 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} \[ \frac{3 (c+d x)^{2/3} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 b d e (e (c+d x))^{2/3}} \]
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+\frac{b}{(c+d x)^{2/3}}\right )}{(c e+d e x)^{5/3}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sin \left (a+\frac{b}{x^{2/3}}\right )}{(e x)^{5/3}} \, dx,x,c+d x\right )}{d}\\ &=\frac{(c+d x)^{2/3} \operatorname{Subst}\left (\int \frac{\sin \left (a+\frac{b}{x^{2/3}}\right )}{x^{5/3}} \, dx,x,c+d x\right )}{d e (e (c+d x))^{2/3}}\\ &=-\frac{\left (3 (c+d x)^{2/3}\right ) \operatorname{Subst}\left (\int \sin (a+b x) \, dx,x,\frac{1}{(c+d x)^{2/3}}\right )}{2 d e (e (c+d x))^{2/3}}\\ &=\frac{3 (c+d x)^{2/3} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 b d e (e (c+d x))^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.0819698, size = 44, normalized size = 0.94 \[ \frac{3 (c+d x)^{5/3} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 b d (e (c+d x))^{5/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.042, size = 0, normalized size = 0. \begin{align*} \int{\sin \left ( a+{b \left ( dx+c \right ) ^{-{\frac{2}{3}}}} \right ) \left ( dex+ce \right ) ^{-{\frac{5}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03258, size = 42, normalized size = 0.89 \begin{align*} \frac{3 \, \cos \left (\frac{{\left (d x + c\right )}^{\frac{2}{3}} a + b}{{\left (d x + c\right )}^{\frac{2}{3}}}\right )}{2 \, b d e^{\frac{5}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46317, size = 157, normalized size = 3.34 \begin{align*} \frac{3 \,{\left (d e x + c e\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} \cos \left (\frac{a d x + a c +{\left (d x + c\right )}^{\frac{1}{3}} b}{d x + c}\right )}{2 \,{\left (b d^{2} e^{2} x + b c d e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \frac{\sin \left (a + \frac{b}{{\left (d x + c\right )}^{\frac{2}{3}}}\right )}{{\left (d e x + c e\right )}^{\frac{5}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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