Optimal. Leaf size=168 \[ \frac{3 \sqrt{2 \pi } \sqrt{b} \cos (a) \sqrt [3]{c+d x} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac{3 \sqrt{2 \pi } \sqrt{b} \sin (a) \sqrt [3]{c+d x} S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac{3 \sin \left (a+b (c+d x)^{2/3}\right )}{d e \sqrt [3]{e (c+d x)}} \]
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Rubi [A] time = 0.16265, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {3435, 3417, 3415, 3387, 3354, 3352, 3351} \[ \frac{3 \sqrt{2 \pi } \sqrt{b} \cos (a) \sqrt [3]{c+d x} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac{3 \sqrt{2 \pi } \sqrt{b} \sin (a) \sqrt [3]{c+d x} S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac{3 \sin \left (a+b (c+d x)^{2/3}\right )}{d e \sqrt [3]{e (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3435
Rule 3417
Rule 3415
Rule 3387
Rule 3354
Rule 3352
Rule 3351
Rubi steps
\begin{align*} \int \frac{\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{4/3}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sin \left (a+b x^{2/3}\right )}{(e x)^{4/3}} \, 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 )}{x^{4/3}} \, dx,x,c+d x\right )}{d e \sqrt [3]{e (c+d x)}}\\ &=\frac{\left (3 \sqrt [3]{c+d x}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (a+b x^2\right )}{x^2} \, dx,x,\sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}\\ &=-\frac{3 \sin \left (a+b (c+d x)^{2/3}\right )}{d e \sqrt [3]{e (c+d x)}}+\frac{\left (6 b \sqrt [3]{c+d x}\right ) \operatorname{Subst}\left (\int \cos \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}\\ &=-\frac{3 \sin \left (a+b (c+d x)^{2/3}\right )}{d e \sqrt [3]{e (c+d x)}}+\frac{\left (6 b \sqrt [3]{c+d x} \cos (a)\right ) \operatorname{Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac{\left (6 b \sqrt [3]{c+d x} \sin (a)\right ) \operatorname{Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}\\ &=\frac{3 \sqrt{b} \sqrt{2 \pi } \sqrt [3]{c+d x} \cos (a) C\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac{3 \sqrt{b} \sqrt{2 \pi } \sqrt [3]{c+d x} S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{d e \sqrt [3]{e (c+d x)}}-\frac{3 \sin \left (a+b (c+d x)^{2/3}\right )}{d e \sqrt [3]{e (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.24195, size = 133, normalized size = 0.79 \[ -\frac{3 \left (\sqrt{2 \pi } \left (-\sqrt{b}\right ) \cos (a) \sqrt [3]{c+d x} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [3]{c+d x}\right )+\sqrt{2 \pi } \sqrt{b} \sin (a) \sqrt [3]{c+d x} S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [3]{c+d x}\right )+\sin \left (a+b (c+d x)^{2/3}\right )\right )}{d e \sqrt [3]{e (c+d x)}} \]
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 ) \left ( dex+ce \right ) ^{-{\frac{4}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: IndexError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (d e x + c e\right )}^{\frac{2}{3}} \sin \left ({\left (d x + c\right )}^{\frac{2}{3}} b + a\right )}{d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}, x\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 ({\left (d x + c\right )}^{\frac{2}{3}} b + a\right )}{{\left (d e x + c e\right )}^{\frac{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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