Optimal. Leaf size=141 \[ -\frac{3 \sqrt{\pi } \sin (a) \sqrt [3]{c+d x} \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b}}{\sqrt [3]{c+d x}}\right )}{\sqrt{2} \sqrt{b} d e \sqrt [3]{e (c+d x)}}-\frac{3 \sqrt{\pi } \cos (a) \sqrt [3]{c+d x} S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{\sqrt{2} \sqrt{b} d e \sqrt [3]{e (c+d x)}} \]
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Rubi [A] time = 0.137753, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3435, 3417, 3383, 3353, 3352, 3351} \[ -\frac{3 \sqrt{\pi } \sin (a) \sqrt [3]{c+d x} \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b}}{\sqrt [3]{c+d x}}\right )}{\sqrt{2} \sqrt{b} d e \sqrt [3]{e (c+d x)}}-\frac{3 \sqrt{\pi } \cos (a) \sqrt [3]{c+d x} S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{\sqrt{2} \sqrt{b} d e \sqrt [3]{e (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3435
Rule 3417
Rule 3383
Rule 3353
Rule 3352
Rule 3351
Rubi steps
\begin{align*} \int \frac{\sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{(c e+d e x)^{4/3}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sin \left (a+\frac{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+\frac{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 \sin \left (a+b x^2\right ) \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d e \sqrt [3]{e (c+d x)}}\\ &=-\frac{\left (3 \sqrt [3]{c+d x} \cos (a)\right ) \operatorname{Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac{\left (3 \sqrt [3]{c+d x} \sin (a)\right ) \operatorname{Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d e \sqrt [3]{e (c+d x)}}\\ &=-\frac{3 \sqrt{\pi } \sqrt [3]{c+d x} \cos (a) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{\sqrt{2} \sqrt{b} d e \sqrt [3]{e (c+d x)}}-\frac{3 \sqrt{\pi } \sqrt [3]{c+d x} C\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{\sqrt{2} \sqrt{b} d e \sqrt [3]{e (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.170769, size = 96, normalized size = 0.68 \[ -\frac{3 \sqrt{\frac{\pi }{2}} (c+d x)^{4/3} \left (\sin (a) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b}}{\sqrt [3]{c+d x}}\right )+\cos (a) S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right )\right )}{\sqrt{b} d (e (c+d x))^{4/3}} \]
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{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 (\frac{a d x + a c +{\left (d x + c\right )}^{\frac{1}{3}} b}{d x + c}\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 (a + \frac{b}{{\left (d x + c\right )}^{\frac{2}{3}}}\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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