Optimal. Leaf size=164 \[ -\frac{3 \sqrt{2 \pi } \sqrt{b} \cos (a) (c+d x)^{2/3} \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b}}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}+\frac{3 \sqrt{2 \pi } \sqrt{b} \sin (a) (c+d x)^{2/3} S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}+\frac{3 (c+d x) \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d (e (c+d x))^{2/3}} \]
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Rubi [A] time = 0.157629, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {3435, 3417, 3415, 3359, 3387, 3354, 3352, 3351} \[ -\frac{3 \sqrt{2 \pi } \sqrt{b} \cos (a) (c+d x)^{2/3} \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b}}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}+\frac{3 \sqrt{2 \pi } \sqrt{b} \sin (a) (c+d x)^{2/3} S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}+\frac{3 (c+d x) \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d (e (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
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Rule 3435
Rule 3417
Rule 3415
Rule 3359
Rule 3387
Rule 3354
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)^{2/3}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sin \left (a+\frac{b}{x^{2/3}}\right )}{(e x)^{2/3}} \, 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^{2/3}} \, dx,x,c+d x\right )}{d (e (c+d x))^{2/3}}\\ &=\frac{\left (3 (c+d x)^{2/3}\right ) \operatorname{Subst}\left (\int \sin \left (a+\frac{b}{x^2}\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d (e (c+d x))^{2/3}}\\ &=-\frac{\left (3 (c+d x)^{2/3}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (a+b x^2\right )}{x^2} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}\\ &=\frac{3 (c+d x) \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d (e (c+d x))^{2/3}}-\frac{\left (6 b (c+d x)^{2/3}\right ) \operatorname{Subst}\left (\int \cos \left (a+b x^2\right ) \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}\\ &=\frac{3 (c+d x) \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d (e (c+d x))^{2/3}}-\frac{\left (6 b (c+d x)^{2/3} \cos (a)\right ) \operatorname{Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}+\frac{\left (6 b (c+d x)^{2/3} \sin (a)\right ) \operatorname{Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}\\ &=-\frac{3 \sqrt{b} \sqrt{2 \pi } (c+d x)^{2/3} \cos (a) C\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}+\frac{3 \sqrt{b} \sqrt{2 \pi } (c+d x)^{2/3} S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{d (e (c+d x))^{2/3}}+\frac{3 (c+d x) \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{d (e (c+d x))^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.327287, size = 136, normalized size = 0.83 \[ \frac{3 \left (\sqrt{2 \pi } \left (-\sqrt{b}\right ) \cos (a) (c+d x)^{2/3} \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{b}}{\sqrt [3]{c+d x}}\right )+\sqrt{2 \pi } \sqrt{b} \sin (a) (c+d x)^{2/3} S\left (\frac{\sqrt{b} \sqrt{\frac{2}{\pi }}}{\sqrt [3]{c+d x}}\right )+(c+d x) \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )\right )}{d (e (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.04, 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{2}{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{\sin \left (\frac{a d x + a c +{\left (d x + c\right )}^{\frac{1}{3}} b}{d x + c}\right )}{{\left (d e x + c e\right )}^{\frac{2}{3}}}, x\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 + \frac{b}{\left (c + d x\right )^{\frac{2}{3}}} \right )}}{\left (e \left (c + d 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 \frac{\sin \left (a + \frac{b}{{\left (d x + c\right )}^{\frac{2}{3}}}\right )}{{\left (d e x + c e\right )}^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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