Optimal. Leaf size=227 \[ -\frac{9 \sqrt{\pi } \sin (a) (e (c+d x))^{2/3} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [3]{c+d x}\right )}{4 \sqrt{2} b^{5/2} d (c+d x)^{2/3}}-\frac{9 \sqrt{\pi } \cos (a) (e (c+d x))^{2/3} S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [3]{c+d x}\right )}{4 \sqrt{2} b^{5/2} d (c+d x)^{2/3}}+\frac{9 (e (c+d x))^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d \sqrt [3]{c+d x}}-\frac{3 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d} \]
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Rubi [A] time = 0.197811, antiderivative size = 227, 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, 3385, 3386, 3353, 3352, 3351} \[ -\frac{9 \sqrt{\pi } \sin (a) (e (c+d x))^{2/3} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [3]{c+d x}\right )}{4 \sqrt{2} b^{5/2} d (c+d x)^{2/3}}-\frac{9 \sqrt{\pi } \cos (a) (e (c+d x))^{2/3} S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [3]{c+d x}\right )}{4 \sqrt{2} b^{5/2} d (c+d x)^{2/3}}+\frac{9 (e (c+d x))^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d \sqrt [3]{c+d x}}-\frac{3 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d} \]
Antiderivative was successfully verified.
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Rule 3435
Rule 3417
Rule 3415
Rule 3385
Rule 3386
Rule 3353
Rule 3352
Rule 3351
Rubi steps
\begin{align*} \int (c e+d e x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right ) \, dx &=\frac{\operatorname{Subst}\left (\int (e x)^{2/3} \sin \left (a+b x^{2/3}\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{(e (c+d x))^{2/3} \operatorname{Subst}\left (\int x^{2/3} \sin \left (a+b x^{2/3}\right ) \, dx,x,c+d x\right )}{d (c+d x)^{2/3}}\\ &=\frac{\left (3 (e (c+d x))^{2/3}\right ) \operatorname{Subst}\left (\int x^4 \sin \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d (c+d x)^{2/3}}\\ &=-\frac{3 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac{\left (9 (e (c+d x))^{2/3}\right ) \operatorname{Subst}\left (\int x^2 \cos \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{2 b d (c+d x)^{2/3}}\\ &=-\frac{3 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac{9 (e (c+d x))^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d \sqrt [3]{c+d x}}-\frac{\left (9 (e (c+d x))^{2/3}\right ) \operatorname{Subst}\left (\int \sin \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{4 b^2 d (c+d x)^{2/3}}\\ &=-\frac{3 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac{9 (e (c+d x))^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d \sqrt [3]{c+d x}}-\frac{\left (9 (e (c+d x))^{2/3} \cos (a)\right ) \operatorname{Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{4 b^2 d (c+d x)^{2/3}}-\frac{\left (9 (e (c+d x))^{2/3} \sin (a)\right ) \operatorname{Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{4 b^2 d (c+d x)^{2/3}}\\ &=-\frac{3 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}-\frac{9 \sqrt{\pi } (e (c+d x))^{2/3} \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [3]{c+d x}\right )}{4 \sqrt{2} b^{5/2} d (c+d x)^{2/3}}-\frac{9 \sqrt{\pi } (e (c+d x))^{2/3} C\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{4 \sqrt{2} b^{5/2} d (c+d x)^{2/3}}+\frac{9 (e (c+d x))^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d \sqrt [3]{c+d x}}\\ \end{align*}
Mathematica [A] time = 0.45467, size = 160, normalized size = 0.7 \[ -\frac{3 (e (c+d x))^{2/3} \left (3 \sqrt{2 \pi } \sin (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [3]{c+d x}\right )+3 \sqrt{2 \pi } \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [3]{c+d x}\right )+2 \sqrt{b} \left (2 b (c+d x) \cos \left (a+b (c+d x)^{2/3}\right )-3 \sqrt [3]{c+d x} \sin \left (a+b (c+d x)^{2/3}\right )\right )\right )}{8 b^{5/2} d (c+d x)^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.033, size = 0, normalized size = 0. \begin{align*} \int \left ( dex+ce \right ) ^{{\frac{2}{3}}}\sin \left ( a+b \left ( dx+c \right ) ^{{\frac{2}{3}}} \right ) \, 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 ({\left (d e x + c e\right )}^{\frac{2}{3}} \sin \left ({\left (d x + c\right )}^{\frac{2}{3}} b + a\right ), 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 \left (e \left (c + d x\right )\right )^{\frac{2}{3}} \sin{\left (a + b \left (c + d x\right )^{\frac{2}{3}} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.2228, size = 284, normalized size = 1.25 \begin{align*} -\frac{3 \,{\left (-\frac{2 i \,{\left (2 i \,{\left (d x e + c e\right )} b e^{\left (-\frac{2}{3}\right )} - 3 \,{\left (d x e + c e\right )}^{\frac{1}{3}}\right )} e^{\left (i \,{\left (d x e + c e\right )}^{\frac{2}{3}} b e^{\left (-\frac{2}{3}\right )} + i \, a + \frac{4}{3}\right )}}{b^{2}} - \frac{2 i \,{\left (2 i \,{\left (d x e + c e\right )} b e^{\left (-\frac{2}{3}\right )} + 3 \,{\left (d x e + c e\right )}^{\frac{1}{3}}\right )} e^{\left (-i \,{\left (d x e + c e\right )}^{\frac{2}{3}} b e^{\left (-\frac{2}{3}\right )} - i \, a + \frac{4}{3}\right )}}{b^{2}} + \frac{3 i \, \sqrt{\pi } \operatorname{erf}\left (-{\left (d x e + c e\right )}^{\frac{1}{3}} \sqrt{-i \, b e^{\left (-\frac{2}{3}\right )}}\right ) e^{\left (i \, a + \frac{4}{3}\right )}}{\sqrt{-i \, b e^{\left (-\frac{2}{3}\right )}} b^{2}} - \frac{3 i \, \sqrt{\pi } \operatorname{erf}\left (-{\left (d x e + c e\right )}^{\frac{1}{3}} \sqrt{i \, b e^{\left (-\frac{2}{3}\right )}}\right ) e^{\left (-i \, a + \frac{4}{3}\right )}}{\sqrt{i \, b e^{\left (-\frac{2}{3}\right )}} b^{2}}\right )} e^{\left (-1\right )}}{16 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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