Optimal. Leaf size=202 \[ \frac{12 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac{72 (e (c+d x))^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d \sqrt [3]{c+d x}}-\frac{72 (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d (c+d x)^{2/3}}+\frac{36 (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac{3 (c+d x)^{2/3} (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d} \]
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Rubi [A] time = 0.177942, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {3431, 15, 3296, 2638} \[ \frac{12 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac{72 (e (c+d x))^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d \sqrt [3]{c+d x}}-\frac{72 (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d (c+d x)^{2/3}}+\frac{36 (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac{3 (c+d x)^{2/3} (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d} \]
Antiderivative was successfully verified.
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Rule 3431
Rule 15
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int (c e+d e x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx &=\frac{3 \operatorname{Subst}\left (\int x^2 \left (e x^3\right )^{2/3} \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d}\\ &=\frac{\left (3 (e (c+d x))^{2/3}\right ) \operatorname{Subst}\left (\int x^4 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d (c+d x)^{2/3}}\\ &=-\frac{3 (c+d x)^{2/3} (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac{\left (12 (e (c+d x))^{2/3}\right ) \operatorname{Subst}\left (\int x^3 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d (c+d x)^{2/3}}\\ &=-\frac{3 (c+d x)^{2/3} (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac{12 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac{\left (36 (e (c+d x))^{2/3}\right ) \operatorname{Subst}\left (\int x^2 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d (c+d x)^{2/3}}\\ &=\frac{36 (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac{3 (c+d x)^{2/3} (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac{12 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac{\left (72 (e (c+d x))^{2/3}\right ) \operatorname{Subst}\left (\int x \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^3 d (c+d x)^{2/3}}\\ &=\frac{36 (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac{3 (c+d x)^{2/3} (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}-\frac{72 (e (c+d x))^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d \sqrt [3]{c+d x}}+\frac{12 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}+\frac{\left (72 (e (c+d x))^{2/3}\right ) \operatorname{Subst}\left (\int \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^4 d (c+d x)^{2/3}}\\ &=\frac{36 (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac{72 (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d (c+d x)^{2/3}}-\frac{3 (c+d x)^{2/3} (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}-\frac{72 (e (c+d x))^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d \sqrt [3]{c+d x}}+\frac{12 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}\\ \end{align*}
Mathematica [A] time = 0.283815, size = 111, normalized size = 0.55 \[ -\frac{3 (e (c+d x))^{2/3} \left (\left (b^4 (c+d x)^{4/3}-12 b^2 (c+d x)^{2/3}+24\right ) \cos \left (a+b \sqrt [3]{c+d x}\right )-4 b \left (b^2 (c+d x)-6 \sqrt [3]{c+d x}\right ) \sin \left (a+b \sqrt [3]{c+d x}\right )\right )}{b^5 d (c+d x)^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.032, size = 0, normalized size = 0. \begin{align*} \int \left ( dex+ce \right ) ^{{\frac{2}{3}}}\sin \left ( a+b\sqrt [3]{dx+c} \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 [A] time = 7.76001, size = 356, normalized size = 1.76 \begin{align*} \frac{3 \,{\left ({\left (12 \, b^{2} d x + 12 \, b^{2} c -{\left (b^{4} d x + b^{4} c\right )}{\left (d x + c\right )}^{\frac{2}{3}} - 24 \,{\left (d x + c\right )}^{\frac{1}{3}}\right )}{\left (d e x + c e\right )}^{\frac{2}{3}} \cos \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right ) - 4 \,{\left (d e x + c e\right )}^{\frac{2}{3}}{\left (6 \,{\left (d x + c\right )}^{\frac{2}{3}} b -{\left (b^{3} d x + b^{3} c\right )}{\left (d x + c\right )}^{\frac{1}{3}}\right )} \sin \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )\right )}}{b^{5} d^{2} x + b^{5} c d} \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 \sqrt [3]{c + d x} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17336, size = 198, normalized size = 0.98 \begin{align*} -\frac{3 \,{\left (\frac{{\left ({\left (d x e + c e\right )}^{\frac{4}{3}} b^{4} e^{\frac{11}{3}} - 12 \,{\left (d x e + c e\right )}^{\frac{2}{3}} b^{2} e^{\frac{13}{3}} + 24 \, e^{5}\right )} \cos \left ({\left ({\left (d x e + c e\right )}^{\frac{1}{3}} b e^{\frac{2}{3}} + a e\right )} e^{\left (-1\right )}\right ) e^{\left (-\frac{10}{3}\right )}}{b^{5}} - \frac{4 \,{\left ({\left (d x e + c e\right )} b^{3} e^{4} - 6 \,{\left (d x e + c e\right )}^{\frac{1}{3}} b e^{\frac{14}{3}}\right )} e^{\left (-\frac{10}{3}\right )} \sin \left ({\left ({\left (d x e + c e\right )}^{\frac{1}{3}} b e^{\frac{2}{3}} + a e\right )} e^{\left (-1\right )}\right )}{b^{5}}\right )} e^{\left (-1\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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