Optimal. Leaf size=172 \[ \frac{18 \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b^4 d e^2 \sqrt [3]{e (c+d x)}}-\frac{9 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b^2 d e^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)}}-\frac{18 \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b^3 d e^2 \sqrt [3]{e (c+d x)}}+\frac{3 \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b d e^2 (c+d x)^{2/3} \sqrt [3]{e (c+d x)}} \]
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Rubi [A] time = 0.147866, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {3431, 15, 3296, 2637} \[ \frac{18 \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b^4 d e^2 \sqrt [3]{e (c+d x)}}-\frac{9 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b^2 d e^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)}}-\frac{18 \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b^3 d e^2 \sqrt [3]{e (c+d x)}}+\frac{3 \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b d e^2 (c+d x)^{2/3} \sqrt [3]{e (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3431
Rule 15
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int \frac{\sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{(c e+d e x)^{7/3}} \, dx &=-\frac{3 \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{\left (\frac{e}{x^3}\right )^{7/3} x^4} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d}\\ &=-\frac{\left (3 \sqrt [3]{c+d x}\right ) \operatorname{Subst}\left (\int x^3 \sin (a+b x) \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d e^2 \sqrt [3]{e (c+d x)}}\\ &=\frac{3 \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b d e^2 (c+d x)^{2/3} \sqrt [3]{e (c+d x)}}-\frac{\left (9 \sqrt [3]{c+d x}\right ) \operatorname{Subst}\left (\int x^2 \cos (a+b x) \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{b d e^2 \sqrt [3]{e (c+d x)}}\\ &=\frac{3 \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b d e^2 (c+d x)^{2/3} \sqrt [3]{e (c+d x)}}-\frac{9 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b^2 d e^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)}}+\frac{\left (18 \sqrt [3]{c+d x}\right ) \operatorname{Subst}\left (\int x \sin (a+b x) \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{b^2 d e^2 \sqrt [3]{e (c+d x)}}\\ &=-\frac{18 \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b^3 d e^2 \sqrt [3]{e (c+d x)}}+\frac{3 \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b d e^2 (c+d x)^{2/3} \sqrt [3]{e (c+d x)}}-\frac{9 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b^2 d e^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)}}+\frac{\left (18 \sqrt [3]{c+d x}\right ) \operatorname{Subst}\left (\int \cos (a+b x) \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{b^3 d e^2 \sqrt [3]{e (c+d x)}}\\ &=-\frac{18 \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b^3 d e^2 \sqrt [3]{e (c+d x)}}+\frac{3 \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b d e^2 (c+d x)^{2/3} \sqrt [3]{e (c+d x)}}-\frac{9 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b^2 d e^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)}}+\frac{18 \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b^4 d e^2 \sqrt [3]{e (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.166364, size = 107, normalized size = 0.62 \[ -\frac{3 \left (3 \sqrt [3]{c+d x} \left (b^2 \sqrt [3]{c+d x}-2 c-2 d x\right ) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )+\left (6 b (c+d x)-b^3 \sqrt [3]{c+d x}\right ) \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )\right )}{b^4 d e (e (c+d x))^{4/3}} \]
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{\frac{1}{\sqrt [3]{dx+c}}}} \right ) \left ( dex+ce \right ) ^{-{\frac{7}{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 [A] time = 8.88699, size = 382, normalized size = 2.22 \begin{align*} \frac{3 \,{\left ({\left ({\left (d x + c\right )}^{\frac{1}{3}} b^{3} - 6 \, b d x - 6 \, b c\right )}{\left (d e x + c e\right )}^{\frac{2}{3}} \cos \left (\frac{a d x + a c +{\left (d x + c\right )}^{\frac{2}{3}} b}{d x + c}\right ) - 3 \,{\left (d e x + c e\right )}^{\frac{2}{3}}{\left ({\left (d x + c\right )}^{\frac{2}{3}} b^{2} - 2 \,{\left (d x + c\right )}^{\frac{4}{3}}\right )} \sin \left (\frac{a d x + a c +{\left (d x + c\right )}^{\frac{2}{3}} b}{d x + c}\right )\right )}}{b^{4} d^{3} e^{3} x^{2} + 2 \, b^{4} c d^{2} e^{3} x + b^{4} c^{2} d e^{3}} \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{1}{3}}}\right )}{{\left (d e x + c e\right )}^{\frac{7}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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