Optimal. Leaf size=91 \[ \frac{3 \sqrt [3]{c+d x} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b d e (e (c+d x))^{2/3}}-\frac{3 (c+d x)^{2/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b^2 d e (e (c+d x))^{2/3}} \]
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Rubi [A] time = 0.0758568, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {3431, 15, 3296, 2637} \[ \frac{3 \sqrt [3]{c+d x} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b d e (e (c+d x))^{2/3}}-\frac{3 (c+d x)^{2/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b^2 d e (e (c+d x))^{2/3}} \]
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)^{5/3}} \, dx &=-\frac{3 \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{\left (\frac{e}{x^3}\right )^{5/3} x^4} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d}\\ &=-\frac{\left (3 (c+d x)^{2/3}\right ) \operatorname{Subst}\left (\int x \sin (a+b x) \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d e (e (c+d x))^{2/3}}\\ &=\frac{3 \sqrt [3]{c+d x} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b d e (e (c+d x))^{2/3}}-\frac{\left (3 (c+d x)^{2/3}\right ) \operatorname{Subst}\left (\int \cos (a+b x) \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{b d e (e (c+d x))^{2/3}}\\ &=\frac{3 \sqrt [3]{c+d x} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b d e (e (c+d x))^{2/3}}-\frac{3 (c+d x)^{2/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b^2 d e (e (c+d x))^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.0877367, size = 72, normalized size = 0.79 \[ \frac{3 (c+d x)^{5/3} \left (\frac{\cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b \sqrt [3]{c+d x}}-\frac{\sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b^2}\right )}{d (e (c+d x))^{5/3}} \]
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{5}{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 = 7.57984, size = 282, normalized size = 3.1 \begin{align*} \frac{3 \,{\left ({\left (d e x + c e\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{1}{3}} b \cos \left (\frac{a d x + a c +{\left (d x + c\right )}^{\frac{2}{3}} b}{d x + c}\right ) -{\left (d e x + c e\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} \sin \left (\frac{a d x + a c +{\left (d x + c\right )}^{\frac{2}{3}} b}{d x + c}\right )\right )}}{b^{2} d^{2} e^{2} x + b^{2} c d e^{2}} \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{5}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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