Optimal. Leaf size=42 \[ -\frac {3 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d (e (c+d x))^{2/3}} \]
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Rubi [A]
time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3512, 15, 2718}
\begin {gather*} -\frac {3 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d (e (c+d x))^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 2718
Rule 3512
Rubi steps
\begin {align*} \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{(c e+d e x)^{2/3}} \, dx &=\frac {3 \text {Subst}\left (\int \frac {x^2 \sin (a+b x)}{\left (e x^3\right )^{2/3}} \, dx,x,\sqrt [3]{c+d x}\right )}{d}\\ &=\frac {\left (3 (c+d x)^{2/3}\right ) \text {Subst}\left (\int \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d (e (c+d x))^{2/3}}\\ &=-\frac {3 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d (e (c+d x))^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 42, normalized size = 1.00 \begin {gather*} -\frac {3 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d (e (c+d x))^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{\left (d e x +c e \right )^{\frac {2}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 22, normalized size = 0.52 \begin {gather*} -\frac {3 \, \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) e^{\left (-\frac {2}{3}\right )}}{b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 34, normalized size = 0.81 \begin {gather*} -\frac {3 \, {\left (d x + c\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) e^{\left (-\frac {2}{3}\right )}}{b d^{2} x + b c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin {\left (a + b \sqrt [3]{c + d x} \right )}}{\left (e \left (c + d x\right )\right )^{\frac {2}{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.51, size = 35, normalized size = 0.83 \begin {gather*} -\frac {3 \, \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 {2}{3}\right )}}{b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sin \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )}{{\left (c\,e+d\,e\,x\right )}^{2/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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