Optimal. Leaf size=44 \[ -\frac {3 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d \sqrt [3]{e (c+d x)}} \]
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Rubi [A] time = 0.06, antiderivative size = 44, normalized size of antiderivative = 1.00, 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 = {3435, 3381, 3379, 2638} \[ -\frac {3 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d \sqrt [3]{e (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3379
Rule 3381
Rule 3435
Rubi steps
\begin {align*} \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{\sqrt [3]{c e+d e x}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sin \left (a+b x^{2/3}\right )}{\sqrt [3]{e x}} \, dx,x,c+d x\right )}{d}\\ &=\frac {\sqrt [3]{c+d x} \operatorname {Subst}\left (\int \frac {\sin \left (a+b x^{2/3}\right )}{\sqrt [3]{x}} \, dx,x,c+d x\right )}{d \sqrt [3]{e (c+d x)}}\\ &=\frac {\left (3 \sqrt [3]{c+d x}\right ) \operatorname {Subst}\left (\int \sin (a+b x) \, dx,x,(c+d x)^{2/3}\right )}{2 d \sqrt [3]{e (c+d x)}}\\ &=-\frac {3 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d \sqrt [3]{e (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 44, normalized size = 1.00 \[ -\frac {3 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d \sqrt [3]{e (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 46, normalized size = 1.05 \[ -\frac {3 \, {\left (d e x + c e\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} \cos \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right )}{2 \, {\left (b d^{2} e x + b c d e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.54, size = 52, normalized size = 1.18 \[ -\frac {3 \, {\left (\cos \left ({\left (d x e + c e\right )}^{\frac {2}{3}} b e^{\left (-\frac {2}{3}\right )} + a\right ) + \cos \left (-{\left (d x e + c e\right )}^{\frac {2}{3}} b e^{\left (-\frac {2}{3}\right )} - a\right )\right )} e^{\left (-\frac {1}{3}\right )}}{4 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{\left (d e x +c e \right )^{\frac {1}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 23, normalized size = 0.52 \[ -\frac {3 \, \cos \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right )}{2 \, b d e^{\frac {1}{3}}} \]
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 \[ \int \frac {\sin \left (a+b\,{\left (c+d\,x\right )}^{2/3}\right )}{{\left (c\,e+d\,e\,x\right )}^{1/3}} \,d x \]
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 \[ \int \frac {\sin {\left (a + b \left (c + d x\right )^{\frac {2}{3}} \right )}}{\sqrt [3]{e \left (c + d x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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