Optimal. Leaf size=47 \[ \frac {3 (c+d x)^{2/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b d e (e (c+d x))^{2/3}} \]
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Rubi [A] time = 0.07, antiderivative size = 47, 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} \[ \frac {3 (c+d x)^{2/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b d e (e (c+d x))^{2/3}} \]
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+\frac {b}{(c+d x)^{2/3}}\right )}{(c e+d e x)^{5/3}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sin \left (a+\frac {b}{x^{2/3}}\right )}{(e x)^{5/3}} \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x)^{2/3} \operatorname {Subst}\left (\int \frac {\sin \left (a+\frac {b}{x^{2/3}}\right )}{x^{5/3}} \, dx,x,c+d x\right )}{d e (e (c+d x))^{2/3}}\\ &=-\frac {\left (3 (c+d x)^{2/3}\right ) \operatorname {Subst}\left (\int \sin (a+b x) \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{2 d e (e (c+d x))^{2/3}}\\ &=\frac {3 (c+d x)^{2/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b d e (e (c+d x))^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 44, normalized size = 0.94 \[ \frac {3 (c+d x)^{5/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b d (e (c+d x))^{5/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 64, normalized size = 1.36 \[ \frac {3 \, {\left (d e x + c e\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} \cos \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {1}{3}} b}{d x + c}\right )}{2 \, {\left (b d^{2} e^{2} x + b c d e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (a + \frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )}{{\left (d e x + c e\right )}^{\frac {5}{3}}}\,{d x} \]
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 \[ \int \frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{\left (d e x +c e \right )^{\frac {5}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 31, normalized size = 0.66 \[ \frac {3 \, \cos \left (\frac {{\left (d x + c\right )}^{\frac {2}{3}} a + b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )}{2 \, b d e^{\frac {5}{3}}} \]
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 \[ \int \frac {\sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{2/3}}\right )}{{\left (c\,e+d\,e\,x\right )}^{5/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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