Optimal. Leaf size=45 \[ \frac {3 \sqrt [3]{c+d x} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b d e \sqrt [3]{e (c+d x)}} \]
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Rubi [A]
time = 0.03, antiderivative size = 45, 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 \sqrt [3]{c+d x} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b d e \sqrt [3]{e (c+d x)}} \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+\frac {b}{\sqrt [3]{c+d x}}\right )}{(c e+d e x)^{4/3}} \, dx &=-\frac {3 \text {Subst}\left (\int \frac {\sin (a+b x)}{\left (\frac {e}{x^3}\right )^{4/3} x^4} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d}\\ &=-\frac {\left (3 \sqrt [3]{c+d x}\right ) \text {Subst}\left (\int \sin (a+b x) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d e \sqrt [3]{e (c+d x)}}\\ &=\frac {3 \sqrt [3]{c+d x} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b d e \sqrt [3]{e (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 42, normalized size = 0.93 \begin {gather*} \frac {3 (c+d x)^{4/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b d (e (c+d x))^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right )}{\left (d e x +c e \right )^{\frac {4}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 30, normalized size = 0.67 \begin {gather*} \frac {3 \, \cos \left (\frac {{\left (d x + c\right )}^{\frac {1}{3}} a + b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) e^{\left (-\frac {4}{3}\right )}}{b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs.
\(2 (22) = 44\).
time = 0.35, size = 48, normalized size = 1.07 \begin {gather*} \frac {3 \, {\left (d x + c\right )} \cos \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {2}{3}} b}{d x + c}\right ) e^{\left (-\frac {4}{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 + \frac {b}{\sqrt [3]{c + d x}} \right )}}{\left (e \left (c + d x\right )\right )^{\frac {4}{3}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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+\frac {b}{{\left (c+d\,x\right )}^{1/3}}\right )}{{\left (c\,e+d\,e\,x\right )}^{4/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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