Optimal. Leaf size=168 \[ \frac {3 \sqrt {b} \sqrt {2 \pi } \sqrt [3]{c+d x} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac {3 \sqrt {b} \sqrt {2 \pi } \sqrt [3]{c+d x} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{d e \sqrt [3]{e (c+d x)}}-\frac {3 \sin \left (a+b (c+d x)^{2/3}\right )}{d e \sqrt [3]{e (c+d x)}} \]
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Rubi [A]
time = 0.12, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {3516, 3498,
3496, 3468, 3435, 3433, 3432} \begin {gather*} \frac {3 \sqrt {2 \pi } \sqrt {b} \cos (a) \sqrt [3]{c+d x} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac {3 \sqrt {2 \pi } \sqrt {b} \sin (a) \sqrt [3]{c+d x} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac {3 \sin \left (a+b (c+d x)^{2/3}\right )}{d e \sqrt [3]{e (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3432
Rule 3433
Rule 3435
Rule 3468
Rule 3496
Rule 3498
Rule 3516
Rubi steps
\begin {align*} \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{4/3}} \, dx &=\frac {\text {Subst}\left (\int \frac {\sin \left (a+b x^{2/3}\right )}{(e x)^{4/3}} \, dx,x,c+d x\right )}{d}\\ &=\frac {\sqrt [3]{c+d x} \text {Subst}\left (\int \frac {\sin \left (a+b x^{2/3}\right )}{x^{4/3}} \, dx,x,c+d x\right )}{d e \sqrt [3]{e (c+d x)}}\\ &=\frac {\left (3 \sqrt [3]{c+d x}\right ) \text {Subst}\left (\int \frac {\sin \left (a+b x^2\right )}{x^2} \, dx,x,\sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}\\ &=-\frac {3 \sin \left (a+b (c+d x)^{2/3}\right )}{d e \sqrt [3]{e (c+d x)}}+\frac {\left (6 b \sqrt [3]{c+d x}\right ) \text {Subst}\left (\int \cos \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}\\ &=-\frac {3 \sin \left (a+b (c+d x)^{2/3}\right )}{d e \sqrt [3]{e (c+d x)}}+\frac {\left (6 b \sqrt [3]{c+d x} \cos (a)\right ) \text {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac {\left (6 b \sqrt [3]{c+d x} \sin (a)\right ) \text {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}\\ &=\frac {3 \sqrt {b} \sqrt {2 \pi } \sqrt [3]{c+d x} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac {3 \sqrt {b} \sqrt {2 \pi } \sqrt [3]{c+d x} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{d e \sqrt [3]{e (c+d x)}}-\frac {3 \sin \left (a+b (c+d x)^{2/3}\right )}{d e \sqrt [3]{e (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 133, normalized size = 0.79 \begin {gather*} -\frac {3 \left (-\sqrt {b} \sqrt {2 \pi } \sqrt [3]{c+d x} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )+\sqrt {b} \sqrt {2 \pi } \sqrt [3]{c+d x} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)+\sin \left (a+b (c+d x)^{2/3}\right )\right )}{d e \sqrt [3]{e (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{\left (d e x +c e \right )^{\frac {4}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.58, size = 379, normalized size = 2.26 \begin {gather*} -\frac {3 \, {\left ({\left ({\left (-i \, \Gamma \left (-\frac {1}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + i \, \Gamma \left (-\frac {1}{2}, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (i \, \Gamma \left (-\frac {1}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) - i \, \Gamma \left (-\frac {1}{2}, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) - {\left (\Gamma \left (-\frac {1}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (-\frac {1}{2}, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (\Gamma \left (-\frac {1}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (-\frac {1}{2}, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right )\right )} \cos \left (a\right ) + {\left ({\left (\Gamma \left (-\frac {1}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (-\frac {1}{2}, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (\Gamma \left (-\frac {1}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (-\frac {1}{2}, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (-i \, \Gamma \left (-\frac {1}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + i \, \Gamma \left (-\frac {1}{2}, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (-i \, \Gamma \left (-\frac {1}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + i \, \Gamma \left (-\frac {1}{2}, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right )\right )} \sin \left (a\right )\right )} \sqrt {{\left (d x + c\right )}^{\frac {2}{3}} b} e^{\left (-\frac {4}{3}\right )}}{8 \, {\left (d x + c\right )}^{\frac {1}{3}} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin {\left (a + b \left (c + d x\right )^{\frac {2}{3}} \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 = -0.01 \begin {gather*} \int \frac {\sin \left (a+b\,{\left (c+d\,x\right )}^{2/3}\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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