Optimal. Leaf size=237 \[ \frac {3 \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b d e^2 \sqrt [3]{c+d x} (e (c+d x))^{2/3}}+\frac {9 \sqrt {\frac {\pi }{2}} (c+d x)^{2/3} \cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{4 b^{5/2} d e^2 (e (c+d x))^{2/3}}+\frac {9 \sqrt {\frac {\pi }{2}} (c+d x)^{2/3} C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{4 b^{5/2} d e^2 (e (c+d x))^{2/3}}-\frac {9 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 b^2 d e^2 (e (c+d x))^{2/3}} \]
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Rubi [A]
time = 0.17, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3516, 3498,
3496, 3490, 3466, 3467, 3434, 3433, 3432} \begin {gather*} \frac {9 \sqrt {\frac {\pi }{2}} \sin (a) (c+d x)^{2/3} \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b}}{\sqrt [3]{c+d x}}\right )}{4 b^{5/2} d e^2 (e (c+d x))^{2/3}}+\frac {9 \sqrt {\frac {\pi }{2}} \cos (a) (c+d x)^{2/3} S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{4 b^{5/2} d e^2 (e (c+d x))^{2/3}}-\frac {9 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 b^2 d e^2 (e (c+d x))^{2/3}}+\frac {3 \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b d e^2 \sqrt [3]{c+d x} (e (c+d x))^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3432
Rule 3433
Rule 3434
Rule 3466
Rule 3467
Rule 3490
Rule 3496
Rule 3498
Rule 3516
Rubi steps
\begin {align*} \int \frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{(c e+d e x)^{8/3}} \, dx &=\frac {\text {Subst}\left (\int \frac {\sin \left (a+\frac {b}{x^{2/3}}\right )}{(e x)^{8/3}} \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x)^{2/3} \text {Subst}\left (\int \frac {\sin \left (a+\frac {b}{x^{2/3}}\right )}{x^{8/3}} \, dx,x,c+d x\right )}{d e^2 (e (c+d x))^{2/3}}\\ &=\frac {\left (3 (c+d x)^{2/3}\right ) \text {Subst}\left (\int \frac {\sin \left (a+\frac {b}{x^2}\right )}{x^6} \, dx,x,\sqrt [3]{c+d x}\right )}{d e^2 (e (c+d x))^{2/3}}\\ &=-\frac {\left (3 (c+d x)^{2/3}\right ) \text {Subst}\left (\int x^4 \sin \left (a+b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d e^2 (e (c+d x))^{2/3}}\\ &=\frac {3 \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b d e^2 \sqrt [3]{c+d x} (e (c+d x))^{2/3}}-\frac {\left (9 (c+d x)^{2/3}\right ) \text {Subst}\left (\int x^2 \cos \left (a+b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{2 b d e^2 (e (c+d x))^{2/3}}\\ &=\frac {3 \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b d e^2 \sqrt [3]{c+d x} (e (c+d x))^{2/3}}-\frac {9 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 b^2 d e^2 (e (c+d x))^{2/3}}+\frac {\left (9 (c+d x)^{2/3}\right ) \text {Subst}\left (\int \sin \left (a+b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{4 b^2 d e^2 (e (c+d x))^{2/3}}\\ &=\frac {3 \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b d e^2 \sqrt [3]{c+d x} (e (c+d x))^{2/3}}-\frac {9 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 b^2 d e^2 (e (c+d x))^{2/3}}+\frac {\left (9 (c+d x)^{2/3} \cos (a)\right ) \text {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{4 b^2 d e^2 (e (c+d x))^{2/3}}+\frac {\left (9 (c+d x)^{2/3} \sin (a)\right ) \text {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{4 b^2 d e^2 (e (c+d x))^{2/3}}\\ &=\frac {3 \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b d e^2 \sqrt [3]{c+d x} (e (c+d x))^{2/3}}+\frac {9 \sqrt {\frac {\pi }{2}} (c+d x)^{2/3} \cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{4 b^{5/2} d e^2 (e (c+d x))^{2/3}}+\frac {9 \sqrt {\frac {\pi }{2}} (c+d x)^{2/3} C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{4 b^{5/2} d e^2 (e (c+d x))^{2/3}}-\frac {9 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 b^2 d e^2 (e (c+d x))^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.62, size = 165, normalized size = 0.70 \begin {gather*} \frac {(c+d x)^{5/3} \left (9 \sqrt {2 \pi } (c+d x) \cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )+9 \sqrt {2 \pi } (c+d x) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)+6 \sqrt {b} \left (2 b \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )-3 (c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )\right )}{8 b^{5/2} d (e (c+d x))^{8/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{\left (d e x +c e \right )^{\frac {8}{3}}}\, dx\]
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 = 0.61, size = 442, normalized size = 1.86 \begin {gather*} \frac {3 \, {\left (d x + c\right )}^{\frac {1}{3}} {\left ({\left ({\left (i \, e^{\frac {1}{3}} \Gamma \left (\frac {5}{2}, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - i \, e^{\frac {1}{3}} \Gamma \left (\frac {5}{2}, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\right )} \cos \left (\frac {5}{4} \, \pi + \frac {5}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (-i \, e^{\frac {1}{3}} \Gamma \left (\frac {5}{2}, -i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + i \, e^{\frac {1}{3}} \Gamma \left (\frac {5}{2}, \frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\right )} \cos \left (-\frac {5}{4} \, \pi + \frac {5}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (e^{\frac {1}{3}} \Gamma \left (\frac {5}{2}, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + e^{\frac {1}{3}} \Gamma \left (\frac {5}{2}, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\right )} \sin \left (\frac {5}{4} \, \pi + \frac {5}{3} \, \arctan \left (0, d x + c\right )\right ) - {\left (e^{\frac {1}{3}} \Gamma \left (\frac {5}{2}, -i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + e^{\frac {1}{3}} \Gamma \left (\frac {5}{2}, \frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\right )} \sin \left (-\frac {5}{4} \, \pi + \frac {5}{3} \, \arctan \left (0, d x + c\right )\right )\right )} \cos \left (a\right ) + {\left ({\left (e^{\frac {1}{3}} \Gamma \left (\frac {5}{2}, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + e^{\frac {1}{3}} \Gamma \left (\frac {5}{2}, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\right )} \cos \left (\frac {5}{4} \, \pi + \frac {5}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (e^{\frac {1}{3}} \Gamma \left (\frac {5}{2}, -i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + e^{\frac {1}{3}} \Gamma \left (\frac {5}{2}, \frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\right )} \cos \left (-\frac {5}{4} \, \pi + \frac {5}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (-i \, e^{\frac {1}{3}} \Gamma \left (\frac {5}{2}, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + i \, e^{\frac {1}{3}} \Gamma \left (\frac {5}{2}, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\right )} \sin \left (\frac {5}{4} \, \pi + \frac {5}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (-i \, e^{\frac {1}{3}} \Gamma \left (\frac {5}{2}, -i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + i \, e^{\frac {1}{3}} \Gamma \left (\frac {5}{2}, \frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\right )} \sin \left (-\frac {5}{4} \, \pi + \frac {5}{3} \, \arctan \left (0, d x + c\right )\right )\right )} \sin \left (a\right )\right )}}{8 \, {\left (d^{3} x^{2} e^{3} + 2 \, c d^{2} x e^{3} + c^{2} d e^{3}\right )} \left (\frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )^{\frac {5}{2}}} \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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 = -0.00 \begin {gather*} \int \frac {\sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{2/3}}\right )}{{\left (c\,e+d\,e\,x\right )}^{8/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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