Optimal. Leaf size=164 \[ -\frac {3 \sqrt {b} \sqrt {2 \pi } (c+d x)^{2/3} \cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}+\frac {3 \sqrt {b} \sqrt {2 \pi } (c+d x)^{2/3} S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{d (e (c+d x))^{2/3}}+\frac {3 (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d (e (c+d x))^{2/3}} \]
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Rubi [A]
time = 0.11, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {3516, 3498,
3496, 3440, 3468, 3435, 3433, 3432} \begin {gather*} -\frac {3 \sqrt {2 \pi } \sqrt {b} \cos (a) (c+d x)^{2/3} \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b}}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}+\frac {3 \sqrt {2 \pi } \sqrt {b} \sin (a) (c+d x)^{2/3} S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}+\frac {3 (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d (e (c+d x))^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3432
Rule 3433
Rule 3435
Rule 3440
Rule 3468
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)^{2/3}} \, dx &=\frac {\text {Subst}\left (\int \frac {\sin \left (a+\frac {b}{x^{2/3}}\right )}{(e x)^{2/3}} \, 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^{2/3}} \, dx,x,c+d x\right )}{d (e (c+d x))^{2/3}}\\ &=\frac {\left (3 (c+d x)^{2/3}\right ) \text {Subst}\left (\int \sin \left (a+\frac {b}{x^2}\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d (e (c+d x))^{2/3}}\\ &=-\frac {\left (3 (c+d x)^{2/3}\right ) \text {Subst}\left (\int \frac {\sin \left (a+b x^2\right )}{x^2} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}\\ &=\frac {3 (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d (e (c+d x))^{2/3}}-\frac {\left (6 b (c+d x)^{2/3}\right ) \text {Subst}\left (\int \cos \left (a+b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}\\ &=\frac {3 (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d (e (c+d x))^{2/3}}-\frac {\left (6 b (c+d x)^{2/3} \cos (a)\right ) \text {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}+\frac {\left (6 b (c+d x)^{2/3} \sin (a)\right ) \text {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}\\ &=-\frac {3 \sqrt {b} \sqrt {2 \pi } (c+d x)^{2/3} \cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}+\frac {3 \sqrt {b} \sqrt {2 \pi } (c+d x)^{2/3} S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{d (e (c+d x))^{2/3}}+\frac {3 (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{d (e (c+d x))^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 136, normalized size = 0.83 \begin {gather*} \frac {3 \left (-\sqrt {b} \sqrt {2 \pi } (c+d x)^{2/3} \cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )+\sqrt {b} \sqrt {2 \pi } (c+d x)^{2/3} S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)+(c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )}{d (e (c+d x))^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{\left (d e x +c e \right )^{\frac {2}{3}}}\, dx\]
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 = 0.57, size = 382, normalized size = 2.33 \begin {gather*} -\frac {3 \, {\left (d x + c\right )}^{\frac {1}{3}} {\left ({\left ({\left (-i \, \Gamma \left (-\frac {1}{2}, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + i \, \Gamma \left (-\frac {1}{2}, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\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 {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - i \, \Gamma \left (-\frac {1}{2}, \frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\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 {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + \Gamma \left (-\frac {1}{2}, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\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 {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + \Gamma \left (-\frac {1}{2}, \frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\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 {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + \Gamma \left (-\frac {1}{2}, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\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 {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + \Gamma \left (-\frac {1}{2}, \frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\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 {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + i \, \Gamma \left (-\frac {1}{2}, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\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 {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + i \, \Gamma \left (-\frac {1}{2}, \frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\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 {\frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}} e^{\left (-\frac {2}{3}\right )}}{8 \, 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 + \frac {b}{\left (c + d x\right )^{\frac {2}{3}}} \right )}}{\left (e \left (c + d x\right )\right )^{\frac {2}{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+\frac {b}{{\left (c+d\,x\right )}^{2/3}}\right )}{{\left (c\,e+d\,e\,x\right )}^{2/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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