Optimal. Leaf size=277 \[ \frac {45 \sqrt {\pi } \cos (a) \sqrt [3]{c+d x} C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{8 \sqrt {2} b^{7/2} d e^3 \sqrt [3]{e (c+d x)}}-\frac {45 \sqrt {\pi } \sin (a) \sqrt [3]{c+d x} S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{8 \sqrt {2} b^{7/2} d e^3 \sqrt [3]{e (c+d x)}}-\frac {45 \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{8 b^3 d e^3 \sqrt [3]{e (c+d x)}}-\frac {15 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 b^2 d e^3 (c+d x)^{2/3} \sqrt [3]{e (c+d x)}}+\frac {3 \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b d e^3 (c+d x)^{4/3} \sqrt [3]{e (c+d x)}} \]
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Rubi [A] time = 0.26, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3435, 3417, 3415, 3409, 3385, 3386, 3354, 3352, 3351} \[ \frac {45 \sqrt {\pi } \cos (a) \sqrt [3]{c+d x} \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b}}{\sqrt [3]{c+d x}}\right )}{8 \sqrt {2} b^{7/2} d e^3 \sqrt [3]{e (c+d x)}}-\frac {45 \sqrt {\pi } \sin (a) \sqrt [3]{c+d x} S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{8 \sqrt {2} b^{7/2} d e^3 \sqrt [3]{e (c+d x)}}-\frac {15 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 b^2 d e^3 (c+d x)^{2/3} \sqrt [3]{e (c+d x)}}-\frac {45 \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{8 b^3 d e^3 \sqrt [3]{e (c+d x)}}+\frac {3 \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b d e^3 (c+d x)^{4/3} \sqrt [3]{e (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3351
Rule 3352
Rule 3354
Rule 3385
Rule 3386
Rule 3409
Rule 3415
Rule 3417
Rule 3435
Rubi steps
\begin {align*} \int \frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{(c e+d e x)^{10/3}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sin \left (a+\frac {b}{x^{2/3}}\right )}{(e x)^{10/3}} \, dx,x,c+d x\right )}{d}\\ &=\frac {\sqrt [3]{c+d x} \operatorname {Subst}\left (\int \frac {\sin \left (a+\frac {b}{x^{2/3}}\right )}{x^{10/3}} \, dx,x,c+d x\right )}{d e^3 \sqrt [3]{e (c+d x)}}\\ &=\frac {\left (3 \sqrt [3]{c+d x}\right ) \operatorname {Subst}\left (\int \frac {\sin \left (a+\frac {b}{x^2}\right )}{x^8} \, dx,x,\sqrt [3]{c+d x}\right )}{d e^3 \sqrt [3]{e (c+d x)}}\\ &=-\frac {\left (3 \sqrt [3]{c+d x}\right ) \operatorname {Subst}\left (\int x^6 \sin \left (a+b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d e^3 \sqrt [3]{e (c+d x)}}\\ &=\frac {3 \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b d e^3 (c+d x)^{4/3} \sqrt [3]{e (c+d x)}}-\frac {\left (15 \sqrt [3]{c+d x}\right ) \operatorname {Subst}\left (\int x^4 \cos \left (a+b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{2 b d e^3 \sqrt [3]{e (c+d x)}}\\ &=\frac {3 \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b d e^3 (c+d x)^{4/3} \sqrt [3]{e (c+d x)}}-\frac {15 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 b^2 d e^3 (c+d x)^{2/3} \sqrt [3]{e (c+d x)}}+\frac {\left (45 \sqrt [3]{c+d x}\right ) \operatorname {Subst}\left (\int x^2 \sin \left (a+b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{4 b^2 d e^3 \sqrt [3]{e (c+d x)}}\\ &=-\frac {45 \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{8 b^3 d e^3 \sqrt [3]{e (c+d x)}}+\frac {3 \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b d e^3 (c+d x)^{4/3} \sqrt [3]{e (c+d x)}}-\frac {15 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 b^2 d e^3 (c+d x)^{2/3} \sqrt [3]{e (c+d x)}}+\frac {\left (45 \sqrt [3]{c+d x}\right ) \operatorname {Subst}\left (\int \cos \left (a+b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{8 b^3 d e^3 \sqrt [3]{e (c+d x)}}\\ &=-\frac {45 \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{8 b^3 d e^3 \sqrt [3]{e (c+d x)}}+\frac {3 \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b d e^3 (c+d x)^{4/3} \sqrt [3]{e (c+d x)}}-\frac {15 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 b^2 d e^3 (c+d x)^{2/3} \sqrt [3]{e (c+d x)}}+\frac {\left (45 \sqrt [3]{c+d x} \cos (a)\right ) \operatorname {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{8 b^3 d e^3 \sqrt [3]{e (c+d x)}}-\frac {\left (45 \sqrt [3]{c+d x} \sin (a)\right ) \operatorname {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{8 b^3 d e^3 \sqrt [3]{e (c+d x)}}\\ &=-\frac {45 \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{8 b^3 d e^3 \sqrt [3]{e (c+d x)}}+\frac {3 \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b d e^3 (c+d x)^{4/3} \sqrt [3]{e (c+d x)}}+\frac {45 \sqrt {\pi } \sqrt [3]{c+d x} \cos (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{8 \sqrt {2} b^{7/2} d e^3 \sqrt [3]{e (c+d x)}}-\frac {45 \sqrt {\pi } \sqrt [3]{c+d x} S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{8 \sqrt {2} b^{7/2} d e^3 \sqrt [3]{e (c+d x)}}-\frac {15 \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 b^2 d e^3 (c+d x)^{2/3} \sqrt [3]{e (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 1.27, size = 192, normalized size = 0.69 \[ \frac {(e (c+d x))^{2/3} \left (-6 \sqrt {b} \left (\left (15 (c+d x)^{4/3}-4 b^2\right ) \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )+10 b (c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )+45 \sqrt {2 \pi } \cos (a) (c+d x)^{5/3} C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )-45 \sqrt {2 \pi } \sin (a) (c+d x)^{5/3} S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )\right )}{16 b^{7/2} d e^4 (c+d x)^{7/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.36, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d e x + c e\right )}^{\frac {2}{3}} \sin \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {1}{3}} b}{d x + c}\right )}{d^{4} e^{4} x^{4} + 4 \, c d^{3} e^{4} x^{3} + 6 \, c^{2} d^{2} e^{4} x^{2} + 4 \, c^{3} d e^{4} x + c^{4} e^{4}}, x\right ) \]
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 \[ \int \frac {\sin \left (a + \frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )}{{\left (d e x + c e\right )}^{\frac {10}{3}}}\,{d x} \]
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 \[ \int \frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{\left (d e x +c e \right )^{\frac {10}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.83, size = 411, normalized size = 1.48 \[ -\frac {{\left ({\left (-3 i \, \Gamma \left (\frac {7}{2}, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + 3 i \, \Gamma \left (\frac {7}{2}, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\right )} \cos \left (\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (3 i \, \Gamma \left (\frac {7}{2}, -i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 3 i \, \Gamma \left (\frac {7}{2}, \frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\right )} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right ) - 3 \, {\left (\Gamma \left (\frac {7}{2}, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + \Gamma \left (\frac {7}{2}, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\right )} \sin \left (\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right ) + 3 \, {\left (\Gamma \left (\frac {7}{2}, -i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + \Gamma \left (\frac {7}{2}, \frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\right )} \sin \left (-\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right )\right )} \cos \relax (a) - {\left (3 \, {\left (\Gamma \left (\frac {7}{2}, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + \Gamma \left (\frac {7}{2}, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\right )} \cos \left (\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right ) + 3 \, {\left (\Gamma \left (\frac {7}{2}, -i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + \Gamma \left (\frac {7}{2}, \frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\right )} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right ) - {\left (3 i \, \Gamma \left (\frac {7}{2}, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 3 i \, \Gamma \left (\frac {7}{2}, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\right )} \sin \left (\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right ) - {\left (3 i \, \Gamma \left (\frac {7}{2}, -i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 3 i \, \Gamma \left (\frac {7}{2}, \frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\right )} \sin \left (-\frac {7}{4} \, \pi + \frac {7}{3} \, \arctan \left (0, d x + c\right )\right )\right )} \sin \relax (a)}{8 \, {\left (d^{3} e^{\frac {10}{3}} x^{2} + 2 \, c d^{2} e^{\frac {10}{3}} x + c^{2} d e^{\frac {10}{3}}\right )} {\left (d x + c\right )}^{\frac {1}{3}} \left (\frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )^{\frac {7}{2}}} \]
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 \[ \int \frac {\sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{2/3}}\right )}{{\left (c\,e+d\,e\,x\right )}^{10/3}} \,d x \]
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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