Optimal. Leaf size=227 \[ -\frac {9 \sqrt {\pi } \sin (a) (e (c+d x))^{2/3} C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{4 \sqrt {2} b^{5/2} d (c+d x)^{2/3}}-\frac {9 \sqrt {\pi } \cos (a) (e (c+d x))^{2/3} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{4 \sqrt {2} b^{5/2} d (c+d x)^{2/3}}+\frac {9 (e (c+d x))^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d \sqrt [3]{c+d x}}-\frac {3 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d} \]
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Rubi [A] time = 0.20, antiderivative size = 227, 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 = {3435, 3417, 3415, 3385, 3386, 3353, 3352, 3351} \[ -\frac {9 \sqrt {\pi } \sin (a) (e (c+d x))^{2/3} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt [3]{c+d x}\right )}{4 \sqrt {2} b^{5/2} d (c+d x)^{2/3}}-\frac {9 \sqrt {\pi } \cos (a) (e (c+d x))^{2/3} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{4 \sqrt {2} b^{5/2} d (c+d x)^{2/3}}+\frac {9 (e (c+d x))^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d \sqrt [3]{c+d x}}-\frac {3 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d} \]
Antiderivative was successfully verified.
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Rule 3351
Rule 3352
Rule 3353
Rule 3385
Rule 3386
Rule 3415
Rule 3417
Rule 3435
Rubi steps
\begin {align*} \int (c e+d e x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right ) \, dx &=\frac {\operatorname {Subst}\left (\int (e x)^{2/3} \sin \left (a+b x^{2/3}\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {(e (c+d x))^{2/3} \operatorname {Subst}\left (\int x^{2/3} \sin \left (a+b x^{2/3}\right ) \, dx,x,c+d x\right )}{d (c+d x)^{2/3}}\\ &=\frac {\left (3 (e (c+d x))^{2/3}\right ) \operatorname {Subst}\left (\int x^4 \sin \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d (c+d x)^{2/3}}\\ &=-\frac {3 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac {\left (9 (e (c+d x))^{2/3}\right ) \operatorname {Subst}\left (\int x^2 \cos \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{2 b d (c+d x)^{2/3}}\\ &=-\frac {3 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac {9 (e (c+d x))^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d \sqrt [3]{c+d x}}-\frac {\left (9 (e (c+d x))^{2/3}\right ) \operatorname {Subst}\left (\int \sin \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{4 b^2 d (c+d x)^{2/3}}\\ &=-\frac {3 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac {9 (e (c+d x))^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d \sqrt [3]{c+d x}}-\frac {\left (9 (e (c+d x))^{2/3} \cos (a)\right ) \operatorname {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{4 b^2 d (c+d x)^{2/3}}-\frac {\left (9 (e (c+d x))^{2/3} \sin (a)\right ) \operatorname {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{4 b^2 d (c+d x)^{2/3}}\\ &=-\frac {3 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}-\frac {9 \sqrt {\pi } (e (c+d x))^{2/3} \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{4 \sqrt {2} b^{5/2} d (c+d x)^{2/3}}-\frac {9 \sqrt {\pi } (e (c+d x))^{2/3} C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{4 \sqrt {2} b^{5/2} d (c+d x)^{2/3}}+\frac {9 (e (c+d x))^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d \sqrt [3]{c+d x}}\\ \end {align*}
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Mathematica [A] time = 0.47, size = 160, normalized size = 0.70 \[ -\frac {3 (e (c+d x))^{2/3} \left (3 \sqrt {2 \pi } \sin (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )+3 \sqrt {2 \pi } \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )+2 \sqrt {b} \left (2 b (c+d x) \cos \left (a+b (c+d x)^{2/3}\right )-3 \sqrt [3]{c+d x} \sin \left (a+b (c+d x)^{2/3}\right )\right )\right )}{8 b^{5/2} d (c+d x)^{2/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d e x + c e\right )}^{\frac {2}{3}} \sin \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.72, size = 321, normalized size = 1.41 \[ -\frac {3 \, {\left (\frac {4 \, c {\left (\cos \left ({\left (d x e + c e\right )}^{\frac {2}{3}} b e^{\left (-\frac {2}{3}\right )} + a\right ) + \cos \left (-{\left (d x e + c e\right )}^{\frac {2}{3}} b e^{\left (-\frac {2}{3}\right )} - a\right )\right )} e^{\frac {2}{3}}}{b} - {\left (\frac {4 \, c e^{\left (i \, {\left (d x e + c e\right )}^{\frac {2}{3}} b e^{\left (-\frac {2}{3}\right )} + i \, a + \frac {5}{3}\right )}}{b} + \frac {4 \, c e^{\left (-i \, {\left (d x e + c e\right )}^{\frac {2}{3}} b e^{\left (-\frac {2}{3}\right )} - i \, a + \frac {5}{3}\right )}}{b} + \frac {2 i \, {\left (2 i \, {\left (d x e + c e\right )} b e^{\left (-\frac {2}{3}\right )} - 3 \, {\left (d x e + c e\right )}^{\frac {1}{3}}\right )} e^{\left (i \, {\left (d x e + c e\right )}^{\frac {2}{3}} b e^{\left (-\frac {2}{3}\right )} + i \, a + \frac {4}{3}\right )}}{b^{2}} + \frac {2 i \, {\left (2 i \, {\left (d x e + c e\right )} b e^{\left (-\frac {2}{3}\right )} + 3 \, {\left (d x e + c e\right )}^{\frac {1}{3}}\right )} e^{\left (-i \, {\left (d x e + c e\right )}^{\frac {2}{3}} b e^{\left (-\frac {2}{3}\right )} - i \, a + \frac {4}{3}\right )}}{b^{2}} - \frac {3 i \, \sqrt {\pi } \operatorname {erf}\left (-{\left (d x e + c e\right )}^{\frac {1}{3}} \sqrt {-i \, b e^{\left (-\frac {2}{3}\right )}}\right ) e^{\left (i \, a + \frac {4}{3}\right )}}{\sqrt {-i \, b e^{\left (-\frac {2}{3}\right )}} b^{2}} + \frac {3 i \, \sqrt {\pi } \operatorname {erf}\left (-{\left (d x e + c e\right )}^{\frac {1}{3}} \sqrt {i \, b e^{\left (-\frac {2}{3}\right )}}\right ) e^{\left (-i \, a + \frac {4}{3}\right )}}{\sqrt {i \, b e^{\left (-\frac {2}{3}\right )}} b^{2}}\right )} e^{\left (-1\right )}\right )}}{16 \, d} \]
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 \[ \int \left (d e x +c e \right )^{\frac {2}{3}} \sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.82, size = 429, normalized size = 1.89 \[ -\frac {{\left (d x + c\right )}^{\frac {2}{3}} {\left ({\left (9 \, {\left (\Gamma \left (\frac {3}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (\frac {3}{2}, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (\frac {3}{4} \, \pi + \arctan \left (0, d x + c\right )\right ) + 9 \, {\left (\Gamma \left (\frac {3}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (\frac {3}{2}, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (-\frac {3}{4} \, \pi + \arctan \left (0, d x + c\right )\right ) - {\left (-9 i \, \Gamma \left (\frac {3}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + 9 i \, \Gamma \left (\frac {3}{2}, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (\frac {3}{4} \, \pi + \arctan \left (0, d x + c\right )\right ) - {\left (-9 i \, \Gamma \left (\frac {3}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + 9 i \, \Gamma \left (\frac {3}{2}, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (-\frac {3}{4} \, \pi + \arctan \left (0, d x + c\right )\right )\right )} \cos \relax (a) - {\left ({\left (-9 i \, \Gamma \left (\frac {3}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + 9 i \, \Gamma \left (\frac {3}{2}, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (\frac {3}{4} \, \pi + \arctan \left (0, d x + c\right )\right ) + {\left (9 i \, \Gamma \left (\frac {3}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) - 9 i \, \Gamma \left (\frac {3}{2}, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (-\frac {3}{4} \, \pi + \arctan \left (0, d x + c\right )\right ) + 9 \, {\left (\Gamma \left (\frac {3}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (\frac {3}{2}, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (\frac {3}{4} \, \pi + \arctan \left (0, d x + c\right )\right ) - 9 \, {\left (\Gamma \left (\frac {3}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (\frac {3}{2}, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (-\frac {3}{4} \, \pi + \arctan \left (0, d x + c\right )\right )\right )} \sin \relax (a)\right )} \sqrt {{\left (d x + c\right )}^{\frac {2}{3}} b} e^{\frac {2}{3}} + 24 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} e^{\frac {2}{3}} \cos \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right )}{16 \, {\left (b^{3} d^{2} x + b^{3} c d\right )}} \]
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 \[ \int \sin \left (a+b\,{\left (c+d\,x\right )}^{2/3}\right )\,{\left (c\,e+d\,e\,x\right )}^{2/3} \,d x \]
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 \[ \int \left (e \left (c + d x\right )\right )^{\frac {2}{3}} \sin {\left (a + b \left (c + d x\right )^{\frac {2}{3}} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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