Optimal. Leaf size=227 \[ -\frac {3 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}-\frac {9 \sqrt {\pi } (e (c+d x))^{2/3} \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{4 \sqrt {2} b^{5/2} d (c+d x)^{2/3}}-\frac {9 \sqrt {\pi } (e (c+d x))^{2/3} C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{4 \sqrt {2} b^{5/2} d (c+d x)^{2/3}}+\frac {9 (e (c+d x))^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d \sqrt [3]{c+d x}} \]
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Rubi [A]
time = 0.14, 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 = {3516, 3498,
3496, 3466, 3467, 3434, 3433, 3432} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 3432
Rule 3433
Rule 3434
Rule 3466
Rule 3467
Rule 3496
Rule 3498
Rule 3516
Rubi steps
\begin {align*} \int (c e+d e x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right ) \, dx &=\frac {\text {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} \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 0.31, size = 160, normalized size = 0.70 \begin {gather*} -\frac {3 (e (c+d x))^{2/3} \left (3 \sqrt {2 \pi } \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )+3 \sqrt {2 \pi } C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)+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}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, 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] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.68, size = 422, normalized size = 1.86 \begin {gather*} -\frac {3 \, {\left (3 \, {\left (d x + c\right )}^{\frac {2}{3}} {\left ({\left ({\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 (\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 (i \, \Gamma \left (\frac {3}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) - 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 (i \, \Gamma \left (\frac {3}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) - 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 \left (a\right ) + {\left ({\left (i \, \Gamma \left (\frac {3}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) - 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 (-i \, \Gamma \left (\frac {3}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + 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 (\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 ) + {\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 \left (a\right )\right )} \sqrt {{\left (d x + c\right )}^{\frac {2}{3}} b} e^{\frac {2}{3}} + 8 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \cos \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right ) e^{\frac {2}{3}}\right )}}{16 \, {\left (b^{3} d^{2} x + b^{3} c d\right )}} \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 \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 5.87, size = 321, normalized size = 1.41 \begin {gather*} -\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} \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 \sin \left (a+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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