Optimal. Leaf size=133 \[ \frac {3 \sqrt {\frac {\pi }{2}} (c+d x)^{2/3} \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{\sqrt {b} d (e (c+d x))^{2/3}}+\frac {3 \sqrt {\frac {\pi }{2}} (c+d x)^{2/3} C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{\sqrt {b} d (e (c+d x))^{2/3}} \]
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Rubi [A]
time = 0.09, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3516, 3498,
3464, 3434, 3433, 3432} \begin {gather*} \frac {3 \sqrt {\frac {\pi }{2}} \sin (a) (c+d x)^{2/3} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt [3]{c+d x}\right )}{\sqrt {b} d (e (c+d x))^{2/3}}+\frac {3 \sqrt {\frac {\pi }{2}} \cos (a) (c+d x)^{2/3} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{\sqrt {b} d (e (c+d x))^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3432
Rule 3433
Rule 3434
Rule 3464
Rule 3498
Rule 3516
Rubi steps
\begin {align*} \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{2/3}} \, dx &=\frac {\text {Subst}\left (\int \frac {\sin \left (a+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+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+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} \cos (a)\right ) \text {Subst}\left (\int \sin \left (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} \sin (a)\right ) \text {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d (e (c+d x))^{2/3}}\\ &=\frac {3 \sqrt {\frac {\pi }{2}} (c+d x)^{2/3} \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{\sqrt {b} d (e (c+d x))^{2/3}}+\frac {3 \sqrt {\frac {\pi }{2}} (c+d x)^{2/3} C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{\sqrt {b} d (e (c+d x))^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 96, normalized size = 0.72 \begin {gather*} \frac {3 \sqrt {\frac {\pi }{2}} (c+d x)^{2/3} \left (\cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )+C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)\right )}{\sqrt {b} d (e (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 \[\int \frac {\sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{\left (d e x +c e \right )^{\frac {2}{3}}}\, dx\]
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 = 0.58, size = 486, normalized size = 3.65 \begin {gather*} \frac {3 \, {\left ({\left ({\left (-i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )} + i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, {\left (d x + c\right )}^{\frac {2}{3}} b}\right ) - 1\right )}\right )} \cos \left (\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )} - i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, {\left (d x + c\right )}^{\frac {2}{3}} b}\right ) - 1\right )}\right )} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )} + \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, {\left (d x + c\right )}^{\frac {2}{3}} b}\right ) - 1\right )}\right )} \sin \left (\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) - {\left (\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )} + \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, {\left (d x + c\right )}^{\frac {2}{3}} b}\right ) - 1\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 (\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )} + \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, {\left (d x + c\right )}^{\frac {2}{3}} b}\right ) - 1\right )}\right )} \cos \left (\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )} + \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, {\left (d x + c\right )}^{\frac {2}{3}} b}\right ) - 1\right )}\right )} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )} - i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, {\left (d x + c\right )}^{\frac {2}{3}} b}\right ) - 1\right )}\right )} \sin \left (\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )} - i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, {\left (d x + c\right )}^{\frac {2}{3}} b}\right ) - 1\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 {{\left (d x + c\right )}^{\frac {2}{3}} b} e^{\left (-\frac {2}{3}\right )}}{8 \, {\left (d x + c\right )}^{\frac {1}{3}} b 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 + 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 [C] Result contains complex when optimal does not.
time = 6.30, size = 84, normalized size = 0.63 \begin {gather*} -\frac {3 \, {\left (-\frac {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\right )}}{\sqrt {-i \, b e^{\left (-\frac {2}{3}\right )}}} + \frac {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\right )}}{\sqrt {i \, b e^{\left (-\frac {2}{3}\right )}}}\right )} e^{\left (-1\right )}}{4 \, 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 = -0.01 \begin {gather*} \int \frac {\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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