Optimal. Leaf size=168 \[ \frac {3 b \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 d}+\frac {3 b^2 \sqrt [3]{e (c+d x)} \text {Ci}\left (\frac {b}{(c+d x)^{2/3}}\right ) \sin (a)}{4 d \sqrt [3]{c+d x}}+\frac {3 (c+d x) \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 d}+\frac {3 b^2 \sqrt [3]{e (c+d x)} \cos (a) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )}{4 d \sqrt [3]{c+d x}} \]
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Rubi [A]
time = 0.13, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {3516, 3462,
3460, 3378, 3384, 3380, 3383} \begin {gather*} \frac {3 b^2 \sin (a) \sqrt [3]{e (c+d x)} \text {CosIntegral}\left (\frac {b}{(c+d x)^{2/3}}\right )}{4 d \sqrt [3]{c+d x}}+\frac {3 b^2 \cos (a) \sqrt [3]{e (c+d x)} \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )}{4 d \sqrt [3]{c+d x}}+\frac {3 (c+d x) \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 d}+\frac {3 b \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rule 3460
Rule 3462
Rule 3516
Rubi steps
\begin {align*} \int \sqrt [3]{c e+d e x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right ) \, dx &=\frac {\text {Subst}\left (\int \sqrt [3]{e x} \sin \left (a+\frac {b}{x^{2/3}}\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {\sqrt [3]{e (c+d x)} \text {Subst}\left (\int \sqrt [3]{x} \sin \left (a+\frac {b}{x^{2/3}}\right ) \, dx,x,c+d x\right )}{d \sqrt [3]{c+d x}}\\ &=-\frac {\left (3 \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int \frac {\sin (a+b x)}{x^3} \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{c+d x}}\\ &=\frac {3 (c+d x) \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 d}-\frac {\left (3 b \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int \frac {\cos (a+b x)}{x^2} \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{4 d \sqrt [3]{c+d x}}\\ &=\frac {3 b \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 d}+\frac {3 (c+d x) \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 d}+\frac {\left (3 b^2 \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int \frac {\sin (a+b x)}{x} \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{4 d \sqrt [3]{c+d x}}\\ &=\frac {3 b \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 d}+\frac {3 (c+d x) \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 d}+\frac {\left (3 b^2 \sqrt [3]{e (c+d x)} \cos (a)\right ) \text {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{4 d \sqrt [3]{c+d x}}+\frac {\left (3 b^2 \sqrt [3]{e (c+d x)} \sin (a)\right ) \text {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{4 d \sqrt [3]{c+d x}}\\ &=\frac {3 b \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 d}+\frac {3 b^2 \sqrt [3]{e (c+d x)} \text {Ci}\left (\frac {b}{(c+d x)^{2/3}}\right ) \sin (a)}{4 d \sqrt [3]{c+d x}}+\frac {3 (c+d x) \sqrt [3]{e (c+d x)} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{4 d}+\frac {3 b^2 \sqrt [3]{e (c+d x)} \cos (a) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )}{4 d \sqrt [3]{c+d x}}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 113, normalized size = 0.67 \begin {gather*} \frac {3 \sqrt [3]{e (c+d x)} \left (b (c+d x)^{2/3} \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )+b^2 \text {Ci}\left (\frac {b}{(c+d x)^{2/3}}\right ) \sin (a)+(c+d x)^{4/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )+b^2 \cos (a) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )\right )}{4 d \sqrt [3]{c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (d e x +c e \right )^{\frac {1}{3}} \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )\, dx\]
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 = 0.51, size = 128, normalized size = 0.76 \begin {gather*} \frac {3 \, {\left ({\left (-i \, \Gamma \left (-2, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + i \, \Gamma \left (-2, -i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - i \, \Gamma \left (-2, \frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + i \, \Gamma \left (-2, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\right )} \cos \left (a\right ) - {\left (\Gamma \left (-2, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + \Gamma \left (-2, -i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + \Gamma \left (-2, \frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (-2, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\right )} \sin \left (a\right )\right )} b^{2} e^{\frac {1}{3}}}{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 \sqrt [3]{e \left (c + d x\right )} \sin {\left (a + \frac {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 [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 \sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{2/3}}\right )\,{\left (c\,e+d\,e\,x\right )}^{1/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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