Optimal. Leaf size=122 \[ -\frac {3 b \sqrt [3]{c+d x} \cos (a) \text {Ci}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac {3 (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac {3 b \sqrt [3]{c+d x} \sin (a) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}} \]
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Rubi [A]
time = 0.10, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps
used = 7, 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 \cos (a) \sqrt [3]{c+d x} \text {CosIntegral}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac {3 b \sin (a) \sqrt [3]{c+d x} \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac {3 (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}} \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 \frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{\sqrt [3]{c e+d e x}} \, dx &=\frac {\text {Subst}\left (\int \frac {\sin \left (a+\frac {b}{x^{2/3}}\right )}{\sqrt [3]{e x}} \, dx,x,c+d x\right )}{d}\\ &=\frac {\sqrt [3]{c+d x} \text {Subst}\left (\int \frac {\sin \left (a+\frac {b}{x^{2/3}}\right )}{\sqrt [3]{x}} \, dx,x,c+d x\right )}{d \sqrt [3]{e (c+d x)}}\\ &=-\frac {\left (3 \sqrt [3]{c+d x}\right ) \text {Subst}\left (\int \frac {\sin (a+b x)}{x^2} \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}}\\ &=\frac {3 (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}}-\frac {\left (3 b \sqrt [3]{c+d x}\right ) \text {Subst}\left (\int \frac {\cos (a+b x)}{x} \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}}\\ &=\frac {3 (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}}-\frac {\left (3 b \sqrt [3]{c+d x} \cos (a)\right ) \text {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac {\left (3 b \sqrt [3]{c+d x} \sin (a)\right ) \text {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}}\\ &=-\frac {3 b \sqrt [3]{c+d x} \cos (a) \text {Ci}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac {3 (c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac {3 b \sqrt [3]{c+d x} \sin (a) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )}{2 d \sqrt [3]{e (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 90, normalized size = 0.74 \begin {gather*} \frac {3 \left (-b \sqrt [3]{c+d x} \cos (a) \text {Ci}\left (\frac {b}{(c+d x)^{2/3}}\right )+(c+d x) \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )+b \sqrt [3]{c+d x} \sin (a) \text {Si}\left (\frac {b}{(c+d x)^{2/3}}\right )\right )}{2 d \sqrt [3]{e (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{\left (d e x +c e \right )^{\frac {1}{3}}}\, dx\]
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 = 0.46, size = 125, normalized size = 1.02 \begin {gather*} -\frac {3 \, {\left ({\left (\Gamma \left (-1, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + \Gamma \left (-1, -i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + \Gamma \left (-1, \frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (-1, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\right )} \cos \left (a\right ) + {\left (-i \, \Gamma \left (-1, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + i \, \Gamma \left (-1, -i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - i \, \Gamma \left (-1, \frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + i \, \Gamma \left (-1, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\right )} \sin \left (a\right )\right )} b e^{\left (-\frac {1}{3}\right )}}{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 \frac {\sin {\left (a + \frac {b}{\left (c + d x\right )^{\frac {2}{3}}} \right )}}{\sqrt [3]{e \left (c + d x\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 \frac {\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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