Optimal. Leaf size=116 \[ -\frac {3 b (c+d x)^{2/3} \cos (a) \text {Ci}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}+\frac {3 (c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}+\frac {3 b (c+d x)^{2/3} \sin (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}} \]
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Rubi [A]
time = 0.09, antiderivative size = 116, 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 = {3512, 15, 3378,
3384, 3380, 3383} \begin {gather*} -\frac {3 b \cos (a) (c+d x)^{2/3} \text {CosIntegral}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}+\frac {3 b \sin (a) (c+d x)^{2/3} \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}+\frac {3 (c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rule 3512
Rubi steps
\begin {align*} \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(c e+d e x)^{2/3}} \, dx &=-\frac {3 \text {Subst}\left (\int \frac {\sin (a+b x)}{\left (\frac {e}{x^3}\right )^{2/3} x^4} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d}\\ &=-\frac {\left (3 (c+d x)^{2/3}\right ) \text {Subst}\left (\int \frac {\sin (a+b x)}{x^2} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}\\ &=\frac {3 (c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}-\frac {\left (3 b (c+d x)^{2/3}\right ) \text {Subst}\left (\int \frac {\cos (a+b x)}{x} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}\\ &=\frac {3 (c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}-\frac {\left (3 b (c+d x)^{2/3} \cos (a)\right ) \text {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}+\frac {\left (3 b (c+d x)^{2/3} \sin (a)\right ) \text {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}\\ &=-\frac {3 b (c+d x)^{2/3} \cos (a) \text {Ci}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}+\frac {3 (c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}+\frac {3 b (c+d x)^{2/3} \sin (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 88, normalized size = 0.76 \begin {gather*} \frac {3 \left (-b (c+d x)^{2/3} \cos (a) \text {Ci}\left (\frac {b}{\sqrt [3]{c+d x}}\right )+(c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )+b (c+d x)^{2/3} \sin (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{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 \[\int \frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right )}{\left (d e x +c e \right )^{\frac {2}{3}}}\, dx\]
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 = 0.57, size = 152, normalized size = 1.31 \begin {gather*} -\frac {3 \, {\left ({\left ({\left ({\rm Ei}\left (i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {1}{3}}}}\right ) + {\rm Ei}\left (-i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {1}{3}}}}\right ) + {\rm Ei}\left (\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + {\rm Ei}\left (-\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} \cos \left (a\right ) + {\left (i \, {\rm Ei}\left (i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {1}{3}}}}\right ) - i \, {\rm Ei}\left (-i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {1}{3}}}}\right ) + i \, {\rm Ei}\left (\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - i \, {\rm Ei}\left (-\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} \sin \left (a\right )\right )} b e^{\frac {1}{3}} - 4 \, {\left (d x + c\right )}^{\frac {1}{3}} e^{\frac {1}{3}} \sin \left (\frac {{\left (d x + c\right )}^{\frac {1}{3}} a + b}{{\left (d x + c\right )}^{\frac {1}{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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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}{\sqrt [3]{c + d x}} \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 [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 )}^{1/3}}\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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