Optimal. Leaf size=168 \[ \frac {3 b (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac {3 b^2 \sqrt [3]{c+d x} \text {Ci}\left (\frac {b}{\sqrt [3]{c+d x}}\right ) \sin (a)}{2 d \sqrt [3]{e (c+d x)}}+\frac {3 (c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac {3 b^2 \sqrt [3]{c+d x} \cos (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d \sqrt [3]{e (c+d x)}} \]
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Rubi [A]
time = 0.11, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps
used = 7, 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^2 \sin (a) \sqrt [3]{c+d x} \text {CosIntegral}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac {3 b^2 \cos (a) \sqrt [3]{c+d x} \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac {3 (c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac {3 b (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d \sqrt [3]{e (c+d x)}} \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 )}{\sqrt [3]{c e+d e x}} \, dx &=-\frac {3 \text {Subst}\left (\int \frac {\sin (a+b x)}{\sqrt [3]{\frac {e}{x^3}} x^4} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d}\\ &=-\frac {\left (3 \sqrt [3]{c+d x}\right ) \text {Subst}\left (\int \frac {\sin (a+b x)}{x^3} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d \sqrt [3]{e (c+d x)}}\\ &=\frac {3 (c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\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^2} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{2 d \sqrt [3]{e (c+d x)}}\\ &=\frac {3 b (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac {3 (c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac {\left (3 b^2 \sqrt [3]{c+d x}\right ) \text {Subst}\left (\int \frac {\sin (a+b x)}{x} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{2 d \sqrt [3]{e (c+d x)}}\\ &=\frac {3 b (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac {3 (c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac {\left (3 b^2 \sqrt [3]{c+d x} \cos (a)\right ) \text {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac {\left (3 b^2 \sqrt [3]{c+d x} \sin (a)\right ) \text {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{2 d \sqrt [3]{e (c+d x)}}\\ &=\frac {3 b (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac {3 b^2 \sqrt [3]{c+d x} \text {Ci}\left (\frac {b}{\sqrt [3]{c+d x}}\right ) \sin (a)}{2 d \sqrt [3]{e (c+d x)}}+\frac {3 (c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac {3 b^2 \sqrt [3]{c+d x} \cos (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d \sqrt [3]{e (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 131, normalized size = 0.78 \begin {gather*} \frac {3 \left (b (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )+b^2 \sqrt [3]{c+d x} \text {Ci}\left (\frac {b}{\sqrt [3]{c+d x}}\right ) \sin (a)+c \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )+d x \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )+b^2 \sqrt [3]{c+d x} \cos (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{2 d \sqrt [3]{e (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right )}{\left (d e x +c e \right )^{\frac {1}{3}}}\, dx\]
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 = 0.45, size = 171, normalized size = 1.02 \begin {gather*} -\frac {3 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} {\left ({\left (-i \, \Gamma \left (-1, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {1}{3}}}}\right ) + i \, \Gamma \left (-1, -i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {1}{3}}}}\right ) - i \, \Gamma \left (-1, \frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + i \, \Gamma \left (-1, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} \cos \left (a\right ) - {\left (\Gamma \left (-1, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {1}{3}}}}\right ) + \Gamma \left (-1, -i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {1}{3}}}}\right ) + \Gamma \left (-1, \frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + \Gamma \left (-1, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} \sin \left (a\right )\right )} b^{2} - 4 \, {\left (d x + c\right )} \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 (-\frac {1}{3}\right )}}{8 \, {\left (d x + c\right )}^{\frac {1}{3}} 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 )}}{\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 )}^{1/3}}\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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