Optimal. Leaf size=175 \[ -\frac {3 b \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {3 b^2 (c+d x)^{2/3} \text {Ci}\left (b \sqrt [3]{c+d x}\right ) \sin (a)}{2 d e (e (c+d x))^{2/3}}-\frac {3 \sin \left (a+b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {3 b^2 (c+d x)^{2/3} \cos (a) \text {Si}\left (b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}} \]
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Rubi [A]
time = 0.12, antiderivative size = 175, 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) (c+d x)^{2/3} \text {CosIntegral}\left (b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {3 b^2 \cos (a) (c+d x)^{2/3} \text {Si}\left (b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {3 \sin \left (a+b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {3 b \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{2 d e (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+b \sqrt [3]{c+d x}\right )}{(c e+d e x)^{5/3}} \, dx &=\frac {3 \text {Subst}\left (\int \frac {x^2 \sin (a+b x)}{\left (e x^3\right )^{5/3}} \, dx,x,\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^3} \, dx,x,\sqrt [3]{c+d x}\right )}{d e (e (c+d x))^{2/3}}\\ &=-\frac {3 \sin \left (a+b \sqrt [3]{c+d x}\right )}{2 d e (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^2} \, dx,x,\sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}\\ &=-\frac {3 b \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {3 \sin \left (a+b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {\left (3 b^2 (c+d x)^{2/3}\right ) \text {Subst}\left (\int \frac {\sin (a+b x)}{x} \, dx,x,\sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}\\ &=-\frac {3 b \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {3 \sin \left (a+b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {\left (3 b^2 (c+d x)^{2/3} \cos (a)\right ) \text {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,\sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {\left (3 b^2 (c+d x)^{2/3} \sin (a)\right ) \text {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,\sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}\\ &=-\frac {3 b \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {3 b^2 (c+d x)^{2/3} \text {Ci}\left (b \sqrt [3]{c+d x}\right ) \sin (a)}{2 d e (e (c+d x))^{2/3}}-\frac {3 \sin \left (a+b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {3 b^2 (c+d x)^{2/3} \cos (a) \text {Si}\left (b \sqrt [3]{c+d x}\right )}{2 d e (e (c+d x))^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 115, normalized size = 0.66 \begin {gather*} -\frac {3 \left (b \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )+b^2 (c+d x)^{2/3} \text {Ci}\left (b \sqrt [3]{c+d x}\right ) \sin (a)+\sin \left (a+b \sqrt [3]{c+d x}\right )+b^2 (c+d x)^{2/3} \cos (a) \text {Si}\left (b \sqrt [3]{c+d x}\right )\right )}{2 d e (e (c+d x))^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{\left (d e x +c e \right )^{\frac {5}{3}}}\, dx\]
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 = 0.51, size = 128, normalized size = 0.73 \begin {gather*} -\frac {3 \, {\left ({\left (-i \, \Gamma \left (-2, i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + i \, \Gamma \left (-2, -i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - i \, \Gamma \left (-2, i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right ) + i \, \Gamma \left (-2, -i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )\right )} \cos \left (a\right ) - {\left (\Gamma \left (-2, i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + \Gamma \left (-2, -i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + \Gamma \left (-2, i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right ) + \Gamma \left (-2, -i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )\right )} \sin \left (a\right )\right )} b^{2} e^{\left (-\frac {5}{3}\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 + b \sqrt [3]{c + d x} \right )}}{\left (e \left (c + d x\right )\right )^{\frac {5}{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+b\,{\left (c+d\,x\right )}^{1/3}\right )}{{\left (c\,e+d\,e\,x\right )}^{5/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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