Optimal. Leaf size=126 \[ \frac {3 b (c+d x)^{2/3} \cos (a) \text {Ci}\left (b (c+d x)^{2/3}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {3 \sin \left (a+b (c+d x)^{2/3}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {3 b (c+d x)^{2/3} \sin (a) \text {Si}\left (b (c+d x)^{2/3}\right )}{2 d e (e (c+d x))^{2/3}} \]
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Rubi [A]
time = 0.10, antiderivative size = 126, 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) (c+d x)^{2/3} \text {CosIntegral}\left (b (c+d x)^{2/3}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {3 b \sin (a) (c+d x)^{2/3} \text {Si}\left (b (c+d x)^{2/3}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {3 \sin \left (a+b (c+d x)^{2/3}\right )}{2 d e (e (c+d x))^{2/3}} \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+b (c+d x)^{2/3}\right )}{(c e+d e x)^{5/3}} \, dx &=\frac {\text {Subst}\left (\int \frac {\sin \left (a+b x^{2/3}\right )}{(e x)^{5/3}} \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x)^{2/3} \text {Subst}\left (\int \frac {\sin \left (a+b x^{2/3}\right )}{x^{5/3}} \, dx,x,c+d x\right )}{d e (e (c+d x))^{2/3}}\\ &=\frac {\left (3 (c+d x)^{2/3}\right ) \text {Subst}\left (\int \frac {\sin (a+b x)}{x^2} \, dx,x,(c+d x)^{2/3}\right )}{2 d e (e (c+d x))^{2/3}}\\ &=-\frac {3 \sin \left (a+b (c+d x)^{2/3}\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} \, dx,x,(c+d x)^{2/3}\right )}{2 d e (e (c+d x))^{2/3}}\\ &=-\frac {3 \sin \left (a+b (c+d x)^{2/3}\right )}{2 d e (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,(c+d x)^{2/3}\right )}{2 d e (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,(c+d x)^{2/3}\right )}{2 d e (e (c+d x))^{2/3}}\\ &=\frac {3 b (c+d x)^{2/3} \cos (a) \text {Ci}\left (b (c+d x)^{2/3}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {3 \sin \left (a+b (c+d x)^{2/3}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {3 b (c+d x)^{2/3} \sin (a) \text {Si}\left (b (c+d x)^{2/3}\right )}{2 d e (e (c+d x))^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 87, normalized size = 0.69 \begin {gather*} -\frac {3 \left (-b (c+d x)^{2/3} \cos (a) \text {Ci}\left (b (c+d x)^{2/3}\right )+\sin \left (a+b (c+d x)^{2/3}\right )+b (c+d x)^{2/3} \sin (a) \text {Si}\left (b (c+d x)^{2/3}\right )\right )}{2 d e (e (c+d x))^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{\left (d e x +c e \right )^{\frac {5}{3}}}\, dx\]
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 = 0.50, size = 125, normalized size = 0.99 \begin {gather*} \frac {3 \, {\left ({\left (\Gamma \left (-1, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (-1, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (-1, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right ) + \Gamma \left (-1, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (a\right ) + {\left (-i \, \Gamma \left (-1, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + i \, \Gamma \left (-1, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) - i \, \Gamma \left (-1, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right ) + i \, \Gamma \left (-1, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (a\right )\right )} b e^{\left (-\frac {5}{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 + b \left (c + d x\right )^{\frac {2}{3}} \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 )}^{2/3}\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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