Optimal. Leaf size=89 \[ -\frac {3 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac {3 \sqrt [3]{e (c+d x)} \sin \left (a+b (c+d x)^{2/3}\right )}{2 b^2 d \sqrt [3]{c+d x}} \]
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Rubi [A]
time = 0.06, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3516, 3462,
3460, 3377, 2717} \begin {gather*} \frac {3 \sqrt [3]{e (c+d x)} \sin \left (a+b (c+d x)^{2/3}\right )}{2 b^2 d \sqrt [3]{c+d x}}-\frac {3 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2717
Rule 3377
Rule 3460
Rule 3462
Rule 3516
Rubi steps
\begin {align*} \int \sqrt [3]{c e+d e x} \sin \left (a+b (c+d x)^{2/3}\right ) \, dx &=\frac {\text {Subst}\left (\int \sqrt [3]{e x} \sin \left (a+b x^{2/3}\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {\sqrt [3]{e (c+d x)} \text {Subst}\left (\int \sqrt [3]{x} \sin \left (a+b x^{2/3}\right ) \, dx,x,c+d x\right )}{d \sqrt [3]{c+d x}}\\ &=\frac {\left (3 \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int x \sin (a+b x) \, dx,x,(c+d x)^{2/3}\right )}{2 d \sqrt [3]{c+d x}}\\ &=-\frac {3 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac {\left (3 \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int \cos (a+b x) \, dx,x,(c+d x)^{2/3}\right )}{2 b d \sqrt [3]{c+d x}}\\ &=-\frac {3 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac {3 \sqrt [3]{e (c+d x)} \sin \left (a+b (c+d x)^{2/3}\right )}{2 b^2 d \sqrt [3]{c+d x}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 72, normalized size = 0.81 \begin {gather*} -\frac {3 \sqrt [3]{e (c+d x)} \left (b (c+d x)^{2/3} \cos \left (a+b (c+d x)^{2/3}\right )-\sin \left (a+b (c+d x)^{2/3}\right )\right )}{2 b^2 d \sqrt [3]{c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (d e x +c e \right )^{\frac {1}{3}} \sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )\, dx\]
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
3.
time = 0.45, size = 128, normalized size = 1.44 \begin {gather*} -\frac {3 \, {\left ({\left (-i \, \Gamma \left (2, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + i \, \Gamma \left (2, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) - i \, \Gamma \left (2, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right ) + i \, \Gamma \left (2, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (a\right ) - {\left (\Gamma \left (2, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (2, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (2, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right ) + \Gamma \left (2, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (a\right )\right )} e^{\frac {1}{3}}}{8 \, b^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.75, size = 71, normalized size = 0.80 \begin {gather*} -\frac {3 \, {\left ({\left (b d x + b c\right )} {\left (d x + c\right )}^{\frac {2}{3}} \cos \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right ) e^{\frac {1}{3}} - {\left (d x + c\right )} e^{\frac {1}{3}} \sin \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right )\right )}}{2 \, {\left (b^{2} d^{2} x + b^{2} c d\right )}} \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 \sqrt [3]{e \left (c + d x\right )} \sin {\left (a + b \left (c + d x\right )^{\frac {2}{3}} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 6.94, size = 265, normalized size = 2.98 \begin {gather*} -\frac {3 \, {\left ({\left (-\frac {i \, \sqrt {\pi } \operatorname {erf}\left (-{\left (d x e + c e\right )}^{\frac {1}{3}} \sqrt {-i \, b e^{\left (-\frac {2}{3}\right )}}\right ) e^{\left (i \, a\right )}}{\sqrt {-i \, b e^{\left (-\frac {2}{3}\right )}}} + \frac {i \, \sqrt {\pi } \operatorname {erf}\left (-{\left (d x e + c e\right )}^{\frac {1}{3}} \sqrt {i \, b e^{\left (-\frac {2}{3}\right )}}\right ) e^{\left (-i \, a\right )}}{\sqrt {i \, b e^{\left (-\frac {2}{3}\right )}}}\right )} c + {\left (\frac {i \, \sqrt {\pi } c \operatorname {erf}\left (-{\left (d x e + c e\right )}^{\frac {1}{3}} \sqrt {-i \, b e^{\left (-\frac {2}{3}\right )}}\right ) e^{\left (i \, a + 1\right )}}{\sqrt {-i \, b e^{\left (-\frac {2}{3}\right )}}} - \frac {i \, \sqrt {\pi } c \operatorname {erf}\left (-{\left (d x e + c e\right )}^{\frac {1}{3}} \sqrt {i \, b e^{\left (-\frac {2}{3}\right )}}\right ) e^{\left (-i \, a + 1\right )}}{\sqrt {i \, b e^{\left (-\frac {2}{3}\right )}}} - \frac {i \, {\left (i \, {\left (d x e + c e\right )}^{\frac {2}{3}} b e^{\left (-\frac {2}{3}\right )} - 1\right )} e^{\left (i \, {\left (d x e + c e\right )}^{\frac {2}{3}} b e^{\left (-\frac {2}{3}\right )} + i \, a + \frac {4}{3}\right )}}{b^{2}} - \frac {i \, {\left (i \, {\left (d x e + c e\right )}^{\frac {2}{3}} b e^{\left (-\frac {2}{3}\right )} + 1\right )} e^{\left (-i \, {\left (d x e + c e\right )}^{\frac {2}{3}} b e^{\left (-\frac {2}{3}\right )} - i \, a + \frac {4}{3}\right )}}{b^{2}}\right )} e^{\left (-1\right )}\right )}}{4 \, d} \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 \sin \left (a+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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