Optimal. Leaf size=160 \[ \frac {18 \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac {3 (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}-\frac {18 \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d \sqrt [3]{c+d x}}+\frac {9 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d} \]
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Rubi [A]
time = 0.09, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3512, 15, 3377,
2717} \begin {gather*} -\frac {18 \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d \sqrt [3]{c+d x}}+\frac {18 \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}+\frac {9 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac {3 (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 2717
Rule 3377
Rule 3512
Rubi steps
\begin {align*} \int \sqrt [3]{c e+d e x} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx &=\frac {3 \text {Subst}\left (\int x^2 \sqrt [3]{e x^3} \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d}\\ &=\frac {\left (3 \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int x^3 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d \sqrt [3]{c+d x}}\\ &=-\frac {3 (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac {\left (9 \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int x^2 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d \sqrt [3]{c+d x}}\\ &=-\frac {3 (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac {9 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac {\left (18 \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int x \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d \sqrt [3]{c+d x}}\\ &=\frac {18 \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac {3 (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac {9 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac {\left (18 \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^3 d \sqrt [3]{c+d x}}\\ &=\frac {18 \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac {3 (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}-\frac {18 \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d \sqrt [3]{c+d x}}+\frac {9 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 97, normalized size = 0.61 \begin {gather*} -\frac {3 \sqrt [3]{e (c+d x)} \left (\left (-6 b \sqrt [3]{c+d x}+b^3 (c+d x)\right ) \cos \left (a+b \sqrt [3]{c+d x}\right )-3 \left (-2+b^2 (c+d x)^{2/3}\right ) \sin \left (a+b \sqrt [3]{c+d x}\right )\right )}{b^4 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 \[\int \left (d e x +c e \right )^{\frac {1}{3}} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\, dx\]
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 = 0.49, size = 155, normalized size = 0.97 \begin {gather*} -\frac {3 \, {\left (4 \, {\left (b^{3} d x + b^{3} c\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - 3 \, {\left (-i \, \Gamma \left (3, i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + i \, \Gamma \left (3, -i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - i \, \Gamma \left (3, i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right ) + i \, \Gamma \left (3, -i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )\right )} \cos \left (a\right ) + 3 \, {\left (\Gamma \left (3, i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + \Gamma \left (3, -i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + \Gamma \left (3, i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right ) + \Gamma \left (3, -i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )\right )} \sin \left (a\right )\right )} e^{\frac {1}{3}}}{4 \, b^{4} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.74, size = 126, normalized size = 0.79 \begin {gather*} \frac {3 \, {\left ({\left (6 \, b d x + 6 \, b c - {\left (b^{3} d x + b^{3} c\right )} {\left (d x + c\right )}^{\frac {2}{3}}\right )} {\left (d x + c\right )}^{\frac {1}{3}} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) e^{\frac {1}{3}} + 3 \, {\left (d x + c\right )}^{\frac {1}{3}} {\left ({\left (b^{2} d x + b^{2} c\right )} {\left (d x + c\right )}^{\frac {1}{3}} - 2 \, {\left (d x + c\right )}^{\frac {2}{3}}\right )} e^{\frac {1}{3}} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )}}{b^{4} d^{2} x + b^{4} c d} \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 \sqrt [3]{c + d x} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.70, size = 196, normalized size = 1.22 \begin {gather*} -\frac {3 \, {\left (\frac {c \cos \left ({\left ({\left (d x e + c e\right )}^{\frac {1}{3}} b e^{\frac {2}{3}} + a e\right )} e^{\left (-1\right )}\right ) e^{\frac {1}{3}}}{b} - {\left (\frac {c \cos \left ({\left ({\left (d x e + c e\right )}^{\frac {1}{3}} b e^{\frac {2}{3}} + a e\right )} e^{\left (-1\right )}\right ) e^{\frac {4}{3}}}{b} - \frac {{\left ({\left (d x e + c e\right )} b^{3} e^{3} - 6 \, {\left (d x e + c e\right )}^{\frac {1}{3}} b e^{\frac {11}{3}}\right )} \cos \left ({\left ({\left (d x e + c e\right )}^{\frac {1}{3}} b e^{\frac {2}{3}} + a e\right )} e^{\left (-1\right )}\right ) e^{\left (-\frac {8}{3}\right )}}{b^{4}} + \frac {3 \, {\left ({\left (d x e + c e\right )}^{\frac {2}{3}} b^{2} e^{\frac {10}{3}} - 2 \, e^{4}\right )} e^{\left (-\frac {8}{3}\right )} \sin \left ({\left ({\left (d x e + c e\right )}^{\frac {1}{3}} b e^{\frac {2}{3}} + a e\right )} e^{\left (-1\right )}\right )}{b^{4}}\right )} e^{\left (-1\right )}\right )}}{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 )}^{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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