Optimal. Leaf size=289 \[ \frac {2160 e \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d \sqrt [3]{c+d x}}-\frac {1080 e \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d}+\frac {90 e (c+d x) \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac {3 e (c+d x)^{5/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac {2160 e \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d}-\frac {360 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d}+\frac {18 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d} \]
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Rubi [A]
time = 0.18, antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3512, 15, 3377,
2718} \begin {gather*} \frac {2160 e \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d \sqrt [3]{c+d x}}+\frac {2160 e \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d}-\frac {1080 e \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d}-\frac {360 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d}+\frac {90 e (c+d x) \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}+\frac {18 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac {3 e (c+d x)^{5/3} \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 2718
Rule 3377
Rule 3512
Rubi steps
\begin {align*} \int (c e+d e x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx &=\frac {3 \text {Subst}\left (\int x^2 \left (e x^3\right )^{4/3} \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d}\\ &=\frac {\left (3 e \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int x^6 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d \sqrt [3]{c+d x}}\\ &=-\frac {3 e (c+d x)^{5/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac {\left (18 e \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int x^5 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d \sqrt [3]{c+d x}}\\ &=-\frac {3 e (c+d x)^{5/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac {18 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac {\left (90 e \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int x^4 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d \sqrt [3]{c+d x}}\\ &=\frac {90 e (c+d x) \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac {3 e (c+d x)^{5/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac {18 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac {\left (360 e \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int x^3 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^3 d \sqrt [3]{c+d x}}\\ &=\frac {90 e (c+d x) \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac {3 e (c+d x)^{5/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}-\frac {360 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d}+\frac {18 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}+\frac {\left (1080 e \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int x^2 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^4 d \sqrt [3]{c+d x}}\\ &=-\frac {1080 e \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d}+\frac {90 e (c+d x) \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac {3 e (c+d x)^{5/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}-\frac {360 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d}+\frac {18 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}+\frac {\left (2160 e \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int x \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^5 d \sqrt [3]{c+d x}}\\ &=-\frac {1080 e \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d}+\frac {90 e (c+d x) \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac {3 e (c+d x)^{5/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac {2160 e \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d}-\frac {360 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d}+\frac {18 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac {\left (2160 e \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^6 d \sqrt [3]{c+d x}}\\ &=\frac {2160 e \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d \sqrt [3]{c+d x}}-\frac {1080 e \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d}+\frac {90 e (c+d x) \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac {3 e (c+d x)^{5/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac {2160 e \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d}-\frac {360 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d}+\frac {18 e (c+d x)^{4/3} \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 = 0.38, size = 226, normalized size = 0.78 \begin {gather*} \frac {3 (e (c+d x))^{4/3} \left (-\cos \left (b \sqrt [3]{c+d x}\right ) \left (\left (-720+360 b^2 (c+d x)^{2/3}-30 b^4 (c+d x)^{4/3}+b^6 (c+d x)^2\right ) \cos (a)-6 b \left (120 \sqrt [3]{c+d x}-20 b^2 (c+d x)+b^4 (c+d x)^{5/3}\right ) \sin (a)\right )+\left (6 b \left (120 \sqrt [3]{c+d x}-20 b^2 (c+d x)+b^4 (c+d x)^{5/3}\right ) \cos (a)+\left (-720+360 b^2 (c+d x)^{2/3}-30 b^4 (c+d x)^{4/3}+b^6 (c+d x)^2\right ) \sin (a)\right ) \sin \left (b \sqrt [3]{c+d x}\right )\right )}{b^7 d (c+d x)^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (d e x +c e \right )^{\frac {4}{3}} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\, dx\]
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 = 195, normalized size = 0.67 \begin {gather*} -\frac {3 \, {\left (2 \, {\left (b^{6} d^{2} x^{2} e + 2 \, b^{6} c d x e + b^{6} c^{2} e\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - 3 \, {\left (e \Gamma \left (6, i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + e \Gamma \left (6, -i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + e \Gamma \left (6, i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right ) + e \Gamma \left (6, -i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )\right )} \cos \left (a\right ) - 3 \, {\left (-i \, e \Gamma \left (6, i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + i \, e \Gamma \left (6, -i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - i \, e \Gamma \left (6, i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right ) + i \, e \Gamma \left (6, -i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )\right )} \sin \left (a\right )\right )} e^{\frac {1}{3}}}{2 \, b^{7} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.76, size = 229, normalized size = 0.79 \begin {gather*} -\frac {3 \, {\left ({\left ({\left (b^{6} d^{2} x^{2} + 2 \, b^{6} c d x + b^{6} c^{2} - 720\right )} {\left (d x + c\right )}^{\frac {2}{3}} e + 360 \, {\left (b^{2} d x + b^{2} c\right )} {\left (d x + c\right )}^{\frac {1}{3}} e - 30 \, {\left (b^{4} d^{2} x^{2} + 2 \, b^{4} c d x + b^{4} c^{2}\right )} e\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}} + 6 \, {\left (20 \, {\left (b^{3} d x + b^{3} c\right )} {\left (d x + c\right )}^{\frac {2}{3}} e - {\left (b^{5} d^{2} x^{2} + 2 \, b^{5} c d x + b^{5} c^{2}\right )} {\left (d x + c\right )}^{\frac {1}{3}} e - 120 \, {\left (b d x + b c\right )} e\right )} {\left (d x + c\right )}^{\frac {1}{3}} e^{\frac {1}{3}} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )}}{b^{7} d^{2} x + b^{7} c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 566 vs.
\(2 (195) = 390\).
time = 4.27, size = 566, normalized size = 1.96 \begin {gather*} \frac {3 \, {\left ({\left (2 \, {\left (\frac {{\left ({\left (d x e + c e\right )} b^{3} c e^{3} - 6 \, {\left (d x e + c e\right )}^{\frac {1}{3}} b c 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} d^{2}} - \frac {3 \, {\left ({\left (d x e + c e\right )}^{\frac {2}{3}} b^{2} c e^{\frac {10}{3}} - 2 \, c 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} d^{2}}\right )} e^{\left (-1\right )} - {\left (\frac {{\left ({\left (d x e + c e\right )}^{2} b^{6} e^{5} - 30 \, {\left (d x e + c e\right )}^{\frac {4}{3}} b^{4} e^{\frac {17}{3}} + 360 \, {\left (d x e + c e\right )}^{\frac {2}{3}} b^{2} e^{\frac {19}{3}} - 720 \, e^{7}\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 {14}{3}\right )}}{b^{7} d^{2}} - \frac {6 \, {\left ({\left (d x e + c e\right )}^{\frac {5}{3}} b^{5} e^{\frac {16}{3}} - 20 \, {\left (d x e + c e\right )} b^{3} e^{6} + 120 \, {\left (d x e + c e\right )}^{\frac {1}{3}} b e^{\frac {20}{3}}\right )} e^{\left (-\frac {14}{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^{7} d^{2}}\right )} e^{\left (-2\right )} - \frac {c^{2} \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 d^{2}}\right )} d^{2} e - \frac {c^{2} \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} + 2 \, {\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 )} c\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 = -0.00 \begin {gather*} \int \sin \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )\,{\left (c\,e+d\,e\,x\right )}^{4/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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