Optimal. Leaf size=202 \[ \frac {36 (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac {72 (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d (c+d x)^{2/3}}-\frac {3 (c+d x)^{2/3} (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}-\frac {72 (e (c+d x))^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d \sqrt [3]{c+d x}}+\frac {12 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d} \]
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Rubi [A]
time = 0.12, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps
used = 7, 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 {72 (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d (c+d x)^{2/3}}-\frac {72 (e (c+d x))^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d \sqrt [3]{c+d x}}+\frac {36 (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}+\frac {12 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac {3 (c+d x)^{2/3} (e (c+d x))^{2/3} \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)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx &=\frac {3 \text {Subst}\left (\int x^2 \left (e x^3\right )^{2/3} \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d}\\ &=\frac {\left (3 (e (c+d x))^{2/3}\right ) \text {Subst}\left (\int x^4 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d (c+d x)^{2/3}}\\ &=-\frac {3 (c+d x)^{2/3} (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac {\left (12 (e (c+d x))^{2/3}\right ) \text {Subst}\left (\int x^3 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d (c+d x)^{2/3}}\\ &=-\frac {3 (c+d x)^{2/3} (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac {12 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac {\left (36 (e (c+d x))^{2/3}\right ) \text {Subst}\left (\int x^2 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d (c+d x)^{2/3}}\\ &=\frac {36 (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac {3 (c+d x)^{2/3} (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac {12 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac {\left (72 (e (c+d x))^{2/3}\right ) \text {Subst}\left (\int x \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^3 d (c+d x)^{2/3}}\\ &=\frac {36 (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac {3 (c+d x)^{2/3} (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}-\frac {72 (e (c+d x))^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d \sqrt [3]{c+d x}}+\frac {12 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}+\frac {\left (72 (e (c+d x))^{2/3}\right ) \text {Subst}\left (\int \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^4 d (c+d x)^{2/3}}\\ &=\frac {36 (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac {72 (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d (c+d x)^{2/3}}-\frac {3 (c+d x)^{2/3} (e (c+d x))^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}-\frac {72 (e (c+d x))^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d \sqrt [3]{c+d x}}+\frac {12 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 111, normalized size = 0.55 \begin {gather*} -\frac {3 (e (c+d x))^{2/3} \left (\left (24-12 b^2 (c+d x)^{2/3}+b^4 (c+d x)^{4/3}\right ) \cos \left (a+b \sqrt [3]{c+d x}\right )-4 b \left (-6 \sqrt [3]{c+d x}+b^2 (c+d x)\right ) \sin \left (a+b \sqrt [3]{c+d x}\right )\right )}{b^5 d (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 \[\int \left (d e x +c e \right )^{\frac {2}{3}} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\, dx\]
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 = 0.48, size = 191, normalized size = 0.95 \begin {gather*} -\frac {3 \, {\left ({\left (b^{4} d x + b^{4} c\right )} {\left (d x + c\right )}^{\frac {1}{3}} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) e^{\frac {2}{3}} + {\left (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 )} \cos \left (a\right ) - 4 \, {\left (b^{3} d x + b^{3} c\right )} \sin \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 )} \sin \left (a\right )\right )} e^{\frac {2}{3}}\right )}}{b^{5} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.80, size = 141, normalized size = 0.70 \begin {gather*} \frac {3 \, {\left ({\left (12 \, b^{2} d x + 12 \, b^{2} c - {\left (b^{4} d x + b^{4} c\right )} {\left (d x + c\right )}^{\frac {2}{3}} - 24 \, {\left (d x + c\right )}^{\frac {1}{3}}\right )} {\left (d x + c\right )}^{\frac {2}{3}} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) e^{\frac {2}{3}} - 4 \, {\left (d x + c\right )}^{\frac {2}{3}} {\left (6 \, {\left (d x + c\right )}^{\frac {2}{3}} b - {\left (b^{3} d x + b^{3} c\right )} {\left (d x + c\right )}^{\frac {1}{3}}\right )} e^{\frac {2}{3}} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )}}{b^{5} d^{2} x + b^{5} 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 \left (e \left (c + d x\right )\right )^{\frac {2}{3}} \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 310 vs.
\(2 (137) = 274\).
time = 3.50, size = 310, normalized size = 1.53 \begin {gather*} -\frac {3 \, {\left ({\left (\frac {{\left (d x e + c e\right )}^{\frac {1}{3}} \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} - \frac {e^{\frac {2}{3}} \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^{2}}\right )} c - {\left ({\left (\frac {{\left (d x e + c e\right )}^{\frac {1}{3}} 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} - \frac {c e^{\frac {2}{3}} \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^{2}}\right )} e - \frac {{\left ({\left (d x e + c e\right )}^{\frac {4}{3}} b^{4} e^{\frac {11}{3}} - 12 \, {\left (d x e + c e\right )}^{\frac {2}{3}} b^{2} e^{\frac {13}{3}} + 24 \, e^{5}\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 {10}{3}\right )}}{b^{5}} + \frac {4 \, {\left ({\left (d x e + c e\right )} b^{3} e^{4} - 6 \, {\left (d x e + c e\right )}^{\frac {1}{3}} b e^{\frac {14}{3}}\right )} e^{\left (-\frac {10}{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^{5}}\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 = -0.00 \begin {gather*} \int \sin \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )\,{\left (c\,e+d\,e\,x\right )}^{2/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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