Optimal. Leaf size=217 \[ -\frac {36 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^3 d e^2 (e (c+d x))^{2/3}}+\frac {3 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b d e^2 (c+d x)^{2/3} (e (c+d x))^{2/3}}+\frac {72 (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^5 d e^2 (e (c+d x))^{2/3}}-\frac {12 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^2 d e^2 \sqrt [3]{c+d x} (e (c+d x))^{2/3}}+\frac {72 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^4 d e^2 (e (c+d x))^{2/3}} \]
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Rubi [A]
time = 0.13, antiderivative size = 217, 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 (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^5 d e^2 (e (c+d x))^{2/3}}+\frac {72 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^4 d e^2 (e (c+d x))^{2/3}}-\frac {36 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^3 d e^2 (e (c+d x))^{2/3}}-\frac {12 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^2 d e^2 \sqrt [3]{c+d x} (e (c+d x))^{2/3}}+\frac {3 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b d e^2 (c+d x)^{2/3} (e (c+d x))^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 2718
Rule 3377
Rule 3512
Rubi steps
\begin {align*} \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(c e+d e x)^{8/3}} \, dx &=-\frac {3 \text {Subst}\left (\int \frac {\sin (a+b x)}{\left (\frac {e}{x^3}\right )^{8/3} x^4} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d}\\ &=-\frac {\left (3 (c+d x)^{2/3}\right ) \text {Subst}\left (\int x^4 \sin (a+b x) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d e^2 (e (c+d x))^{2/3}}\\ &=\frac {3 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b d e^2 (c+d x)^{2/3} (e (c+d x))^{2/3}}-\frac {\left (12 (c+d x)^{2/3}\right ) \text {Subst}\left (\int x^3 \cos (a+b x) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{b d e^2 (e (c+d x))^{2/3}}\\ &=\frac {3 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b d e^2 (c+d x)^{2/3} (e (c+d x))^{2/3}}-\frac {12 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^2 d e^2 \sqrt [3]{c+d x} (e (c+d x))^{2/3}}+\frac {\left (36 (c+d x)^{2/3}\right ) \text {Subst}\left (\int x^2 \sin (a+b x) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{b^2 d e^2 (e (c+d x))^{2/3}}\\ &=-\frac {36 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^3 d e^2 (e (c+d x))^{2/3}}+\frac {3 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b d e^2 (c+d x)^{2/3} (e (c+d x))^{2/3}}-\frac {12 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^2 d e^2 \sqrt [3]{c+d x} (e (c+d x))^{2/3}}+\frac {\left (72 (c+d x)^{2/3}\right ) \text {Subst}\left (\int x \cos (a+b x) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{b^3 d e^2 (e (c+d x))^{2/3}}\\ &=-\frac {36 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^3 d e^2 (e (c+d x))^{2/3}}+\frac {3 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b d e^2 (c+d x)^{2/3} (e (c+d x))^{2/3}}-\frac {12 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^2 d e^2 \sqrt [3]{c+d x} (e (c+d x))^{2/3}}+\frac {72 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^4 d e^2 (e (c+d x))^{2/3}}-\frac {\left (72 (c+d x)^{2/3}\right ) \text {Subst}\left (\int \sin (a+b x) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{b^4 d e^2 (e (c+d x))^{2/3}}\\ &=-\frac {36 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^3 d e^2 (e (c+d x))^{2/3}}+\frac {3 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b d e^2 (c+d x)^{2/3} (e (c+d x))^{2/3}}+\frac {72 (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^5 d e^2 (e (c+d x))^{2/3}}-\frac {12 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^2 d e^2 \sqrt [3]{c+d x} (e (c+d x))^{2/3}}+\frac {72 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^4 d e^2 (e (c+d x))^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 112, normalized size = 0.52 \begin {gather*} \frac {(c+d x)^{4/3} \left (3 \left (b^4-12 b^2 (c+d x)^{2/3}+24 (c+d x)^{4/3}\right ) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )+12 b \left (6 c+6 d x-b^2 \sqrt [3]{c+d x}\right ) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{b^5 d (e (c+d x))^{8/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right )}{\left (d e x +c e \right )^{\frac {8}{3}}}\, dx\]
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
3.
time = 1.08, size = 935, normalized size = 4.31 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.90, size = 172, normalized size = 0.79 \begin {gather*} \frac {3 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b^{4} - 12 \, b^{2} d x - 12 \, b^{2} c + 24 \, {\left (d x + c\right )}^{\frac {5}{3}}\right )} {\left (d x + c\right )}^{\frac {1}{3}} \cos \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {2}{3}} b}{d x + c}\right ) e^{\frac {1}{3}} - 4 \, {\left ({\left (d x + c\right )}^{\frac {2}{3}} b^{3} - 6 \, {\left (b d x + b c\right )} {\left (d x + c\right )}^{\frac {1}{3}}\right )} {\left (d x + c\right )}^{\frac {1}{3}} e^{\frac {1}{3}} \sin \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {2}{3}} b}{d x + c}\right )\right )} e^{\left (-3\right )}}{b^{5} d^{3} x^{2} + 2 \, b^{5} c d^{2} x + b^{5} c^{2} 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 [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 = -0.00 \begin {gather*} \int \frac {\sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{1/3}}\right )}{{\left (c\,e+d\,e\,x\right )}^{8/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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