Optimal. Leaf size=95 \[ \frac {3 \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b d e^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)}}-\frac {3 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b^2 d e^2 \sqrt [3]{e (c+d x)}} \]
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Rubi [A] time = 0.10, antiderivative size = 95, 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 = {3435, 3381, 3379, 3296, 2637} \[ \frac {3 \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b d e^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)}}-\frac {3 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b^2 d e^2 \sqrt [3]{e (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 3379
Rule 3381
Rule 3435
Rubi steps
\begin {align*} \int \frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{(c e+d e x)^{7/3}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sin \left (a+\frac {b}{x^{2/3}}\right )}{(e x)^{7/3}} \, dx,x,c+d x\right )}{d}\\ &=\frac {\sqrt [3]{c+d x} \operatorname {Subst}\left (\int \frac {\sin \left (a+\frac {b}{x^{2/3}}\right )}{x^{7/3}} \, dx,x,c+d x\right )}{d e^2 \sqrt [3]{e (c+d x)}}\\ &=-\frac {\left (3 \sqrt [3]{c+d x}\right ) \operatorname {Subst}\left (\int x \sin (a+b x) \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{2 d e^2 \sqrt [3]{e (c+d x)}}\\ &=\frac {3 \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b d e^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)}}-\frac {\left (3 \sqrt [3]{c+d x}\right ) \operatorname {Subst}\left (\int \cos (a+b x) \, dx,x,\frac {1}{(c+d x)^{2/3}}\right )}{2 b d e^2 \sqrt [3]{e (c+d x)}}\\ &=\frac {3 \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b d e^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)}}-\frac {3 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{2 b^2 d e^2 \sqrt [3]{e (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 72, normalized size = 0.76 \[ -\frac {3 (c+d x)^{5/3} \left ((c+d x)^{2/3} \sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )-b \cos \left (a+\frac {b}{(c+d x)^{2/3}}\right )\right )}{2 b^2 d (e (c+d x))^{7/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.61, size = 133, normalized size = 1.40 \[ \frac {3 \, {\left ({\left (d e x + c e\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b \cos \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {1}{3}} b}{d x + c}\right ) - {\left (d e x + c e\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {4}{3}} \sin \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {1}{3}} b}{d x + c}\right )\right )}}{2 \, {\left (b^{2} d^{3} e^{3} x^{2} + 2 \, b^{2} c d^{2} e^{3} x + b^{2} c^{2} d e^{3}\right )}} \]
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 \[ \int \frac {\sin \left (a + \frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )}{{\left (d e x + c e\right )}^{\frac {7}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{\left (d e x +c e \right )^{\frac {7}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.66, size = 129, normalized size = 1.36 \[ -\frac {{\left (3 i \, \Gamma \left (2, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 3 i \, \Gamma \left (2, -i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + 3 i \, \Gamma \left (2, \frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right ) - 3 i \, \Gamma \left (2, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\right )} \cos \relax (a) + 3 \, {\left (\Gamma \left (2, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + \Gamma \left (2, -i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) + \Gamma \left (2, \frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (2, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )\right )} \sin \relax (a)}{8 \, b^{2} d e^{\frac {7}{3}}} \]
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 \[ \int \frac {\sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{2/3}}\right )}{{\left (c\,e+d\,e\,x\right )}^{7/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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