Optimal. Leaf size=141 \[ -\frac {3 \sqrt {\pi } \sin (a) \sqrt [3]{c+d x} C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{\sqrt {2} \sqrt {b} d e \sqrt [3]{e (c+d x)}}-\frac {3 \sqrt {\pi } \cos (a) \sqrt [3]{c+d x} S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{\sqrt {2} \sqrt {b} d e \sqrt [3]{e (c+d x)}} \]
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Rubi [A] time = 0.14, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3435, 3417, 3383, 3353, 3352, 3351} \[ -\frac {3 \sqrt {\pi } \sin (a) \sqrt [3]{c+d x} \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b}}{\sqrt [3]{c+d x}}\right )}{\sqrt {2} \sqrt {b} d e \sqrt [3]{e (c+d x)}}-\frac {3 \sqrt {\pi } \cos (a) \sqrt [3]{c+d x} S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{\sqrt {2} \sqrt {b} d e \sqrt [3]{e (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3351
Rule 3352
Rule 3353
Rule 3383
Rule 3417
Rule 3435
Rubi steps
\begin {align*} \int \frac {\sin \left (a+\frac {b}{(c+d x)^{2/3}}\right )}{(c e+d e x)^{4/3}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sin \left (a+\frac {b}{x^{2/3}}\right )}{(e x)^{4/3}} \, 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^{4/3}} \, dx,x,c+d x\right )}{d e \sqrt [3]{e (c+d x)}}\\ &=-\frac {\left (3 \sqrt [3]{c+d x}\right ) \operatorname {Subst}\left (\int \sin \left (a+b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d e \sqrt [3]{e (c+d x)}}\\ &=-\frac {\left (3 \sqrt [3]{c+d x} \cos (a)\right ) \operatorname {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac {\left (3 \sqrt [3]{c+d x} \sin (a)\right ) \operatorname {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d e \sqrt [3]{e (c+d x)}}\\ &=-\frac {3 \sqrt {\pi } \sqrt [3]{c+d x} \cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )}{\sqrt {2} \sqrt {b} d e \sqrt [3]{e (c+d x)}}-\frac {3 \sqrt {\pi } \sqrt [3]{c+d x} C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right ) \sin (a)}{\sqrt {2} \sqrt {b} d e \sqrt [3]{e (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 96, normalized size = 0.68 \[ -\frac {3 \sqrt {\frac {\pi }{2}} (c+d x)^{4/3} \left (\sin (a) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )+\cos (a) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }}}{\sqrt [3]{c+d x}}\right )\right )}{\sqrt {b} d (e (c+d x))^{4/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d e x + c e\right )}^{\frac {2}{3}} \sin \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {1}{3}} b}{d x + c}\right )}{d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}, x\right ) \]
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 \[ \int \frac {\sin \left (a + \frac {b}{{\left (d x + c\right )}^{\frac {2}{3}}}\right )}{{\left (d e x + c e\right )}^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {2}{3}}}\right )}{\left (d e x +c e \right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.12, size = 487, normalized size = 3.45 \[ \text {result too large to display} \]
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 \[ \int \frac {\sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{2/3}}\right )}{{\left (c\,e+d\,e\,x\right )}^{4/3}} \,d x \]
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 \[ \int \frac {\sin {\left (a + \frac {b}{\left (c + d x\right )^{\frac {2}{3}}} \right )}}{\left (e \left (c + d x\right )\right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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