Optimal. Leaf size=184 \[ -\frac {32}{315} \sqrt {2 \pi } b^{9/2} \sin (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )-\frac {32}{315} \sqrt {2 \pi } b^{9/2} \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )-\frac {32 b^4 \cos \left (a+b \sqrt [3]{x}\right )}{315 \sqrt [6]{x}}-\frac {16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{315 \sqrt {x}}+\frac {8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{105 x^{5/6}}+\frac {4 b \sin \left (a+b \sqrt [3]{x}\right )}{21 x^{7/6}}-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}} \]
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Rubi [A] time = 0.24, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3416, 3297, 3306, 3305, 3351, 3304, 3352} \[ -\frac {32}{315} \sqrt {2 \pi } b^{9/2} \sin (a) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt [6]{x}\right )-\frac {32}{315} \sqrt {2 \pi } b^{9/2} \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )+\frac {8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{105 x^{5/6}}-\frac {16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{315 \sqrt {x}}-\frac {32 b^4 \cos \left (a+b \sqrt [3]{x}\right )}{315 \sqrt [6]{x}}+\frac {4 b \sin \left (a+b \sqrt [3]{x}\right )}{21 x^{7/6}}-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3304
Rule 3305
Rule 3306
Rule 3351
Rule 3352
Rule 3416
Rubi steps
\begin {align*} \int \frac {\cos \left (a+b \sqrt [3]{x}\right )}{x^{5/2}} \, dx &=3 \operatorname {Subst}\left (\int \frac {\cos (a+b x)}{x^{11/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}}-\frac {1}{3} (2 b) \operatorname {Subst}\left (\int \frac {\sin (a+b x)}{x^{9/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}}+\frac {4 b \sin \left (a+b \sqrt [3]{x}\right )}{21 x^{7/6}}-\frac {1}{21} \left (4 b^2\right ) \operatorname {Subst}\left (\int \frac {\cos (a+b x)}{x^{7/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}}+\frac {8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{105 x^{5/6}}+\frac {4 b \sin \left (a+b \sqrt [3]{x}\right )}{21 x^{7/6}}+\frac {1}{105} \left (8 b^3\right ) \operatorname {Subst}\left (\int \frac {\sin (a+b x)}{x^{5/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}}+\frac {8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{105 x^{5/6}}+\frac {4 b \sin \left (a+b \sqrt [3]{x}\right )}{21 x^{7/6}}-\frac {16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{315 \sqrt {x}}+\frac {1}{315} \left (16 b^4\right ) \operatorname {Subst}\left (\int \frac {\cos (a+b x)}{x^{3/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}}+\frac {8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{105 x^{5/6}}-\frac {32 b^4 \cos \left (a+b \sqrt [3]{x}\right )}{315 \sqrt [6]{x}}+\frac {4 b \sin \left (a+b \sqrt [3]{x}\right )}{21 x^{7/6}}-\frac {16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{315 \sqrt {x}}-\frac {1}{315} \left (32 b^5\right ) \operatorname {Subst}\left (\int \frac {\sin (a+b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}}+\frac {8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{105 x^{5/6}}-\frac {32 b^4 \cos \left (a+b \sqrt [3]{x}\right )}{315 \sqrt [6]{x}}+\frac {4 b \sin \left (a+b \sqrt [3]{x}\right )}{21 x^{7/6}}-\frac {16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{315 \sqrt {x}}-\frac {1}{315} \left (32 b^5 \cos (a)\right ) \operatorname {Subst}\left (\int \frac {\sin (b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )-\frac {1}{315} \left (32 b^5 \sin (a)\right ) \operatorname {Subst}\left (\int \frac {\cos (b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}}+\frac {8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{105 x^{5/6}}-\frac {32 b^4 \cos \left (a+b \sqrt [3]{x}\right )}{315 \sqrt [6]{x}}+\frac {4 b \sin \left (a+b \sqrt [3]{x}\right )}{21 x^{7/6}}-\frac {16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{315 \sqrt {x}}-\frac {1}{315} \left (64 b^5 \cos (a)\right ) \operatorname {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\sqrt [6]{x}\right )-\frac {1}{315} \left (64 b^5 \sin (a)\right ) \operatorname {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\sqrt [6]{x}\right )\\ &=-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}}+\frac {8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{105 x^{5/6}}-\frac {32 b^4 \cos \left (a+b \sqrt [3]{x}\right )}{315 \sqrt [6]{x}}-\frac {32}{315} b^{9/2} \sqrt {2 \pi } \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )-\frac {32}{315} b^{9/2} \sqrt {2 \pi } C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right ) \sin (a)+\frac {4 b \sin \left (a+b \sqrt [3]{x}\right )}{21 x^{7/6}}-\frac {16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{315 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 180, normalized size = 0.98 \[ -\frac {2 \left (16 \sqrt {2 \pi } b^{9/2} x^{3/2} \sin (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )+16 \sqrt {2 \pi } b^{9/2} x^{3/2} \cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )+16 b^4 x^{4/3} \cos \left (a+b \sqrt [3]{x}\right )+8 b^3 x \sin \left (a+b \sqrt [3]{x}\right )-12 b^2 x^{2/3} \cos \left (a+b \sqrt [3]{x}\right )-30 b \sqrt [3]{x} \sin \left (a+b \sqrt [3]{x}\right )+105 \cos \left (a+b \sqrt [3]{x}\right )\right )}{315 x^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 134, normalized size = 0.73 \[ -\frac {2 \, {\left (16 \, \sqrt {2} \pi b^{4} x^{2} \sqrt {\frac {b}{\pi }} \cos \relax (a) \operatorname {S}\left (\sqrt {2} x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) + 16 \, \sqrt {2} \pi b^{4} x^{2} \sqrt {\frac {b}{\pi }} \operatorname {C}\left (\sqrt {2} x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) \sin \relax (a) + {\left (16 \, b^{4} x^{\frac {11}{6}} - 12 \, b^{2} x^{\frac {7}{6}} + 105 \, \sqrt {x}\right )} \cos \left (b x^{\frac {1}{3}} + a\right ) + 2 \, {\left (4 \, b^{3} x^{\frac {3}{2}} - 15 \, b x^{\frac {5}{6}}\right )} \sin \left (b x^{\frac {1}{3}} + a\right )\right )}}{315 \, x^{2}} \]
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 \[ \int \frac {\cos \left (b x^{\frac {1}{3}} + a\right )}{x^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 129, normalized size = 0.70 \[ -\frac {2 \cos \left (a +b \,x^{\frac {1}{3}}\right )}{3 x^{\frac {3}{2}}}-\frac {4 b \left (-\frac {\sin \left (a +b \,x^{\frac {1}{3}}\right )}{7 x^{\frac {7}{6}}}+\frac {2 b \left (-\frac {\cos \left (a +b \,x^{\frac {1}{3}}\right )}{5 x^{\frac {5}{6}}}-\frac {2 b \left (-\frac {\sin \left (a +b \,x^{\frac {1}{3}}\right )}{3 \sqrt {x}}+\frac {2 b \left (-\frac {\cos \left (a +b \,x^{\frac {1}{3}}\right )}{x^{\frac {1}{6}}}-\sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \relax (a ) \mathrm {S}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )+\sin \relax (a ) \FresnelC \left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )\right )}{3}\right )}{5}\right )}{7}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.62, size = 76, normalized size = 0.41 \[ \frac {3 \, {\left ({\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {9}{2}, i \, b x^{\frac {1}{3}}\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {9}{2}, -i \, b x^{\frac {1}{3}}\right )\right )} \cos \relax (a) + {\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {9}{2}, i \, b x^{\frac {1}{3}}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {9}{2}, -i \, b x^{\frac {1}{3}}\right )\right )} \sin \relax (a)\right )} \sqrt {b x^{\frac {1}{3}}} b^{4}}{4 \, x^{\frac {1}{6}}} \]
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 \[ \int \frac {\cos \left (a+b\,x^{1/3}\right )}{x^{5/2}} \,d x \]
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 \[ \int \frac {\cos {\left (a + b \sqrt [3]{x} \right )}}{x^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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