Optimal. Leaf size=169 \[ \frac {315 \sqrt {\frac {\pi }{2}} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )}{8 b^{9/2}}-\frac {315 \sqrt {\frac {\pi }{2}} \sin (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )}{8 b^{9/2}}-\frac {315 \sqrt [6]{x} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}-\frac {105 \sqrt {x} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac {21 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}+\frac {3 x^{7/6} \sin \left (a+b \sqrt [3]{x}\right )}{b} \]
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Rubi [A] time = 0.20, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3416, 3296, 3306, 3305, 3351, 3304, 3352} \[ \frac {315 \sqrt {\frac {\pi }{2}} \cos (a) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt [6]{x}\right )}{8 b^{9/2}}-\frac {315 \sqrt {\frac {\pi }{2}} \sin (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )}{8 b^{9/2}}+\frac {21 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}-\frac {105 \sqrt {x} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}-\frac {315 \sqrt [6]{x} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}+\frac {3 x^{7/6} \sin \left (a+b \sqrt [3]{x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 3296
Rule 3304
Rule 3305
Rule 3306
Rule 3351
Rule 3352
Rule 3416
Rubi steps
\begin {align*} \int \sqrt {x} \cos \left (a+b \sqrt [3]{x}\right ) \, dx &=3 \operatorname {Subst}\left (\int x^{7/2} \cos (a+b x) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3 x^{7/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}-\frac {21 \operatorname {Subst}\left (\int x^{5/2} \sin (a+b x) \, dx,x,\sqrt [3]{x}\right )}{2 b}\\ &=\frac {21 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}+\frac {3 x^{7/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}-\frac {105 \operatorname {Subst}\left (\int x^{3/2} \cos (a+b x) \, dx,x,\sqrt [3]{x}\right )}{4 b^2}\\ &=\frac {21 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}-\frac {105 \sqrt {x} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac {3 x^{7/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}+\frac {315 \operatorname {Subst}\left (\int \sqrt {x} \sin (a+b x) \, dx,x,\sqrt [3]{x}\right )}{8 b^3}\\ &=-\frac {315 \sqrt [6]{x} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}+\frac {21 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}-\frac {105 \sqrt {x} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac {3 x^{7/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}+\frac {315 \operatorname {Subst}\left (\int \frac {\cos (a+b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )}{16 b^4}\\ &=-\frac {315 \sqrt [6]{x} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}+\frac {21 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}-\frac {105 \sqrt {x} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac {3 x^{7/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}+\frac {(315 \cos (a)) \operatorname {Subst}\left (\int \frac {\cos (b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )}{16 b^4}-\frac {(315 \sin (a)) \operatorname {Subst}\left (\int \frac {\sin (b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )}{16 b^4}\\ &=-\frac {315 \sqrt [6]{x} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}+\frac {21 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}-\frac {105 \sqrt {x} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac {3 x^{7/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}+\frac {(315 \cos (a)) \operatorname {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{8 b^4}-\frac {(315 \sin (a)) \operatorname {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{8 b^4}\\ &=-\frac {315 \sqrt [6]{x} \cos \left (a+b \sqrt [3]{x}\right )}{8 b^4}+\frac {21 x^{5/6} \cos \left (a+b \sqrt [3]{x}\right )}{2 b^2}+\frac {315 \sqrt {\frac {\pi }{2}} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )}{8 b^{9/2}}-\frac {315 \sqrt {\frac {\pi }{2}} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right ) \sin (a)}{8 b^{9/2}}-\frac {105 \sqrt {x} \sin \left (a+b \sqrt [3]{x}\right )}{4 b^3}+\frac {3 x^{7/6} \sin \left (a+b \sqrt [3]{x}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 141, normalized size = 0.83 \[ \frac {6 \sqrt {b} \sqrt [6]{x} \left (2 b \sqrt [3]{x} \left (4 b^2 x^{2/3}-35\right ) \sin \left (a+b \sqrt [3]{x}\right )+7 \left (4 b^2 x^{2/3}-15\right ) \cos \left (a+b \sqrt [3]{x}\right )\right )+315 \sqrt {2 \pi } \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )-315 \sqrt {2 \pi } \sin (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )}{16 b^{9/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 118, normalized size = 0.70 \[ \frac {3 \, {\left (105 \, \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \cos \relax (a) \operatorname {C}\left (\sqrt {2} x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) - 105 \, \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\sqrt {2} x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) \sin \relax (a) + 14 \, {\left (4 \, b^{3} x^{\frac {5}{6}} - 15 \, b x^{\frac {1}{6}}\right )} \cos \left (b x^{\frac {1}{3}} + a\right ) + 4 \, {\left (4 \, b^{4} x^{\frac {7}{6}} - 35 \, b^{2} \sqrt {x}\right )} \sin \left (b x^{\frac {1}{3}} + a\right )\right )}}{16 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.50, size = 193, normalized size = 1.14 \[ -\frac {3 \, {\left (8 i \, b^{3} x^{\frac {7}{6}} - 28 \, b^{2} x^{\frac {5}{6}} - 70 i \, b \sqrt {x} + 105 \, x^{\frac {1}{6}}\right )} e^{\left (i \, b x^{\frac {1}{3}} + i \, a\right )}}{16 \, b^{4}} - \frac {3 \, {\left (-8 i \, b^{3} x^{\frac {7}{6}} - 28 \, b^{2} x^{\frac {5}{6}} + 70 i \, b \sqrt {x} + 105 \, x^{\frac {1}{6}}\right )} e^{\left (-i \, b x^{\frac {1}{3}} - i \, a\right )}}{16 \, b^{4}} - \frac {315 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} x^{\frac {1}{6}} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (i \, a\right )}}{32 \, b^{4} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} - \frac {315 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} x^{\frac {1}{6}} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (-i \, a\right )}}{32 \, b^{4} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 131, normalized size = 0.78 \[ \frac {3 x^{\frac {7}{6}} \sin \left (a +b \,x^{\frac {1}{3}}\right )}{b}-\frac {21 \left (-\frac {x^{\frac {5}{6}} \cos \left (a +b \,x^{\frac {1}{3}}\right )}{2 b}+\frac {\frac {5 \sqrt {x}\, \sin \left (a +b \,x^{\frac {1}{3}}\right )}{4 b}-\frac {15 \left (-\frac {x^{\frac {1}{6}} \cos \left (a +b \,x^{\frac {1}{3}}\right )}{2 b}+\frac {\sqrt {2}\, \sqrt {\pi }\, \left (\cos \relax (a ) \FresnelC \left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \relax (a ) \mathrm {S}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{4 b^{\frac {3}{2}}}\right )}{4 b}}{b}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.24, size = 111, normalized size = 0.66 \[ -\frac {3 \, {\left (\sqrt {2} \sqrt {\pi } {\left ({\left (\left (105 i - 105\right ) \, \cos \relax (a) + \left (105 i + 105\right ) \, \sin \relax (a)\right )} \operatorname {erf}\left (\sqrt {i \, b} x^{\frac {1}{6}}\right ) + {\left (-\left (105 i + 105\right ) \, \cos \relax (a) - \left (105 i - 105\right ) \, \sin \relax (a)\right )} \operatorname {erf}\left (\sqrt {-i \, b} x^{\frac {1}{6}}\right )\right )} b^{\frac {3}{2}} - 56 \, {\left (4 \, b^{4} x^{\frac {5}{6}} - 15 \, b^{2} x^{\frac {1}{6}}\right )} \cos \left (b x^{\frac {1}{3}} + a\right ) - 16 \, {\left (4 \, b^{5} x^{\frac {7}{6}} - 35 \, b^{3} \sqrt {x}\right )} \sin \left (b x^{\frac {1}{3}} + a\right )\right )}}{64 \, b^{6}} \]
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 \[ \int \sqrt {x}\,\cos \left (a+b\,x^{1/3}\right ) \,d x \]
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 \[ \int \sqrt {x} \cos {\left (a + b \sqrt [3]{x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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