Optimal. Leaf size=218 \[ \frac {315 \sqrt {\pi } \cos (2 a) C\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{512 b^{9/2}}-\frac {315 \sqrt {\pi } \sin (2 a) S\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{512 b^{9/2}}-\frac {315 \sqrt [6]{x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}-\frac {105 \sqrt {x} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {21 x^{5/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {3 x^{7/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {315 \sqrt [6]{x}}{256 b^4}-\frac {21 x^{5/6}}{16 b^2}+\frac {x^{3/2}}{3} \]
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Rubi [A] time = 0.25, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3416, 3311, 30, 3312, 3306, 3305, 3351, 3304, 3352} \[ \frac {315 \sqrt {\pi } \cos (2 a) \text {FresnelC}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{512 b^{9/2}}-\frac {315 \sqrt {\pi } \sin (2 a) S\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{512 b^{9/2}}+\frac {21 x^{5/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}-\frac {315 \sqrt [6]{x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}-\frac {105 \sqrt {x} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {3 x^{7/6} \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{2 b}-\frac {21 x^{5/6}}{16 b^2}+\frac {315 \sqrt [6]{x}}{256 b^4}+\frac {x^{3/2}}{3} \]
Antiderivative was successfully verified.
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Rule 30
Rule 3304
Rule 3305
Rule 3306
Rule 3311
Rule 3312
Rule 3351
Rule 3352
Rule 3416
Rubi steps
\begin {align*} \int \sqrt {x} \cos ^2\left (a+b \sqrt [3]{x}\right ) \, dx &=3 \operatorname {Subst}\left (\int x^{7/2} \cos ^2(a+b x) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {21 x^{5/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {3 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {3}{2} \operatorname {Subst}\left (\int x^{7/2} \, dx,x,\sqrt [3]{x}\right )-\frac {105 \operatorname {Subst}\left (\int x^{3/2} \cos ^2(a+b x) \, dx,x,\sqrt [3]{x}\right )}{16 b^2}\\ &=\frac {x^{3/2}}{3}-\frac {315 \sqrt [6]{x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac {21 x^{5/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}-\frac {105 \sqrt {x} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {3 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {315 \operatorname {Subst}\left (\int \frac {\cos ^2(a+b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )}{256 b^4}-\frac {105 \operatorname {Subst}\left (\int x^{3/2} \, dx,x,\sqrt [3]{x}\right )}{32 b^2}\\ &=-\frac {21 x^{5/6}}{16 b^2}+\frac {x^{3/2}}{3}-\frac {315 \sqrt [6]{x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac {21 x^{5/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}-\frac {105 \sqrt {x} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {3 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {315 \operatorname {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}+\frac {\cos (2 a+2 b x)}{2 \sqrt {x}}\right ) \, dx,x,\sqrt [3]{x}\right )}{256 b^4}\\ &=\frac {315 \sqrt [6]{x}}{256 b^4}-\frac {21 x^{5/6}}{16 b^2}+\frac {x^{3/2}}{3}-\frac {315 \sqrt [6]{x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac {21 x^{5/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}-\frac {105 \sqrt {x} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {3 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {315 \operatorname {Subst}\left (\int \frac {\cos (2 a+2 b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )}{512 b^4}\\ &=\frac {315 \sqrt [6]{x}}{256 b^4}-\frac {21 x^{5/6}}{16 b^2}+\frac {x^{3/2}}{3}-\frac {315 \sqrt [6]{x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac {21 x^{5/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}-\frac {105 \sqrt {x} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {3 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {(315 \cos (2 a)) \operatorname {Subst}\left (\int \frac {\cos (2 b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )}{512 b^4}-\frac {(315 \sin (2 a)) \operatorname {Subst}\left (\int \frac {\sin (2 b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )}{512 b^4}\\ &=\frac {315 \sqrt [6]{x}}{256 b^4}-\frac {21 x^{5/6}}{16 b^2}+\frac {x^{3/2}}{3}-\frac {315 \sqrt [6]{x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac {21 x^{5/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}-\frac {105 \sqrt {x} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {3 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}+\frac {(315 \cos (2 a)) \operatorname {Subst}\left (\int \cos \left (2 b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{256 b^4}-\frac {(315 \sin (2 a)) \operatorname {Subst}\left (\int \sin \left (2 b x^2\right ) \, dx,x,\sqrt [6]{x}\right )}{256 b^4}\\ &=\frac {315 \sqrt [6]{x}}{256 b^4}-\frac {21 x^{5/6}}{16 b^2}+\frac {x^{3/2}}{3}-\frac {315 \sqrt [6]{x} \cos ^2\left (a+b \sqrt [3]{x}\right )}{128 b^4}+\frac {21 x^{5/6} \cos ^2\left (a+b \sqrt [3]{x}\right )}{8 b^2}+\frac {315 \sqrt {\pi } \cos (2 a) C\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{512 b^{9/2}}-\frac {315 \sqrt {\pi } S\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right ) \sin (2 a)}{512 b^{9/2}}-\frac {105 \sqrt {x} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{32 b^3}+\frac {3 x^{7/6} \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.44, size = 148, normalized size = 0.68 \[ \frac {2 \sqrt {b} \sqrt [6]{x} \left (63 \left (16 b^2 x^{2/3}-15\right ) \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )+4 b \sqrt [3]{x} \left (9 \left (16 b^2 x^{2/3}-35\right ) \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )+64 b^3 x\right )\right )+945 \sqrt {\pi } \cos (2 a) C\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )-945 \sqrt {\pi } \sin (2 a) S\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{1536 b^{9/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 144, normalized size = 0.66 \[ \frac {512 \, b^{5} x^{\frac {3}{2}} - 2016 \, b^{3} x^{\frac {5}{6}} + 945 \, \pi \sqrt {\frac {b}{\pi }} \cos \left (2 \, a\right ) \operatorname {C}\left (2 \, x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) - 945 \, \pi \sqrt {\frac {b}{\pi }} \operatorname {S}\left (2 \, x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) \sin \left (2 \, a\right ) + 252 \, {\left (16 \, b^{3} x^{\frac {5}{6}} - 15 \, b x^{\frac {1}{6}}\right )} \cos \left (b x^{\frac {1}{3}} + a\right )^{2} + 144 \, {\left (16 \, b^{4} x^{\frac {7}{6}} - 35 \, b^{2} \sqrt {x}\right )} \cos \left (b x^{\frac {1}{3}} + a\right ) \sin \left (b x^{\frac {1}{3}} + a\right ) + 1890 \, b x^{\frac {1}{6}}}{1536 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.45, size = 176, normalized size = 0.81 \[ \frac {1}{3} \, x^{\frac {3}{2}} - \frac {3 \, {\left (64 i \, b^{3} x^{\frac {7}{6}} - 112 \, b^{2} x^{\frac {5}{6}} - 140 i \, b \sqrt {x} + 105 \, x^{\frac {1}{6}}\right )} e^{\left (2 i \, b x^{\frac {1}{3}} + 2 i \, a\right )}}{512 \, b^{4}} - \frac {3 \, {\left (-64 i \, b^{3} x^{\frac {7}{6}} - 112 \, b^{2} x^{\frac {5}{6}} + 140 i \, b \sqrt {x} + 105 \, x^{\frac {1}{6}}\right )} e^{\left (-2 i \, b x^{\frac {1}{3}} - 2 i \, a\right )}}{512 \, b^{4}} - \frac {315 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {b} x^{\frac {1}{6}} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (2 i \, a\right )}}{1024 \, b^{\frac {9}{2}} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}} - \frac {315 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {b} x^{\frac {1}{6}} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (-2 i \, a\right )}}{1024 \, b^{\frac {9}{2}} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 145, normalized size = 0.67 \[ \frac {x^{\frac {3}{2}}}{3}+\frac {3 x^{\frac {7}{6}} \sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}-\frac {21 \left (-\frac {x^{\frac {5}{6}} \cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}+\frac {\frac {5 \sqrt {x}\, \sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{16 b}-\frac {15 \left (-\frac {x^{\frac {1}{6}} \cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{4 b}+\frac {\sqrt {\pi }\, \left (\cos \left (2 a \right ) \FresnelC \left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )-\sin \left (2 a \right ) \mathrm {S}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )\right )}{8 b^{\frac {3}{2}}}\right )}{16 b}}{b}\right )}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.24, size = 137, normalized size = 0.63 \[ \frac {4096 \, b^{6} x^{\frac {3}{2}} - 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } {\left ({\left (\left (945 i - 945\right ) \, \cos \left (2 \, a\right ) + \left (945 i + 945\right ) \, \sin \left (2 \, a\right )\right )} \operatorname {erf}\left (\sqrt {2 i \, b} x^{\frac {1}{6}}\right ) + {\left (-\left (945 i + 945\right ) \, \cos \left (2 \, a\right ) - \left (945 i - 945\right ) \, \sin \left (2 \, a\right )\right )} \operatorname {erf}\left (\sqrt {-2 i \, b} x^{\frac {1}{6}}\right )\right )} b^{\frac {3}{2}} + 1008 \, {\left (16 \, b^{4} x^{\frac {5}{6}} - 15 \, b^{2} x^{\frac {1}{6}}\right )} \cos \left (2 \, b x^{\frac {1}{3}} + 2 \, a\right ) + 576 \, {\left (16 \, b^{5} x^{\frac {7}{6}} - 35 \, b^{3} \sqrt {x}\right )} \sin \left (2 \, b x^{\frac {1}{3}} + 2 \, a\right )}{12288 \, b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {x}\,{\cos \left (a+b\,x^{1/3}\right )}^2 \,d x \]
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 \[ \int \sqrt {x} \cos ^{2}{\left (a + b \sqrt [3]{x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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