Optimal. Leaf size=228 \[ -\frac {512}{315} \sqrt {\pi } b^{9/2} \sin (2 a) C\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )-\frac {512}{315} \sqrt {\pi } b^{9/2} \cos (2 a) S\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )-\frac {512 b^4 \cos ^2\left (a+b \sqrt [3]{x}\right )}{315 \sqrt [6]{x}}-\frac {128 b^3 \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{315 \sqrt {x}}+\frac {32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{105 x^{5/6}}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{21 x^{7/6}}+\frac {256 b^4}{315 \sqrt [6]{x}}-\frac {16 b^2}{105 x^{5/6}} \]
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Rubi [A] time = 0.25, antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3416, 3314, 30, 3313, 12, 3306, 3305, 3351, 3304, 3352} \[ -\frac {512}{315} \sqrt {\pi } b^{9/2} \sin (2 a) \text {FresnelC}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )-\frac {512}{315} \sqrt {\pi } b^{9/2} \cos (2 a) S\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )+\frac {32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{105 x^{5/6}}-\frac {512 b^4 \cos ^2\left (a+b \sqrt [3]{x}\right )}{315 \sqrt [6]{x}}-\frac {128 b^3 \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{315 \sqrt {x}}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{21 x^{7/6}}-\frac {16 b^2}{105 x^{5/6}}+\frac {256 b^4}{315 \sqrt [6]{x}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 3304
Rule 3305
Rule 3306
Rule 3313
Rule 3314
Rule 3351
Rule 3352
Rule 3416
Rubi steps
\begin {align*} \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{5/2}} \, dx &=3 \operatorname {Subst}\left (\int \frac {\cos ^2(a+b x)}{x^{11/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}}+\frac {8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{21 x^{7/6}}+\frac {1}{21} \left (8 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^{7/2}} \, dx,x,\sqrt [3]{x}\right )-\frac {1}{21} \left (16 b^2\right ) \operatorname {Subst}\left (\int \frac {\cos ^2(a+b x)}{x^{7/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {16 b^2}{105 x^{5/6}}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}}+\frac {32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{105 x^{5/6}}+\frac {8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{21 x^{7/6}}-\frac {128 b^3 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{315 \sqrt {x}}-\frac {1}{315} \left (128 b^4\right ) \operatorname {Subst}\left (\int \frac {1}{x^{3/2}} \, dx,x,\sqrt [3]{x}\right )+\frac {1}{315} \left (256 b^4\right ) \operatorname {Subst}\left (\int \frac {\cos ^2(a+b x)}{x^{3/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {16 b^2}{105 x^{5/6}}+\frac {256 b^4}{315 \sqrt [6]{x}}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}}+\frac {32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{105 x^{5/6}}-\frac {512 b^4 \cos ^2\left (a+b \sqrt [3]{x}\right )}{315 \sqrt [6]{x}}+\frac {8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{21 x^{7/6}}-\frac {128 b^3 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{315 \sqrt {x}}+\frac {1}{315} \left (1024 b^5\right ) \operatorname {Subst}\left (\int -\frac {\sin (2 a+2 b x)}{2 \sqrt {x}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {16 b^2}{105 x^{5/6}}+\frac {256 b^4}{315 \sqrt [6]{x}}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}}+\frac {32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{105 x^{5/6}}-\frac {512 b^4 \cos ^2\left (a+b \sqrt [3]{x}\right )}{315 \sqrt [6]{x}}+\frac {8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{21 x^{7/6}}-\frac {128 b^3 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{315 \sqrt {x}}-\frac {1}{315} \left (512 b^5\right ) \operatorname {Subst}\left (\int \frac {\sin (2 a+2 b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {16 b^2}{105 x^{5/6}}+\frac {256 b^4}{315 \sqrt [6]{x}}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}}+\frac {32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{105 x^{5/6}}-\frac {512 b^4 \cos ^2\left (a+b \sqrt [3]{x}\right )}{315 \sqrt [6]{x}}+\frac {8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{21 x^{7/6}}-\frac {128 b^3 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{315 \sqrt {x}}-\frac {1}{315} \left (512 b^5 \cos (2 a)\right ) \operatorname {Subst}\left (\int \frac {\sin (2 b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )-\frac {1}{315} \left (512 b^5 \sin (2 a)\right ) \operatorname {Subst}\left (\int \frac {\cos (2 b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {16 b^2}{105 x^{5/6}}+\frac {256 b^4}{315 \sqrt [6]{x}}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}}+\frac {32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{105 x^{5/6}}-\frac {512 b^4 \cos ^2\left (a+b \sqrt [3]{x}\right )}{315 \sqrt [6]{x}}+\frac {8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{21 x^{7/6}}-\frac {128 b^3 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{315 \sqrt {x}}-\frac {1}{315} \left (1024 b^5 \cos (2 a)\right ) \operatorname {Subst}\left (\int \sin \left (2 b x^2\right ) \, dx,x,\sqrt [6]{x}\right )-\frac {1}{315} \left (1024 b^5 \sin (2 a)\right ) \operatorname {Subst}\left (\int \cos \left (2 b x^2\right ) \, dx,x,\sqrt [6]{x}\right )\\ &=-\frac {16 b^2}{105 x^{5/6}}+\frac {256 b^4}{315 \sqrt [6]{x}}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{3 x^{3/2}}+\frac {32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{105 x^{5/6}}-\frac {512 b^4 \cos ^2\left (a+b \sqrt [3]{x}\right )}{315 \sqrt [6]{x}}-\frac {512}{315} b^{9/2} \sqrt {\pi } \cos (2 a) S\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )-\frac {512}{315} b^{9/2} \sqrt {\pi } C\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right ) \sin (2 a)+\frac {8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{21 x^{7/6}}-\frac {128 b^3 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{315 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 185, normalized size = 0.81 \[ \frac {-512 \sqrt {\pi } b^{9/2} x^{3/2} \sin (2 a) C\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )-512 \sqrt {\pi } b^{9/2} x^{3/2} \cos (2 a) S\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )-256 b^4 x^{4/3} \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )-64 b^3 x \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )+48 b^2 x^{2/3} \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )+60 b \sqrt [3]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )-105 \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )-105}{315 x^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 154, normalized size = 0.68 \[ -\frac {2 \, {\left (256 \, \pi b^{4} x^{2} \sqrt {\frac {b}{\pi }} \cos \left (2 \, a\right ) \operatorname {S}\left (2 \, x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) + 256 \, \pi b^{4} x^{2} \sqrt {\frac {b}{\pi }} \operatorname {C}\left (2 \, x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) \sin \left (2 \, a\right ) - 128 \, b^{4} x^{\frac {11}{6}} + 24 \, b^{2} x^{\frac {7}{6}} + {\left (256 \, b^{4} x^{\frac {11}{6}} - 48 \, b^{2} x^{\frac {7}{6}} + 105 \, \sqrt {x}\right )} \cos \left (b x^{\frac {1}{3}} + a\right )^{2} + 4 \, {\left (16 \, b^{3} x^{\frac {3}{2}} - 15 \, b x^{\frac {5}{6}}\right )} \cos \left (b x^{\frac {1}{3}} + a\right ) \sin \left (b x^{\frac {1}{3}} + a\right )\right )}}{315 \, x^{2}} \]
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 \[ \int \frac {\cos \left (b x^{\frac {1}{3}} + a\right )^{2}}{x^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 146, normalized size = 0.64 \[ -\frac {1}{3 x^{\frac {3}{2}}}-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{3 x^{\frac {3}{2}}}-\frac {4 b \left (-\frac {\sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{7 x^{\frac {7}{6}}}+\frac {4 b \left (-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{5 x^{\frac {5}{6}}}-\frac {4 b \left (-\frac {\sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{3 \sqrt {x}}+\frac {4 b \left (-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{x^{\frac {1}{6}}}-2 \sqrt {b}\, \sqrt {\pi }\, \left (\cos \left (2 a \right ) \mathrm {S}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )+\sin \left (2 a \right ) \FresnelC \left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )\right )\right )}{3}\right )}{5}\right )}{7}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.71, size = 89, normalized size = 0.39 \[ \frac {\sqrt {2} {\left ({\left (-\left (18 i + 18\right ) \, \sqrt {2} \Gamma \left (-\frac {9}{2}, 2 i \, b x^{\frac {1}{3}}\right ) + \left (18 i - 18\right ) \, \sqrt {2} \Gamma \left (-\frac {9}{2}, -2 i \, b x^{\frac {1}{3}}\right )\right )} \cos \left (2 \, a\right ) + {\left (\left (18 i - 18\right ) \, \sqrt {2} \Gamma \left (-\frac {9}{2}, 2 i \, b x^{\frac {1}{3}}\right ) - \left (18 i + 18\right ) \, \sqrt {2} \Gamma \left (-\frac {9}{2}, -2 i \, b x^{\frac {1}{3}}\right )\right )} \sin \left (2 \, a\right )\right )} \sqrt {b x^{\frac {1}{3}}} b^{4} x^{\frac {4}{3}} - 1}{3 \, x^{\frac {3}{2}}} \]
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 \[ \int \frac {{\cos \left (a+b\,x^{1/3}\right )}^2}{x^{5/2}} \,d x \]
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 \[ \int \frac {\cos ^{2}{\left (a + b \sqrt [3]{x} \right )}}{x^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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