Optimal. Leaf size=116 \[ -8 \sqrt {\pi } b^{3/2} \cos (2 a) C\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )+8 \sqrt {\pi } b^{3/2} \sin (2 a) S\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt {x}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}} \]
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Rubi [A] time = 0.19, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3416, 3314, 30, 3312, 3306, 3305, 3351, 3304, 3352} \[ -8 \sqrt {\pi } b^{3/2} \cos (2 a) \text {FresnelC}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )+8 \sqrt {\pi } b^{3/2} \sin (2 a) S\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt {x}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}} \]
Antiderivative was successfully verified.
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Rule 30
Rule 3304
Rule 3305
Rule 3306
Rule 3312
Rule 3314
Rule 3351
Rule 3352
Rule 3416
Rubi steps
\begin {align*} \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{3/2}} \, dx &=3 \operatorname {Subst}\left (\int \frac {\cos ^2(a+b x)}{x^{5/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt {x}}+\frac {8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}+\left (8 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )-\left (16 b^2\right ) \operatorname {Subst}\left (\int \frac {\cos ^2(a+b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )\\ &=16 b^2 \sqrt [6]{x}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt {x}}+\frac {8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}-\left (16 b^2\right ) \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 )\\ &=-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt {x}}+\frac {8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}-\left (8 b^2\right ) \operatorname {Subst}\left (\int \frac {\cos (2 a+2 b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt {x}}+\frac {8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}-\left (8 b^2 \cos (2 a)\right ) \operatorname {Subst}\left (\int \frac {\cos (2 b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )+\left (8 b^2 \sin (2 a)\right ) \operatorname {Subst}\left (\int \frac {\sin (2 b x)}{\sqrt {x}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt {x}}+\frac {8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}-\left (16 b^2 \cos (2 a)\right ) \operatorname {Subst}\left (\int \cos \left (2 b x^2\right ) \, dx,x,\sqrt [6]{x}\right )+\left (16 b^2 \sin (2 a)\right ) \operatorname {Subst}\left (\int \sin \left (2 b x^2\right ) \, dx,x,\sqrt [6]{x}\right )\\ &=-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt {x}}-8 b^{3/2} \sqrt {\pi } \cos (2 a) C\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )+8 b^{3/2} \sqrt {\pi } S\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 )}{\sqrt [6]{x}}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 116, normalized size = 1.00 \[ \frac {8 \sqrt {\pi } b^{3/2} \sqrt {x} \sin (2 a) S\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )+4 b \sqrt [3]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )-\cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )-1}{\sqrt {x}}-8 \sqrt {\pi } b^{3/2} \cos (2 a) C\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 100, normalized size = 0.86 \[ -\frac {2 \, {\left (4 \, \pi b x \sqrt {\frac {b}{\pi }} \cos \left (2 \, a\right ) \operatorname {C}\left (2 \, x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) - 4 \, \pi b x \sqrt {\frac {b}{\pi }} \operatorname {S}\left (2 \, x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) \sin \left (2 \, a\right ) - 4 \, b x^{\frac {5}{6}} \cos \left (b x^{\frac {1}{3}} + a\right ) \sin \left (b x^{\frac {1}{3}} + a\right ) + \sqrt {x} \cos \left (b x^{\frac {1}{3}} + a\right )^{2}\right )}}{x} \]
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 \[ \int \frac {\cos \left (b x^{\frac {1}{3}} + a\right )^{2}}{x^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 87, normalized size = 0.75 \[ -\frac {1}{\sqrt {x}}-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{\sqrt {x}}-4 b \left (-\frac {\sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{x^{\frac {1}{6}}}+2 \sqrt {b}\, \sqrt {\pi }\, \left (\cos \left (2 a \right ) \FresnelC \left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )-\sin \left (2 a \right ) \mathrm {S}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.30, size = 87, normalized size = 0.75 \[ -\frac {\sqrt {2} {\left ({\left (\left (3 i - 3\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, 2 i \, b x^{\frac {1}{3}}\right ) - \left (3 i + 3\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -2 i \, b x^{\frac {1}{3}}\right )\right )} \cos \left (2 \, a\right ) + {\left (\left (3 i + 3\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, 2 i \, b x^{\frac {1}{3}}\right ) - \left (3 i - 3\right ) \, \sqrt {2} \Gamma \left (-\frac {3}{2}, -2 i \, b x^{\frac {1}{3}}\right )\right )} \sin \left (2 \, a\right )\right )} \sqrt {b x^{\frac {1}{3}}} b x^{\frac {1}{3}} + 4}{4 \, \sqrt {x}} \]
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 \[ \int \frac {{\cos \left (a+b\,x^{1/3}\right )}^2}{x^{3/2}} \,d x \]
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 \[ \int \frac {\cos ^{2}{\left (a + b \sqrt [3]{x} \right )}}{x^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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