Optimal. Leaf size=117 \[ -\frac {\cos \left (2 \left (a+b x^2\right )\right )}{\sqrt {x}}-\frac {i e^{2 i a} b x^{3/2} \Gamma \left (\frac {3}{4},-2 i b x^2\right )}{2^{3/4} \left (-i b x^2\right )^{3/4}}+\frac {i e^{-2 i a} b x^{3/2} \Gamma \left (\frac {3}{4},2 i b x^2\right )}{2^{3/4} \left (i b x^2\right )^{3/4}}-\frac {1}{\sqrt {x}} \]
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Rubi [A] time = 0.16, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3402, 3404, 3388, 3389, 2218} \[ -\frac {i e^{2 i a} b x^{3/2} \text {Gamma}\left (\frac {3}{4},-2 i b x^2\right )}{2^{3/4} \left (-i b x^2\right )^{3/4}}+\frac {i e^{-2 i a} b x^{3/2} \text {Gamma}\left (\frac {3}{4},2 i b x^2\right )}{2^{3/4} \left (i b x^2\right )^{3/4}}-\frac {\cos \left (2 \left (a+b x^2\right )\right )}{\sqrt {x}}-\frac {1}{\sqrt {x}} \]
Antiderivative was successfully verified.
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Rule 2218
Rule 3388
Rule 3389
Rule 3402
Rule 3404
Rubi steps
\begin {align*} \int \frac {\cos ^2\left (a+b x^2\right )}{x^{3/2}} \, dx &=2 \operatorname {Subst}\left (\int \frac {\cos ^2\left (a+b x^4\right )}{x^2} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {1}{2 x^2}+\frac {\cos \left (2 a+2 b x^4\right )}{2 x^2}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {1}{\sqrt {x}}+\operatorname {Subst}\left (\int \frac {\cos \left (2 a+2 b x^4\right )}{x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {1}{\sqrt {x}}-\frac {\cos \left (2 \left (a+b x^2\right )\right )}{\sqrt {x}}-(8 b) \operatorname {Subst}\left (\int x^2 \sin \left (2 a+2 b x^4\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {1}{\sqrt {x}}-\frac {\cos \left (2 \left (a+b x^2\right )\right )}{\sqrt {x}}-(4 i b) \operatorname {Subst}\left (\int e^{-2 i a-2 i b x^4} x^2 \, dx,x,\sqrt {x}\right )+(4 i b) \operatorname {Subst}\left (\int e^{2 i a+2 i b x^4} x^2 \, dx,x,\sqrt {x}\right )\\ &=-\frac {1}{\sqrt {x}}-\frac {\cos \left (2 \left (a+b x^2\right )\right )}{\sqrt {x}}-\frac {i b e^{2 i a} x^{3/2} \Gamma \left (\frac {3}{4},-2 i b x^2\right )}{2^{3/4} \left (-i b x^2\right )^{3/4}}+\frac {i b e^{-2 i a} x^{3/2} \Gamma \left (\frac {3}{4},2 i b x^2\right )}{2^{3/4} \left (i b x^2\right )^{3/4}}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 137, normalized size = 1.17 \[ \frac {-4 \left (b^2 x^4\right )^{3/4} \cos ^2\left (a+b x^2\right )+\sqrt [4]{2} b x^2 \left (i b x^2\right )^{3/4} (\sin (2 a)-i \cos (2 a)) \Gamma \left (\frac {3}{4},-2 i b x^2\right )+i \sqrt [4]{2} \left (-i b x^2\right )^{7/4} (\sin (2 a)+i \cos (2 a)) \Gamma \left (\frac {3}{4},2 i b x^2\right )}{2 \sqrt {x} \left (b^2 x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 59, normalized size = 0.50 \[ \frac {\left (2 i \, b\right )^{\frac {1}{4}} x e^{\left (-2 i \, a\right )} \Gamma \left (\frac {3}{4}, 2 i \, b x^{2}\right ) + \left (-2 i \, b\right )^{\frac {1}{4}} x e^{\left (2 i \, a\right )} \Gamma \left (\frac {3}{4}, -2 i \, b x^{2}\right ) - 4 \, \sqrt {x} \cos \left (b x^{2} + a\right )^{2}}{2 \, x} \]
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 \[ \int \frac {\cos \left (b x^{2} + a\right )^{2}}{x^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {\cos ^{2}\left (b \,x^{2}+a \right )}{x^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.87, size = 143, normalized size = 1.22 \[ -\frac {2^{\frac {1}{4}} \left (b x^{2}\right )^{\frac {1}{4}} {\left ({\left (\sqrt {\sqrt {2} + 2} {\left (\Gamma \left (-\frac {1}{4}, 2 i \, b x^{2}\right ) + \Gamma \left (-\frac {1}{4}, -2 i \, b x^{2}\right )\right )} + \sqrt {-\sqrt {2} + 2} {\left (i \, \Gamma \left (-\frac {1}{4}, 2 i \, b x^{2}\right ) - i \, \Gamma \left (-\frac {1}{4}, -2 i \, b x^{2}\right )\right )}\right )} \cos \left (2 \, a\right ) + {\left (\sqrt {-\sqrt {2} + 2} {\left (\Gamma \left (-\frac {1}{4}, 2 i \, b x^{2}\right ) + \Gamma \left (-\frac {1}{4}, -2 i \, b x^{2}\right )\right )} + \sqrt {\sqrt {2} + 2} {\left (-i \, \Gamma \left (-\frac {1}{4}, 2 i \, b x^{2}\right ) + i \, \Gamma \left (-\frac {1}{4}, -2 i \, b x^{2}\right )\right )}\right )} \sin \left (2 \, a\right )\right )} + 16}{16 \, \sqrt {x}} \]
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 \[ \int \frac {{\cos \left (b\,x^2+a\right )}^2}{x^{3/2}} \,d x \]
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 \[ \int \frac {\cos ^{2}{\left (a + b x^{2} \right )}}{x^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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