Optimal. Leaf size=116 \[ -\frac {2 \cos ^2\left (a+b x^2\right )}{3 x^{3/2}}-\frac {i b e^{2 i a} \sqrt {x} \text {Gamma}\left (\frac {1}{4},-2 i b x^2\right )}{3 \sqrt [4]{2} \sqrt [4]{-i b x^2}}+\frac {i b e^{-2 i a} \sqrt {x} \text {Gamma}\left (\frac {1}{4},2 i b x^2\right )}{3 \sqrt [4]{2} \sqrt [4]{i b x^2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3483, 3475,
4669, 3454, 3436, 2239} \begin {gather*} -\frac {i e^{2 i a} b \sqrt {x} \text {Gamma}\left (\frac {1}{4},-2 i b x^2\right )}{3 \sqrt [4]{2} \sqrt [4]{-i b x^2}}+\frac {i e^{-2 i a} b \sqrt {x} \text {Gamma}\left (\frac {1}{4},2 i b x^2\right )}{3 \sqrt [4]{2} \sqrt [4]{i b x^2}}-\frac {2 \cos ^2\left (a+b x^2\right )}{3 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2239
Rule 3436
Rule 3454
Rule 3475
Rule 3483
Rule 4669
Rubi steps
\begin {align*} \int \frac {\cos ^2\left (a+b x^2\right )}{x^{5/2}} \, dx &=2 \text {Subst}\left (\int \frac {\cos ^2\left (a+b x^4\right )}{x^4} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 \cos ^2\left (a+b x^2\right )}{3 x^{3/2}}-\frac {1}{3} (16 b) \text {Subst}\left (\int \cos \left (a+b x^4\right ) \sin \left (a+b x^4\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 \cos ^2\left (a+b x^2\right )}{3 x^{3/2}}-\frac {1}{3} (8 b) \text {Subst}\left (\int \sin \left (2 \left (a+b x^4\right )\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 \cos ^2\left (a+b x^2\right )}{3 x^{3/2}}-\frac {1}{3} (8 b) \text {Subst}\left (\int \sin \left (2 a+2 b x^4\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 \cos ^2\left (a+b x^2\right )}{3 x^{3/2}}-\frac {1}{3} (4 i b) \text {Subst}\left (\int e^{-2 i a-2 i b x^4} \, dx,x,\sqrt {x}\right )+\frac {1}{3} (4 i b) \text {Subst}\left (\int e^{2 i a+2 i b x^4} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 \cos ^2\left (a+b x^2\right )}{3 x^{3/2}}-\frac {i b e^{2 i a} \sqrt {x} \Gamma \left (\frac {1}{4},-2 i b x^2\right )}{3 \sqrt [4]{2} \sqrt [4]{-i b x^2}}+\frac {i b e^{-2 i a} \sqrt {x} \Gamma \left (\frac {1}{4},2 i b x^2\right )}{3 \sqrt [4]{2} \sqrt [4]{i b x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 137, normalized size = 1.18 \begin {gather*} \frac {-4 \sqrt [4]{b^2 x^4} \cos ^2\left (a+b x^2\right )+2^{3/4} b x^2 \sqrt [4]{i b x^2} \text {Gamma}\left (\frac {1}{4},-2 i b x^2\right ) (-i \cos (2 a)+\sin (2 a))+i 2^{3/4} \left (-i b x^2\right )^{5/4} \text {Gamma}\left (\frac {1}{4},2 i b x^2\right ) (i \cos (2 a)+\sin (2 a))}{6 x^{3/2} \sqrt [4]{b^2 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {\cos ^{2}\left (b \,x^{2}+a \right )}{x^{\frac {5}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.13, size = 146, normalized size = 1.26 \begin {gather*} -\frac {3 \cdot 2^{\frac {3}{4}} \left (b x^{2}\right )^{\frac {3}{4}} {\left ({\left (\sqrt {-\sqrt {2} + 2} {\left (\Gamma \left (-\frac {3}{4}, 2 i \, b x^{2}\right ) + \Gamma \left (-\frac {3}{4}, -2 i \, b x^{2}\right )\right )} - \sqrt {\sqrt {2} + 2} {\left (-i \, \Gamma \left (-\frac {3}{4}, 2 i \, b x^{2}\right ) + i \, \Gamma \left (-\frac {3}{4}, -2 i \, b x^{2}\right )\right )}\right )} \cos \left (2 \, a\right ) + {\left (\sqrt {\sqrt {2} + 2} {\left (\Gamma \left (-\frac {3}{4}, 2 i \, b x^{2}\right ) + \Gamma \left (-\frac {3}{4}, -2 i \, b x^{2}\right )\right )} - \sqrt {-\sqrt {2} + 2} {\left (i \, \Gamma \left (-\frac {3}{4}, 2 i \, b x^{2}\right ) - i \, \Gamma \left (-\frac {3}{4}, -2 i \, b x^{2}\right )\right )}\right )} \sin \left (2 \, a\right )\right )} + 16}{48 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.11, size = 63, normalized size = 0.54 \begin {gather*} \frac {\left (2 i \, b\right )^{\frac {3}{4}} x^{2} e^{\left (-2 i \, a\right )} \Gamma \left (\frac {1}{4}, 2 i \, b x^{2}\right ) + \left (-2 i \, b\right )^{\frac {3}{4}} x^{2} e^{\left (2 i \, a\right )} \Gamma \left (\frac {1}{4}, -2 i \, b x^{2}\right ) - 4 \, \sqrt {x} \cos \left (b x^{2} + a\right )^{2}}{6 \, x^{2}} \end {gather*}
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 \begin {gather*} \int \frac {\cos ^{2}{\left (a + b x^{2} \right )}}{x^{\frac {5}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\cos \left (b\,x^2+a\right )}^2}{x^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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