Optimal. Leaf size=96 \[ \sqrt {x}-\frac {e^{2 i a} \sqrt {x} \text {Gamma}\left (\frac {1}{4},-2 i b x^2\right )}{8 \sqrt [4]{2} \sqrt [4]{-i b x^2}}-\frac {e^{-2 i a} \sqrt {x} \text {Gamma}\left (\frac {1}{4},2 i b x^2\right )}{8 \sqrt [4]{2} \sqrt [4]{i b x^2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3483, 3439,
3437, 2239} \begin {gather*} -\frac {e^{2 i a} \sqrt {x} \text {Gamma}\left (\frac {1}{4},-2 i b x^2\right )}{8 \sqrt [4]{2} \sqrt [4]{-i b x^2}}-\frac {e^{-2 i a} \sqrt {x} \text {Gamma}\left (\frac {1}{4},2 i b x^2\right )}{8 \sqrt [4]{2} \sqrt [4]{i b x^2}}+\sqrt {x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2239
Rule 3437
Rule 3439
Rule 3483
Rubi steps
\begin {align*} \int \frac {\cos ^2\left (a+b x^2\right )}{\sqrt {x}} \, dx &=2 \text {Subst}\left (\int \cos ^2\left (a+b x^4\right ) \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (\frac {1}{2}+\frac {1}{2} \cos \left (2 a+2 b x^4\right )\right ) \, dx,x,\sqrt {x}\right )\\ &=\sqrt {x}+\text {Subst}\left (\int \cos \left (2 a+2 b x^4\right ) \, dx,x,\sqrt {x}\right )\\ &=\sqrt {x}+\frac {1}{2} \text {Subst}\left (\int e^{-2 i a-2 i b x^4} \, dx,x,\sqrt {x}\right )+\frac {1}{2} \text {Subst}\left (\int e^{2 i a+2 i b x^4} \, dx,x,\sqrt {x}\right )\\ &=\sqrt {x}-\frac {e^{2 i a} \sqrt {x} \Gamma \left (\frac {1}{4},-2 i b x^2\right )}{8 \sqrt [4]{2} \sqrt [4]{-i b x^2}}-\frac {e^{-2 i a} \sqrt {x} \Gamma \left (\frac {1}{4},2 i b x^2\right )}{8 \sqrt [4]{2} \sqrt [4]{i b x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 120, normalized size = 1.25 \begin {gather*} -\frac {\sqrt {x} \left (-16 \sqrt [4]{b^2 x^4}+2^{3/4} \sqrt [4]{-i b x^2} \text {Gamma}\left (\frac {1}{4},2 i b x^2\right ) (\cos (2 a)-i \sin (2 a))+2^{3/4} \sqrt [4]{i b x^2} \text {Gamma}\left (\frac {1}{4},-2 i b x^2\right ) (\cos (2 a)+i \sin (2 a))\right )}{16 \sqrt [4]{b^2 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {\cos ^{2}\left (b \,x^{2}+a \right )}{\sqrt {x}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 159 vs. \(2 (60) = 120\).
time = 0.11, size = 159, normalized size = 1.66 \begin {gather*} -\frac {2^{\frac {3}{4}} {\left ({\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 )} \sqrt {x} - 16 \cdot 2^{\frac {1}{4}} \left (b x^{2}\right )^{\frac {1}{4}} \sqrt {x}\right )}}{32 \, \left (b x^{2}\right )^{\frac {1}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.12, size = 50, normalized size = 0.52 \begin {gather*} \frac {i \, \left (2 i \, b\right )^{\frac {3}{4}} e^{\left (-2 i \, a\right )} \Gamma \left (\frac {1}{4}, 2 i \, b x^{2}\right ) - i \, \left (-2 i \, b\right )^{\frac {3}{4}} e^{\left (2 i \, a\right )} \Gamma \left (\frac {1}{4}, -2 i \, b x^{2}\right ) + 16 \, b \sqrt {x}}{16 \, b} \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 )}}{\sqrt {x}}\, 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}{\sqrt {x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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