Optimal. Leaf size=52 \[ \frac {i e^{-i a} c x^3}{4 \sqrt {c x^2}}-\frac {i e^{i a} x \log (x)}{2 \sqrt {c x^2}} \]
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Rubi [A] time = 0.04, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4483, 4489} \[ \frac {i e^{-i a} c x^3}{4 \sqrt {c x^2}}-\frac {i e^{i a} x \log (x)}{2 \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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Rule 4483
Rule 4489
Rubi steps
\begin {align*} \int \sin \left (a+\frac {1}{2} i \log \left (c x^2\right )\right ) \, dx &=\frac {x \operatorname {Subst}\left (\int \frac {\sin \left (a+\frac {1}{2} i \log (x)\right )}{\sqrt {x}} \, dx,x,c x^2\right )}{2 \sqrt {c x^2}}\\ &=-\frac {(i x) \operatorname {Subst}\left (\int \left (-e^{-i a}+\frac {e^{i a}}{x}\right ) \, dx,x,c x^2\right )}{4 \sqrt {c x^2}}\\ &=\frac {i c e^{-i a} x^3}{4 \sqrt {c x^2}}-\frac {i e^{i a} x \log (x)}{2 \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 44, normalized size = 0.85 \[ \frac {x \left (\sin (a) \left (c x^2+2 \log (x)\right )+i \cos (a) \left (c x^2-2 \log (x)\right )\right )}{4 \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 24, normalized size = 0.46 \[ \frac {{\left (i \, c x^{2} - 2 i \, e^{\left (2 i \, a\right )} \log \relax (x)\right )} e^{\left (-i \, a\right )}}{4 \, \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 29, normalized size = 0.56 \[ -\frac {-i \, c^{\frac {3}{2}} x^{2} e^{\left (-i \, a\right )} + 2 i \, \sqrt {c} e^{\left (i \, a\right )} \log \relax (x)}{4 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 106, normalized size = 2.04 \[ \frac {\frac {i x}{2}-\frac {i x \left (\tan ^{2}\left (\frac {a}{2}+\frac {i \ln \left (c \,x^{2}\right )}{4}\right )\right )}{2}+\frac {x \ln \left (c \,x^{2}\right ) \tan \left (\frac {a}{2}+\frac {i \ln \left (c \,x^{2}\right )}{4}\right )}{2}-\frac {i x \ln \left (c \,x^{2}\right )}{4}+\frac {i x \ln \left (c \,x^{2}\right ) \left (\tan ^{2}\left (\frac {a}{2}+\frac {i \ln \left (c \,x^{2}\right )}{4}\right )\right )}{4}}{1+\tan ^{2}\left (\frac {a}{2}+\frac {i \ln \left (c \,x^{2}\right )}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 31, normalized size = 0.60 \[ \frac {c x^{2} {\left (i \, \cos \relax (a) + \sin \relax (a)\right )} - 2 \, {\left (i \, \cos \relax (a) - \sin \relax (a)\right )} \log \relax (x)}{4 \, \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \sin \left (a+\frac {\ln \left (c\,x^2\right )\,1{}\mathrm {i}}{2}\right ) \,d x \]
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 \[ \int \sin {\left (a + \frac {i \log {\left (c x^{2} \right )}}{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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