Optimal. Leaf size=53 \[ -\frac {e^{2 i a} x \log (x)}{4 \sqrt {c x^2}}-\frac {e^{-2 i a} c x^3}{8 \sqrt {c x^2}}+\frac {x}{2} \]
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Rubi [A] time = 0.05, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {4483, 4489} \[ -\frac {e^{-2 i a} c x^3}{8 \sqrt {c x^2}}-\frac {e^{2 i a} x \log (x)}{4 \sqrt {c x^2}}+\frac {x}{2} \]
Antiderivative was successfully verified.
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Rule 4483
Rule 4489
Rubi steps
\begin {align*} \int \sin ^2\left (a+\frac {1}{4} i \log \left (c x^2\right )\right ) \, dx &=\frac {x \operatorname {Subst}\left (\int \frac {\sin ^2\left (a+\frac {1}{4} i \log (x)\right )}{\sqrt {x}} \, dx,x,c x^2\right )}{2 \sqrt {c x^2}}\\ &=-\frac {x \operatorname {Subst}\left (\int \left (e^{-2 i a}+\frac {e^{2 i a}}{x}-\frac {2}{\sqrt {x}}\right ) \, dx,x,c x^2\right )}{8 \sqrt {c x^2}}\\ &=\frac {x}{2}-\frac {c e^{-2 i a} x^3}{8 \sqrt {c x^2}}-\frac {e^{2 i a} x \log (x)}{4 \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 60, normalized size = 1.13 \[ \frac {x \left (i \sin (2 a) \left (c x^2-2 \log (x)\right )-\cos (2 a) \left (c x^2+2 \log (x)\right )+4 \sqrt {c x^2}\right )}{8 \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.83, size = 145, normalized size = 2.74 \[ \frac {{\left (4 \, x^{2} e^{\left (2 i \, a\right )} - \frac {x e^{\left (4 i \, a\right )} \log \left (\frac {{\left (\sqrt {c x^{2}} {\left (x^{2} + 1\right )} e^{\left (2 i \, a\right )} + \frac {{\left (c x^{3} - c x\right )} e^{\left (2 i \, a\right )}}{\sqrt {c}}\right )} e^{\left (-2 i \, a\right )}}{8 \, x^{2}}\right )}{\sqrt {c}} + \frac {x e^{\left (4 i \, a\right )} \log \left (\frac {{\left (\sqrt {c x^{2}} {\left (x^{2} + 1\right )} e^{\left (2 i \, a\right )} - \frac {{\left (c x^{3} - c x\right )} e^{\left (2 i \, a\right )}}{\sqrt {c}}\right )} e^{\left (-2 i \, a\right )}}{8 \, x^{2}}\right )}{\sqrt {c}} - \sqrt {c x^{2}} {\left (x^{2} - 1\right )}\right )} e^{\left (-2 i \, a\right )}}{8 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 32, normalized size = 0.60 \[ \frac {1}{2} \, x - \frac {c^{\frac {3}{2}} x^{2} e^{\left (-2 i \, a\right )} + 2 \, \sqrt {c} e^{\left (2 i \, a\right )} \log \relax (x)}{8 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 173, normalized size = 3.26 \[ \frac {\frac {x}{4}+\frac {5 x \left (\tan ^{2}\left (\frac {a}{2}+\frac {i \ln \left (c \,x^{2}\right )}{8}\right )\right )}{2}+\frac {x \left (\tan ^{4}\left (\frac {a}{2}+\frac {i \ln \left (c \,x^{2}\right )}{8}\right )\right )}{4}-\frac {x \ln \left (c \,x^{2}\right )}{8}+\frac {3 x \ln \left (c \,x^{2}\right ) \left (\tan ^{2}\left (\frac {a}{2}+\frac {i \ln \left (c \,x^{2}\right )}{8}\right )\right )}{4}-\frac {x \ln \left (c \,x^{2}\right ) \left (\tan ^{4}\left (\frac {a}{2}+\frac {i \ln \left (c \,x^{2}\right )}{8}\right )\right )}{8}-\frac {i x \ln \left (c \,x^{2}\right ) \tan \left (\frac {a}{2}+\frac {i \ln \left (c \,x^{2}\right )}{8}\right )}{2}+\frac {i x \ln \left (c \,x^{2}\right ) \left (\tan ^{3}\left (\frac {a}{2}+\frac {i \ln \left (c \,x^{2}\right )}{8}\right )\right )}{2}}{\left (1+\tan ^{2}\left (\frac {a}{2}+\frac {i \ln \left (c \,x^{2}\right )}{8}\right )\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 48, normalized size = 0.91 \[ \frac {4 \, c x - {\left (c x^{2} {\left (\cos \left (2 \, a\right ) - i \, \sin \left (2 \, a\right )\right )} + {\left (2 \, \cos \left (2 \, a\right ) + 2 i \, \sin \left (2 \, a\right )\right )} \log \relax (x)\right )} \sqrt {c}}{8 \, c} \]
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 \[ \int {\sin \left (a+\frac {\ln \left (c\,x^2\right )\,1{}\mathrm {i}}{4}\right )}^2 \,d x \]
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 \[ \int \sin ^{2}{\left (a + \frac {i \log {\left (c x^{2} \right )}}{4} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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