Optimal. Leaf size=11 \[ -\frac {1}{2} \log \left (\csc ^2(x)-4\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 17, normalized size of antiderivative = 1.55, number of steps used = 5, number of rules used = 5, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {4356, 266, 36, 31, 29} \[ \log (\sin (x))-\frac {1}{2} \log \left (1-4 \sin ^2(x)\right ) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 4356
Rubi steps
\begin {align*} \int \cos (x) \cot (x) \sec (3 x) \, dx &=\operatorname {Subst}\left (\int \frac {1}{x \left (1-4 x^2\right )} \, dx,x,\sin (x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(1-4 x) x} \, dx,x,\sin ^2(x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\sin ^2(x)\right )+2 \operatorname {Subst}\left (\int \frac {1}{1-4 x} \, dx,x,\sin ^2(x)\right )\\ &=\log (\sin (x))-\frac {1}{2} \log \left (1-4 \sin ^2(x)\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 1.55 \[ \log (\sin (x))-\frac {1}{2} \log \left (1-4 \sin ^2(x)\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos (x) \cot (x) \sec (3 x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.81, size = 17, normalized size = 1.55 \[ -\frac {1}{2} \, \log \left (4 \, \cos \relax (x)^{2} - 3\right ) + \log \left (\frac {1}{2} \, \sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.64, size = 24, normalized size = 2.18 \[ \frac {1}{2} \, \log \left (-\cos \relax (x)^{2} + 1\right ) - \frac {1}{2} \, \log \left ({\left | 4 \, \cos \relax (x)^{2} - 3 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.15, size = 27, normalized size = 2.45
method | result | size |
default | \(\frac {\ln \left (1+\cos \relax (x )\right )}{2}-\frac {\ln \left (4 \left (\cos ^{2}\relax (x )\right )-3\right )}{2}+\frac {\ln \left (-1+\cos \relax (x )\right )}{2}\) | \(27\) |
risch | \(\ln \left ({\mathrm e}^{2 i x}-1\right )-\frac {\ln \left ({\mathrm e}^{4 i x}-{\mathrm e}^{2 i x}+1\right )}{2}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 92, normalized size = 8.36 \[ -\frac {1}{4} \, \log \left (-2 \, {\left (\cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) + \cos \left (4 \, x\right )^{2} + \cos \left (2 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} - 2 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1\right ) + \frac {1}{2} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) + \frac {1}{2} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.62, size = 25, normalized size = 2.27 \[ \ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )-\frac {\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^4-14\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}{2} \]
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 \[ \int \frac {\cos ^{2}{\relax (x )}}{\sin {\relax (x )} \cos {\left (3 x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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