Optimal. Leaf size=12 \[ \tanh ^{-1}(\sin (x))-\frac {8 \sin ^3(x)}{3} \]
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Rubi [A] time = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4364, 1153, 206} \[ \tanh ^{-1}(\sin (x))-\frac {8 \sin ^3(x)}{3} \]
Antiderivative was successfully verified.
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Rule 206
Rule 1153
Rule 4364
Rubi steps
\begin {align*} \int \cos (4 x) \sec (x) \, dx &=\operatorname {Subst}\left (\int \frac {1-8 x^2+8 x^4}{1-x^2} \, dx,x,\sin (x)\right )\\ &=\operatorname {Subst}\left (\int \left (-8 x^2+\frac {1}{1-x^2}\right ) \, dx,x,\sin (x)\right )\\ &=-\frac {8}{3} \sin ^3(x)+\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (x)\right )\\ &=\tanh ^{-1}(\sin (x))-\frac {8 \sin ^3(x)}{3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 12, normalized size = 1.00 \[ \tanh ^{-1}(\sin (x))-\frac {8 \sin ^3(x)}{3} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos (4 x) \sec (x) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.80, size = 27, normalized size = 2.25 \[ \frac {8}{3} \, {\left (\cos \relax (x)^{2} - 1\right )} \sin \relax (x) + \frac {1}{2} \, \log \left (\sin \relax (x) + 1\right ) - \frac {1}{2} \, \log \left (-\sin \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.60, size = 23, normalized size = 1.92 \[ -\frac {8}{3} \, \sin \relax (x)^{3} + \frac {1}{2} \, \log \left (\sin \relax (x) + 1\right ) - \frac {1}{2} \, \log \left (-\sin \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.33, size = 22, normalized size = 1.83
method | result | size |
default | \(\ln \left (\sec \relax (x )+\tan \relax (x )\right )+\frac {8 \left (2+\cos ^{2}\relax (x )\right ) \sin \relax (x )}{3}-8 \sin \relax (x )\) | \(22\) |
risch | \(i {\mathrm e}^{i x}-i {\mathrm e}^{-i x}+\ln \left ({\mathrm e}^{i x}+i\right )-\ln \left ({\mathrm e}^{i x}-i\right )+\frac {2 \sin \left (3 x \right )}{3}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 21, normalized size = 1.75 \[ -\frac {8}{3} \, \sin \relax (x)^{3} + \frac {1}{2} \, \log \left (\sin \relax (x) + 1\right ) - \frac {1}{2} \, \log \left (\sin \relax (x) - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 10, normalized size = 0.83 \[ \mathrm {atanh}\left (\sin \relax (x)\right )-\frac {8\,{\sin \relax (x)}^3}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.70, size = 24, normalized size = 2.00 \[ - \frac {\log {\left (\sin {\relax (x )} - 1 \right )}}{2} + \frac {\log {\left (\sin {\relax (x )} + 1 \right )}}{2} - \frac {8 \sin ^{3}{\relax (x )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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