Optimal. Leaf size=26 \[ \frac {35}{8} \tanh ^{-1}(\sin (x))+\frac {1}{4} \tan (x) \sec ^3(x)-\frac {29}{8} \tan (x) \sec (x) \]
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Rubi [A] time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4364, 1157, 385, 206} \[ \frac {35}{8} \tanh ^{-1}(\sin (x))+\frac {1}{4} \tan (x) \sec ^3(x)-\frac {29}{8} \tan (x) \sec (x) \]
Antiderivative was successfully verified.
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Rule 206
Rule 385
Rule 1157
Rule 4364
Rubi steps
\begin {align*} \int \cos (4 x) \sec ^5(x) \, dx &=\operatorname {Subst}\left (\int \frac {1-8 x^2+8 x^4}{\left (1-x^2\right )^3} \, dx,x,\sin (x)\right )\\ &=\frac {1}{4} \sec ^3(x) \tan (x)-\frac {1}{4} \operatorname {Subst}\left (\int \frac {-3+32 x^2}{\left (1-x^2\right )^2} \, dx,x,\sin (x)\right )\\ &=-\frac {29}{8} \sec (x) \tan (x)+\frac {1}{4} \sec ^3(x) \tan (x)+\frac {35}{8} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (x)\right )\\ &=\frac {35}{8} \tanh ^{-1}(\sin (x))-\frac {29}{8} \sec (x) \tan (x)+\frac {1}{4} \sec ^3(x) \tan (x)\\ \end {align*}
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Mathematica [A] time = 0.05, size = 26, normalized size = 1.00 \[ \frac {1}{8} \left (35 \tanh ^{-1}(\sin (x))-27 \tan (x) \sec ^3(x)+29 \tan ^3(x) \sec (x)\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos (4 x) \sec ^5(x) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.91, size = 43, normalized size = 1.65 \[ \frac {35 \, \cos \relax (x)^{4} \log \left (\sin \relax (x) + 1\right ) - 35 \, \cos \relax (x)^{4} \log \left (-\sin \relax (x) + 1\right ) - 2 \, {\left (29 \, \cos \relax (x)^{2} - 2\right )} \sin \relax (x)}{16 \, \cos \relax (x)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.59, size = 38, normalized size = 1.46 \[ \frac {29 \, \sin \relax (x)^{3} - 27 \, \sin \relax (x)}{8 \, {\left (\sin \relax (x)^{2} - 1\right )}^{2}} + \frac {35}{16} \, \log \left (\sin \relax (x) + 1\right ) - \frac {35}{16} \, \log \left (-\sin \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 31, normalized size = 1.19
method | result | size |
default | \(-\left (-\frac {\left (\sec ^{3}\relax (x )\right )}{4}-\frac {3 \sec \relax (x )}{8}\right ) \tan \relax (x )+\frac {35 \ln \left (\sec \relax (x )+\tan \relax (x )\right )}{8}-4 \sec \relax (x ) \tan \relax (x )\) | \(31\) |
risch | \(\frac {i \left (29 \,{\mathrm e}^{7 i x}+21 \,{\mathrm e}^{5 i x}-21 \,{\mathrm e}^{3 i x}-29 \,{\mathrm e}^{i x}\right )}{4 \left (1+{\mathrm e}^{2 i x}\right )^{4}}+\frac {35 \ln \left ({\mathrm e}^{i x}+i\right )}{8}-\frac {35 \ln \left ({\mathrm e}^{i x}-i\right )}{8}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 54, normalized size = 2.08 \[ \frac {5 \, \sin \relax (x)^{3} - 3 \, \sin \relax (x)}{8 \, {\left (\sin \relax (x)^{4} - 2 \, \sin \relax (x)^{2} + 1\right )}} + \frac {3 \, \sin \relax (x)}{\sin \relax (x)^{2} - 1} + \frac {35}{16} \, \log \left (\sin \relax (x) + 1\right ) - \frac {35}{16} \, \log \left (\sin \relax (x) - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 33, normalized size = 1.27 \[ \frac {35\,\mathrm {atanh}\left (\sin \relax (x)\right )}{8}-\frac {\frac {27\,\sin \relax (x)}{8}-\frac {29\,{\sin \relax (x)}^3}{8}}{{\sin \relax (x)}^4-2\,{\sin \relax (x)}^2+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 17.15, size = 75, normalized size = 2.88 \[ - \frac {35 \log {\left (\sin {\relax (x )} - 1 \right )}}{16} + \frac {35 \log {\left (\sin {\relax (x )} + 1 \right )}}{16} - \frac {3 \sin ^{3}{\relax (x )}}{8 \sin ^{4}{\relax (x )} - 16 \sin ^{2}{\relax (x )} + 8} + \frac {5 \sin {\relax (x )}}{8 \sin ^{4}{\relax (x )} - 16 \sin ^{2}{\relax (x )} + 8} + \frac {8 \sin {\relax (x )}}{2 \sin ^{2}{\relax (x )} - 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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