Optimal. Leaf size=16 \[ 16 x+\frac {5 \tan ^3(x)}{3}-15 \tan (x) \]
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Rubi [A] time = 0.04, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1153, 203} \[ 16 x+\frac {5 \tan ^3(x)}{3}-15 \tan (x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 1153
Rubi steps
\begin {align*} \int \cos (5 x) \sec ^5(x) \, dx &=\operatorname {Subst}\left (\int \frac {1-10 x^2+5 x^4}{1+x^2} \, dx,x,\tan (x)\right )\\ &=\operatorname {Subst}\left (\int \left (-15+5 x^2+\frac {16}{1+x^2}\right ) \, dx,x,\tan (x)\right )\\ &=-15 \tan (x)+\frac {5 \tan ^3(x)}{3}+16 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (x)\right )\\ &=16 x-15 \tan (x)+\frac {5 \tan ^3(x)}{3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 20, normalized size = 1.25 \[ 16 x-\frac {50 \tan (x)}{3}+\frac {5}{3} \tan (x) \sec ^2(x) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos (5 x) \sec ^5(x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.68, size = 26, normalized size = 1.62 \[ \frac {48 \, x \cos \relax (x)^{3} - 5 \, {\left (10 \, \cos \relax (x)^{2} - 1\right )} \sin \relax (x)}{3 \, \cos \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.62, size = 14, normalized size = 0.88 \[ \frac {5}{3} \, \tan \relax (x)^{3} + 16 \, x - 15 \, \tan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 21, normalized size = 1.31
method | result | size |
default | \(16 x -5 \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\relax (x )\right )}{3}\right ) \tan \relax (x )-20 \tan \relax (x )\) | \(21\) |
risch | \(16 x -\frac {20 i \left (6 \,{\mathrm e}^{4 i x}+9 \,{\mathrm e}^{2 i x}+5\right )}{3 \left (1+{\mathrm e}^{2 i x}\right )^{3}}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 14, normalized size = 0.88 \[ \frac {5}{3} \, \tan \relax (x)^{3} + 16 \, x - 15 \, \tan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 26, normalized size = 1.62 \[ \frac {48\,x\,{\cos \relax (x)}^3-50\,\sin \relax (x)\,{\cos \relax (x)}^2+5\,\sin \relax (x)}{3\,{\cos \relax (x)}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 17.46, size = 24, normalized size = 1.50 \[ 16 x - \frac {20 \sin {\relax (x )}}{\cos {\relax (x )}} + \frac {5 \tan ^{3}{\relax (x )}}{3} + 5 \tan {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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