Optimal. Leaf size=30 \[ x+\frac {2 \cos ^3(x)}{3 (1-\sin (x))^3}-\frac {2 \cos (x)}{1-\sin (x)} \]
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Rubi [A] time = 0.10, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4391, 2670, 2680, 8} \[ x+\frac {2 \cos ^3(x)}{3 (1-\sin (x))^3}-\frac {2 \cos (x)}{1-\sin (x)} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2670
Rule 2680
Rule 4391
Rubi steps
\begin {align*} \int (\sec (x)+\tan (x))^4 \, dx &=\int \sec ^4(x) (1+\sin (x))^4 \, dx\\ &=\int \frac {\cos ^4(x)}{(1-\sin (x))^4} \, dx\\ &=\frac {2 \cos ^3(x)}{3 (1-\sin (x))^3}-\int \frac {\cos ^2(x)}{(1-\sin (x))^2} \, dx\\ &=\frac {2 \cos ^3(x)}{3 (1-\sin (x))^3}-\frac {2 \cos (x)}{1-\sin (x)}+\int 1 \, dx\\ &=x+\frac {2 \cos ^3(x)}{3 (1-\sin (x))^3}-\frac {2 \cos (x)}{1-\sin (x)}\\ \end {align*}
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Mathematica [B] time = 0.13, size = 64, normalized size = 2.13 \[ -\frac {-3 (3 x+8) \cos \left (\frac {x}{2}\right )+(3 x+16) \cos \left (\frac {3 x}{2}\right )+6 \sin \left (\frac {x}{2}\right ) (2 x+x \cos (x)+4)}{6 \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.91, size = 61, normalized size = 2.03 \[ \frac {{\left (3 \, x + 8\right )} \cos \relax (x)^{2} - {\left (3 \, x - 4\right )} \cos \relax (x) + {\left ({\left (3 \, x - 8\right )} \cos \relax (x) + 6 \, x - 4\right )} \sin \relax (x) - 6 \, x - 4}{3 \, {\left (\cos \relax (x)^{2} + {\left (\cos \relax (x) + 2\right )} \sin \relax (x) - \cos \relax (x) - 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 20, normalized size = 0.67 \[ x - \frac {8 \, {\left (3 \, \tan \left (\frac {1}{2} \, x\right ) - 1\right )}}{3 \, {\left (\tan \left (\frac {1}{2} \, x\right ) - 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 71, normalized size = 2.37 \[ -\left (-\frac {2}{3}-\frac {\left (\sec ^{2}\relax (x )\right )}{3}\right ) \tan \relax (x )+\frac {4}{3 \cos \relax (x )^{3}}+\frac {2 \left (\sin ^{3}\relax (x )\right )}{\cos \relax (x )^{3}}+\frac {4 \left (\sin ^{4}\relax (x )\right )}{3 \cos \relax (x )^{3}}-\frac {4 \left (\sin ^{4}\relax (x )\right )}{3 \cos \relax (x )}-\frac {4 \left (2+\sin ^{2}\relax (x )\right ) \cos \relax (x )}{3}+\frac {\left (\tan ^{3}\relax (x )\right )}{3}-\tan \relax (x )+x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 28, normalized size = 0.93 \[ \frac {8}{3} \, \tan \relax (x)^{3} + x - \frac {4 \, {\left (3 \, \cos \relax (x)^{2} - 1\right )}}{3 \, \cos \relax (x)^{3}} + \frac {4}{3 \, \cos \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.37, size = 20, normalized size = 0.67 \[ x-\frac {8\,\mathrm {tan}\left (\frac {x}{2}\right )-\frac {8}{3}}{{\left (\mathrm {tan}\left (\frac {x}{2}\right )-1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.10, size = 44, normalized size = 1.47 \[ x + \frac {\sin ^{3}{\relax (x )}}{3 \cos ^{3}{\relax (x )}} - \frac {\sin {\relax (x )}}{\cos {\relax (x )}} + \frac {7 \tan ^{3}{\relax (x )}}{3} + \tan {\relax (x )} + \frac {8 \sec ^{3}{\relax (x )}}{3} - 4 \sec {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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