Optimal. Leaf size=17 \[ \frac {\tan ^8(x)}{8}+\frac {\tan ^6(x)}{6} \]
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Rubi [A] time = 0.07, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4120, 2607, 14} \[ \frac {\tan ^8(x)}{8}+\frac {\tan ^6(x)}{6} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2607
Rule 4120
Rubi steps
\begin {align*} \int \sec ^4(x) \left (-1+\sec ^2(x)\right )^2 \tan (x) \, dx &=\int \sec ^4(x) \tan ^5(x) \, dx\\ &=\operatorname {Subst}\left (\int x^5 \left (1+x^2\right ) \, dx,x,\tan (x)\right )\\ &=\operatorname {Subst}\left (\int \left (x^5+x^7\right ) \, dx,x,\tan (x)\right )\\ &=\frac {\tan ^6(x)}{6}+\frac {\tan ^8(x)}{8}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 25, normalized size = 1.47 \[ \frac {\sec ^8(x)}{8}-\frac {\sec ^6(x)}{3}+\frac {\sec ^4(x)}{4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 20, normalized size = 1.18 \[ \frac {6 \, \cos \relax (x)^{4} - 8 \, \cos \relax (x)^{2} + 3}{24 \, \cos \relax (x)^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 20, normalized size = 1.18 \[ \frac {6 \, \cos \relax (x)^{4} - 8 \, \cos \relax (x)^{2} + 3}{24 \, \cos \relax (x)^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 20, normalized size = 1.18 \[ \frac {\left (\sec ^{8}\relax (x )\right )}{8}-\frac {\left (\sec ^{6}\relax (x )\right )}{3}+\frac {\left (\sec ^{4}\relax (x )\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 42, normalized size = 2.47 \[ \frac {6 \, \sin \relax (x)^{4} - 4 \, \sin \relax (x)^{2} + 1}{24 \, {\left (\sin \relax (x)^{8} - 4 \, \sin \relax (x)^{6} + 6 \, \sin \relax (x)^{4} - 4 \, \sin \relax (x)^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.92, size = 14, normalized size = 0.82 \[ \frac {{\mathrm {tan}\relax (x)}^6\,\left (3\,{\mathrm {tan}\relax (x)}^2+4\right )}{24} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.97, size = 19, normalized size = 1.12 \[ \frac {\sec ^{8}{\relax (x )}}{8} - \frac {\sec ^{6}{\relax (x )}}{3} + \frac {\sec ^{4}{\relax (x )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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