Optimal. Leaf size=17 \[ -\frac {1}{6} \sec ^6(x)+\frac {\sec ^8(x)}{8} \]
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Rubi [A]
time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2686, 14}
\begin {gather*} \frac {\sec ^8(x)}{8}-\frac {\sec ^6(x)}{6} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2686
Rubi steps
\begin {align*} \int \sec ^6(x) \tan ^3(x) \, dx &=\text {Subst}\left (\int x^5 \left (-1+x^2\right ) \, dx,x,\sec (x)\right )\\ &=\text {Subst}\left (\int \left (-x^5+x^7\right ) \, dx,x,\sec (x)\right )\\ &=-\frac {1}{6} \sec ^6(x)+\frac {\sec ^8(x)}{8}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} -\frac {1}{6} \sec ^6(x)+\frac {\sec ^8(x)}{8} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(31\) vs.
\(2(13)=26\).
time = 0.04, size = 32, normalized size = 1.88
method | result | size |
risch | \(-\frac {32 \left ({\mathrm e}^{10 i x}-{\mathrm e}^{8 i x}+{\mathrm e}^{6 i x}\right )}{3 \left ({\mathrm e}^{2 i x}+1\right )^{8}}\) | \(30\) |
default | \(\frac {\sin ^{4}\left (x \right )}{8 \cos \left (x \right )^{8}}+\frac {\sin ^{4}\left (x \right )}{12 \cos \left (x \right )^{6}}+\frac {\sin ^{4}\left (x \right )}{24 \cos \left (x \right )^{4}}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs.
\(2 (13) = 26\).
time = 1.72, size = 36, normalized size = 2.12 \begin {gather*} \frac {4 \, \sin \left (x\right )^{2} - 1}{24 \, {\left (\sin \left (x\right )^{8} - 4 \, \sin \left (x\right )^{6} + 6 \, \sin \left (x\right )^{4} - 4 \, \sin \left (x\right )^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.79, size = 14, normalized size = 0.82 \begin {gather*} -\frac {4 \, \cos \left (x\right )^{2} - 3}{24 \, \cos \left (x\right )^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 14, normalized size = 0.82 \begin {gather*} \frac {3 - 4 \cos ^{2}{\left (x \right )}}{24 \cos ^{8}{\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.16, size = 14, normalized size = 0.82 \begin {gather*} -\frac {4 \, \cos \left (x\right )^{2} - 3}{24 \, \cos \left (x\right )^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 20, normalized size = 1.18 \begin {gather*} \frac {{\mathrm {tan}\left (x\right )}^4\,\left (3\,{\mathrm {tan}\left (x\right )}^4+8\,{\mathrm {tan}\left (x\right )}^2+6\right )}{24} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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