Optimal. Leaf size=17 \[ \frac {\tan ^3(x)}{3}+\frac {\tan ^5(x)}{5} \]
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Rubi [A]
time = 0.01, 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 = {2687, 14}
\begin {gather*} \frac {\tan ^5(x)}{5}+\frac {\tan ^3(x)}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2687
Rubi steps
\begin {align*} \int \sec ^4(x) \tan ^2(x) \, dx &=\text {Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,\tan (x)\right )\\ &=\text {Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,\tan (x)\right )\\ &=\frac {\tan ^3(x)}{3}+\frac {\tan ^5(x)}{5}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 27, normalized size = 1.59 \begin {gather*} -\frac {2 \tan (x)}{15}-\frac {1}{15} \sec ^2(x) \tan (x)+\frac {1}{5} \sec ^4(x) \tan (x) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.84, size = 17, normalized size = 1.00 \begin {gather*} \frac {-2 \text {Tan}\left [x\right ]^5}{15}+\frac {\text {Sin}\left [x\right ]^3}{3 \text {Cos}\left [x\right ]^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 22, normalized size = 1.29
method | result | size |
default | \(\frac {\sin ^{3}\left (x \right )}{5 \cos \left (x \right )^{5}}+\frac {2 \left (\sin ^{3}\left (x \right )\right )}{15 \cos \left (x \right )^{3}}\) | \(22\) |
risch | \(-\frac {4 i \left (15 \,{\mathrm e}^{6 i x}-5 \,{\mathrm e}^{4 i x}+5 \,{\mathrm e}^{2 i x}+1\right )}{15 \left ({\mathrm e}^{2 i x}+1\right )^{5}}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{5} \, \tan \left (x\right )^{5} + \frac {1}{3} \, \tan \left (x\right )^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 20, normalized size = 1.18 \begin {gather*} -\frac {{\left (2 \, \cos \left (x\right )^{4} + \cos \left (x\right )^{2} - 3\right )} \sin \left (x\right )}{15 \, \cos \left (x\right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs.
\(2 (12) = 24\)
time = 0.03, size = 29, normalized size = 1.71 \begin {gather*} - \frac {2 \sin {\left (x \right )}}{15 \cos {\left (x \right )}} - \frac {\sin {\left (x \right )}}{15 \cos ^{3}{\left (x \right )}} + \frac {\sin {\left (x \right )}}{5 \cos ^{5}{\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 15, normalized size = 0.88 \begin {gather*} \frac {\tan ^{5}x}{5}+\frac {\tan ^{3}x}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 13, normalized size = 0.76 \begin {gather*} \frac {{\mathrm {tan}\left (x\right )}^5}{5}+\frac {{\mathrm {tan}\left (x\right )}^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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