Optimal. Leaf size=17 \[ -\frac {1}{5} \cot ^5(x)-\frac {\cot ^7(x)}{7} \]
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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 = {2687, 14}
\begin {gather*} -\frac {1}{7} \cot ^7(x)-\frac {\cot ^5(x)}{5} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2687
Rubi steps
\begin {align*} \int \cot ^4(x) \csc ^4(x) \, dx &=\text {Subst}\left (\int x^4 \left (1+x^2\right ) \, dx,x,-\cot (x)\right )\\ &=\text {Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,-\cot (x)\right )\\ &=-\frac {1}{5} \cot ^5(x)-\frac {\cot ^7(x)}{7}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(37\) vs. \(2(17)=34\).
time = 0.02, size = 37, normalized size = 2.18 \begin {gather*} -\frac {2 \cot (x)}{35}-\frac {1}{35} \cot (x) \csc ^2(x)+\frac {8}{35} \cot (x) \csc ^4(x)-\frac {1}{7} \cot (x) \csc ^6(x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 22, normalized size = 1.29
method | result | size |
default | \(-\frac {\cos ^{5}\left (x \right )}{7 \sin \left (x \right )^{7}}-\frac {2 \left (\cos ^{5}\left (x \right )\right )}{35 \sin \left (x \right )^{5}}\) | \(22\) |
risch | \(\frac {4 i \left (35 \,{\mathrm e}^{10 i x}+35 \,{\mathrm e}^{8 i x}+70 \,{\mathrm e}^{6 i x}+14 \,{\mathrm e}^{4 i x}+7 \,{\mathrm e}^{2 i x}-1\right )}{35 \left ({\mathrm e}^{2 i x}-1\right )^{7}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.16, size = 14, normalized size = 0.82 \begin {gather*} -\frac {7 \, \tan \left (x\right )^{2} + 5}{35 \, \tan \left (x\right )^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 39 vs.
\(2 (13) = 26\).
time = 0.58, size = 39, normalized size = 2.29 \begin {gather*} -\frac {2 \, \cos \left (x\right )^{7} - 7 \, \cos \left (x\right )^{5}}{35 \, {\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \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. 41 vs.
\(2 (14) = 28\).
time = 0.01, size = 41, normalized size = 2.41 \begin {gather*} - \frac {2 \cos {\left (x \right )}}{35 \sin {\left (x \right )}} - \frac {\cos {\left (x \right )}}{35 \sin ^{3}{\left (x \right )}} + \frac {8 \cos {\left (x \right )}}{35 \sin ^{5}{\left (x \right )}} - \frac {\cos {\left (x \right )}}{7 \sin ^{7}{\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.86, size = 14, normalized size = 0.82 \begin {gather*} -\frac {7 \, \tan \left (x\right )^{2} + 5}{35 \, \tan \left (x\right )^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 14, normalized size = 0.82 \begin {gather*} -\frac {{\mathrm {cot}\left (x\right )}^5\,\left (5\,{\mathrm {cot}\left (x\right )}^2+7\right )}{35} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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