Optimal. Leaf size=21 \[ -x+\cot (a) \log (\cos (a-x))-\cot (a) \log (\cos (x)) \]
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Rubi [A]
time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4708, 4706,
3556} \begin {gather*} \cot (a) \log (\cos (a-x))-\cot (a) \log (\cos (x))-x \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 4706
Rule 4708
Rubi steps
\begin {align*} \int -\tan (a-x) \tan (x) \, dx &=-x+\cos (a) \int \sec (a-x) \sec (x) \, dx\\ &=-x+\cot (a) \int \tan (a-x) \, dx+\cot (a) \int \tan (x) \, dx\\ &=-x+\cot (a) \log (\cos (a-x))-\cot (a) \log (\cos (x))\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 21, normalized size = 1.00 \begin {gather*} -x+\cot (a) \log (\cos (a-x))-\cot (a) \log (\cos (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 20, normalized size = 0.95
method | result | size |
derivativedivides | \(-\arctan \left (\tan \left (x \right )\right )+\frac {\ln \left (1+\tan \left (x \right ) \tan \left (a \right )\right )}{\tan \left (a \right )}\) | \(20\) |
default | \(-\arctan \left (\tan \left (x \right )\right )+\frac {\ln \left (1+\tan \left (x \right ) \tan \left (a \right )\right )}{\tan \left (a \right )}\) | \(20\) |
risch | \(-x +\frac {i \ln \left ({\mathrm e}^{2 i a}+{\mathrm e}^{2 i x}\right ) {\mathrm e}^{2 i a}}{{\mathrm e}^{2 i a}-1}+\frac {i \ln \left ({\mathrm e}^{2 i a}+{\mathrm e}^{2 i x}\right )}{{\mathrm e}^{2 i a}-1}-\frac {i \ln \left ({\mathrm e}^{2 i x}+1\right ) {\mathrm e}^{2 i a}}{{\mathrm e}^{2 i a}-1}-\frac {i \ln \left ({\mathrm e}^{2 i x}+1\right )}{{\mathrm e}^{2 i a}-1}\) | \(103\) |
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. 186 vs.
\(2 (21) = 42\).
time = 6.22, size = 186, normalized size = 8.86 \begin {gather*} -\frac {{\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2} - 2 \, \cos \left (2 \, a\right ) + 1\right )} x + {\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2} - 1\right )} \arctan \left (\sin \left (2 \, a\right ) + \sin \left (2 \, x\right ), \cos \left (2 \, a\right ) + \cos \left (2 \, x\right )\right ) - {\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2} - 1\right )} \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right ) + 1\right ) - \log \left (\cos \left (2 \, a\right )^{2} + 2 \, \cos \left (2 \, a\right ) \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + \sin \left (2 \, a\right )^{2} + 2 \, \sin \left (2 \, a\right ) \sin \left (2 \, x\right ) + \sin \left (2 \, x\right )^{2}\right ) \sin \left (2 \, a\right ) + \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right ) \sin \left (2 \, a\right )}{\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2} - 2 \, \cos \left (2 \, a\right ) + 1} \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. 89 vs.
\(2 (21) = 42\).
time = 0.50, size = 89, normalized size = 4.24 \begin {gather*} \frac {{\left (\cos \left (2 \, a\right ) + 1\right )} \log \left (-\frac {{\left (\cos \left (2 \, a\right ) - 1\right )} \tan \left (x\right )^{2} - 2 \, \sin \left (2 \, a\right ) \tan \left (x\right ) - \cos \left (2 \, a\right ) - 1}{{\left (\cos \left (2 \, a\right ) + 1\right )} \tan \left (x\right )^{2} + \cos \left (2 \, a\right ) + 1}\right ) - {\left (\cos \left (2 \, a\right ) + 1\right )} \log \left (\frac {1}{\tan \left (x\right )^{2} + 1}\right ) - 2 \, x \sin \left (2 \, a\right )}{2 \, \sin \left (2 \, a\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. 66 vs.
\(2 (19) = 38\).
time = 0.69, size = 138, normalized size = 6.57 \begin {gather*} - \left (\begin {cases} \frac {2 x \tan {\left (a \right )}}{2 \tan ^{2}{\left (a \right )} + 2} - \frac {2 \log {\left (\tan {\left (x \right )} + \frac {1}{\tan {\left (a \right )}} \right )}}{2 \tan ^{2}{\left (a \right )} + 2} + \frac {\log {\left (\tan ^{2}{\left (x \right )} + 1 \right )}}{2 \tan ^{2}{\left (a \right )} + 2} & \text {for}\: a \neq 0 \\\frac {\log {\left (\tan ^{2}{\left (x \right )} + 1 \right )}}{2} & \text {otherwise} \end {cases}\right ) \tan {\left (a \right )} + \begin {cases} - \frac {2 x \tan {\left (a \right )}}{2 \tan ^{3}{\left (a \right )} + 2 \tan {\left (a \right )}} + \frac {2 \log {\left (\tan {\left (x \right )} + \frac {1}{\tan {\left (a \right )}} \right )}}{2 \tan ^{3}{\left (a \right )} + 2 \tan {\left (a \right )}} + \frac {\log {\left (\tan ^{2}{\left (x \right )} + 1 \right )} \tan ^{2}{\left (a \right )}}{2 \tan ^{3}{\left (a \right )} + 2 \tan {\left (a \right )}} & \text {for}\: a \neq 0 \\- x + \tan {\left (x \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.10, size = 18, normalized size = 0.86 \begin {gather*} -x + \frac {\log \left ({\left | \tan \left (a\right ) \tan \left (x\right ) + 1 \right |}\right )}{\tan \left (a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.33, size = 118, normalized size = 5.62 \begin {gather*} -x-\frac {\frac {\sin \left (2\,a\right )\,\ln \left ({\sin \left (2\,a+x\right )}^2\,2{}\mathrm {i}+{\sin \left (2\,a\right )}^2\,2{}\mathrm {i}-{\sin \left (x\right )}^2\,2{}\mathrm {i}+\sin \left (4\,a\right )-\sin \left (2\,x\right )+\sin \left (4\,a+2\,x\right )\right )}{2}-\frac {\sin \left (2\,a\right )\,\ln \left (\sin \left (2\,a\right )\,\left (2\,{\sin \left (a\right )}^2-1\right )-{\sin \left (2\,a\right )}^2\,1{}\mathrm {i}+\sin \left (2\,a\right )\,\left (2\,{\sin \left (x\right )}^2-1\right )-\sin \left (2\,a\right )\,\sin \left (2\,x\right )\,1{}\mathrm {i}\right )}{2}}{{\sin \left (a\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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