Optimal. Leaf size=48 \[ \frac {(a \text {a1}+b \text {b1}) x}{a^2+b^2}-\frac {(\text {a1} b-a \text {b1}) \log (b \cos (x)+a \sin (x))}{a^2+b^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {3212}
\begin {gather*} \frac {x (a \text {a1}+b \text {b1})}{a^2+b^2}-\frac {(\text {a1} b-a \text {b1}) \log (a \sin (x)+b \cos (x))}{a^2+b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3212
Rubi steps
\begin {align*} \int \frac {\text {b1} \cos (x)+\text {a1} \sin (x)}{b \cos (x)+a \sin (x)} \, dx &=\frac {(a \text {a1}+b \text {b1}) x}{a^2+b^2}-\frac {(\text {a1} b-a \text {b1}) \log (b \cos (x)+a \sin (x))}{a^2+b^2}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 39, normalized size = 0.81 \begin {gather*} \frac {(a \text {a1}+b \text {b1}) x+(-\text {a1} b+a \text {b1}) \log (b \cos (x)+a \sin (x))}{a^2+b^2} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 5.65, size = 209, normalized size = 4.35 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\text {DirectedInfinity}\left [\text {a1} \text {Log}\left [\text {Cos}\left [x\right ]\right ]-\text {b1} x\right ],a\text {==}0\text {\&\&}b\text {==}0\right \},\left \{\frac {-\text {a1} \text {Log}\left [\text {Cos}\left [x\right ]\right ]+\text {b1} x}{b},a\text {==}0\right \},\left \{\frac {\left (-I \text {a1} \text {Sin}\left [x\right ]+I \text {a1} x \text {Cos}\left [x\right ]+\text {a1} x \text {Sin}\left [x\right ]+\text {b1} \text {Sin}\left [x\right ]-I \text {b1} x \text {Sin}\left [x\right ]+\text {b1} x \text {Cos}\left [x\right ]\right ) E^{I x}}{2 b},a\text {==}-I b\right \},\left \{\frac {\left (I \text {a1} \text {Sin}\left [x\right ]-I \text {a1} x \text {Cos}\left [x\right ]+\text {a1} x \text {Sin}\left [x\right ]+\text {b1} \text {Sin}\left [x\right ]+I \text {b1} x \text {Sin}\left [x\right ]+\text {b1} x \text {Cos}\left [x\right ]\right ) E^{-I x}}{2 b},a\text {==}I b\right \}\right \},\frac {a \text {a1} x}{a^2+b^2}+\frac {a \text {b1} \text {Log}\left [\frac {b \text {Cos}\left [x\right ]}{a}+\text {Sin}\left [x\right ]\right ]}{a^2+b^2}-\frac {\text {a1} b \text {Log}\left [\frac {b \text {Cos}\left [x\right ]}{a}+\text {Sin}\left [x\right ]\right ]}{a^2+b^2}+\frac {b \text {b1} x}{a^2+b^2}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.10, size = 66, normalized size = 1.38
method | result | size |
default | \(\frac {\frac {\left (-a \mathit {b1} +\mathit {a1} b \right ) \ln \left (1+\tan ^{2}\left (x \right )\right )}{2}+\left (a \mathit {a1} +b \mathit {b1} \right ) \arctan \left (\tan \left (x \right )\right )}{a^{2}+b^{2}}+\frac {\left (a \mathit {b1} -\mathit {a1} b \right ) \ln \left (a \tan \left (x \right )+b \right )}{a^{2}+b^{2}}\) | \(66\) |
norman | \(\frac {\frac {\left (a \mathit {a1} +b \mathit {b1} \right ) x}{a^{2}+b^{2}}+\frac {\left (a \mathit {a1} +b \mathit {b1} \right ) x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a^{2}+b^{2}}}{1+\tan ^{2}\left (\frac {x}{2}\right )}+\frac {\left (a \mathit {b1} -\mathit {a1} b \right ) \ln \left (-b \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 a \tan \left (\frac {x}{2}\right )+b \right )}{a^{2}+b^{2}}-\frac {\left (a \mathit {b1} -\mathit {a1} b \right ) \ln \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )}{a^{2}+b^{2}}\) | \(121\) |
risch | \(\frac {i x \mathit {b1}}{i b +a}+\frac {x \mathit {a1}}{i b +a}-\frac {2 i x a \mathit {b1}}{a^{2}+b^{2}}+\frac {2 i x \mathit {a1} b}{a^{2}+b^{2}}+\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {i b -a}{i b +a}\right ) a \mathit {b1}}{a^{2}+b^{2}}-\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {i b -a}{i b +a}\right ) \mathit {a1} b}{a^{2}+b^{2}}\) | \(129\) |
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. 181 vs.
\(2 (48) = 96\).
time = 0.37, size = 181, normalized size = 3.77 \begin {gather*} a_{1} {\left (\frac {2 \, a \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{2} + b^{2}} - \frac {b \log \left (-b - \frac {2 \, a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {b \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}{a^{2} + b^{2}} + \frac {b \log \left (\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}{a^{2} + b^{2}}\right )} + b_{1} {\left (\frac {2 \, b \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{2} + b^{2}} + \frac {a \log \left (-b - \frac {2 \, a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {b \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}{a^{2} + b^{2}} - \frac {a \log \left (\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}{a^{2} + b^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 60, normalized size = 1.25 \begin {gather*} \frac {2 \, {\left (a a_{1} + b b_{1}\right )} x - {\left (a_{1} b - a b_{1}\right )} \log \left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) - {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + a^{2}\right )}{2 \, {\left (a^{2} + b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.46, size = 360, normalized size = 7.50 \begin {gather*} \begin {cases} \tilde {\infty } \left (- a_{1} \log {\left (\cos {\left (x \right )} \right )} + b_{1} x\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {a_{1} x \sin {\left (x \right )}}{- 2 i b \sin {\left (x \right )} + 2 b \cos {\left (x \right )}} + \frac {i a_{1} x \cos {\left (x \right )}}{- 2 i b \sin {\left (x \right )} + 2 b \cos {\left (x \right )}} - \frac {a_{1} \cos {\left (x \right )}}{- 2 i b \sin {\left (x \right )} + 2 b \cos {\left (x \right )}} - \frac {i b_{1} x \sin {\left (x \right )}}{- 2 i b \sin {\left (x \right )} + 2 b \cos {\left (x \right )}} + \frac {b_{1} x \cos {\left (x \right )}}{- 2 i b \sin {\left (x \right )} + 2 b \cos {\left (x \right )}} - \frac {i b_{1} \cos {\left (x \right )}}{- 2 i b \sin {\left (x \right )} + 2 b \cos {\left (x \right )}} & \text {for}\: a = - i b \\\frac {a_{1} x \sin {\left (x \right )}}{2 i b \sin {\left (x \right )} + 2 b \cos {\left (x \right )}} - \frac {i a_{1} x \cos {\left (x \right )}}{2 i b \sin {\left (x \right )} + 2 b \cos {\left (x \right )}} - \frac {a_{1} \cos {\left (x \right )}}{2 i b \sin {\left (x \right )} + 2 b \cos {\left (x \right )}} + \frac {i b_{1} x \sin {\left (x \right )}}{2 i b \sin {\left (x \right )} + 2 b \cos {\left (x \right )}} + \frac {b_{1} x \cos {\left (x \right )}}{2 i b \sin {\left (x \right )} + 2 b \cos {\left (x \right )}} + \frac {i b_{1} \cos {\left (x \right )}}{2 i b \sin {\left (x \right )} + 2 b \cos {\left (x \right )}} & \text {for}\: a = i b \\\frac {a_{1} x + b_{1} \log {\left (\sin {\left (x \right )} \right )}}{a} & \text {for}\: b = 0 \\\frac {a a_{1} x}{a^{2} + b^{2}} + \frac {a b_{1} \log {\left (\frac {a \sin {\left (x \right )}}{b} + \cos {\left (x \right )} \right )}}{a^{2} + b^{2}} - \frac {a_{1} b \log {\left (\frac {a \sin {\left (x \right )}}{b} + \cos {\left (x \right )} \right )}}{a^{2} + b^{2}} + \frac {b b_{1} x}{a^{2} + b^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.01, size = 83, normalized size = 1.73 \begin {gather*} 2 \left (\frac {\left (-a_{1} a b+b_{1} a^{2}\right ) \ln \left |\tan x\cdot a+b\right |}{2 a^{3}+2 a b^{2}}+\frac {\left (a_{1} b-b_{1} a\right ) \ln \left (\tan ^{2}x+1\right )}{4 a^{2}+4 b^{2}}+\frac {\left (a_{1} a+b_{1} b\right ) x}{2 \left (a^{2}+b^{2}\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 10.61, size = 2034, normalized size = 42.38
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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