Optimal. Leaf size=46 \[ \frac {b x}{a^2+b^2}-\frac {a \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right ) d} \]
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Rubi [A]
time = 0.04, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3612, 3611}
\begin {gather*} \frac {b x}{a^2+b^2}-\frac {a \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 3611
Rule 3612
Rubi steps
\begin {align*} \int \frac {\tan (c+d x)}{a+b \tan (c+d x)} \, dx &=\frac {b x}{a^2+b^2}-\frac {a \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^2+b^2}\\ &=\frac {b x}{a^2+b^2}-\frac {a \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right ) d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.13, size = 66, normalized size = 1.43 \begin {gather*} \frac {2 (-i a+b) (c+d x)+2 i a \text {ArcTan}(\tan (c+d x))-a \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right )}{2 \left (a^2+b^2\right ) d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 63, normalized size = 1.37
method | result | size |
derivativedivides | \(\frac {\frac {\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+b \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}-\frac {a \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{2}+b^{2}}}{d}\) | \(63\) |
default | \(\frac {\frac {\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+b \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}-\frac {a \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{2}+b^{2}}}{d}\) | \(63\) |
norman | \(\frac {b x}{a^{2}+b^{2}}+\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{2}+b^{2}\right )}-\frac {a \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )}\) | \(66\) |
risch | \(\frac {i x}{i b -a}+\frac {2 i a x}{a^{2}+b^{2}}+\frac {2 i a c}{d \left (a^{2}+b^{2}\right )}-\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{2}+b^{2}\right )}\) | \(91\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 68, normalized size = 1.48 \begin {gather*} \frac {\frac {2 \, {\left (d x + c\right )} b}{a^{2} + b^{2}} - \frac {2 \, a \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{2} + b^{2}} + \frac {a \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.41, size = 63, normalized size = 1.37 \begin {gather*} \frac {2 \, b d x - a \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, {\left (a^{2} + b^{2}\right )} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.46, size = 255, normalized size = 5.54 \begin {gather*} \begin {cases} \tilde {\infty } x & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac {d x \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} - \frac {i d x}{2 b d \tan {\left (c + d x \right )} - 2 i b d} - \frac {1}{2 b d \tan {\left (c + d x \right )} - 2 i b d} & \text {for}\: a = - i b \\\frac {d x \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} + \frac {i d x}{2 b d \tan {\left (c + d x \right )} + 2 i b d} - \frac {1}{2 b d \tan {\left (c + d x \right )} + 2 i b d} & \text {for}\: a = i b \\\frac {x \tan {\left (c \right )}}{a + b \tan {\left (c \right )}} & \text {for}\: d = 0 \\\frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a d} & \text {for}\: b = 0 \\- \frac {2 a \log {\left (\frac {a}{b} + \tan {\left (c + d x \right )} \right )}}{2 a^{2} d + 2 b^{2} d} + \frac {a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a^{2} d + 2 b^{2} d} + \frac {2 b d x}{2 a^{2} d + 2 b^{2} d} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.52, size = 73, normalized size = 1.59 \begin {gather*} -\frac {\frac {2 \, a b \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{2} b + b^{3}} - \frac {2 \, {\left (d x + c\right )} b}{a^{2} + b^{2}} - \frac {a \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.03, size = 76, normalized size = 1.65 \begin {gather*} \frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,d\,\left (a-b\,1{}\mathrm {i}\right )}-\frac {a\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}{d\,\left (a^2+b^2\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-b+a\,1{}\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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