Optimal. Leaf size=82 \[ \frac {2 a b x}{\left (a^2+b^2\right )^2}-\frac {\left (a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {a}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))} \]
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Rubi [A]
time = 0.07, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3610, 3612,
3611} \begin {gather*} \frac {a}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {\left (a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^2}+\frac {2 a b x}{\left (a^2+b^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3610
Rule 3611
Rule 3612
Rubi steps
\begin {align*} \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx &=\frac {a}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\int \frac {b+a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^2+b^2}\\ &=\frac {2 a b x}{\left (a^2+b^2\right )^2}+\frac {a}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {\left (a^2-b^2\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{\left (a^2+b^2\right )^2}\\ &=\frac {2 a b x}{\left (a^2+b^2\right )^2}-\frac {\left (a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {a}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.49, size = 181, normalized size = 2.21 \begin {gather*} \frac {a \left ((a-i b)^2 \log (i-\tan (c+d x))+(a+i b)^2 \log (i+\tan (c+d x))+2 \left (a^2+b^2+\left (-a^2+b^2\right ) \log (a+b \tan (c+d x))\right )\right )+b \left ((a-i b)^2 \log (i-\tan (c+d x))+(a+i b)^2 \log (i+\tan (c+d x))+2 \left (-a^2+b^2\right ) \log (a+b \tan (c+d x))\right ) \tan (c+d x)}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 104, normalized size = 1.27
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+2 a b \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {a}{\left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}-\frac {\left (a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\) | \(104\) |
default | \(\frac {\frac {\frac {\left (a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+2 a b \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {a}{\left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}-\frac {\left (a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\) | \(104\) |
norman | \(\frac {-\frac {b \tan \left (d x +c \right )}{d \left (a^{2}+b^{2}\right )}+\frac {2 a^{2} b x}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {2 b^{2} a x \tan \left (d x +c \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}}{a +b \tan \left (d x +c \right )}+\frac {\left (a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\) | \(175\) |
risch | \(\frac {i x}{2 i a b -a^{2}+b^{2}}+\frac {2 i a^{2} x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {2 i b^{2} x}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {2 i a^{2} c}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {2 i b^{2} c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 i a b}{\left (i b +a \right ) d \left (-i b +a \right )^{2} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right )}-\frac {a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) b^{2}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\) | \(287\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.96, size = 139, normalized size = 1.70 \begin {gather*} \frac {\frac {4 \, {\left (d x + c\right )} a b}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left (a^{2} - b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, a}{a^{3} + a b^{2} + {\left (a^{2} b + b^{3}\right )} \tan \left (d x + c\right )}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.85, size = 157, normalized size = 1.91 \begin {gather*} \frac {4 \, a^{2} b d x + 2 \, a b^{2} - {\left (a^{3} - a b^{2} + {\left (a^{2} b - b^{3}\right )} \tan \left (d x + c\right )\right )} \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 (2 \, a b^{2} d x - a^{2} b\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d \tan \left (d x + c\right ) + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d\right )}} \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 = 0.76, size = 1476, normalized size = 18.00 \begin {gather*} \begin {cases} \frac {\tilde {\infty } x}{\tan {\left (c \right )}} & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a^{2} d} & \text {for}\: b = 0 \\\frac {i d x \tan ^{2}{\left (c + d x \right )}}{4 b^{2} d \tan ^{2}{\left (c + d x \right )} - 8 i b^{2} d \tan {\left (c + d x \right )} - 4 b^{2} d} + \frac {2 d x \tan {\left (c + d x \right )}}{4 b^{2} d \tan ^{2}{\left (c + d x \right )} - 8 i b^{2} d \tan {\left (c + d x \right )} - 4 b^{2} d} - \frac {i d x}{4 b^{2} d \tan ^{2}{\left (c + d x \right )} - 8 i b^{2} d \tan {\left (c + d x \right )} - 4 b^{2} d} + \frac {i \tan {\left (c + d x \right )}}{4 b^{2} d \tan ^{2}{\left (c + d x \right )} - 8 i b^{2} d \tan {\left (c + d x \right )} - 4 b^{2} d} & \text {for}\: a = - i b \\- \frac {i d x \tan ^{2}{\left (c + d x \right )}}{4 b^{2} d \tan ^{2}{\left (c + d x \right )} + 8 i b^{2} d \tan {\left (c + d x \right )} - 4 b^{2} d} + \frac {2 d x \tan {\left (c + d x \right )}}{4 b^{2} d \tan ^{2}{\left (c + d x \right )} + 8 i b^{2} d \tan {\left (c + d x \right )} - 4 b^{2} d} + \frac {i d x}{4 b^{2} d \tan ^{2}{\left (c + d x \right )} + 8 i b^{2} d \tan {\left (c + d x \right )} - 4 b^{2} d} - \frac {i \tan {\left (c + d x \right )}}{4 b^{2} d \tan ^{2}{\left (c + d x \right )} + 8 i b^{2} d \tan {\left (c + d x \right )} - 4 b^{2} d} & \text {for}\: a = i b \\\frac {x \tan {\left (c \right )}}{\left (a + b \tan {\left (c \right )}\right )^{2}} & \text {for}\: d = 0 \\- \frac {2 a^{3} \log {\left (\frac {a}{b} + \tan {\left (c + d x \right )} \right )}}{2 a^{5} d + 2 a^{4} b d \tan {\left (c + d x \right )} + 4 a^{3} b^{2} d + 4 a^{2} b^{3} d \tan {\left (c + d x \right )} + 2 a b^{4} d + 2 b^{5} d \tan {\left (c + d x \right )}} + \frac {a^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a^{5} d + 2 a^{4} b d \tan {\left (c + d x \right )} + 4 a^{3} b^{2} d + 4 a^{2} b^{3} d \tan {\left (c + d x \right )} + 2 a b^{4} d + 2 b^{5} d \tan {\left (c + d x \right )}} + \frac {2 a^{3}}{2 a^{5} d + 2 a^{4} b d \tan {\left (c + d x \right )} + 4 a^{3} b^{2} d + 4 a^{2} b^{3} d \tan {\left (c + d x \right )} + 2 a b^{4} d + 2 b^{5} d \tan {\left (c + d x \right )}} + \frac {4 a^{2} b d x}{2 a^{5} d + 2 a^{4} b d \tan {\left (c + d x \right )} + 4 a^{3} b^{2} d + 4 a^{2} b^{3} d \tan {\left (c + d x \right )} + 2 a b^{4} d + 2 b^{5} d \tan {\left (c + d x \right )}} - \frac {2 a^{2} b \log {\left (\frac {a}{b} + \tan {\left (c + d x \right )} \right )} \tan {\left (c + d x \right )}}{2 a^{5} d + 2 a^{4} b d \tan {\left (c + d x \right )} + 4 a^{3} b^{2} d + 4 a^{2} b^{3} d \tan {\left (c + d x \right )} + 2 a b^{4} d + 2 b^{5} d \tan {\left (c + d x \right )}} + \frac {a^{2} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan {\left (c + d x \right )}}{2 a^{5} d + 2 a^{4} b d \tan {\left (c + d x \right )} + 4 a^{3} b^{2} d + 4 a^{2} b^{3} d \tan {\left (c + d x \right )} + 2 a b^{4} d + 2 b^{5} d \tan {\left (c + d x \right )}} + \frac {4 a b^{2} d x \tan {\left (c + d x \right )}}{2 a^{5} d + 2 a^{4} b d \tan {\left (c + d x \right )} + 4 a^{3} b^{2} d + 4 a^{2} b^{3} d \tan {\left (c + d x \right )} + 2 a b^{4} d + 2 b^{5} d \tan {\left (c + d x \right )}} + \frac {2 a b^{2} \log {\left (\frac {a}{b} + \tan {\left (c + d x \right )} \right )}}{2 a^{5} d + 2 a^{4} b d \tan {\left (c + d x \right )} + 4 a^{3} b^{2} d + 4 a^{2} b^{3} d \tan {\left (c + d x \right )} + 2 a b^{4} d + 2 b^{5} d \tan {\left (c + d x \right )}} - \frac {a b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a^{5} d + 2 a^{4} b d \tan {\left (c + d x \right )} + 4 a^{3} b^{2} d + 4 a^{2} b^{3} d \tan {\left (c + d x \right )} + 2 a b^{4} d + 2 b^{5} d \tan {\left (c + d x \right )}} + \frac {2 a b^{2}}{2 a^{5} d + 2 a^{4} b d \tan {\left (c + d x \right )} + 4 a^{3} b^{2} d + 4 a^{2} b^{3} d \tan {\left (c + d x \right )} + 2 a b^{4} d + 2 b^{5} d \tan {\left (c + d x \right )}} + \frac {2 b^{3} \log {\left (\frac {a}{b} + \tan {\left (c + d x \right )} \right )} \tan {\left (c + d x \right )}}{2 a^{5} d + 2 a^{4} b d \tan {\left (c + d x \right )} + 4 a^{3} b^{2} d + 4 a^{2} b^{3} d \tan {\left (c + d x \right )} + 2 a b^{4} d + 2 b^{5} d \tan {\left (c + d x \right )}} - \frac {b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan {\left (c + d x \right )}}{2 a^{5} d + 2 a^{4} b d \tan {\left (c + d x \right )} + 4 a^{3} b^{2} d + 4 a^{2} b^{3} d \tan {\left (c + d x \right )} + 2 a b^{4} d + 2 b^{5} d \tan {\left (c + d x \right )}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 173 vs.
\(2 (82) = 164\).
time = 0.59, size = 173, normalized size = 2.11 \begin {gather*} \frac {\frac {4 \, {\left (d x + c\right )} a b}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left (a^{2} b - b^{3}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{4} b + 2 \, a^{2} b^{3} + b^{5}} + \frac {2 \, {\left (a^{2} b \tan \left (d x + c\right ) - b^{3} \tan \left (d x + c\right ) + 2 \, a^{3}\right )}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.17, size = 133, normalized size = 1.62 \begin {gather*} \frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (a^2+a\,b\,2{}\mathrm {i}-b^2\right )}-\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {1}{a^2+b^2}-\frac {2\,b^2}{{\left (a^2+b^2\right )}^2}\right )}{d}+\frac {a}{d\,\left (a^2+b^2\right )\,\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a^2\,1{}\mathrm {i}+2\,a\,b-b^2\,1{}\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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