Integrand size = 19, antiderivative size = 58 \[ \int \tan (c+d x) (a+b \tan (c+d x))^2 \, dx=-2 a b x-\frac {\left (a^2-b^2\right ) \log (\cos (c+d x))}{d}+\frac {a b \tan (c+d x)}{d}+\frac {(a+b \tan (c+d x))^2}{2 d} \]
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Time = 0.05 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3609, 3606, 3556} \[ \int \tan (c+d x) (a+b \tan (c+d x))^2 \, dx=-\frac {\left (a^2-b^2\right ) \log (\cos (c+d x))}{d}+\frac {(a+b \tan (c+d x))^2}{2 d}+\frac {a b \tan (c+d x)}{d}-2 a b x \]
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Rule 3556
Rule 3606
Rule 3609
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b \tan (c+d x))^2}{2 d}+\int (-b+a \tan (c+d x)) (a+b \tan (c+d x)) \, dx \\ & = -2 a b x+\frac {a b \tan (c+d x)}{d}+\frac {(a+b \tan (c+d x))^2}{2 d}+\left (a^2-b^2\right ) \int \tan (c+d x) \, dx \\ & = -2 a b x-\frac {\left (a^2-b^2\right ) \log (\cos (c+d x))}{d}+\frac {a b \tan (c+d x)}{d}+\frac {(a+b \tan (c+d x))^2}{2 d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 162, normalized size of antiderivative = 2.79 \[ \int \tan (c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {a^2 \log (i-\tan (c+d x))}{2 d}+\frac {i a b \log (i-\tan (c+d x))}{d}-\frac {b^2 \log (i-\tan (c+d x))}{2 d}+\frac {a^2 \log (i+\tan (c+d x))}{2 d}-\frac {i a b \log (i+\tan (c+d x))}{d}-\frac {b^2 \log (i+\tan (c+d x))}{2 d}+\frac {2 a b \tan (c+d x)}{d}+\frac {b^2 \tan ^2(c+d x)}{2 d} \]
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Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.05
method | result | size |
norman | \(-2 a b x +\frac {b^{2} \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {2 a b \tan \left (d x +c \right )}{d}+\frac {\left (a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(61\) |
derivativedivides | \(\frac {\frac {b^{2} \left (\tan ^{2}\left (d x +c \right )\right )}{2}+2 a b \tan \left (d x +c \right )+\frac {\left (a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-2 a b \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(62\) |
default | \(\frac {\frac {b^{2} \left (\tan ^{2}\left (d x +c \right )\right )}{2}+2 a b \tan \left (d x +c \right )+\frac {\left (a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-2 a b \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(62\) |
parallelrisch | \(\frac {-4 a b d x +b^{2} \left (\tan ^{2}\left (d x +c \right )\right )+\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{2}-\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) b^{2}+4 a b \tan \left (d x +c \right )}{2 d}\) | \(66\) |
parts | \(\frac {b^{2} \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}+\frac {a^{2} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {2 a b \left (\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(75\) |
risch | \(-2 a b x +i a^{2} x -i b^{2} x +\frac {2 i a^{2} c}{d}-\frac {2 i b^{2} c}{d}+\frac {2 i b \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 a \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) a^{2}}{d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) b^{2}}{d}\) | \(129\) |
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Time = 0.24 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00 \[ \int \tan (c+d x) (a+b \tan (c+d x))^2 \, dx=-\frac {4 \, a b d x - b^{2} \tan \left (d x + c\right )^{2} - 4 \, a b \tan \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, d} \]
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Time = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.47 \[ \int \tan (c+d x) (a+b \tan (c+d x))^2 \, dx=\begin {cases} \frac {a^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - 2 a b x + \frac {2 a b \tan {\left (c + d x \right )}}{d} - \frac {b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {b^{2} \tan ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\left (c \right )}\right )^{2} \tan {\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.32 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00 \[ \int \tan (c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {b^{2} \tan \left (d x + c\right )^{2} - 4 \, {\left (d x + c\right )} a b + 4 \, a b \tan \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (56) = 112\).
Time = 0.62 (sec) , antiderivative size = 494, normalized size of antiderivative = 8.52 \[ \int \tan (c+d x) (a+b \tan (c+d x))^2 \, dx=-\frac {4 \, a b d x \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + a^{2} \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} + \tan \left (c\right )^{2} + 1}\right ) \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - b^{2} \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} + \tan \left (c\right )^{2} + 1}\right ) \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 8 \, a b d x \tan \left (d x\right ) \tan \left (c\right ) - b^{2} \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, a^{2} \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} + \tan \left (c\right )^{2} + 1}\right ) \tan \left (d x\right ) \tan \left (c\right ) + 2 \, b^{2} \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} + \tan \left (c\right )^{2} + 1}\right ) \tan \left (d x\right ) \tan \left (c\right ) + 4 \, a b \tan \left (d x\right )^{2} \tan \left (c\right ) + 4 \, a b \tan \left (d x\right ) \tan \left (c\right )^{2} + 4 \, a b d x - b^{2} \tan \left (d x\right )^{2} - b^{2} \tan \left (c\right )^{2} + a^{2} \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} + \tan \left (c\right )^{2} + 1}\right ) - b^{2} \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} + \tan \left (c\right )^{2} + 1}\right ) - 4 \, a b \tan \left (d x\right ) - 4 \, a b \tan \left (c\right ) - b^{2}}{2 \, {\left (d \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, d \tan \left (d x\right ) \tan \left (c\right ) + d\right )}} \]
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Time = 5.48 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.98 \[ \int \tan (c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )\,\left (\frac {a^2}{2}-\frac {b^2}{2}\right )+\frac {b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}+2\,a\,b\,\mathrm {tan}\left (c+d\,x\right )-2\,a\,b\,d\,x}{d} \]
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