Integrand size = 19, antiderivative size = 28 \[ \int \sec ^2(c+d x) (a+b \tan (c+d x)) \, dx=\frac {b \sec ^2(c+d x)}{2 d}+\frac {a \tan (c+d x)}{d} \]
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Time = 0.04 (sec) , antiderivative size = 28, 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 = {3567, 3852, 8} \[ \int \sec ^2(c+d x) (a+b \tan (c+d x)) \, dx=\frac {a \tan (c+d x)}{d}+\frac {b \sec ^2(c+d x)}{2 d} \]
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Rule 8
Rule 3567
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {b \sec ^2(c+d x)}{2 d}+a \int \sec ^2(c+d x) \, dx \\ & = \frac {b \sec ^2(c+d x)}{2 d}-\frac {a \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d} \\ & = \frac {b \sec ^2(c+d x)}{2 d}+\frac {a \tan (c+d x)}{d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \sec ^2(c+d x) (a+b \tan (c+d x)) \, dx=\frac {b \sec ^2(c+d x)}{2 d}+\frac {a \tan (c+d x)}{d} \]
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Time = 0.86 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {\frac {b \left (\tan ^{2}\left (d x +c \right )\right )}{2}+a \tan \left (d x +c \right )}{d}\) | \(25\) |
default | \(\frac {\frac {b \left (\tan ^{2}\left (d x +c \right )\right )}{2}+a \tan \left (d x +c \right )}{d}\) | \(25\) |
risch | \(\frac {2 i a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 i a}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}\) | \(48\) |
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Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \sec ^2(c+d x) (a+b \tan (c+d x)) \, dx=\frac {2 \, a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b}{2 \, d \cos \left (d x + c\right )^{2}} \]
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Time = 0.67 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \sec ^2(c+d x) (a+b \tan (c+d x)) \, dx=\begin {cases} \frac {a \tan {\left (c + d x \right )} + \frac {b \tan ^{2}{\left (c + d x \right )}}{2}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\left (c \right )}\right ) \sec ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \sec ^2(c+d x) (a+b \tan (c+d x)) \, dx=\frac {{\left (b \tan \left (d x + c\right ) + a\right )}^{2}}{2 \, b d} \]
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Time = 0.35 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \sec ^2(c+d x) (a+b \tan (c+d x)) \, dx=\frac {b \tan \left (d x + c\right )^{2} + 2 \, a \tan \left (d x + c\right )}{2 \, d} \]
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Time = 4.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \sec ^2(c+d x) (a+b \tan (c+d x)) \, dx=\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (2\,a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}{2\,d} \]
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