Integrand size = 19, antiderivative size = 44 \[ \int \sec ^4(c+d x) (a+b \tan (c+d x)) \, dx=\frac {b \sec ^4(c+d x)}{4 d}+\frac {a \tan (c+d x)}{d}+\frac {a \tan ^3(c+d x)}{3 d} \]
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Time = 0.04 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3567, 3852} \[ \int \sec ^4(c+d x) (a+b \tan (c+d x)) \, dx=\frac {a \tan ^3(c+d x)}{3 d}+\frac {a \tan (c+d x)}{d}+\frac {b \sec ^4(c+d x)}{4 d} \]
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Rule 3567
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {b \sec ^4(c+d x)}{4 d}+a \int \sec ^4(c+d x) \, dx \\ & = \frac {b \sec ^4(c+d x)}{4 d}-\frac {a \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d} \\ & = \frac {b \sec ^4(c+d x)}{4 d}+\frac {a \tan (c+d x)}{d}+\frac {a \tan ^3(c+d x)}{3 d} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.93 \[ \int \sec ^4(c+d x) (a+b \tan (c+d x)) \, dx=\frac {b \sec ^4(c+d x)}{4 d}+\frac {a \left (\tan (c+d x)+\frac {1}{3} \tan ^3(c+d x)\right )}{d} \]
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Time = 3.54 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.07
method | result | size |
derivativedivides | \(\frac {\frac {\left (\tan ^{4}\left (d x +c \right )\right ) b}{4}+\frac {a \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\frac {b \left (\tan ^{2}\left (d x +c \right )\right )}{2}+a \tan \left (d x +c \right )}{d}\) | \(47\) |
default | \(\frac {\frac {\left (\tan ^{4}\left (d x +c \right )\right ) b}{4}+\frac {a \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\frac {b \left (\tan ^{2}\left (d x +c \right )\right )}{2}+a \tan \left (d x +c \right )}{d}\) | \(47\) |
risch | \(\frac {4 i a \,{\mathrm e}^{4 i \left (d x +c \right )}+4 b \,{\mathrm e}^{4 i \left (d x +c \right )}+\frac {16 i a \,{\mathrm e}^{2 i \left (d x +c \right )}}{3}+\frac {4 i a}{3}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}\) | \(62\) |
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Time = 0.24 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.02 \[ \int \sec ^4(c+d x) (a+b \tan (c+d x)) \, dx=\frac {4 \, {\left (2 \, a \cos \left (d x + c\right )^{3} + a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 3 \, b}{12 \, d \cos \left (d x + c\right )^{4}} \]
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Time = 1.04 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \sec ^4(c+d x) (a+b \tan (c+d x)) \, dx=\begin {cases} \frac {a \left (\frac {\tan ^{3}{\left (c + d x \right )}}{3} + \tan {\left (c + d x \right )}\right ) + \frac {b \sec ^{4}{\left (c + d x \right )}}{4}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\left (c \right )}\right ) \sec ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.39 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.09 \[ \int \sec ^4(c+d x) (a+b \tan (c+d x)) \, dx=\frac {3 \, b \tan \left (d x + c\right )^{4} + 4 \, a \tan \left (d x + c\right )^{3} + 6 \, b \tan \left (d x + c\right )^{2} + 12 \, a \tan \left (d x + c\right )}{12 \, d} \]
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Time = 0.39 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.09 \[ \int \sec ^4(c+d x) (a+b \tan (c+d x)) \, dx=\frac {3 \, b \tan \left (d x + c\right )^{4} + 4 \, a \tan \left (d x + c\right )^{3} + 6 \, b \tan \left (d x + c\right )^{2} + 12 \, a \tan \left (d x + c\right )}{12 \, d} \]
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Time = 4.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.05 \[ \int \sec ^4(c+d x) (a+b \tan (c+d x)) \, dx=\frac {\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^4}{4}+\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3}+\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}+a\,\mathrm {tan}\left (c+d\,x\right )}{d} \]
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