Integrand size = 26, antiderivative size = 33 \[ \int \sec ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx=\frac {a \sec ^2(c+d x)}{2 d}+\frac {b \sec ^3(c+d x)}{3 d} \]
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Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4462, 12, 2686, 30} \[ \int \sec ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx=\frac {a \sec ^2(c+d x)}{2 d}+\frac {b \sec ^3(c+d x)}{3 d} \]
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Rule 12
Rule 30
Rule 2686
Rule 4462
Rubi steps \begin{align*} \text {integral}& = a \int \sec ^2(c+d x) \tan (c+d x) \, dx+\int b \sec ^3(c+d x) \tan (c+d x) \, dx \\ & = b \int \sec ^3(c+d x) \tan (c+d x) \, dx+\frac {a \text {Subst}(\int x \, dx,x,\sec (c+d x))}{d} \\ & = \frac {a \sec ^2(c+d x)}{2 d}+\frac {b \text {Subst}\left (\int x^2 \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {a \sec ^2(c+d x)}{2 d}+\frac {b \sec ^3(c+d x)}{3 d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \sec ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx=\frac {a \sec ^2(c+d x)}{2 d}+\frac {b \sec ^3(c+d x)}{3 d} \]
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Time = 0.86 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(\frac {\frac {b \sec \left (d x +c \right )^{3}}{3}+\frac {a \sec \left (d x +c \right )^{2}}{2}}{d}\) | \(28\) |
default | \(\frac {\frac {b \sec \left (d x +c \right )^{3}}{3}+\frac {a \sec \left (d x +c \right )^{2}}{2}}{d}\) | \(28\) |
risch | \(\frac {2 a \,{\mathrm e}^{4 i \left (d x +c \right )}+\frac {8 b \,{\mathrm e}^{3 i \left (d x +c \right )}}{3}+2 \,{\mathrm e}^{2 i \left (d x +c \right )} a}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}\) | \(56\) |
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Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \sec ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx=\frac {3 \, a \cos \left (d x + c\right ) + 2 \, b}{6 \, d \cos \left (d x + c\right )^{3}} \]
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\[ \int \sec ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx=\int \left (a \sin {\left (c + d x \right )} + b \tan {\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}\, dx \]
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none
Time = 0.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \sec ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx=-\frac {\frac {3 \, a}{\sin \left (d x + c\right )^{2} - 1} - \frac {2 \, b}{\cos \left (d x + c\right )^{3}}}{6 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (29) = 58\).
Time = 0.37 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.94 \[ \int \sec ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx=\frac {2 \, {\left (b - \frac {3 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {3 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{3 \, d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{3}} \]
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Time = 22.44 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \sec ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx=\frac {a}{2\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {b}{3\,d\,{\cos \left (c+d\,x\right )}^3} \]
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