Integrand size = 19, antiderivative size = 55 \[ \int (a+b \sec (c+d x)) \tan ^3(c+d x) \, dx=\frac {a \log (\cos (c+d x))}{d}-\frac {2 b \sec (c+d x)}{3 d}+\frac {(3 a+2 b \sec (c+d x)) \tan ^2(c+d x)}{6 d} \]
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Time = 0.06 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3966, 3969, 3556, 2686, 8} \[ \int (a+b \sec (c+d x)) \tan ^3(c+d x) \, dx=\frac {\tan ^2(c+d x) (3 a+2 b \sec (c+d x))}{6 d}+\frac {a \log (\cos (c+d x))}{d}-\frac {2 b \sec (c+d x)}{3 d} \]
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Rule 8
Rule 2686
Rule 3556
Rule 3966
Rule 3969
Rubi steps \begin{align*} \text {integral}& = \frac {(3 a+2 b \sec (c+d x)) \tan ^2(c+d x)}{6 d}-\frac {1}{3} \int (3 a+2 b \sec (c+d x)) \tan (c+d x) \, dx \\ & = \frac {(3 a+2 b \sec (c+d x)) \tan ^2(c+d x)}{6 d}-a \int \tan (c+d x) \, dx-\frac {1}{3} (2 b) \int \sec (c+d x) \tan (c+d x) \, dx \\ & = \frac {a \log (\cos (c+d x))}{d}+\frac {(3 a+2 b \sec (c+d x)) \tan ^2(c+d x)}{6 d}-\frac {(2 b) \text {Subst}(\int 1 \, dx,x,\sec (c+d x))}{3 d} \\ & = \frac {a \log (\cos (c+d x))}{d}-\frac {2 b \sec (c+d x)}{3 d}+\frac {(3 a+2 b \sec (c+d x)) \tan ^2(c+d x)}{6 d} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int (a+b \sec (c+d x)) \tan ^3(c+d x) \, dx=-\frac {b \sec (c+d x)}{d}+\frac {b \sec ^3(c+d x)}{3 d}+\frac {a \left (2 \log (\cos (c+d x))+\tan ^2(c+d x)\right )}{2 d} \]
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Time = 0.84 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(\frac {\frac {b \sec \left (d x +c \right )^{3}}{3}+\frac {a \sec \left (d x +c \right )^{2}}{2}-b \sec \left (d x +c \right )-a \ln \left (\sec \left (d x +c \right )\right )}{d}\) | \(47\) |
default | \(\frac {\frac {b \sec \left (d x +c \right )^{3}}{3}+\frac {a \sec \left (d x +c \right )^{2}}{2}-b \sec \left (d x +c \right )-a \ln \left (\sec \left (d x +c \right )\right )}{d}\) | \(47\) |
parts | \(\frac {a \left (\frac {\tan \left (d x +c \right )^{2}}{2}-\frac {\ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}\right )}{d}+\frac {b \left (\frac {\sec \left (d x +c \right )^{3}}{3}-\sec \left (d x +c \right )\right )}{d}\) | \(55\) |
risch | \(-i a x -\frac {2 i a c}{d}-\frac {2 \left (3 b \,{\mathrm e}^{5 i \left (d x +c \right )}-3 a \,{\mathrm e}^{4 i \left (d x +c \right )}+2 b \,{\mathrm e}^{3 i \left (d x +c \right )}-3 a \,{\mathrm e}^{2 i \left (d x +c \right )}+3 b \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(111\) |
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Time = 0.29 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.04 \[ \int (a+b \sec (c+d x)) \tan ^3(c+d x) \, dx=\frac {6 \, a \cos \left (d x + c\right )^{3} \log \left (-\cos \left (d x + c\right )\right ) - 6 \, b \cos \left (d x + c\right )^{2} + 3 \, a \cos \left (d x + c\right ) + 2 \, b}{6 \, d \cos \left (d x + c\right )^{3}} \]
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Time = 0.19 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.38 \[ \int (a+b \sec (c+d x)) \tan ^3(c+d x) \, dx=\begin {cases} - \frac {a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a \tan ^{2}{\left (c + d x \right )}}{2 d} + \frac {b \tan ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{3 d} - \frac {2 b \sec {\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a + b \sec {\left (c \right )}\right ) \tan ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.91 \[ \int (a+b \sec (c+d x)) \tan ^3(c+d x) \, dx=\frac {6 \, a \log \left (\cos \left (d x + c\right )\right ) - \frac {6 \, b \cos \left (d x + c\right )^{2} - 3 \, a \cos \left (d x + c\right ) - 2 \, b}{\cos \left (d x + c\right )^{3}}}{6 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (51) = 102\).
Time = 0.62 (sec) , antiderivative size = 179, normalized size of antiderivative = 3.25 \[ \int (a+b \sec (c+d x)) \tan ^3(c+d x) \, dx=-\frac {6 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 6 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {11 \, a + 8 \, b + \frac {45 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {24 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {45 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {11 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{3}}}{6 \, d} \]
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Time = 15.54 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.85 \[ \int (a+b \sec (c+d x)) \tan ^3(c+d x) \, dx=\frac {2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (-2\,a-4\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {4\,b}{3}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}-\frac {2\,a\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d} \]
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