Integrand size = 19, antiderivative size = 84 \[ \int (a+b \sec (c+d x)) \tan ^5(c+d x) \, dx=-\frac {a \log (\cos (c+d x))}{d}+\frac {8 b \sec (c+d x)}{15 d}-\frac {(15 a+8 b \sec (c+d x)) \tan ^2(c+d x)}{30 d}+\frac {(5 a+4 b \sec (c+d x)) \tan ^4(c+d x)}{20 d} \]
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Time = 0.11 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3966, 3969, 3556, 2686, 8} \[ \int (a+b \sec (c+d x)) \tan ^5(c+d x) \, dx=\frac {\tan ^4(c+d x) (5 a+4 b \sec (c+d x))}{20 d}-\frac {\tan ^2(c+d x) (15 a+8 b \sec (c+d x))}{30 d}-\frac {a \log (\cos (c+d x))}{d}+\frac {8 b \sec (c+d x)}{15 d} \]
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Rule 8
Rule 2686
Rule 3556
Rule 3966
Rule 3969
Rubi steps \begin{align*} \text {integral}& = \frac {(5 a+4 b \sec (c+d x)) \tan ^4(c+d x)}{20 d}-\frac {1}{5} \int (5 a+4 b \sec (c+d x)) \tan ^3(c+d x) \, dx \\ & = -\frac {(15 a+8 b \sec (c+d x)) \tan ^2(c+d x)}{30 d}+\frac {(5 a+4 b \sec (c+d x)) \tan ^4(c+d x)}{20 d}+\frac {1}{15} \int (15 a+8 b \sec (c+d x)) \tan (c+d x) \, dx \\ & = -\frac {(15 a+8 b \sec (c+d x)) \tan ^2(c+d x)}{30 d}+\frac {(5 a+4 b \sec (c+d x)) \tan ^4(c+d x)}{20 d}+a \int \tan (c+d x) \, dx+\frac {1}{15} (8 b) \int \sec (c+d x) \tan (c+d x) \, dx \\ & = -\frac {a \log (\cos (c+d x))}{d}-\frac {(15 a+8 b \sec (c+d x)) \tan ^2(c+d x)}{30 d}+\frac {(5 a+4 b \sec (c+d x)) \tan ^4(c+d x)}{20 d}+\frac {(8 b) \text {Subst}(\int 1 \, dx,x,\sec (c+d x))}{15 d} \\ & = -\frac {a \log (\cos (c+d x))}{d}+\frac {8 b \sec (c+d x)}{15 d}-\frac {(15 a+8 b \sec (c+d x)) \tan ^2(c+d x)}{30 d}+\frac {(5 a+4 b \sec (c+d x)) \tan ^4(c+d x)}{20 d} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.98 \[ \int (a+b \sec (c+d x)) \tan ^5(c+d x) \, dx=\frac {b \sec (c+d x)}{d}-\frac {2 b \sec ^3(c+d x)}{3 d}+\frac {b \sec ^5(c+d x)}{5 d}-\frac {a \left (4 \log (\cos (c+d x))+2 \tan ^2(c+d x)-\tan ^4(c+d x)\right )}{4 d} \]
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Time = 1.10 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(\frac {\frac {b \sec \left (d x +c \right )^{5}}{5}+\frac {a \sec \left (d x +c \right )^{4}}{4}-\frac {2 b \sec \left (d x +c \right )^{3}}{3}-a \sec \left (d x +c \right )^{2}+b \sec \left (d x +c \right )+a \ln \left (\sec \left (d x +c \right )\right )}{d}\) | \(67\) |
default | \(\frac {\frac {b \sec \left (d x +c \right )^{5}}{5}+\frac {a \sec \left (d x +c \right )^{4}}{4}-\frac {2 b \sec \left (d x +c \right )^{3}}{3}-a \sec \left (d x +c \right )^{2}+b \sec \left (d x +c \right )+a \ln \left (\sec \left (d x +c \right )\right )}{d}\) | \(67\) |
parts | \(\frac {a \left (\frac {\tan \left (d x +c \right )^{4}}{4}-\frac {\tan \left (d x +c \right )^{2}}{2}+\frac {\ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}\right )}{d}+\frac {b \left (\frac {\sec \left (d x +c \right )^{5}}{5}-\frac {2 \sec \left (d x +c \right )^{3}}{3}+\sec \left (d x +c \right )\right )}{d}\) | \(73\) |
risch | \(i a x +\frac {2 i a c}{d}+\frac {2 b \,{\mathrm e}^{9 i \left (d x +c \right )}-4 a \,{\mathrm e}^{8 i \left (d x +c \right )}+\frac {8 b \,{\mathrm e}^{7 i \left (d x +c \right )}}{3}-8 a \,{\mathrm e}^{6 i \left (d x +c \right )}+\frac {116 b \,{\mathrm e}^{5 i \left (d x +c \right )}}{15}-8 a \,{\mathrm e}^{4 i \left (d x +c \right )}+\frac {8 b \,{\mathrm e}^{3 i \left (d x +c \right )}}{3}-4 a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}-\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(160\) |
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Time = 0.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.94 \[ \int (a+b \sec (c+d x)) \tan ^5(c+d x) \, dx=-\frac {60 \, a \cos \left (d x + c\right )^{5} \log \left (-\cos \left (d x + c\right )\right ) - 60 \, b \cos \left (d x + c\right )^{4} + 60 \, a \cos \left (d x + c\right )^{3} + 40 \, b \cos \left (d x + c\right )^{2} - 15 \, a \cos \left (d x + c\right ) - 12 \, b}{60 \, d \cos \left (d x + c\right )^{5}} \]
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Time = 0.40 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.33 \[ \int (a+b \sec (c+d x)) \tan ^5(c+d x) \, dx=\begin {cases} \frac {a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac {a \tan ^{2}{\left (c + d x \right )}}{2 d} + \frac {b \tan ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{5 d} - \frac {4 b \tan ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{15 d} + \frac {8 b \sec {\left (c + d x \right )}}{15 d} & \text {for}\: d \neq 0 \\x \left (a + b \sec {\left (c \right )}\right ) \tan ^{5}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.86 \[ \int (a+b \sec (c+d x)) \tan ^5(c+d x) \, dx=-\frac {60 \, a \log \left (\cos \left (d x + c\right )\right ) - \frac {60 \, b \cos \left (d x + c\right )^{4} - 60 \, a \cos \left (d x + c\right )^{3} - 40 \, b \cos \left (d x + c\right )^{2} + 15 \, a \cos \left (d x + c\right ) + 12 \, b}{\cos \left (d x + c\right )^{5}}}{60 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (78) = 156\).
Time = 1.53 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.95 \[ \int (a+b \sec (c+d x)) \tan ^5(c+d x) \, dx=\frac {60 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 60 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {137 \, a + 64 \, b + \frac {805 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {320 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {1970 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {640 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1970 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {805 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {137 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{5}}}{60 \, d} \]
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Time = 19.03 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.93 \[ \int (a+b \sec (c+d x)) \tan ^5(c+d x) \, dx=\frac {2\,a\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d}-\frac {2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-10\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (10\,a+\frac {32\,b}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (-2\,a-\frac {16\,b}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {16\,b}{15}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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