Integrand size = 19, antiderivative size = 74 \[ \int \sec ^5(c+d x) (a+b \tan (c+d x)) \, dx=\frac {3 a \text {arctanh}(\sin (c+d x))}{8 d}+\frac {b \sec ^5(c+d x)}{5 d}+\frac {3 a \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a \sec ^3(c+d x) \tan (c+d x)}{4 d} \]
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Time = 0.06 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3567, 3853, 3855} \[ \int \sec ^5(c+d x) (a+b \tan (c+d x)) \, dx=\frac {3 a \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {3 a \tan (c+d x) \sec (c+d x)}{8 d}+\frac {b \sec ^5(c+d x)}{5 d} \]
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Rule 3567
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {b \sec ^5(c+d x)}{5 d}+a \int \sec ^5(c+d x) \, dx \\ & = \frac {b \sec ^5(c+d x)}{5 d}+\frac {a \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{4} (3 a) \int \sec ^3(c+d x) \, dx \\ & = \frac {b \sec ^5(c+d x)}{5 d}+\frac {3 a \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{8} (3 a) \int \sec (c+d x) \, dx \\ & = \frac {3 a \text {arctanh}(\sin (c+d x))}{8 d}+\frac {b \sec ^5(c+d x)}{5 d}+\frac {3 a \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a \sec ^3(c+d x) \tan (c+d x)}{4 d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int \sec ^5(c+d x) (a+b \tan (c+d x)) \, dx=\frac {3 a \text {arctanh}(\sin (c+d x))}{8 d}+\frac {b \sec ^5(c+d x)}{5 d}+\frac {3 a \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a \sec ^3(c+d x) \tan (c+d x)}{4 d} \]
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Time = 8.66 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(\frac {a \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {b}{5 \cos \left (d x +c \right )^{5}}}{d}\) | \(63\) |
default | \(\frac {a \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {b}{5 \cos \left (d x +c \right )^{5}}}{d}\) | \(63\) |
risch | \(\frac {-15 i a \,{\mathrm e}^{9 i \left (d x +c \right )}-70 i a \,{\mathrm e}^{7 i \left (d x +c \right )}+128 b \,{\mathrm e}^{5 i \left (d x +c \right )}+70 i a \,{\mathrm e}^{3 i \left (d x +c \right )}+15 i a \,{\mathrm e}^{i \left (d x +c \right )}}{20 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}\) | \(123\) |
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Time = 0.26 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.19 \[ \int \sec ^5(c+d x) (a+b \tan (c+d x)) \, dx=\frac {15 \, a \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, a \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 10 \, {\left (3 \, a \cos \left (d x + c\right )^{3} + 2 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 16 \, b}{80 \, d \cos \left (d x + c\right )^{5}} \]
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\[ \int \sec ^5(c+d x) (a+b \tan (c+d x)) \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right ) \sec ^{5}{\left (c + d x \right )}\, dx \]
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Time = 0.35 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.16 \[ \int \sec ^5(c+d x) (a+b \tan (c+d x)) \, dx=-\frac {5 \, a {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - \frac {16 \, b}{\cos \left (d x + c\right )^{5}}}{80 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (66) = 132\).
Time = 0.41 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.91 \[ \int \sec ^5(c+d x) (a+b \tan (c+d x)) \, dx=\frac {15 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (25 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 40 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 10 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 80 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 10 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 25 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, b\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5}}}{40 \, d} \]
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Time = 7.98 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.36 \[ \int \sec ^5(c+d x) (a+b \tan (c+d x)) \, dx=\frac {3\,a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{4\,d}-\frac {-\frac {5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4}+2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2}+4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}+\frac {5\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {2\,b}{5}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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