Integrand size = 21, antiderivative size = 105 \[ \int \frac {\tan ^8(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {x}{a}-\frac {5 \text {arctanh}(\sin (c+d x))}{16 a d}-\frac {(16-5 \sec (c+d x)) \tan (c+d x)}{16 a d}+\frac {(8-5 \sec (c+d x)) \tan ^3(c+d x)}{24 a d}-\frac {(6-5 \sec (c+d x)) \tan ^5(c+d x)}{30 a d} \]
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Time = 0.17 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3973, 3966, 3855} \[ \int \frac {\tan ^8(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {5 \text {arctanh}(\sin (c+d x))}{16 a d}-\frac {\tan ^5(c+d x) (6-5 \sec (c+d x))}{30 a d}+\frac {\tan ^3(c+d x) (8-5 \sec (c+d x))}{24 a d}-\frac {\tan (c+d x) (16-5 \sec (c+d x))}{16 a d}+\frac {x}{a} \]
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Rule 3855
Rule 3966
Rule 3973
Rubi steps \begin{align*} \text {integral}& = \frac {\int (-a+a \sec (c+d x)) \tan ^6(c+d x) \, dx}{a^2} \\ & = -\frac {(6-5 \sec (c+d x)) \tan ^5(c+d x)}{30 a d}-\frac {\int (-6 a+5 a \sec (c+d x)) \tan ^4(c+d x) \, dx}{6 a^2} \\ & = \frac {(8-5 \sec (c+d x)) \tan ^3(c+d x)}{24 a d}-\frac {(6-5 \sec (c+d x)) \tan ^5(c+d x)}{30 a d}+\frac {\int (-24 a+15 a \sec (c+d x)) \tan ^2(c+d x) \, dx}{24 a^2} \\ & = -\frac {(16-5 \sec (c+d x)) \tan (c+d x)}{16 a d}+\frac {(8-5 \sec (c+d x)) \tan ^3(c+d x)}{24 a d}-\frac {(6-5 \sec (c+d x)) \tan ^5(c+d x)}{30 a d}-\frac {\int (-48 a+15 a \sec (c+d x)) \, dx}{48 a^2} \\ & = \frac {x}{a}-\frac {(16-5 \sec (c+d x)) \tan (c+d x)}{16 a d}+\frac {(8-5 \sec (c+d x)) \tan ^3(c+d x)}{24 a d}-\frac {(6-5 \sec (c+d x)) \tan ^5(c+d x)}{30 a d}-\frac {5 \int \sec (c+d x) \, dx}{16 a} \\ & = \frac {x}{a}-\frac {5 \text {arctanh}(\sin (c+d x))}{16 a d}-\frac {(16-5 \sec (c+d x)) \tan (c+d x)}{16 a d}+\frac {(8-5 \sec (c+d x)) \tan ^3(c+d x)}{24 a d}-\frac {(6-5 \sec (c+d x)) \tan ^5(c+d x)}{30 a d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(301\) vs. \(2(105)=210\).
Time = 2.33 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.87 \[ \int \frac {\tan ^8(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \left (2400 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\sec (c) \sec ^6(c+d x) (2400 d x \cos (c)+1800 d x \cos (c+2 d x)+1800 d x \cos (3 c+2 d x)+720 d x \cos (3 c+4 d x)+720 d x \cos (5 c+4 d x)+120 d x \cos (5 c+6 d x)+120 d x \cos (7 c+6 d x)+3680 \sin (c)+450 \sin (d x)+450 \sin (2 c+d x)-3360 \sin (c+2 d x)+2160 \sin (3 c+2 d x)-25 \sin (2 c+3 d x)-25 \sin (4 c+3 d x)-1488 \sin (3 c+4 d x)+720 \sin (5 c+4 d x)+165 \sin (4 c+5 d x)+165 \sin (6 c+5 d x)-368 \sin (5 c+6 d x))\right )}{3840 a d (1+\sec (c+d x))} \]
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Result contains complex when optimal does not.
Time = 1.19 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.85
method | result | size |
risch | \(\frac {x}{a}-\frac {i \left (165 \,{\mathrm e}^{11 i \left (d x +c \right )}+720 \,{\mathrm e}^{10 i \left (d x +c \right )}-25 \,{\mathrm e}^{9 i \left (d x +c \right )}+2160 \,{\mathrm e}^{8 i \left (d x +c \right )}+450 \,{\mathrm e}^{7 i \left (d x +c \right )}+3680 \,{\mathrm e}^{6 i \left (d x +c \right )}-450 \,{\mathrm e}^{5 i \left (d x +c \right )}+3360 \,{\mathrm e}^{4 i \left (d x +c \right )}+25 \,{\mathrm e}^{3 i \left (d x +c \right )}+1488 \,{\mathrm e}^{2 i \left (d x +c \right )}-165 \,{\mathrm e}^{i \left (d x +c \right )}+368\right )}{120 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{16 a d}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{16 a d}\) | \(194\) |
derivativedivides | \(\frac {\frac {1}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{6}}+\frac {7}{10 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}+\frac {3}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {5}{12 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {9}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {21}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16}-\frac {1}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {7}{10 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {3}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {5}{12 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {9}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {21}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16}+2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}\) | \(230\) |
default | \(\frac {\frac {1}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{6}}+\frac {7}{10 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}+\frac {3}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {5}{12 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {9}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {21}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16}-\frac {1}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {7}{10 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {3}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {5}{12 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {9}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {21}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16}+2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}\) | \(230\) |
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Time = 0.29 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.21 \[ \int \frac {\tan ^8(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {480 \, d x \cos \left (d x + c\right )^{6} - 75 \, \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) + 75 \, \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (368 \, \cos \left (d x + c\right )^{5} - 165 \, \cos \left (d x + c\right )^{4} - 176 \, \cos \left (d x + c\right )^{3} + 130 \, \cos \left (d x + c\right )^{2} + 48 \, \cos \left (d x + c\right ) - 40\right )} \sin \left (d x + c\right )}{480 \, a d \cos \left (d x + c\right )^{6}} \]
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\[ \int \frac {\tan ^8(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\int \frac {\tan ^{8}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 329 vs. \(2 (97) = 194\).
Time = 0.31 (sec) , antiderivative size = 329, normalized size of antiderivative = 3.13 \[ \int \frac {\tan ^8(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {2 \, {\left (\frac {165 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1095 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3138 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5118 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {1945 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {315 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}\right )}}{a - \frac {6 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {20 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {6 \, a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {a \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} - \frac {480 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {75 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} - \frac {75 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a}}{240 \, d} \]
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Time = 4.80 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.42 \[ \int \frac {\tan ^8(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\frac {240 \, {\left (d x + c\right )}}{a} - \frac {75 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a} + \frac {75 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a} + \frac {2 \, {\left (315 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 1945 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 5118 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 3138 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1095 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 165 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{6} a}}{240 \, d} \]
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Time = 15.58 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.84 \[ \int \frac {\tan ^8(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {x}{a}-\frac {5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,a\,d}-\frac {-\frac {21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+\frac {389\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}-\frac {853\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{20}+\frac {523\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{20}-\frac {73\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8}+\frac {11\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )} \]
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