Integrand size = 23, antiderivative size = 120 \[ \int \frac {\tan ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {a^{7/2} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{(a+b)^{9/2} d}-\frac {a^3 \tan (c+d x)}{(a+b)^4 d}+\frac {a^2 \tan ^3(c+d x)}{3 (a+b)^3 d}-\frac {a \tan ^5(c+d x)}{5 (a+b)^2 d}+\frac {\tan ^7(c+d x)}{7 (a+b) d} \]
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Time = 0.21 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3274, 308, 211} \[ \int \frac {\tan ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {a^{7/2} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{d (a+b)^{9/2}}-\frac {a^3 \tan (c+d x)}{d (a+b)^4}+\frac {a^2 \tan ^3(c+d x)}{3 d (a+b)^3}+\frac {\tan ^7(c+d x)}{7 d (a+b)}-\frac {a \tan ^5(c+d x)}{5 d (a+b)^2} \]
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Rule 211
Rule 308
Rule 3274
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^8}{a+(a+b) x^2} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {a^3}{(a+b)^4}+\frac {a^2 x^2}{(a+b)^3}-\frac {a x^4}{(a+b)^2}+\frac {x^6}{a+b}+\frac {a^4}{(a+b)^4 \left (a+(a+b) x^2\right )}\right ) \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {a^3 \tan (c+d x)}{(a+b)^4 d}+\frac {a^2 \tan ^3(c+d x)}{3 (a+b)^3 d}-\frac {a \tan ^5(c+d x)}{5 (a+b)^2 d}+\frac {\tan ^7(c+d x)}{7 (a+b) d}+\frac {a^4 \text {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\tan (c+d x)\right )}{(a+b)^4 d} \\ & = \frac {a^{7/2} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{(a+b)^{9/2} d}-\frac {a^3 \tan (c+d x)}{(a+b)^4 d}+\frac {a^2 \tan ^3(c+d x)}{3 (a+b)^3 d}-\frac {a \tan ^5(c+d x)}{5 (a+b)^2 d}+\frac {\tan ^7(c+d x)}{7 (a+b) d} \\ \end{align*}
Time = 4.55 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.22 \[ \int \frac {\tan ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {a^{7/2} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{(a+b)^{9/2} d}+\frac {\left (-176 a^3-122 a^2 b-66 a b^2-15 b^3+\left (122 a^3+254 a^2 b+177 a b^2+45 b^3\right ) \sec ^2(c+d x)-3 (a+b)^2 (22 a+15 b) \sec ^4(c+d x)+15 (a+b)^3 \sec ^6(c+d x)\right ) \tan (c+d x)}{105 (a+b)^4 d} \]
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Time = 6.45 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (a +b \right ) \left (a^{2}+2 a b +b^{2}\right ) \left (\tan ^{7}\left (d x +c \right )\right )}{7}-\frac {a \left (a^{2}+2 a b +b^{2}\right ) \left (\tan ^{5}\left (d x +c \right )\right )}{5}+\frac {a^{2} \left (\tan ^{3}\left (d x +c \right )\right ) \left (a +b \right )}{3}-a^{3} \tan \left (d x +c \right )}{\left (a +b \right )^{4}}+\frac {a^{4} \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{\left (a +b \right )^{4} \sqrt {a \left (a +b \right )}}}{d}\) | \(120\) |
default | \(\frac {\frac {\frac {\left (a +b \right ) \left (a^{2}+2 a b +b^{2}\right ) \left (\tan ^{7}\left (d x +c \right )\right )}{7}-\frac {a \left (a^{2}+2 a b +b^{2}\right ) \left (\tan ^{5}\left (d x +c \right )\right )}{5}+\frac {a^{2} \left (\tan ^{3}\left (d x +c \right )\right ) \left (a +b \right )}{3}-a^{3} \tan \left (d x +c \right )}{\left (a +b \right )^{4}}+\frac {a^{4} \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{\left (a +b \right )^{4} \sqrt {a \left (a +b \right )}}}{d}\) | \(120\) |
risch | \(-\frac {2 i \left (122 a^{2} b +66 a \,b^{2}+15 b^{3}+176 a^{3}+1260 a^{3} {\mathrm e}^{10 i \left (d x +c \right )}+105 b^{3} {\mathrm e}^{12 i \left (d x +c \right )}+812 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+2436 a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+315 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+3080 a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+420 a^{3} {\mathrm e}^{12 i \left (d x +c \right )}+3080 a^{3} {\mathrm e}^{8 i \left (d x +c \right )}+525 b^{3} {\mathrm e}^{8 i \left (d x +c \right )}+420 a \,b^{2} {\mathrm e}^{12 i \left (d x +c \right )}+840 a^{2} b \,{\mathrm e}^{10 i \left (d x +c \right )}+1400 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+2870 a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}+1890 a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+1176 a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+210 a \,b^{2} {\mathrm e}^{10 i \left (d x +c \right )}+224 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+630 a^{2} b \,{\mathrm e}^{12 i \left (d x +c \right )}+42 a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+420 a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+1722 a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}\right )}{105 d \left (a +b \right )^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{7}}+\frac {\sqrt {-a \left (a +b \right )}\, a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {-a \left (a +b \right )}-2 a -b}{b}\right )}{2 \left (a +b \right )^{5} d}-\frac {\sqrt {-a \left (a +b \right )}\, a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {-a \left (a +b \right )}+2 a +b}{b}\right )}{2 \left (a +b \right )^{5} d}\) | \(462\) |
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Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (106) = 212\).
Time = 0.37 (sec) , antiderivative size = 602, normalized size of antiderivative = 5.02 \[ \int \frac {\tan ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\left [\frac {105 \, a^{3} \sqrt {-\frac {a}{a + b}} \cos \left (d x + c\right )^{7} \log \left (\frac {{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (4 \, a^{2} + 5 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} - 4 \, {\left ({\left (2 \, a^{2} + 3 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{3} - {\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt {-\frac {a}{a + b}} \sin \left (d x + c\right ) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos \left (d x + c\right )^{4} - 2 \, {\left (a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}}\right ) - 4 \, {\left ({\left (176 \, a^{3} + 122 \, a^{2} b + 66 \, a b^{2} + 15 \, b^{3}\right )} \cos \left (d x + c\right )^{6} - {\left (122 \, a^{3} + 254 \, a^{2} b + 177 \, a b^{2} + 45 \, b^{3}\right )} \cos \left (d x + c\right )^{4} - 15 \, a^{3} - 45 \, a^{2} b - 45 \, a b^{2} - 15 \, b^{3} + 3 \, {\left (22 \, a^{3} + 59 \, a^{2} b + 52 \, a b^{2} + 15 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{420 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} d \cos \left (d x + c\right )^{7}}, -\frac {105 \, a^{3} \sqrt {\frac {a}{a + b}} \arctan \left (\frac {{\left ({\left (2 \, a + b\right )} \cos \left (d x + c\right )^{2} - a - b\right )} \sqrt {\frac {a}{a + b}}}{2 \, a \cos \left (d x + c\right ) \sin \left (d x + c\right )}\right ) \cos \left (d x + c\right )^{7} + 2 \, {\left ({\left (176 \, a^{3} + 122 \, a^{2} b + 66 \, a b^{2} + 15 \, b^{3}\right )} \cos \left (d x + c\right )^{6} - {\left (122 \, a^{3} + 254 \, a^{2} b + 177 \, a b^{2} + 45 \, b^{3}\right )} \cos \left (d x + c\right )^{4} - 15 \, a^{3} - 45 \, a^{2} b - 45 \, a b^{2} - 15 \, b^{3} + 3 \, {\left (22 \, a^{3} + 59 \, a^{2} b + 52 \, a b^{2} + 15 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{210 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} d \cos \left (d x + c\right )^{7}}\right ] \]
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\[ \int \frac {\tan ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\int \frac {\tan ^{8}{\left (c + d x \right )}}{a + b \sin ^{2}{\left (c + d x \right )}}\, dx \]
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Time = 0.31 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.50 \[ \int \frac {\tan ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {\frac {105 \, a^{4} \arctan \left (\frac {{\left (a + b\right )} \tan \left (d x + c\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \sqrt {{\left (a + b\right )} a}} + \frac {15 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \tan \left (d x + c\right )^{7} - 21 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \tan \left (d x + c\right )^{5} - 105 \, a^{3} \tan \left (d x + c\right ) + 35 \, {\left (a^{3} + a^{2} b\right )} \tan \left (d x + c\right )^{3}}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}}}{105 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 472 vs. \(2 (106) = 212\).
Time = 5.09 (sec) , antiderivative size = 472, normalized size of antiderivative = 3.93 \[ \int \frac {\tan ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {\frac {105 \, {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt {a^{2} + a b}}\right )\right )} a^{4}}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \sqrt {a^{2} + a b}} + \frac {15 \, a^{6} \tan \left (d x + c\right )^{7} + 90 \, a^{5} b \tan \left (d x + c\right )^{7} + 225 \, a^{4} b^{2} \tan \left (d x + c\right )^{7} + 300 \, a^{3} b^{3} \tan \left (d x + c\right )^{7} + 225 \, a^{2} b^{4} \tan \left (d x + c\right )^{7} + 90 \, a b^{5} \tan \left (d x + c\right )^{7} + 15 \, b^{6} \tan \left (d x + c\right )^{7} - 21 \, a^{6} \tan \left (d x + c\right )^{5} - 105 \, a^{5} b \tan \left (d x + c\right )^{5} - 210 \, a^{4} b^{2} \tan \left (d x + c\right )^{5} - 210 \, a^{3} b^{3} \tan \left (d x + c\right )^{5} - 105 \, a^{2} b^{4} \tan \left (d x + c\right )^{5} - 21 \, a b^{5} \tan \left (d x + c\right )^{5} + 35 \, a^{6} \tan \left (d x + c\right )^{3} + 140 \, a^{5} b \tan \left (d x + c\right )^{3} + 210 \, a^{4} b^{2} \tan \left (d x + c\right )^{3} + 140 \, a^{3} b^{3} \tan \left (d x + c\right )^{3} + 35 \, a^{2} b^{4} \tan \left (d x + c\right )^{3} - 105 \, a^{6} \tan \left (d x + c\right ) - 315 \, a^{5} b \tan \left (d x + c\right ) - 315 \, a^{4} b^{2} \tan \left (d x + c\right ) - 105 \, a^{3} b^{3} \tan \left (d x + c\right )}{a^{7} + 7 \, a^{6} b + 21 \, a^{5} b^{2} + 35 \, a^{4} b^{3} + 35 \, a^{3} b^{4} + 21 \, a^{2} b^{5} + 7 \, a b^{6} + b^{7}}}{105 \, d} \]
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Time = 14.83 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.18 \[ \int \frac {\tan ^8(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {{\mathrm {tan}\left (c+d\,x\right )}^7}{7\,d\,\left (a+b\right )}+\frac {a^2\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3\,d\,{\left (a+b\right )}^3}+\frac {a^{7/2}\,\mathrm {atan}\left (\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (2\,a+2\,b\right )\,\left (a^4+4\,a^3\,b+6\,a^2\,b^2+4\,a\,b^3+b^4\right )}{2\,\sqrt {a}\,{\left (a+b\right )}^{9/2}}\right )}{d\,{\left (a+b\right )}^{9/2}}-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^5}{5\,d\,{\left (a+b\right )}^2}-\frac {a^3\,\mathrm {tan}\left (c+d\,x\right )}{d\,{\left (a+b\right )}^4} \]
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