Integrand size = 23, antiderivative size = 128 \[ \int \frac {\tan ^7(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {a^3 \log (\cos (c+d x))}{(a+b)^4 d}-\frac {a^3 \log \left (a+b \sin ^2(c+d x)\right )}{2 (a+b)^4 d}+\frac {\left (3 a^2+3 a b+b^2\right ) \sec ^2(c+d x)}{2 (a+b)^3 d}-\frac {(3 a+2 b) \sec ^4(c+d x)}{4 (a+b)^2 d}+\frac {\sec ^6(c+d x)}{6 (a+b) d} \]
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Time = 0.15 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3273, 90} \[ \int \frac {\tan ^7(c+d x)}{a+b \sin ^2(c+d x)} \, dx=-\frac {a^3 \log \left (a+b \sin ^2(c+d x)\right )}{2 d (a+b)^4}+\frac {a^3 \log (\cos (c+d x))}{d (a+b)^4}+\frac {\left (3 a^2+3 a b+b^2\right ) \sec ^2(c+d x)}{2 d (a+b)^3}+\frac {\sec ^6(c+d x)}{6 d (a+b)}-\frac {(3 a+2 b) \sec ^4(c+d x)}{4 d (a+b)^2} \]
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Rule 90
Rule 3273
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^3}{(1-x)^4 (a+b x)} \, dx,x,\sin ^2(c+d x)\right )}{2 d} \\ & = \frac {\text {Subst}\left (\int \left (\frac {1}{(a+b) (-1+x)^4}+\frac {3 a+2 b}{(a+b)^2 (-1+x)^3}+\frac {3 a^2+3 a b+b^2}{(a+b)^3 (-1+x)^2}+\frac {a^3}{(a+b)^4 (-1+x)}-\frac {a^3 b}{(a+b)^4 (a+b x)}\right ) \, dx,x,\sin ^2(c+d x)\right )}{2 d} \\ & = \frac {a^3 \log (\cos (c+d x))}{(a+b)^4 d}-\frac {a^3 \log \left (a+b \sin ^2(c+d x)\right )}{2 (a+b)^4 d}+\frac {\left (3 a^2+3 a b+b^2\right ) \sec ^2(c+d x)}{2 (a+b)^3 d}-\frac {(3 a+2 b) \sec ^4(c+d x)}{4 (a+b)^2 d}+\frac {\sec ^6(c+d x)}{6 (a+b) d} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.88 \[ \int \frac {\tan ^7(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {\frac {12 a^3 \log (\cos (c+d x))}{(a+b)^4}-\frac {6 a^3 \log \left (a+b \sin ^2(c+d x)\right )}{(a+b)^4}+\frac {6 \left (3 a^2+3 a b+b^2\right ) \sec ^2(c+d x)}{(a+b)^3}-\frac {3 (3 a+2 b) \sec ^4(c+d x)}{(a+b)^2}+\frac {2 \sec ^6(c+d x)}{a+b}}{12 d} \]
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Time = 4.58 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {-\frac {a^{3} \ln \left (a +b -b \left (\cos ^{2}\left (d x +c \right )\right )\right )}{2 \left (a +b \right )^{4}}+\frac {a^{3} \ln \left (\cos \left (d x +c \right )\right )}{\left (a +b \right )^{4}}-\frac {3 a +2 b}{4 \left (a +b \right )^{2} \cos \left (d x +c \right )^{4}}-\frac {-3 a^{2}-3 a b -b^{2}}{2 \left (a +b \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {1}{6 \left (a +b \right ) \cos \left (d x +c \right )^{6}}}{d}\) | \(114\) |
default | \(\frac {-\frac {a^{3} \ln \left (a +b -b \left (\cos ^{2}\left (d x +c \right )\right )\right )}{2 \left (a +b \right )^{4}}+\frac {a^{3} \ln \left (\cos \left (d x +c \right )\right )}{\left (a +b \right )^{4}}-\frac {3 a +2 b}{4 \left (a +b \right )^{2} \cos \left (d x +c \right )^{4}}-\frac {-3 a^{2}-3 a b -b^{2}}{2 \left (a +b \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {1}{6 \left (a +b \right ) \cos \left (d x +c \right )^{6}}}{d}\) | \(114\) |
risch | \(\frac {6 a^{2} {\mathrm e}^{10 i \left (d x +c \right )}+6 a b \,{\mathrm e}^{10 i \left (d x +c \right )}+2 b^{2} {\mathrm e}^{10 i \left (d x +c \right )}+12 a^{2} {\mathrm e}^{8 i \left (d x +c \right )}+4 a b \,{\mathrm e}^{8 i \left (d x +c \right )}+\frac {68 a^{2} {\mathrm e}^{6 i \left (d x +c \right )}}{3}+\frac {52 a b \,{\mathrm e}^{6 i \left (d x +c \right )}}{3}+\frac {20 b^{2} {\mathrm e}^{6 i \left (d x +c \right )}}{3}+12 a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+4 a b \,{\mathrm e}^{4 i \left (d x +c \right )}+6 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+6 a b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{d \left (a +b \right )^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}+\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right )}-\frac {a^{3} \ln \left ({\mathrm e}^{4 i \left (d x +c \right )}-\frac {2 \left (2 a +b \right ) {\mathrm e}^{2 i \left (d x +c \right )}}{b}+1\right )}{2 d \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right )}\) | \(319\) |
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none
Time = 0.56 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.40 \[ \int \frac {\tan ^7(c+d x)}{a+b \sin ^2(c+d x)} \, dx=-\frac {6 \, a^{3} \cos \left (d x + c\right )^{6} \log \left (-b \cos \left (d x + c\right )^{2} + a + b\right ) - 12 \, a^{3} \cos \left (d x + c\right )^{6} \log \left (-\cos \left (d x + c\right )\right ) - 6 \, {\left (3 \, a^{3} + 6 \, a^{2} b + 4 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{4} - 2 \, a^{3} - 6 \, a^{2} b - 6 \, a b^{2} - 2 \, b^{3} + 3 \, {\left (3 \, a^{3} + 8 \, a^{2} b + 7 \, a b^{2} + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2}}{12 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} d \cos \left (d x + c\right )^{6}} \]
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\[ \int \frac {\tan ^7(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\int \frac {\tan ^{7}{\left (c + d x \right )}}{a + b \sin ^{2}{\left (c + d x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (120) = 240\).
Time = 0.27 (sec) , antiderivative size = 273, normalized size of antiderivative = 2.13 \[ \int \frac {\tan ^7(c+d x)}{a+b \sin ^2(c+d x)} \, dx=-\frac {\frac {6 \, a^{3} \log \left (b \sin \left (d x + c\right )^{2} + a\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} - \frac {6 \, a^{3} \log \left (\sin \left (d x + c\right )^{2} - 1\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac {6 \, {\left (3 \, a^{2} + 3 \, a b + b^{2}\right )} \sin \left (d x + c\right )^{4} - 3 \, {\left (9 \, a^{2} + 7 \, a b + 2 \, b^{2}\right )} \sin \left (d x + c\right )^{2} + 11 \, a^{2} + 7 \, a b + 2 \, b^{2}}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sin \left (d x + c\right )^{6} - 3 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sin \left (d x + c\right )^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sin \left (d x + c\right )^{2}}}{12 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 603 vs. \(2 (120) = 240\).
Time = 4.01 (sec) , antiderivative size = 603, normalized size of antiderivative = 4.71 \[ \int \frac {\tan ^7(c+d x)}{a+b \sin ^2(c+d x)} \, dx=-\frac {\frac {30 \, a^{3} \log \left (a - \frac {2 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {4 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} - \frac {60 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac {147 \, a^{3} + \frac {1002 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {120 \, a^{2} b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {2925 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {960 \, a^{2} b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {240 \, a b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {4780 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3600 \, a^{2} b {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2400 \, a b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {640 \, b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2925 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {960 \, a^{2} b {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {240 \, a b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1002 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {120 \, a^{2} b {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {147 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{6}}}{60 \, d} \]
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Time = 13.91 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.90 \[ \int \frac {\tan ^7(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {{\mathrm {tan}\left (c+d\,x\right )}^6}{6\,d\,\left (a+b\right )}+\frac {a^2\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2\,d\,{\left (a+b\right )}^3}-\frac {a^3\,\ln \left (\left (a+b\right )\,{\mathrm {tan}\left (c+d\,x\right )}^2+a\right )}{d\,\left (2\,a^4+8\,a^3\,b+12\,a^2\,b^2+8\,a\,b^3+2\,b^4\right )}-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^4}{4\,d\,{\left (a+b\right )}^2} \]
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