Integrand size = 21, antiderivative size = 234 \[ \int \frac {\cot ^5(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {\log (\cos (c+d x))}{a d}+\frac {\left (8 a^2+21 a b+15 b^2\right ) \log (1-\sec (c+d x))}{16 (a+b)^3 d}+\frac {\left (8 a^2-21 a b+15 b^2\right ) \log (1+\sec (c+d x))}{16 (a-b)^3 d}-\frac {b^6 \log (a+b \sec (c+d x))}{a \left (a^2-b^2\right )^3 d}-\frac {1}{16 (a+b) d (1-\sec (c+d x))^2}-\frac {5 a+7 b}{16 (a+b)^2 d (1-\sec (c+d x))}-\frac {1}{16 (a-b) d (1+\sec (c+d x))^2}-\frac {5 a-7 b}{16 (a-b)^2 d (1+\sec (c+d x))} \]
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Time = 0.37 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3970, 908} \[ \int \frac {\cot ^5(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {\left (8 a^2+21 a b+15 b^2\right ) \log (1-\sec (c+d x))}{16 d (a+b)^3}+\frac {\left (8 a^2-21 a b+15 b^2\right ) \log (\sec (c+d x)+1)}{16 d (a-b)^3}-\frac {b^6 \log (a+b \sec (c+d x))}{a d \left (a^2-b^2\right )^3}-\frac {5 a+7 b}{16 d (a+b)^2 (1-\sec (c+d x))}-\frac {5 a-7 b}{16 d (a-b)^2 (\sec (c+d x)+1)}-\frac {1}{16 d (a+b) (1-\sec (c+d x))^2}-\frac {1}{16 d (a-b) (\sec (c+d x)+1)^2}+\frac {\log (\cos (c+d x))}{a d} \]
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Rule 908
Rule 3970
Rubi steps \begin{align*} \text {integral}& = -\frac {b^6 \text {Subst}\left (\int \frac {1}{x (a+x) \left (b^2-x^2\right )^3} \, dx,x,b \sec (c+d x)\right )}{d} \\ & = -\frac {b^6 \text {Subst}\left (\int \left (\frac {1}{8 b^4 (a+b) (b-x)^3}+\frac {5 a+7 b}{16 b^5 (a+b)^2 (b-x)^2}+\frac {8 a^2+21 a b+15 b^2}{16 b^6 (a+b)^3 (b-x)}+\frac {1}{a b^6 x}+\frac {1}{a (a-b)^3 (a+b)^3 (a+x)}+\frac {1}{8 b^4 (-a+b) (b+x)^3}+\frac {-5 a+7 b}{16 (a-b)^2 b^5 (b+x)^2}+\frac {8 a^2-21 a b+15 b^2}{16 b^6 (-a+b)^3 (b+x)}\right ) \, dx,x,b \sec (c+d x)\right )}{d} \\ & = \frac {\log (\cos (c+d x))}{a d}+\frac {\left (8 a^2+21 a b+15 b^2\right ) \log (1-\sec (c+d x))}{16 (a+b)^3 d}+\frac {\left (8 a^2-21 a b+15 b^2\right ) \log (1+\sec (c+d x))}{16 (a-b)^3 d}-\frac {b^6 \log (a+b \sec (c+d x))}{a \left (a^2-b^2\right )^3 d}-\frac {1}{16 (a+b) d (1-\sec (c+d x))^2}-\frac {5 a+7 b}{16 (a+b)^2 d (1-\sec (c+d x))}-\frac {1}{16 (a-b) d (1+\sec (c+d x))^2}-\frac {5 a-7 b}{16 (a-b)^2 d (1+\sec (c+d x))} \\ \end{align*}
Time = 4.50 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.94 \[ \int \frac {\cot ^5(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {b^6 \left (-\frac {16 \log (\cos (c+d x))}{a b^6}-\frac {\left (8 a^2+21 a b+15 b^2\right ) \log (1-\sec (c+d x))}{b^6 (a+b)^3}-\frac {\left (8 a^2-21 a b+15 b^2\right ) \log (1+\sec (c+d x))}{(a-b)^3 b^6}+\frac {16 \log (a+b \sec (c+d x))}{a (a-b)^3 (a+b)^3}+\frac {1}{b^6 (a+b) (-1+\sec (c+d x))^2}+\frac {-5 a-7 b}{b^6 (a+b)^2 (-1+\sec (c+d x))}+\frac {1}{(a-b) b^6 (1+\sec (c+d x))^2}+\frac {5 a-7 b}{(a-b)^2 b^6 (1+\sec (c+d x))}\right )}{16 d} \]
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Time = 1.22 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {-\frac {b^{6} \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3} a}-\frac {1}{2 \left (8 a +8 b \right ) \left (\cos \left (d x +c \right )-1\right )^{2}}-\frac {7 a +9 b}{16 \left (a +b \right )^{2} \left (\cos \left (d x +c \right )-1\right )}+\frac {\left (8 a^{2}+21 a b +15 b^{2}\right ) \ln \left (\cos \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{3}}-\frac {1}{2 \left (8 a -8 b \right ) \left (\cos \left (d x +c \right )+1\right )^{2}}-\frac {-7 a +9 b}{16 \left (a -b \right )^{2} \left (\cos \left (d x +c \right )+1\right )}+\frac {\left (8 a^{2}-21 a b +15 b^{2}\right ) \ln \left (\cos \left (d x +c \right )+1\right )}{16 \left (a -b \right )^{3}}}{d}\) | \(193\) |
default | \(\frac {-\frac {b^{6} \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3} a}-\frac {1}{2 \left (8 a +8 b \right ) \left (\cos \left (d x +c \right )-1\right )^{2}}-\frac {7 a +9 b}{16 \left (a +b \right )^{2} \left (\cos \left (d x +c \right )-1\right )}+\frac {\left (8 a^{2}+21 a b +15 b^{2}\right ) \ln \left (\cos \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{3}}-\frac {1}{2 \left (8 a -8 b \right ) \left (\cos \left (d x +c \right )+1\right )^{2}}-\frac {-7 a +9 b}{16 \left (a -b \right )^{2} \left (\cos \left (d x +c \right )+1\right )}+\frac {\left (8 a^{2}-21 a b +15 b^{2}\right ) \ln \left (\cos \left (d x +c \right )+1\right )}{16 \left (a -b \right )^{3}}}{d}\) | \(193\) |
risch | \(\frac {i x}{a}-\frac {i a^{2} c}{d \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}+\frac {2 i b^{6} c}{d a \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}+\frac {21 i a b c}{8 d \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}-\frac {i a^{2} x}{a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}}+\frac {2 i b^{6} x}{a \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}-\frac {15 i b^{2} x}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}-\frac {i a^{2} x}{a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}}-\frac {21 i a b x}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {21 i a b c}{8 d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {15 i b^{2} c}{8 d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {15 i b^{2} c}{8 d \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}-\frac {i a^{2} c}{d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {15 i b^{2} x}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {21 i a b x}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}+\frac {5 a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}-9 b^{3} {\mathrm e}^{7 i \left (d x +c \right )}-16 a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+24 a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+3 a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}+b^{3} {\mathrm e}^{5 i \left (d x +c \right )}+16 a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-32 a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+3 a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}+b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-16 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+24 a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+5 a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}-9 b^{3} {\mathrm e}^{i \left (d x +c \right )}}{4 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) a^{2}}{d \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}-\frac {21 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) a b}{8 d \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}+\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{8 d \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) a^{2}}{d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {21 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) a b}{8 d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{8 d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {b^{6} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{d a \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}\) | \(998\) |
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Leaf count of result is larger than twice the leaf count of optimal. 576 vs. \(2 (218) = 436\).
Time = 0.43 (sec) , antiderivative size = 576, normalized size of antiderivative = 2.46 \[ \int \frac {\cot ^5(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {12 \, a^{6} - 32 \, a^{4} b^{2} + 20 \, a^{2} b^{4} + 2 \, {\left (5 \, a^{5} b - 14 \, a^{3} b^{3} + 9 \, a b^{5}\right )} \cos \left (d x + c\right )^{3} - 8 \, {\left (2 \, a^{6} - 5 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left (3 \, a^{5} b - 10 \, a^{3} b^{3} + 7 \, a b^{5}\right )} \cos \left (d x + c\right ) - 16 \, {\left (b^{6} \cos \left (d x + c\right )^{4} - 2 \, b^{6} \cos \left (d x + c\right )^{2} + b^{6}\right )} \log \left (a \cos \left (d x + c\right ) + b\right ) + {\left (8 \, a^{6} + 3 \, a^{5} b - 24 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 24 \, a^{2} b^{4} + 15 \, a b^{5} + {\left (8 \, a^{6} + 3 \, a^{5} b - 24 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 24 \, a^{2} b^{4} + 15 \, a b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (8 \, a^{6} + 3 \, a^{5} b - 24 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 24 \, a^{2} b^{4} + 15 \, a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (8 \, a^{6} - 3 \, a^{5} b - 24 \, a^{4} b^{2} + 10 \, a^{3} b^{3} + 24 \, a^{2} b^{4} - 15 \, a b^{5} + {\left (8 \, a^{6} - 3 \, a^{5} b - 24 \, a^{4} b^{2} + 10 \, a^{3} b^{3} + 24 \, a^{2} b^{4} - 15 \, a b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (8 \, a^{6} - 3 \, a^{5} b - 24 \, a^{4} b^{2} + 10 \, a^{3} b^{3} + 24 \, a^{2} b^{4} - 15 \, a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{16 \, {\left ({\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} d \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} d \cos \left (d x + c\right )^{2} + {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} d\right )}} \]
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\[ \int \frac {\cot ^5(c+d x)}{a+b \sec (c+d x)} \, dx=\int \frac {\cot ^{5}{\left (c + d x \right )}}{a + b \sec {\left (c + d x \right )}}\, dx \]
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none
Time = 0.21 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.24 \[ \int \frac {\cot ^5(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {\frac {16 \, b^{6} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}} - \frac {{\left (8 \, a^{2} - 21 \, a b + 15 \, b^{2}\right )} \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac {{\left (8 \, a^{2} + 21 \, a b + 15 \, b^{2}\right )} \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac {2 \, {\left ({\left (5 \, a^{2} b - 9 \, b^{3}\right )} \cos \left (d x + c\right )^{3} + 6 \, a^{3} - 10 \, a b^{2} - 4 \, {\left (2 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} - {\left (3 \, a^{2} b - 7 \, b^{3}\right )} \cos \left (d x + c\right )\right )}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} + a^{4} - 2 \, a^{2} b^{2} + b^{4} - 2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2}}}{16 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 649 vs. \(2 (218) = 436\).
Time = 0.39 (sec) , antiderivative size = 649, normalized size of antiderivative = 2.77 \[ \int \frac {\cot ^5(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {\frac {4 \, {\left (8 \, a^{2} + 21 \, a b + 15 \, b^{2}\right )} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac {32 \, {\left (a^{5} - 3 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \log \left ({\left | a + b - \frac {2 \, 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}} + \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} \right |}\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} - \frac {\frac {12 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {16 \, 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}} - \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{a^{2} - 2 \, a b + b^{2}} - \frac {{\left (a^{2} + 2 \, a b + b^{2} + \frac {12 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {28 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {16 \, b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {48 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {126 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {90 \, b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}} - \frac {32 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - 2 \, b^{6}\right )} \log \left (\frac {{\left | 2 \, b + \frac {2 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {2 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - 2 \, {\left | a \right |} \right |}}{{\left | 2 \, b + \frac {2 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {2 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + 2 \, {\left | a \right |} \right |}}\right )}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left | a \right |}}}{64 \, d} \]
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Time = 15.27 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.24 \[ \int \frac {\cot ^5(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (8\,a^2+21\,a\,b+15\,b^2\right )}{d\,\left (8\,a^3+24\,a^2\,b+24\,a\,b^2+8\,b^3\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{4\,d\,\left (16\,a-16\,b\right )}-\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {16\,b}{{\left (16\,a-16\,b\right )}^2}-\frac {3}{16\,a-16\,b}\right )}{d}-\frac {\frac {a^2-2\,a\,b+b^2}{4\,\left (a+b\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (-3\,a^3+2\,a^2\,b+5\,a\,b^2-4\,b^3\right )}{{\left (a+b\right )}^2}}{d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (16\,a^2-32\,a\,b+16\,b^2\right )}-\frac {b^6\,\ln \left (a+b-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{a\,d\,{\left (a^2-b^2\right )}^3} \]
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