Integrand size = 19, antiderivative size = 130 \[ \int \cot ^7(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {(16 a+5 b) \log (1-\cos (c+d x))}{32 d}-\frac {(16 a-5 b) \log (1+\cos (c+d x))}{32 d}-\frac {\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}+\frac {\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{24 d}-\frac {\cot ^2(c+d x) (8 a+5 b \sec (c+d x))}{16 d} \]
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Time = 0.20 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3967, 3968, 2747, 647, 31} \[ \int \cot ^7(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {(16 a+5 b) \log (1-\cos (c+d x))}{32 d}-\frac {(16 a-5 b) \log (\cos (c+d x)+1)}{32 d}-\frac {\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}+\frac {\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{24 d}-\frac {\cot ^2(c+d x) (8 a+5 b \sec (c+d x))}{16 d} \]
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Rule 31
Rule 647
Rule 2747
Rule 3967
Rule 3968
Rubi steps \begin{align*} \text {integral}& = -\frac {\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}+\frac {1}{6} \int \cot ^5(c+d x) (-6 a-5 b \sec (c+d x)) \, dx \\ & = -\frac {\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}+\frac {\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{24 d}+\frac {1}{24} \int \cot ^3(c+d x) (24 a+15 b \sec (c+d x)) \, dx \\ & = -\frac {\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}+\frac {\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{24 d}-\frac {\cot ^2(c+d x) (8 a+5 b \sec (c+d x))}{16 d}+\frac {1}{48} \int \cot (c+d x) (-48 a-15 b \sec (c+d x)) \, dx \\ & = -\frac {\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}+\frac {\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{24 d}-\frac {\cot ^2(c+d x) (8 a+5 b \sec (c+d x))}{16 d}+\frac {1}{48} \int (-15 b-48 a \cos (c+d x)) \csc (c+d x) \, dx \\ & = -\frac {\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}+\frac {\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{24 d}-\frac {\cot ^2(c+d x) (8 a+5 b \sec (c+d x))}{16 d}+\frac {a \text {Subst}\left (\int \frac {-15 b+x}{2304 a^2-x^2} \, dx,x,-48 a \cos (c+d x)\right )}{d} \\ & = -\frac {\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}+\frac {\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{24 d}-\frac {\cot ^2(c+d x) (8 a+5 b \sec (c+d x))}{16 d}+\frac {(16 a-5 b) \text {Subst}\left (\int \frac {1}{48 a-x} \, dx,x,-48 a \cos (c+d x)\right )}{32 d}+\frac {(16 a+5 b) \text {Subst}\left (\int \frac {1}{-48 a-x} \, dx,x,-48 a \cos (c+d x)\right )}{32 d} \\ & = -\frac {(16 a+5 b) \log (1-\cos (c+d x))}{32 d}-\frac {(16 a-5 b) \log (1+\cos (c+d x))}{32 d}-\frac {\cot ^6(c+d x) (a+b \sec (c+d x))}{6 d}+\frac {\cot ^4(c+d x) (6 a+5 b \sec (c+d x))}{24 d}-\frac {\cot ^2(c+d x) (8 a+5 b \sec (c+d x))}{16 d} \\ \end{align*}
Time = 0.49 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.66 \[ \int \cot ^7(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {11 b \csc ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {b \csc ^4\left (\frac {1}{2} (c+d x)\right )}{32 d}-\frac {b \csc ^6\left (\frac {1}{2} (c+d x)\right )}{384 d}+\frac {5 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {5 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {a \left (6 \cot ^2(c+d x)-3 \cot ^4(c+d x)+2 \cot ^6(c+d x)+12 \log (\cos (c+d x))+12 \log (\tan (c+d x))\right )}{12 d}+\frac {11 b \sec ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {b \sec ^4\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {b \sec ^6\left (\frac {1}{2} (c+d x)\right )}{384 d} \]
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Time = 0.91 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.16
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {\cot \left (d x +c \right )^{6}}{6}+\frac {\cot \left (d x +c \right )^{4}}{4}-\frac {\cot \left (d x +c \right )^{2}}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+b \left (-\frac {\cos \left (d x +c \right )^{7}}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{16 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{16}-\frac {5 \cos \left (d x +c \right )^{3}}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{16}\right )}{d}\) | \(151\) |
default | \(\frac {a \left (-\frac {\cot \left (d x +c \right )^{6}}{6}+\frac {\cot \left (d x +c \right )^{4}}{4}-\frac {\cot \left (d x +c \right )^{2}}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+b \left (-\frac {\cos \left (d x +c \right )^{7}}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{16 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{16}-\frac {5 \cos \left (d x +c \right )^{3}}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{16}\right )}{d}\) | \(151\) |
risch | \(i a x +\frac {2 i a c}{d}+\frac {{\mathrm e}^{i \left (d x +c \right )} \left (33 b \,{\mathrm e}^{10 i \left (d x +c \right )}+144 a \,{\mathrm e}^{9 i \left (d x +c \right )}+5 b \,{\mathrm e}^{8 i \left (d x +c \right )}-288 a \,{\mathrm e}^{7 i \left (d x +c \right )}+90 b \,{\mathrm e}^{6 i \left (d x +c \right )}+544 a \,{\mathrm e}^{5 i \left (d x +c \right )}+90 b \,{\mathrm e}^{4 i \left (d x +c \right )}-288 a \,{\mathrm e}^{3 i \left (d x +c \right )}+5 b \,{\mathrm e}^{2 i \left (d x +c \right )}+144 \,{\mathrm e}^{i \left (d x +c \right )} a +33 b \right )}{24 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b}{16 d}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b}{16 d}\) | \(258\) |
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Time = 0.27 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.82 \[ \int \cot ^7(c+d x) (a+b \sec (c+d x)) \, dx=\frac {66 \, b \cos \left (d x + c\right )^{5} + 144 \, a \cos \left (d x + c\right )^{4} - 80 \, b \cos \left (d x + c\right )^{3} - 216 \, a \cos \left (d x + c\right )^{2} + 30 \, b \cos \left (d x + c\right ) - 3 \, {\left ({\left (16 \, a - 5 \, b\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (16 \, a - 5 \, b\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (16 \, a - 5 \, b\right )} \cos \left (d x + c\right )^{2} - 16 \, a + 5 \, b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, {\left ({\left (16 \, a + 5 \, b\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (16 \, a + 5 \, b\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (16 \, a + 5 \, b\right )} \cos \left (d x + c\right )^{2} - 16 \, a - 5 \, b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 88 \, a}{96 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]
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\[ \int \cot ^7(c+d x) (a+b \sec (c+d x)) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right ) \cot ^{7}{\left (c + d x \right )}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.02 \[ \int \cot ^7(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {3 \, {\left (16 \, a - 5 \, b\right )} \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, {\left (16 \, a + 5 \, b\right )} \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (33 \, b \cos \left (d x + c\right )^{5} + 72 \, a \cos \left (d x + c\right )^{4} - 40 \, b \cos \left (d x + c\right )^{3} - 108 \, a \cos \left (d x + c\right )^{2} + 15 \, b \cos \left (d x + c\right ) + 44 \, a\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1}}{96 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 358 vs. \(2 (120) = 240\).
Time = 0.36 (sec) , antiderivative size = 358, normalized size of antiderivative = 2.75 \[ \int \cot ^7(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {12 \, {\left (16 \, a + 5 \, b\right )} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 384 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - \frac {{\left (a + b + \frac {12 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {9 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {87 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {45 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {352 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {110 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{3}} - \frac {87 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {45 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {12 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {9 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{384 \, d} \]
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Time = 14.09 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.31 \[ \int \cot ^7(c+d x) (a+b \sec (c+d x)) \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a}{32}-\frac {3\,b}{128}\right )}{d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\left (\frac {29\,a}{2}+\frac {15\,b}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (-2\,a-\frac {3\,b}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {a}{6}+\frac {b}{6}\right )}{64\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {29\,a}{128}-\frac {15\,b}{128}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {a}{384}-\frac {b}{384}\right )}{d}+\frac {a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (a+\frac {5\,b}{16}\right )}{d} \]
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