Integrand size = 19, antiderivative size = 163 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {a^4}{24 d (a-a \cos (c+d x))^3}-\frac {5 a^3}{32 d (a-a \cos (c+d x))^2}-\frac {a^2}{2 d (a-a \cos (c+d x))}-\frac {a^3}{32 d (a+a \cos (c+d x))^2}-\frac {3 a^2}{16 d (a+a \cos (c+d x))}+\frac {21 a \log (1-\cos (c+d x))}{32 d}-\frac {a \log (\cos (c+d x))}{d}+\frac {11 a \log (1+\cos (c+d x))}{32 d} \]
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Time = 0.19 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {3957, 2915, 12, 90} \[ \int \csc ^7(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {a^4}{24 d (a-a \cos (c+d x))^3}-\frac {5 a^3}{32 d (a-a \cos (c+d x))^2}-\frac {a^3}{32 d (a \cos (c+d x)+a)^2}-\frac {a^2}{2 d (a-a \cos (c+d x))}-\frac {3 a^2}{16 d (a \cos (c+d x)+a)}+\frac {21 a \log (1-\cos (c+d x))}{32 d}-\frac {a \log (\cos (c+d x))}{d}+\frac {11 a \log (\cos (c+d x)+1)}{32 d} \]
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Rule 12
Rule 90
Rule 2915
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int (-a-a \cos (c+d x)) \csc ^7(c+d x) \sec (c+d x) \, dx \\ & = \frac {a^7 \text {Subst}\left (\int \frac {a}{(-a-x)^4 x (-a+x)^3} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a^8 \text {Subst}\left (\int \frac {1}{(-a-x)^4 x (-a+x)^3} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a^8 \text {Subst}\left (\int \left (-\frac {1}{16 a^5 (a-x)^3}-\frac {3}{16 a^6 (a-x)^2}-\frac {11}{32 a^7 (a-x)}-\frac {1}{a^7 x}+\frac {1}{8 a^4 (a+x)^4}+\frac {5}{16 a^5 (a+x)^3}+\frac {1}{2 a^6 (a+x)^2}+\frac {21}{32 a^7 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = -\frac {a^4}{24 d (a-a \cos (c+d x))^3}-\frac {5 a^3}{32 d (a-a \cos (c+d x))^2}-\frac {a^2}{2 d (a-a \cos (c+d x))}-\frac {a^3}{32 d (a+a \cos (c+d x))^2}-\frac {3 a^2}{16 d (a+a \cos (c+d x))}+\frac {21 a \log (1-\cos (c+d x))}{32 d}-\frac {a \log (\cos (c+d x))}{d}+\frac {11 a \log (1+\cos (c+d x))}{32 d} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.42 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x)) \, dx=-\frac {5 a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a \csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a \csc ^6\left (\frac {1}{2} (c+d x)\right )}{384 d}-\frac {a \csc ^2(c+d x)}{2 d}-\frac {a \csc ^4(c+d x)}{4 d}-\frac {a \csc ^6(c+d x)}{6 d}-\frac {5 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {a \log (\cos (c+d x))}{d}+\frac {5 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}+\frac {a \log (\sin (c+d x))}{d}+\frac {5 a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {a \sec ^6\left (\frac {1}{2} (c+d x)\right )}{384 d} \]
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Time = 1.12 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.63
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {1}{6 \sin \left (d x +c \right )^{6}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a \left (\left (-\frac {\csc \left (d x +c \right )^{5}}{6}-\frac {5 \csc \left (d x +c \right )^{3}}{24}-\frac {5 \csc \left (d x +c \right )}{16}\right ) \cot \left (d x +c \right )+\frac {5 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{16}\right )}{d}\) | \(103\) |
default | \(\frac {a \left (-\frac {1}{6 \sin \left (d x +c \right )^{6}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )+a \left (\left (-\frac {\csc \left (d x +c \right )^{5}}{6}-\frac {5 \csc \left (d x +c \right )^{3}}{24}-\frac {5 \csc \left (d x +c \right )}{16}\right ) \cot \left (d x +c \right )+\frac {5 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{16}\right )}{d}\) | \(103\) |
parallelrisch | \(-\frac {a \left (\cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {21 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2}+66 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+21 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-252 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+192 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+192 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )\right )}{192 d}\) | \(111\) |
norman | \(\frac {-\frac {a}{192 d}-\frac {7 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{128 d}-\frac {11 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{32 d}-\frac {7 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{64 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{128 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {21 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(141\) |
risch | \(\frac {a \left (15 \,{\mathrm e}^{9 i \left (d x +c \right )}+18 \,{\mathrm e}^{8 i \left (d x +c \right )}-136 \,{\mathrm e}^{7 i \left (d x +c \right )}-34 \,{\mathrm e}^{6 i \left (d x +c \right )}+402 \,{\mathrm e}^{5 i \left (d x +c \right )}-34 \,{\mathrm e}^{4 i \left (d x +c \right )}-136 \,{\mathrm e}^{3 i \left (d x +c \right )}+18 \,{\mathrm e}^{2 i \left (d x +c \right )}+15 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{24 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{6} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{4}}+\frac {21 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}+\frac {11 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}-\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(188\) |
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Leaf count of result is larger than twice the leaf count of optimal. 307 vs. \(2 (152) = 304\).
Time = 0.30 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.88 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x)) \, dx=\frac {30 \, a \cos \left (d x + c\right )^{4} + 18 \, a \cos \left (d x + c\right )^{3} - 98 \, a \cos \left (d x + c\right )^{2} - 22 \, a \cos \left (d x + c\right ) - 96 \, {\left (a \cos \left (d x + c\right )^{5} - a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{3} + 2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - a\right )} \log \left (-\cos \left (d x + c\right )\right ) + 33 \, {\left (a \cos \left (d x + c\right )^{5} - a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{3} + 2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 63 \, {\left (a \cos \left (d x + c\right )^{5} - a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{3} + 2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 88 \, a}{96 \, {\left (d \cos \left (d x + c\right )^{5} - d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} + 2 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right ) - d\right )}} \]
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Timed out. \[ \int \csc ^7(c+d x) (a+a \sec (c+d x)) \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.83 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x)) \, dx=\frac {33 \, a \log \left (\cos \left (d x + c\right ) + 1\right ) + 63 \, a \log \left (\cos \left (d x + c\right ) - 1\right ) - 96 \, a \log \left (\cos \left (d x + c\right )\right ) + \frac {2 \, {\left (15 \, a \cos \left (d x + c\right )^{4} + 9 \, a \cos \left (d x + c\right )^{3} - 49 \, a \cos \left (d x + c\right )^{2} - 11 \, a \cos \left (d x + c\right ) + 44 \, a\right )}}{\cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} + 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) - 1}}{96 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.20 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x)) \, dx=\frac {252 \, a \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 384 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {{\left (2 \, a - \frac {21 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {132 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {462 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{3}} + \frac {42 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {3 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{384 \, d} \]
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Time = 13.58 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.87 \[ \int \csc ^7(c+d x) (a+a \sec (c+d x)) \, dx=\frac {\frac {5\,a\,{\cos \left (c+d\,x\right )}^4}{16}+\frac {3\,a\,{\cos \left (c+d\,x\right )}^3}{16}-\frac {49\,a\,{\cos \left (c+d\,x\right )}^2}{48}-\frac {11\,a\,\cos \left (c+d\,x\right )}{48}+\frac {11\,a}{12}}{d\,\left ({\cos \left (c+d\,x\right )}^5-{\cos \left (c+d\,x\right )}^4-2\,{\cos \left (c+d\,x\right )}^3+2\,{\cos \left (c+d\,x\right )}^2+\cos \left (c+d\,x\right )-1\right )}-\frac {a\,\ln \left (\cos \left (c+d\,x\right )\right )}{d}+\frac {21\,a\,\ln \left (\cos \left (c+d\,x\right )-1\right )}{32\,d}+\frac {11\,a\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{32\,d} \]
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