Integrand size = 19, antiderivative size = 140 \[ \int \csc ^7(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {5 a \text {arctanh}(\cos (c+d x))}{16 d}-\frac {3 b \cot ^2(c+d x)}{2 d}-\frac {3 b \cot ^4(c+d x)}{4 d}-\frac {b \cot ^6(c+d x)}{6 d}-\frac {5 a \cot (c+d x) \csc (c+d x)}{16 d}-\frac {5 a \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {b \log (\tan (c+d x))}{d} \]
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Time = 0.18 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {3957, 2913, 2700, 272, 45, 3853, 3855} \[ \int \csc ^7(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {5 a \text {arctanh}(\cos (c+d x))}{16 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac {5 a \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {5 a \cot (c+d x) \csc (c+d x)}{16 d}-\frac {b \cot ^6(c+d x)}{6 d}-\frac {3 b \cot ^4(c+d x)}{4 d}-\frac {3 b \cot ^2(c+d x)}{2 d}+\frac {b \log (\tan (c+d x))}{d} \]
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Rule 45
Rule 272
Rule 2700
Rule 2913
Rule 3853
Rule 3855
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int (-b-a \cos (c+d x)) \csc ^7(c+d x) \sec (c+d x) \, dx \\ & = a \int \csc ^7(c+d x) \, dx+b \int \csc ^7(c+d x) \sec (c+d x) \, dx \\ & = -\frac {a \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {1}{6} (5 a) \int \csc ^5(c+d x) \, dx+\frac {b \text {Subst}\left (\int \frac {\left (1+x^2\right )^3}{x^7} \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {5 a \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {1}{8} (5 a) \int \csc ^3(c+d x) \, dx+\frac {b \text {Subst}\left (\int \frac {(1+x)^3}{x^4} \, dx,x,\tan ^2(c+d x)\right )}{2 d} \\ & = -\frac {5 a \cot (c+d x) \csc (c+d x)}{16 d}-\frac {5 a \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {1}{16} (5 a) \int \csc (c+d x) \, dx+\frac {b \text {Subst}\left (\int \left (\frac {1}{x^4}+\frac {3}{x^3}+\frac {3}{x^2}+\frac {1}{x}\right ) \, dx,x,\tan ^2(c+d x)\right )}{2 d} \\ & = -\frac {5 a \text {arctanh}(\cos (c+d x))}{16 d}-\frac {3 b \cot ^2(c+d x)}{2 d}-\frac {3 b \cot ^4(c+d x)}{4 d}-\frac {b \cot ^6(c+d x)}{6 d}-\frac {5 a \cot (c+d x) \csc (c+d x)}{16 d}-\frac {5 a \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {b \log (\tan (c+d x))}{d} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.66 \[ \int \csc ^7(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {5 a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a \csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a \csc ^6\left (\frac {1}{2} (c+d x)\right )}{384 d}-\frac {b \csc ^2(c+d x)}{2 d}-\frac {b \csc ^4(c+d x)}{4 d}-\frac {b \csc ^6(c+d x)}{6 d}-\frac {5 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {b \log (\cos (c+d x))}{d}+\frac {5 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}+\frac {b \log (\sin (c+d x))}{d}+\frac {5 a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {a \sec ^6\left (\frac {1}{2} (c+d x)\right )}{384 d} \]
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Time = 1.24 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.74
method | result | size |
derivativedivides | \(\frac {a \left (\left (-\frac {\csc \left (d x +c \right )^{5}}{6}-\frac {5 \csc \left (d x +c \right )^{3}}{24}-\frac {5 \csc \left (d x +c \right )}{16}\right ) \cot \left (d x +c \right )+\frac {5 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{16}\right )+b \left (-\frac {1}{6 \sin \left (d x +c \right )^{6}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(103\) |
default | \(\frac {a \left (\left (-\frac {\csc \left (d x +c \right )^{5}}{6}-\frac {5 \csc \left (d x +c \right )^{3}}{24}-\frac {5 \csc \left (d x +c \right )}{16}\right ) \cot \left (d x +c \right )+\frac {5 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{16}\right )+b \left (-\frac {1}{6 \sin \left (d x +c \right )^{6}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{2 \sin \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(103\) |
parallelrisch | \(\frac {-384 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-384 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (120 a +384 b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-a -b \right ) \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (-9 a -12 b \right ) \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-45 a -87 b \right ) \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (a -b \right )+\left (9 a -12 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+45 a -87 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{384 d}\) | \(167\) |
norman | \(\frac {-\frac {a +b}{384 d}+\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{384 d}+\frac {\left (3 a -4 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{128 d}-\frac {\left (3 a +4 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{128 d}+\frac {\left (15 a -29 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{128 d}-\frac {\left (15 a +29 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{128 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}+\frac {\left (5 a +16 b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}\) | \(194\) |
risch | \(\frac {15 a \,{\mathrm e}^{11 i \left (d x +c \right )}+48 b \,{\mathrm e}^{10 i \left (d x +c \right )}-85 a \,{\mathrm e}^{9 i \left (d x +c \right )}-288 b \,{\mathrm e}^{8 i \left (d x +c \right )}+198 a \,{\mathrm e}^{7 i \left (d x +c \right )}+736 b \,{\mathrm e}^{6 i \left (d x +c \right )}+198 a \,{\mathrm e}^{5 i \left (d x +c \right )}-288 b \,{\mathrm e}^{4 i \left (d x +c \right )}-85 a \,{\mathrm e}^{3 i \left (d x +c \right )}+48 b \,{\mathrm e}^{2 i \left (d x +c \right )}+15 \,{\mathrm e}^{i \left (d x +c \right )} a}{24 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}-\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b}{d}+\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b}{d}-\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(241\) |
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Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (126) = 252\).
Time = 0.27 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.03 \[ \int \csc ^7(c+d x) (a+b \sec (c+d x)) \, dx=\frac {30 \, a \cos \left (d x + c\right )^{5} + 48 \, b \cos \left (d x + c\right )^{4} - 80 \, a \cos \left (d x + c\right )^{3} - 120 \, b \cos \left (d x + c\right )^{2} + 66 \, a \cos \left (d x + c\right ) - 96 \, {\left (b \cos \left (d x + c\right )^{6} - 3 \, b \cos \left (d x + c\right )^{4} + 3 \, b \cos \left (d x + c\right )^{2} - b\right )} \log \left (-\cos \left (d x + c\right )\right ) - 3 \, {\left ({\left (5 \, a - 16 \, b\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (5 \, a - 16 \, b\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (5 \, a - 16 \, b\right )} \cos \left (d x + c\right )^{2} - 5 \, a + 16 \, b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 3 \, {\left ({\left (5 \, a + 16 \, b\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (5 \, a + 16 \, b\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (5 \, a + 16 \, b\right )} \cos \left (d x + c\right )^{2} - 5 \, a - 16 \, b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 88 \, b}{96 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]
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Timed out. \[ \int \csc ^7(c+d x) (a+b \sec (c+d x)) \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.02 \[ \int \csc ^7(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {3 \, {\left (5 \, a - 16 \, b\right )} \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, {\left (5 \, a + 16 \, b\right )} \log \left (\cos \left (d x + c\right ) - 1\right ) + 96 \, b \log \left (\cos \left (d x + c\right )\right ) - \frac {2 \, {\left (15 \, a \cos \left (d x + c\right )^{5} + 24 \, b \cos \left (d x + c\right )^{4} - 40 \, a \cos \left (d x + c\right )^{3} - 60 \, b \cos \left (d x + c\right )^{2} + 33 \, a \cos \left (d x + c\right ) + 44 \, b\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1}}{96 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (126) = 252\).
Time = 0.31 (sec) , antiderivative size = 357, normalized size of antiderivative = 2.55 \[ \int \csc ^7(c+d x) (a+b \sec (c+d x)) \, dx=\frac {12 \, {\left (5 \, a + 16 \, b\right )} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 384 \, b \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {{\left (a + b - \frac {9 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {12 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {45 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {87 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {110 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {352 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{3}} - \frac {45 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {87 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {9 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {12 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{384 \, d} \]
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Time = 13.47 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.06 \[ \int \csc ^7(c+d x) (a+b \sec (c+d x)) \, dx=\frac {\frac {5\,a\,{\cos \left (c+d\,x\right )}^5}{16}+\frac {b\,{\cos \left (c+d\,x\right )}^4}{2}-\frac {5\,a\,{\cos \left (c+d\,x\right )}^3}{6}-\frac {5\,b\,{\cos \left (c+d\,x\right )}^2}{4}+\frac {11\,a\,\cos \left (c+d\,x\right )}{16}+\frac {11\,b}{12}}{d\,\left ({\cos \left (c+d\,x\right )}^6-3\,{\cos \left (c+d\,x\right )}^4+3\,{\cos \left (c+d\,x\right )}^2-1\right )}+\frac {\ln \left (\cos \left (c+d\,x\right )-1\right )\,\left (\frac {5\,a}{32}+\frac {b}{2}\right )}{d}-\frac {\ln \left (\cos \left (c+d\,x\right )+1\right )\,\left (\frac {5\,a}{32}-\frac {b}{2}\right )}{d}-\frac {b\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]
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