Integrand size = 19, antiderivative size = 101 \[ \int \csc ^6(c+d x) (a+b \sec (c+d x)) \, dx=\frac {b \text {arctanh}(\sin (c+d x))}{d}-\frac {a \cot (c+d x)}{d}-\frac {2 a \cot ^3(c+d x)}{3 d}-\frac {a \cot ^5(c+d x)}{5 d}-\frac {b \csc (c+d x)}{d}-\frac {b \csc ^3(c+d x)}{3 d}-\frac {b \csc ^5(c+d x)}{5 d} \]
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Time = 0.14 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3957, 2917, 2701, 308, 213, 3852} \[ \int \csc ^6(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {a \cot ^5(c+d x)}{5 d}-\frac {2 a \cot ^3(c+d x)}{3 d}-\frac {a \cot (c+d x)}{d}+\frac {b \text {arctanh}(\sin (c+d x))}{d}-\frac {b \csc ^5(c+d x)}{5 d}-\frac {b \csc ^3(c+d x)}{3 d}-\frac {b \csc (c+d x)}{d} \]
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Rule 213
Rule 308
Rule 2701
Rule 2917
Rule 3852
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int (-b-a \cos (c+d x)) \csc ^6(c+d x) \sec (c+d x) \, dx \\ & = a \int \csc ^6(c+d x) \, dx+b \int \csc ^6(c+d x) \sec (c+d x) \, dx \\ & = -\frac {a \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac {b \text {Subst}\left (\int \frac {x^6}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d} \\ & = -\frac {a \cot (c+d x)}{d}-\frac {2 a \cot ^3(c+d x)}{3 d}-\frac {a \cot ^5(c+d x)}{5 d}-\frac {b \text {Subst}\left (\int \left (1+x^2+x^4+\frac {1}{-1+x^2}\right ) \, dx,x,\csc (c+d x)\right )}{d} \\ & = -\frac {a \cot (c+d x)}{d}-\frac {2 a \cot ^3(c+d x)}{3 d}-\frac {a \cot ^5(c+d x)}{5 d}-\frac {b \csc (c+d x)}{d}-\frac {b \csc ^3(c+d x)}{3 d}-\frac {b \csc ^5(c+d x)}{5 d}-\frac {b \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d} \\ & = \frac {b \text {arctanh}(\sin (c+d x))}{d}-\frac {a \cot (c+d x)}{d}-\frac {2 a \cot ^3(c+d x)}{3 d}-\frac {a \cot ^5(c+d x)}{5 d}-\frac {b \csc (c+d x)}{d}-\frac {b \csc ^3(c+d x)}{3 d}-\frac {b \csc ^5(c+d x)}{5 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.90 \[ \int \csc ^6(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {8 a \cot (c+d x)}{15 d}-\frac {4 a \cot (c+d x) \csc ^2(c+d x)}{15 d}-\frac {a \cot (c+d x) \csc ^4(c+d x)}{5 d}-\frac {b \csc ^5(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},\sin ^2(c+d x)\right )}{5 d} \]
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Time = 1.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {8}{15}-\frac {\csc \left (d x +c \right )^{4}}{5}-\frac {4 \csc \left (d x +c \right )^{2}}{15}\right ) \cot \left (d x +c \right )+b \left (-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )}{d}\) | \(83\) |
default | \(\frac {a \left (-\frac {8}{15}-\frac {\csc \left (d x +c \right )^{4}}{5}-\frac {4 \csc \left (d x +c \right )^{2}}{15}\right ) \cot \left (d x +c \right )+b \left (-\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )}{d}\) | \(83\) |
parallelrisch | \(\frac {-960 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+960 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \left (a \cos \left (5 d x +5 c \right )+10 a \cos \left (d x +c \right )-5 a \cos \left (3 d x +3 c \right )-20 \cos \left (2 d x +2 c \right ) b +\frac {15 b \cos \left (4 d x +4 c \right )}{4}+\frac {89 b}{4}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{960 d}\) | \(121\) |
risch | \(-\frac {2 i \left (15 b \,{\mathrm e}^{9 i \left (d x +c \right )}-80 b \,{\mathrm e}^{7 i \left (d x +c \right )}+178 b \,{\mathrm e}^{5 i \left (d x +c \right )}+80 a \,{\mathrm e}^{4 i \left (d x +c \right )}-80 b \,{\mathrm e}^{3 i \left (d x +c \right )}-40 a \,{\mathrm e}^{2 i \left (d x +c \right )}+15 b \,{\mathrm e}^{i \left (d x +c \right )}+8 a \right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}-\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}\) | \(146\) |
norman | \(\frac {-\frac {a +b}{160 d}+\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{160 d}+\frac {\left (5 a -11 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{16 d}+\frac {\left (5 a -7 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{96 d}-\frac {\left (5 a +7 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{96 d}-\frac {\left (5 a +11 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{16 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(171\) |
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Time = 0.28 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.72 \[ \int \csc ^6(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {16 \, a \cos \left (d x + c\right )^{5} + 30 \, b \cos \left (d x + c\right )^{4} - 40 \, a \cos \left (d x + c\right )^{3} - 70 \, b \cos \left (d x + c\right )^{2} - 15 \, {\left (b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + b\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 15 \, {\left (b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + b\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 30 \, a \cos \left (d x + c\right ) + 46 \, b}{30 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
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\[ \int \csc ^6(c+d x) (a+b \sec (c+d x)) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right ) \csc ^{6}{\left (c + d x \right )}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.95 \[ \int \csc ^6(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {b {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{4} + 5 \, \sin \left (d x + c\right )^{2} + 3\right )}}{\sin \left (d x + c\right )^{5}} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + \frac {2 \, {\left (15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} + 3\right )} a}{\tan \left (d x + c\right )^{5}}}{30 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (93) = 186\).
Time = 0.31 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.92 \[ \int \csc ^6(c+d x) (a+b \sec (c+d x)) \, dx=\frac {3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 25 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 35 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 480 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 480 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 150 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 330 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {150 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 330 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 25 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, a + 3 \, b}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{480 \, d} \]
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Time = 14.36 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.41 \[ \int \csc ^6(c+d x) (a+b \sec (c+d x)) \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {5\,a}{96}-\frac {7\,b}{96}\right )}{d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\left (10\,a+22\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (\frac {5\,a}{3}+\frac {7\,b}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {a}{5}+\frac {b}{5}\right )}{32\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {a}{160}-\frac {b}{160}\right )}{d}+\frac {2\,b\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {5\,a}{16}-\frac {11\,b}{16}\right )}{d} \]
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