Integrand size = 8, antiderivative size = 76 \[ \int \csc ^7(a+b x) \, dx=-\frac {5 \text {arctanh}(\cos (a+b x))}{16 b}-\frac {5 \cot (a+b x) \csc (a+b x)}{16 b}-\frac {5 \cot (a+b x) \csc ^3(a+b x)}{24 b}-\frac {\cot (a+b x) \csc ^5(a+b x)}{6 b} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3853, 3855} \[ \int \csc ^7(a+b x) \, dx=-\frac {5 \text {arctanh}(\cos (a+b x))}{16 b}-\frac {\cot (a+b x) \csc ^5(a+b x)}{6 b}-\frac {5 \cot (a+b x) \csc ^3(a+b x)}{24 b}-\frac {5 \cot (a+b x) \csc (a+b x)}{16 b} \]
[In]
[Out]
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {\cot (a+b x) \csc ^5(a+b x)}{6 b}+\frac {5}{6} \int \csc ^5(a+b x) \, dx \\ & = -\frac {5 \cot (a+b x) \csc ^3(a+b x)}{24 b}-\frac {\cot (a+b x) \csc ^5(a+b x)}{6 b}+\frac {5}{8} \int \csc ^3(a+b x) \, dx \\ & = -\frac {5 \cot (a+b x) \csc (a+b x)}{16 b}-\frac {5 \cot (a+b x) \csc ^3(a+b x)}{24 b}-\frac {\cot (a+b x) \csc ^5(a+b x)}{6 b}+\frac {5}{16} \int \csc (a+b x) \, dx \\ & = -\frac {5 \text {arctanh}(\cos (a+b x))}{16 b}-\frac {5 \cot (a+b x) \csc (a+b x)}{16 b}-\frac {5 \cot (a+b x) \csc ^3(a+b x)}{24 b}-\frac {\cot (a+b x) \csc ^5(a+b x)}{6 b} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.99 \[ \int \csc ^7(a+b x) \, dx=-\frac {5 \csc ^2\left (\frac {1}{2} (a+b x)\right )}{64 b}-\frac {\csc ^4\left (\frac {1}{2} (a+b x)\right )}{64 b}-\frac {\csc ^6\left (\frac {1}{2} (a+b x)\right )}{384 b}-\frac {5 \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )}{16 b}+\frac {5 \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )}{16 b}+\frac {5 \sec ^2\left (\frac {1}{2} (a+b x)\right )}{64 b}+\frac {\sec ^4\left (\frac {1}{2} (a+b x)\right )}{64 b}+\frac {\sec ^6\left (\frac {1}{2} (a+b x)\right )}{384 b} \]
[In]
[Out]
Time = 0.64 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.79
method | result | size |
derivativedivides | \(\frac {\left (-\frac {\csc \left (x b +a \right )^{5}}{6}-\frac {5 \csc \left (x b +a \right )^{3}}{24}-\frac {5 \csc \left (x b +a \right )}{16}\right ) \cot \left (x b +a \right )+\frac {5 \ln \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}{16}}{b}\) | \(60\) |
default | \(\frac {\left (-\frac {\csc \left (x b +a \right )^{5}}{6}-\frac {5 \csc \left (x b +a \right )^{3}}{24}-\frac {5 \csc \left (x b +a \right )}{16}\right ) \cot \left (x b +a \right )+\frac {5 \ln \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}{16}}{b}\) | \(60\) |
parallelrisch | \(\frac {-\cot \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}+\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}-9 \cot \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}+9 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}-45 \cot \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}+45 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}+120 \ln \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )\right )}{384 b}\) | \(95\) |
norman | \(\frac {-\frac {1}{384 b}-\frac {3 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{128 b}-\frac {15 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{128 b}+\frac {15 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{8}}{128 b}+\frac {3 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{10}}{128 b}+\frac {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{12}}{384 b}}{\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}+\frac {5 \ln \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )\right )}{16 b}\) | \(115\) |
risch | \(\frac {15 \,{\mathrm e}^{11 i \left (x b +a \right )}-85 \,{\mathrm e}^{9 i \left (x b +a \right )}+198 \,{\mathrm e}^{7 i \left (x b +a \right )}+198 \,{\mathrm e}^{5 i \left (x b +a \right )}-85 \,{\mathrm e}^{3 i \left (x b +a \right )}+15 \,{\mathrm e}^{i \left (x b +a \right )}}{24 b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{6}}-\frac {5 \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right )}{16 b}+\frac {5 \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{16 b}\) | \(121\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (68) = 136\).
Time = 0.25 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.04 \[ \int \csc ^7(a+b x) \, dx=\frac {30 \, \cos \left (b x + a\right )^{5} - 80 \, \cos \left (b x + a\right )^{3} - 15 \, {\left (\cos \left (b x + a\right )^{6} - 3 \, \cos \left (b x + a\right )^{4} + 3 \, \cos \left (b x + a\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (b x + a\right )^{6} - 3 \, \cos \left (b x + a\right )^{4} + 3 \, \cos \left (b x + a\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + 66 \, \cos \left (b x + a\right )}{96 \, {\left (b \cos \left (b x + a\right )^{6} - 3 \, b \cos \left (b x + a\right )^{4} + 3 \, b \cos \left (b x + a\right )^{2} - b\right )}} \]
[In]
[Out]
\[ \int \csc ^7(a+b x) \, dx=\int \csc ^{7}{\left (a + b x \right )}\, dx \]
[In]
[Out]
none
Time = 0.18 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.20 \[ \int \csc ^7(a+b x) \, dx=\frac {\frac {2 \, {\left (15 \, \cos \left (b x + a\right )^{5} - 40 \, \cos \left (b x + a\right )^{3} + 33 \, \cos \left (b x + a\right )\right )}}{\cos \left (b x + a\right )^{6} - 3 \, \cos \left (b x + a\right )^{4} + 3 \, \cos \left (b x + a\right )^{2} - 1} - 15 \, \log \left (\cos \left (b x + a\right ) + 1\right ) + 15 \, \log \left (\cos \left (b x + a\right ) - 1\right )}{96 \, b} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (68) = 136\).
Time = 0.26 (sec) , antiderivative size = 182, normalized size of antiderivative = 2.39 \[ \int \csc ^7(a+b x) \, dx=-\frac {\frac {{\left (\frac {9 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {45 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {110 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} - 1\right )} {\left (\cos \left (b x + a\right ) + 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) - 1\right )}^{3}} + \frac {45 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {9 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {{\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} - 60 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{384 \, b} \]
[In]
[Out]
Time = 21.40 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.03 \[ \int \csc ^7(a+b x) \, dx=\frac {\frac {5\,{\cos \left (a+b\,x\right )}^5}{16}-\frac {5\,{\cos \left (a+b\,x\right )}^3}{6}+\frac {11\,\cos \left (a+b\,x\right )}{16}}{b\,\left ({\cos \left (a+b\,x\right )}^6-3\,{\cos \left (a+b\,x\right )}^4+3\,{\cos \left (a+b\,x\right )}^2-1\right )}-\frac {5\,\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{16\,b} \]
[In]
[Out]