Integrand size = 15, antiderivative size = 57 \[ \int \csc (a+b x) \sec ^7(a+b x) \, dx=\frac {\log (\tan (a+b x))}{b}+\frac {3 \tan ^2(a+b x)}{2 b}+\frac {3 \tan ^4(a+b x)}{4 b}+\frac {\tan ^6(a+b x)}{6 b} \]
Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.21 \[ \int \csc (a+b x) \sec ^7(a+b x) \, dx=-\frac {\log (\cos (a+b x))}{b}+\frac {\log (\sin (a+b x))}{b}+\frac {\sec ^2(a+b x)}{2 b}+\frac {\sec ^4(a+b x)}{4 b}+\frac {\sec ^6(a+b x)}{6 b} \]
-(Log[Cos[a + b*x]]/b) + Log[Sin[a + b*x]]/b + Sec[a + b*x]^2/(2*b) + Sec[ a + b*x]^4/(4*b) + Sec[a + b*x]^6/(6*b)
Time = 0.20 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.89, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3100, 243, 49, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \csc (a+b x) \sec ^7(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \csc (a+b x) \sec (a+b x)^7dx\) |
\(\Big \downarrow \) 3100 |
\(\displaystyle \frac {\int \cot (a+b x) \left (\tan ^2(a+b x)+1\right )^3d\tan (a+b x)}{b}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\int \cot (a+b x) \left (\tan ^2(a+b x)+1\right )^3d\tan ^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {\int \left (\tan ^4(a+b x)+3 \tan ^2(a+b x)+\cot (a+b x)+3\right )d\tan ^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {1}{3} \tan ^6(a+b x)+\frac {3}{2} \tan ^4(a+b x)+3 \tan ^2(a+b x)+\log \left (\tan ^2(a+b x)\right )}{2 b}\) |
3.2.32.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Simp[1/f Subst[Int[(1 + x^2)^((m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]] , x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]
Time = 0.35 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(\frac {\frac {1}{6 \cos \left (b x +a \right )^{6}}+\frac {1}{4 \cos \left (b x +a \right )^{4}}+\frac {1}{2 \cos \left (b x +a \right )^{2}}+\ln \left (\tan \left (b x +a \right )\right )}{b}\) | \(43\) |
default | \(\frac {\frac {1}{6 \cos \left (b x +a \right )^{6}}+\frac {1}{4 \cos \left (b x +a \right )^{4}}+\frac {1}{2 \cos \left (b x +a \right )^{2}}+\ln \left (\tan \left (b x +a \right )\right )}{b}\) | \(43\) |
risch | \(\frac {2 \,{\mathrm e}^{10 i \left (b x +a \right )}+12 \,{\mathrm e}^{8 i \left (b x +a \right )}+\frac {92 \,{\mathrm e}^{6 i \left (b x +a \right )}}{3}+12 \,{\mathrm e}^{4 i \left (b x +a \right )}+2 \,{\mathrm e}^{2 i \left (b x +a \right )}}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{6}}-\frac {\ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}{b}+\frac {\ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}{b}\) | \(109\) |
norman | \(\frac {\frac {6 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}+\frac {6 \left (\tan ^{10}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}-\frac {12 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}-\frac {12 \left (\tan ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}+\frac {68 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 b}}{\left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{6}}+\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}-\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}{b}-\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}{b}\) | \(147\) |
parallelrisch | \(\frac {\left (-180 \cos \left (2 b x +2 a \right )-72 \cos \left (4 b x +4 a \right )-12 \cos \left (6 b x +6 a \right )-120\right ) \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )+\left (-180 \cos \left (2 b x +2 a \right )-72 \cos \left (4 b x +4 a \right )-12 \cos \left (6 b x +6 a \right )-120\right ) \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )+\left (180 \cos \left (2 b x +2 a \right )+72 \cos \left (4 b x +4 a \right )+12 \cos \left (6 b x +6 a \right )+120\right ) \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )-21 \cos \left (2 b x +2 a \right )-42 \cos \left (4 b x +4 a \right )-11 \cos \left (6 b x +6 a \right )+74}{12 b \left (\cos \left (6 b x +6 a \right )+6 \cos \left (4 b x +4 a \right )+15 \cos \left (2 b x +2 a \right )+10\right )}\) | \(218\) |
Time = 0.31 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.35 \[ \int \csc (a+b x) \sec ^7(a+b x) \, dx=-\frac {6 \, \cos \left (b x + a\right )^{6} \log \left (\cos \left (b x + a\right )^{2}\right ) - 6 \, \cos \left (b x + a\right )^{6} \log \left (-\frac {1}{4} \, \cos \left (b x + a\right )^{2} + \frac {1}{4}\right ) - 6 \, \cos \left (b x + a\right )^{4} - 3 \, \cos \left (b x + a\right )^{2} - 2}{12 \, b \cos \left (b x + a\right )^{6}} \]
-1/12*(6*cos(b*x + a)^6*log(cos(b*x + a)^2) - 6*cos(b*x + a)^6*log(-1/4*co s(b*x + a)^2 + 1/4) - 6*cos(b*x + a)^4 - 3*cos(b*x + a)^2 - 2)/(b*cos(b*x + a)^6)
\[ \int \csc (a+b x) \sec ^7(a+b x) \, dx=\int \frac {\sec ^{7}{\left (a + b x \right )}}{\sin {\left (a + b x \right )}}\, dx \]
Time = 0.19 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.49 \[ \int \csc (a+b x) \sec ^7(a+b x) \, dx=-\frac {\frac {6 \, \sin \left (b x + a\right )^{4} - 15 \, \sin \left (b x + a\right )^{2} + 11}{\sin \left (b x + a\right )^{6} - 3 \, \sin \left (b x + a\right )^{4} + 3 \, \sin \left (b x + a\right )^{2} - 1} + 6 \, \log \left (\sin \left (b x + a\right )^{2} - 1\right ) - 6 \, \log \left (\sin \left (b x + a\right )^{2}\right )}{12 \, b} \]
-1/12*((6*sin(b*x + a)^4 - 15*sin(b*x + a)^2 + 11)/(sin(b*x + a)^6 - 3*sin (b*x + a)^4 + 3*sin(b*x + a)^2 - 1) + 6*log(sin(b*x + a)^2 - 1) - 6*log(si n(b*x + a)^2))/b
Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (51) = 102\).
Time = 0.34 (sec) , antiderivative size = 214, normalized size of antiderivative = 3.75 \[ \int \csc (a+b x) \sec ^7(a+b x) \, dx=\frac {\frac {\frac {522 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {1485 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {1580 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} + \frac {1485 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + \frac {522 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}} + \frac {147 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{6}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{6}} + 147}{{\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1\right )}^{6}} + 30 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right ) - 60 \, \log \left ({\left | -\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1 \right |}\right )}{60 \, b} \]
1/60*((522*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) + 1485*(cos(b*x + a) - 1) ^2/(cos(b*x + a) + 1)^2 + 1580*(cos(b*x + a) - 1)^3/(cos(b*x + a) + 1)^3 + 1485*(cos(b*x + a) - 1)^4/(cos(b*x + a) + 1)^4 + 522*(cos(b*x + a) - 1)^5 /(cos(b*x + a) + 1)^5 + 147*(cos(b*x + a) - 1)^6/(cos(b*x + a) + 1)^6 + 14 7)/((cos(b*x + a) - 1)/(cos(b*x + a) + 1) + 1)^6 + 30*log(abs(-cos(b*x + a ) + 1)/abs(cos(b*x + a) + 1)) - 60*log(abs(-(cos(b*x + a) - 1)/(cos(b*x + a) + 1) - 1)))/b
Time = 0.13 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.98 \[ \int \csc (a+b x) \sec ^7(a+b x) \, dx=\frac {\frac {\ln \left ({\sin \left (a+b\,x\right )}^2\right )}{2}-\ln \left (\cos \left (a+b\,x\right )\right )+\frac {\frac {{\cos \left (a+b\,x\right )}^4}{2}+\frac {{\cos \left (a+b\,x\right )}^2}{4}+\frac {1}{6}}{{\cos \left (a+b\,x\right )}^6}}{b} \]