Integrand size = 17, antiderivative size = 58 \[ \int \csc ^5(a+b x) \sec ^3(a+b x) \, dx=-\frac {3 \cot ^2(a+b x)}{2 b}-\frac {\cot ^4(a+b x)}{4 b}+\frac {3 \log (\tan (a+b x))}{b}+\frac {\tan ^2(a+b x)}{2 b} \]
Time = 0.22 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.93 \[ \int \csc ^5(a+b x) \sec ^3(a+b x) \, dx=-\frac {4 \csc ^2(a+b x)+\csc ^4(a+b x)+12 \log (\cos (a+b x))-12 \log (\sin (a+b x))-2 \sec ^2(a+b x)}{4 b} \]
-1/4*(4*Csc[a + b*x]^2 + Csc[a + b*x]^4 + 12*Log[Cos[a + b*x]] - 12*Log[Si n[a + b*x]] - 2*Sec[a + b*x]^2)/b
Time = 0.21 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.81, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, 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.
\(\displaystyle \int \csc ^5(a+b x) \sec ^3(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \csc (a+b x)^5 \sec (a+b x)^3dx\) |
\(\Big \downarrow \) 3100 |
\(\displaystyle \frac {\int \cot ^5(a+b x) \left (\tan ^2(a+b x)+1\right )^3d\tan (a+b x)}{b}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\int \cot ^3(a+b x) \left (\tan ^2(a+b x)+1\right )^3d\tan ^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {\int \left (\cot ^3(a+b x)+3 \cot ^2(a+b x)+3 \cot (a+b x)+1\right )d\tan ^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\tan ^2(a+b x)-\frac {1}{2} \cot ^2(a+b x)-3 \cot (a+b x)+3 \log \left (\tan ^2(a+b x)\right )}{2 b}\) |
3.2.82.3.1 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 = 0.31 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.05
method | result | size |
derivativedivides | \(\frac {-\frac {1}{4 \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{4}}+\frac {3}{4 \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{2}}-\frac {3}{2 \sin \left (b x +a \right )^{2}}+3 \ln \left (\tan \left (b x +a \right )\right )}{b}\) | \(61\) |
default | \(\frac {-\frac {1}{4 \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{4}}+\frac {3}{4 \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{2}}-\frac {3}{2 \sin \left (b x +a \right )^{2}}+3 \ln \left (\tan \left (b x +a \right )\right )}{b}\) | \(61\) |
risch | \(\frac {6 \,{\mathrm e}^{10 i \left (b x +a \right )}-12 \,{\mathrm e}^{8 i \left (b x +a \right )}-4 \,{\mathrm e}^{6 i \left (b x +a \right )}-12 \,{\mathrm e}^{4 i \left (b x +a \right )}+6 \,{\mathrm e}^{2 i \left (b x +a \right )}}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{4} \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{2}}+\frac {3 \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}{b}-\frac {3 \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}{b}\) | \(123\) |
norman | \(\frac {-\frac {1}{64 b}-\frac {9 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{32 b}-\frac {9 \left (\tan ^{10}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{32 b}-\frac {\tan ^{12}\left (\frac {b x}{2}+\frac {a}{2}\right )}{64 b}+\frac {83 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{32 b}}{\left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}}+\frac {3 \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}-\frac {3 \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}{b}-\frac {3 \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}{b}\) | \(148\) |
parallelrisch | \(\frac {\left (-192 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+384 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-192\right ) \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )+\left (-192 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+384 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-192\right ) \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )+\left (192 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-384 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+192\right ) \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\left (\tan ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-18 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\left (\cot ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-18 \left (\cot ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+166 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{64 b \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{2} \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )^{2}}\) | \(219\) |
1/b*(-1/4/cos(b*x+a)^2/sin(b*x+a)^4+3/4/cos(b*x+a)^2/sin(b*x+a)^2-3/2/sin( b*x+a)^2+3*ln(tan(b*x+a)))
Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (52) = 104\).
Time = 0.31 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.38 \[ \int \csc ^5(a+b x) \sec ^3(a+b x) \, dx=\frac {6 \, \cos \left (b x + a\right )^{4} - 9 \, \cos \left (b x + a\right )^{2} - 6 \, {\left (\cos \left (b x + a\right )^{6} - 2 \, \cos \left (b x + a\right )^{4} + \cos \left (b x + a\right )^{2}\right )} \log \left (\cos \left (b x + a\right )^{2}\right ) + 6 \, {\left (\cos \left (b x + a\right )^{6} - 2 \, \cos \left (b x + a\right )^{4} + \cos \left (b x + a\right )^{2}\right )} \log \left (-\frac {1}{4} \, \cos \left (b x + a\right )^{2} + \frac {1}{4}\right ) + 2}{4 \, {\left (b \cos \left (b x + a\right )^{6} - 2 \, b \cos \left (b x + a\right )^{4} + b \cos \left (b x + a\right )^{2}\right )}} \]
1/4*(6*cos(b*x + a)^4 - 9*cos(b*x + a)^2 - 6*(cos(b*x + a)^6 - 2*cos(b*x + a)^4 + cos(b*x + a)^2)*log(cos(b*x + a)^2) + 6*(cos(b*x + a)^6 - 2*cos(b* x + a)^4 + cos(b*x + a)^2)*log(-1/4*cos(b*x + a)^2 + 1/4) + 2)/(b*cos(b*x + a)^6 - 2*b*cos(b*x + a)^4 + b*cos(b*x + a)^2)
\[ \int \csc ^5(a+b x) \sec ^3(a+b x) \, dx=\int \frac {\sec ^{3}{\left (a + b x \right )}}{\sin ^{5}{\left (a + b x \right )}}\, dx \]
Time = 0.19 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.28 \[ \int \csc ^5(a+b x) \sec ^3(a+b x) \, dx=-\frac {\frac {6 \, \sin \left (b x + a\right )^{4} - 3 \, \sin \left (b x + a\right )^{2} - 1}{\sin \left (b x + a\right )^{6} - \sin \left (b x + a\right )^{4}} + 6 \, \log \left (\sin \left (b x + a\right )^{2} - 1\right ) - 6 \, \log \left (\sin \left (b x + a\right )^{2}\right )}{4 \, b} \]
-1/4*((6*sin(b*x + a)^4 - 3*sin(b*x + a)^2 - 1)/(sin(b*x + a)^6 - sin(b*x + a)^4) + 6*log(sin(b*x + a)^2 - 1) - 6*log(sin(b*x + a)^2))/b
Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (52) = 104\).
Time = 0.51 (sec) , antiderivative size = 232, normalized size of antiderivative = 4.00 \[ \int \csc ^5(a+b x) \sec ^3(a+b x) \, dx=\frac {\frac {20 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {\frac {18 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {111 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {36 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} + \frac {72 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} - 1}{{\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + \frac {{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}}\right )}^{2}} + 96 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right ) - 192 \, \log \left ({\left | -\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1 \right |}\right )}{64 \, b} \]
1/64*(20*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) - (cos(b*x + a) - 1)^2/(cos (b*x + a) + 1)^2 + (18*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) + 111*(cos(b* x + a) - 1)^2/(cos(b*x + a) + 1)^2 + 36*(cos(b*x + a) - 1)^3/(cos(b*x + a) + 1)^3 + 72*(cos(b*x + a) - 1)^4/(cos(b*x + a) + 1)^4 - 1)/((cos(b*x + a) - 1)/(cos(b*x + a) + 1) + (cos(b*x + a) - 1)^2/(cos(b*x + a) + 1)^2)^2 + 96*log(abs(-cos(b*x + a) + 1)/abs(cos(b*x + a) + 1)) - 192*log(abs(-(cos(b *x + a) - 1)/(cos(b*x + a) + 1) - 1)))/b
Time = 0.11 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.41 \[ \int \csc ^5(a+b x) \sec ^3(a+b x) \, dx=\frac {3\,\ln \left ({\sin \left (a+b\,x\right )}^2\right )}{2\,b}-\frac {3\,\ln \left (\cos \left (a+b\,x\right )\right )}{b}+\frac {\frac {3\,{\cos \left (a+b\,x\right )}^4}{2}-\frac {9\,{\cos \left (a+b\,x\right )}^2}{4}+\frac {1}{2}}{b\,\left ({\cos \left (a+b\,x\right )}^6-2\,{\cos \left (a+b\,x\right )}^4+{\cos \left (a+b\,x\right )}^2\right )} \]