Integrand size = 17, antiderivative size = 58 \[ \int \csc ^5(a+b x) \sec ^3(a+b x) \, dx=-\frac {3 \cot ^2(a+b x)}{2 b}-\frac {\cot ^4(a+b x)}{4 b}+\frac {3 \log (\tan (a+b x))}{b}+\frac {\tan ^2(a+b x)}{2 b} \] Output:
-3/2*cot(b*x+a)^2/b-1/4*cot(b*x+a)^4/b+3*ln(tan(b*x+a))/b+1/2*tan(b*x+a)^2 /b
Time = 0.36 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.93 \[ \int \csc ^5(a+b x) \sec ^3(a+b x) \, dx=-\frac {4 \csc ^2(a+b x)+\csc ^4(a+b x)+12 \log (\cos (a+b x))-12 \log (\sin (a+b x))-2 \sec ^2(a+b x)}{4 b} \] Input:
Integrate[Csc[a + b*x]^5*Sec[a + b*x]^3,x]
Output:
-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 = 0.39 (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 |
\(\displaystyle \frac {\tan ^2(a+b x)-\frac {1}{2} \cot ^2(a+b x)-3 \cot (a+b x)+3 \log \left (\tan ^2(a+b x)\right )}{2 b}\) |
Input:
Int[Csc[a + b*x]^5*Sec[a + b*x]^3,x]
Output:
(-3*Cot[a + b*x] - Cot[a + b*x]^2/2 + 3*Log[Tan[a + b*x]^2] + Tan[a + b*x] ^2)/(2*b)
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 = 2.42 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.05
method | result | size |
derivativedivides | \(\frac {-\frac {1}{4 \sin \left (b x +a \right )^{4} \cos \left (b x +a \right )^{2}}+\frac {3}{4 \sin \left (b x +a \right )^{2} \cos \left (b x +a \right )^{2}}-\frac {3}{2 \sin \left (b x +a \right )^{2}}+3 \ln \left (\tan \left (b x +a \right )\right )}{b}\) | \(61\) |
default | \(\frac {-\frac {1}{4 \sin \left (b x +a \right )^{4} \cos \left (b x +a \right )^{2}}+\frac {3}{4 \sin \left (b x +a \right )^{2} \cos \left (b x +a \right )^{2}}-\frac {3}{2 \sin \left (b x +a \right )^{2}}+3 \ln \left (\tan \left (b x +a \right )\right )}{b}\) | \(61\) |
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 | \(\frac {-\frac {1}{64 b}-\frac {9 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}{32 b}-\frac {9 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{10}}{32 b}-\frac {\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{12}}{64 b}+\frac {83 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{6}}{32 b}}{\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4} \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right )^{2}}+\frac {3 \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}-\frac {3 \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}{b}-\frac {3 \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}{b}\) | \(148\) |
parallelrisch | \(\frac {\left (-192 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+384 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-192\right ) \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )+\left (-192 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+384 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-192\right ) \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )+\left (192 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}-384 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+192\right ) \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{8}-18 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{6}-\cot \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}-18 \cot \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+166 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}{64 b \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{2} \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )^{2}}\) | \(219\) |
Input:
int(csc(b*x+a)^5*sec(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
1/b*(-1/4/sin(b*x+a)^4/cos(b*x+a)^2+3/4/sin(b*x+a)^2/cos(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 = 0.10 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.38 \[ \int \csc ^5(a+b x) \sec ^3(a+b x) \, dx=\frac {6 \, \cos \left (b x + a\right )^{4} - 9 \, \cos \left (b x + a\right )^{2} - 6 \, {\left (\cos \left (b x + a\right )^{6} - 2 \, \cos \left (b x + a\right )^{4} + \cos \left (b x + a\right )^{2}\right )} \log \left (\cos \left (b x + a\right )^{2}\right ) + 6 \, {\left (\cos \left (b x + a\right )^{6} - 2 \, \cos \left (b x + a\right )^{4} + \cos \left (b x + a\right )^{2}\right )} \log \left (-\frac {1}{4} \, \cos \left (b x + a\right )^{2} + \frac {1}{4}\right ) + 2}{4 \, {\left (b \cos \left (b x + a\right )^{6} - 2 \, b \cos \left (b x + a\right )^{4} + b \cos \left (b x + a\right )^{2}\right )}} \] Input:
integrate(csc(b*x+a)^5*sec(b*x+a)^3,x, algorithm="fricas")
Output:
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)
\[ \int \csc ^5(a+b x) \sec ^3(a+b x) \, dx=\int \csc ^{5}{\left (a + b x \right )} \sec ^{3}{\left (a + b x \right )}\, dx \] Input:
integrate(csc(b*x+a)**5*sec(b*x+a)**3,x)
Output:
Integral(csc(a + b*x)**5*sec(a + b*x)**3, x)
Time = 0.05 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.28 \[ \int \csc ^5(a+b x) \sec ^3(a+b x) \, dx=-\frac {\frac {6 \, \sin \left (b x + a\right )^{4} - 3 \, \sin \left (b x + a\right )^{2} - 1}{\sin \left (b x + a\right )^{6} - \sin \left (b x + a\right )^{4}} + 6 \, \log \left (\sin \left (b x + a\right )^{2} - 1\right ) - 6 \, \log \left (\sin \left (b x + a\right )^{2}\right )}{4 \, b} \] Input:
integrate(csc(b*x+a)^5*sec(b*x+a)^3,x, algorithm="maxima")
Output:
-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
Time = 0.13 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.34 \[ \int \csc ^5(a+b x) \sec ^3(a+b x) \, dx=-\frac {3 \, \log \left ({\left | \sin \left (b x + a\right )^{2} - 1 \right |}\right )}{2 \, b} + \frac {3 \, \log \left ({\left | \sin \left (b x + a\right ) \right |}\right )}{b} - \frac {6 \, \sin \left (b x + a\right )^{4} - 3 \, \sin \left (b x + a\right )^{2} - 1}{4 \, {\left (\sin \left (b x + a\right )^{2} - 1\right )} b \sin \left (b x + a\right )^{4}} \] Input:
integrate(csc(b*x+a)^5*sec(b*x+a)^3,x, algorithm="giac")
Output:
-3/2*log(abs(sin(b*x + a)^2 - 1))/b + 3*log(abs(sin(b*x + a)))/b - 1/4*(6* sin(b*x + a)^4 - 3*sin(b*x + a)^2 - 1)/((sin(b*x + a)^2 - 1)*b*sin(b*x + a )^4)
Time = 0.11 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.41 \[ \int \csc ^5(a+b x) \sec ^3(a+b x) \, dx=\frac {3\,\ln \left ({\sin \left (a+b\,x\right )}^2\right )}{2\,b}-\frac {3\,\ln \left (\cos \left (a+b\,x\right )\right )}{b}+\frac {\frac {3\,{\cos \left (a+b\,x\right )}^4}{2}-\frac {9\,{\cos \left (a+b\,x\right )}^2}{4}+\frac {1}{2}}{b\,\left ({\cos \left (a+b\,x\right )}^6-2\,{\cos \left (a+b\,x\right )}^4+{\cos \left (a+b\,x\right )}^2\right )} \] Input:
int(1/(cos(a + b*x)^3*sin(a + b*x)^5),x)
Output:
(3*log(sin(a + b*x)^2))/(2*b) - (3*log(cos(a + b*x)))/b + ((3*cos(a + b*x) ^4)/2 - (9*cos(a + b*x)^2)/4 + 1/2)/(b*(cos(a + b*x)^2 - 2*cos(a + b*x)^4 + cos(a + b*x)^6))
Time = 0.16 (sec) , antiderivative size = 185, normalized size of antiderivative = 3.19 \[ \int \csc ^5(a+b x) \sec ^3(a+b x) \, dx=\frac {-192 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right ) \sin \left (b x +a \right )^{6}+192 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right ) \sin \left (b x +a \right )^{4}-192 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right ) \sin \left (b x +a \right )^{6}+192 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right ) \sin \left (b x +a \right )^{4}+192 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \sin \left (b x +a \right )^{6}-192 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \sin \left (b x +a \right )^{4}-33 \sin \left (b x +a \right )^{6}-63 \sin \left (b x +a \right )^{4}+48 \sin \left (b x +a \right )^{2}+16}{64 \sin \left (b x +a \right )^{4} b \left (\sin \left (b x +a \right )^{2}-1\right )} \] Input:
int(csc(b*x+a)^5*sec(b*x+a)^3,x)
Output:
( - 192*log(tan((a + b*x)/2) - 1)*sin(a + b*x)**6 + 192*log(tan((a + b*x)/ 2) - 1)*sin(a + b*x)**4 - 192*log(tan((a + b*x)/2) + 1)*sin(a + b*x)**6 + 192*log(tan((a + b*x)/2) + 1)*sin(a + b*x)**4 + 192*log(tan((a + b*x)/2))* sin(a + b*x)**6 - 192*log(tan((a + b*x)/2))*sin(a + b*x)**4 - 33*sin(a + b *x)**6 - 63*sin(a + b*x)**4 + 48*sin(a + b*x)**2 + 16)/(64*sin(a + b*x)**4 *b*(sin(a + b*x)**2 - 1))