Integrand size = 15, antiderivative size = 57 \[ \int \csc (a+b x) \sec ^7(a+b x) \, dx=\frac {\log (\tan (a+b x))}{b}+\frac {3 \tan ^2(a+b x)}{2 b}+\frac {3 \tan ^4(a+b x)}{4 b}+\frac {\tan ^6(a+b x)}{6 b} \] Output:
ln(tan(b*x+a))/b+3/2*tan(b*x+a)^2/b+3/4*tan(b*x+a)^4/b+1/6*tan(b*x+a)^6/b
Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.21 \[ \int \csc (a+b x) \sec ^7(a+b x) \, dx=-\frac {\log (\cos (a+b x))}{b}+\frac {\log (\sin (a+b x))}{b}+\frac {\sec ^2(a+b x)}{2 b}+\frac {\sec ^4(a+b x)}{4 b}+\frac {\sec ^6(a+b x)}{6 b} \] Input:
Integrate[Csc[a + b*x]*Sec[a + b*x]^7,x]
Output:
-(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 = 0.36 (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 |
\(\displaystyle \frac {\frac {1}{3} \tan ^6(a+b x)+\frac {3}{2} \tan ^4(a+b x)+3 \tan ^2(a+b x)+\log \left (\tan ^2(a+b x)\right )}{2 b}\) |
Input:
Int[Csc[a + b*x]*Sec[a + b*x]^7,x]
Output:
(Log[Tan[a + b*x]^2] + 3*Tan[a + b*x]^2 + (3*Tan[a + b*x]^4)/2 + Tan[a + b *x]^6/3)/(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 = 3.15 (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 | \(\frac {\frac {6 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}{b}+\frac {6 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{10}}{b}-\frac {12 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}}{b}-\frac {12 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{8}}{b}+\frac {68 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{6}}{3 b}}{\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right )^{6}}+\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}-\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}{b}-\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}{b}\) | \(147\) |
parallelrisch | \(\frac {\left (-180 \cos \left (2 b x +2 a \right )-72 \cos \left (4 b x +4 a \right )-12 \cos \left (6 b x +6 a \right )-120\right ) \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )+\left (-180 \cos \left (2 b x +2 a \right )-72 \cos \left (4 b x +4 a \right )-12 \cos \left (6 b x +6 a \right )-120\right ) \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )+\left (180 \cos \left (2 b x +2 a \right )+72 \cos \left (4 b x +4 a \right )+12 \cos \left (6 b x +6 a \right )+120\right ) \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )-21 \cos \left (2 b x +2 a \right )-42 \cos \left (4 b x +4 a \right )-11 \cos \left (6 b x +6 a \right )+74}{12 b \left (\cos \left (6 b x +6 a \right )+6 \cos \left (4 b x +4 a \right )+15 \cos \left (2 b x +2 a \right )+10\right )}\) | \(218\) |
Input:
int(csc(b*x+a)*sec(b*x+a)^7,x,method=_RETURNVERBOSE)
Output:
1/b*(1/6/cos(b*x+a)^6+1/4/cos(b*x+a)^4+1/2/cos(b*x+a)^2+ln(tan(b*x+a)))
Time = 0.11 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.35 \[ \int \csc (a+b x) \sec ^7(a+b x) \, dx=-\frac {6 \, \cos \left (b x + a\right )^{6} \log \left (\cos \left (b x + a\right )^{2}\right ) - 6 \, \cos \left (b x + a\right )^{6} \log \left (-\frac {1}{4} \, \cos \left (b x + a\right )^{2} + \frac {1}{4}\right ) - 6 \, \cos \left (b x + a\right )^{4} - 3 \, \cos \left (b x + a\right )^{2} - 2}{12 \, b \cos \left (b x + a\right )^{6}} \] Input:
integrate(csc(b*x+a)*sec(b*x+a)^7,x, algorithm="fricas")
Output:
-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)
\[ \int \csc (a+b x) \sec ^7(a+b x) \, dx=\int \csc {\left (a + b x \right )} \sec ^{7}{\left (a + b x \right )}\, dx \] Input:
integrate(csc(b*x+a)*sec(b*x+a)**7,x)
Output:
Integral(csc(a + b*x)*sec(a + b*x)**7, x)
Time = 0.04 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.49 \[ \int \csc (a+b x) \sec ^7(a+b x) \, dx=-\frac {\frac {6 \, \sin \left (b x + a\right )^{4} - 15 \, \sin \left (b x + a\right )^{2} + 11}{\sin \left (b x + a\right )^{6} - 3 \, \sin \left (b x + a\right )^{4} + 3 \, \sin \left (b x + a\right )^{2} - 1} + 6 \, \log \left (\sin \left (b x + a\right )^{2} - 1\right ) - 6 \, \log \left (\sin \left (b x + a\right )^{2}\right )}{12 \, b} \] Input:
integrate(csc(b*x+a)*sec(b*x+a)^7,x, algorithm="maxima")
Output:
-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
Time = 0.13 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.16 \[ \int \csc (a+b x) \sec ^7(a+b x) \, dx=\frac {\log \left ({\left | \cos \left (b x + a\right )^{2} - 1 \right |}\right )}{2 \, b} - \frac {\log \left ({\left | \cos \left (b x + a\right ) \right |}\right )}{b} + \frac {6 \, \cos \left (b x + a\right )^{4} + 3 \, \cos \left (b x + a\right )^{2} + 2}{12 \, b \cos \left (b x + a\right )^{6}} \] Input:
integrate(csc(b*x+a)*sec(b*x+a)^7,x, algorithm="giac")
Output:
1/2*log(abs(cos(b*x + a)^2 - 1))/b - log(abs(cos(b*x + a)))/b + 1/12*(6*co s(b*x + a)^4 + 3*cos(b*x + a)^2 + 2)/(b*cos(b*x + a)^6)
Time = 25.30 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.98 \[ \int \csc (a+b x) \sec ^7(a+b x) \, dx=\frac {\frac {\ln \left ({\sin \left (a+b\,x\right )}^2\right )}{2}-\ln \left (\cos \left (a+b\,x\right )\right )+\frac {\frac {{\cos \left (a+b\,x\right )}^4}{2}+\frac {{\cos \left (a+b\,x\right )}^2}{4}+\frac {1}{6}}{{\cos \left (a+b\,x\right )}^6}}{b} \] Input:
int(1/(cos(a + b*x)^7*sin(a + b*x)),x)
Output:
(log(sin(a + b*x)^2)/2 - log(cos(a + b*x)) + (cos(a + b*x)^2/4 + cos(a + b *x)^4/2 + 1/6)/cos(a + b*x)^6)/b
Time = 0.27 (sec) , antiderivative size = 300, normalized size of antiderivative = 5.26 \[ \int \csc (a+b x) \sec ^7(a+b x) \, dx=\frac {-12 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right ) \sin \left (b x +a \right )^{6}+36 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right ) \sin \left (b x +a \right )^{4}-36 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right ) \sin \left (b x +a \right )^{2}+12 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )-12 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right ) \sin \left (b x +a \right )^{6}+36 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right ) \sin \left (b x +a \right )^{4}-36 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right ) \sin \left (b x +a \right )^{2}+12 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )+12 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \sin \left (b x +a \right )^{6}-36 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \sin \left (b x +a \right )^{4}+36 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \sin \left (b x +a \right )^{2}-12 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )-11 \sin \left (b x +a \right )^{6}+27 \sin \left (b x +a \right )^{4}-18 \sin \left (b x +a \right )^{2}}{12 b \left (\sin \left (b x +a \right )^{6}-3 \sin \left (b x +a \right )^{4}+3 \sin \left (b x +a \right )^{2}-1\right )} \] Input:
int(csc(b*x+a)*sec(b*x+a)^7,x)
Output:
( - 12*log(tan((a + b*x)/2) - 1)*sin(a + b*x)**6 + 36*log(tan((a + b*x)/2) - 1)*sin(a + b*x)**4 - 36*log(tan((a + b*x)/2) - 1)*sin(a + b*x)**2 + 12* log(tan((a + b*x)/2) - 1) - 12*log(tan((a + b*x)/2) + 1)*sin(a + b*x)**6 + 36*log(tan((a + b*x)/2) + 1)*sin(a + b*x)**4 - 36*log(tan((a + b*x)/2) + 1)*sin(a + b*x)**2 + 12*log(tan((a + b*x)/2) + 1) + 12*log(tan((a + b*x)/2 ))*sin(a + b*x)**6 - 36*log(tan((a + b*x)/2))*sin(a + b*x)**4 + 36*log(tan ((a + b*x)/2))*sin(a + b*x)**2 - 12*log(tan((a + b*x)/2)) - 11*sin(a + b*x )**6 + 27*sin(a + b*x)**4 - 18*sin(a + b*x)**2)/(12*b*(sin(a + b*x)**6 - 3 *sin(a + b*x)**4 + 3*sin(a + b*x)**2 - 1))