Integrand size = 17, antiderivative size = 69 \[ \int \csc ^5(a+b x) \sec ^5(a+b x) \, dx=-\frac {2 \cot ^2(a+b x)}{b}-\frac {\cot ^4(a+b x)}{4 b}+\frac {6 \log (\tan (a+b x))}{b}+\frac {2 \tan ^2(a+b x)}{b}+\frac {\tan ^4(a+b x)}{4 b} \] Output:
-2*cot(b*x+a)^2/b-1/4*cot(b*x+a)^4/b+6*ln(tan(b*x+a))/b+2*tan(b*x+a)^2/b+1 /4*tan(b*x+a)^4/b
Time = 0.11 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.32 \[ \int \csc ^5(a+b x) \sec ^5(a+b x) \, dx=32 \left (-\frac {3 \csc ^2(a+b x)}{64 b}-\frac {\csc ^4(a+b x)}{128 b}-\frac {3 \log (\cos (a+b x))}{16 b}+\frac {3 \log (\sin (a+b x))}{16 b}+\frac {3 \sec ^2(a+b x)}{64 b}+\frac {\sec ^4(a+b x)}{128 b}\right ) \] Input:
Integrate[Csc[a + b*x]^5*Sec[a + b*x]^5,x]
Output:
32*((-3*Csc[a + b*x]^2)/(64*b) - Csc[a + b*x]^4/(128*b) - (3*Log[Cos[a + b *x]])/(16*b) + (3*Log[Sin[a + b*x]])/(16*b) + (3*Sec[a + b*x]^2)/(64*b) + Sec[a + b*x]^4/(128*b))
Time = 0.40 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.88, 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.
\(\displaystyle \int \csc ^5(a+b x) \sec ^5(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \csc (a+b x)^5 \sec (a+b x)^5dx\) |
\(\Big \downarrow \) 3100 |
\(\displaystyle \frac {\int \cot ^5(a+b x) \left (\tan ^2(a+b x)+1\right )^4d\tan (a+b x)}{b}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\int \cot ^3(a+b x) \left (\tan ^2(a+b x)+1\right )^4d\tan ^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {\int \left (\cot ^3(a+b x)+4 \cot ^2(a+b x)+6 \cot (a+b x)+\tan ^2(a+b x)+4\right )d\tan ^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {1}{2} \tan ^4(a+b x)+4 \tan ^2(a+b x)-\frac {1}{2} \cot ^2(a+b x)-4 \cot (a+b x)+6 \log \left (\tan ^2(a+b x)\right )}{2 b}\) |
Input:
Int[Csc[a + b*x]^5*Sec[a + b*x]^5,x]
Output:
(-4*Cot[a + b*x] - Cot[a + b*x]^2/2 + 6*Log[Tan[a + b*x]^2] + 4*Tan[a + b* x]^2 + Tan[a + b*x]^4/2)/(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 = 2.55 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.14
method | result | size |
derivativedivides | \(\frac {\frac {1}{4 \sin \left (b x +a \right )^{4} \cos \left (b x +a \right )^{4}}-\frac {1}{2 \sin \left (b x +a \right )^{4} \cos \left (b x +a \right )^{2}}+\frac {3}{2 \sin \left (b x +a \right )^{2} \cos \left (b x +a \right )^{2}}-\frac {3}{\sin \left (b x +a \right )^{2}}+6 \ln \left (\tan \left (b x +a \right )\right )}{b}\) | \(79\) |
default | \(\frac {\frac {1}{4 \sin \left (b x +a \right )^{4} \cos \left (b x +a \right )^{4}}-\frac {1}{2 \sin \left (b x +a \right )^{4} \cos \left (b x +a \right )^{2}}+\frac {3}{2 \sin \left (b x +a \right )^{2} \cos \left (b x +a \right )^{2}}-\frac {3}{\sin \left (b x +a \right )^{2}}+6 \ln \left (\tan \left (b x +a \right )\right )}{b}\) | \(79\) |
risch | \(\frac {12 \,{\mathrm e}^{14 i \left (b x +a \right )}-44 \,{\mathrm e}^{10 i \left (b x +a \right )}-44 \,{\mathrm e}^{6 i \left (b x +a \right )}+12 \,{\mathrm e}^{2 i \left (b x +a \right )}}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{4} \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{4}}-\frac {6 \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}{b}+\frac {6 \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}{b}\) | \(112\) |
parallelrisch | \(\frac {\left (-384 \cos \left (2 b x +2 a \right )-96 \cos \left (4 b x +4 a \right )-288\right ) \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )+\left (-384 \cos \left (2 b x +2 a \right )-96 \cos \left (4 b x +4 a \right )-288\right ) \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )+\left (384 \cos \left (2 b x +2 a \right )+96 \cos \left (4 b x +4 a \right )+288\right ) \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\left (3384 \cos \left (b x +a \right )-1284 \cos \left (2 b x +2 a \right )+328 \cos \left (3 b x +3 a \right )-41 \cos \left (4 b x +4 a \right )-3227\right ) \cot \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+592 \cot \left (\frac {b x}{2}+\frac {a}{2}\right )^{2} \csc \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-2 \csc \left (\frac {b x}{2}+\frac {a}{2}\right )^{4} \left (\sec \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+16 \sec \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-140\right )}{16 b \left (\cos \left (4 b x +4 a \right )+4 \cos \left (2 b x +2 a \right )+3\right )}\) | \(258\) |
Input:
int(csc(b*x+a)^5*sec(b*x+a)^5,x,method=_RETURNVERBOSE)
Output:
1/b*(1/4/sin(b*x+a)^4/cos(b*x+a)^4-1/2/sin(b*x+a)^4/cos(b*x+a)^2+3/2/sin(b *x+a)^2/cos(b*x+a)^2-3/sin(b*x+a)^2+6*ln(tan(b*x+a)))
Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (65) = 130\).
Time = 0.10 (sec) , antiderivative size = 148, normalized size of antiderivative = 2.14 \[ \int \csc ^5(a+b x) \sec ^5(a+b x) \, dx=\frac {12 \, \cos \left (b x + a\right )^{6} - 18 \, \cos \left (b x + a\right )^{4} + 4 \, \cos \left (b x + a\right )^{2} - 12 \, {\left (\cos \left (b x + a\right )^{8} - 2 \, \cos \left (b x + a\right )^{6} + \cos \left (b x + a\right )^{4}\right )} \log \left (\cos \left (b x + a\right )^{2}\right ) + 12 \, {\left (\cos \left (b x + a\right )^{8} - 2 \, \cos \left (b x + a\right )^{6} + \cos \left (b x + a\right )^{4}\right )} \log \left (-\frac {1}{4} \, \cos \left (b x + a\right )^{2} + \frac {1}{4}\right ) + 1}{4 \, {\left (b \cos \left (b x + a\right )^{8} - 2 \, b \cos \left (b x + a\right )^{6} + b \cos \left (b x + a\right )^{4}\right )}} \] Input:
integrate(csc(b*x+a)^5*sec(b*x+a)^5,x, algorithm="fricas")
Output:
1/4*(12*cos(b*x + a)^6 - 18*cos(b*x + a)^4 + 4*cos(b*x + a)^2 - 12*(cos(b* x + a)^8 - 2*cos(b*x + a)^6 + cos(b*x + a)^4)*log(cos(b*x + a)^2) + 12*(co s(b*x + a)^8 - 2*cos(b*x + a)^6 + cos(b*x + a)^4)*log(-1/4*cos(b*x + a)^2 + 1/4) + 1)/(b*cos(b*x + a)^8 - 2*b*cos(b*x + a)^6 + b*cos(b*x + a)^4)
Timed out. \[ \int \csc ^5(a+b x) \sec ^5(a+b x) \, dx=\text {Timed out} \] Input:
integrate(csc(b*x+a)**5*sec(b*x+a)**5,x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.33 \[ \int \csc ^5(a+b x) \sec ^5(a+b x) \, dx=-\frac {\frac {12 \, \sin \left (b x + a\right )^{6} - 18 \, \sin \left (b x + a\right )^{4} + 4 \, \sin \left (b x + a\right )^{2} + 1}{\sin \left (b x + a\right )^{8} - 2 \, \sin \left (b x + a\right )^{6} + \sin \left (b x + a\right )^{4}} + 12 \, \log \left (\sin \left (b x + a\right )^{2} - 1\right ) - 12 \, \log \left (\sin \left (b x + a\right )^{2}\right )}{4 \, b} \] Input:
integrate(csc(b*x+a)^5*sec(b*x+a)^5,x, algorithm="maxima")
Output:
-1/4*((12*sin(b*x + a)^6 - 18*sin(b*x + a)^4 + 4*sin(b*x + a)^2 + 1)/(sin( b*x + a)^8 - 2*sin(b*x + a)^6 + sin(b*x + a)^4) + 12*log(sin(b*x + a)^2 - 1) - 12*log(sin(b*x + a)^2))/b
Time = 0.13 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.29 \[ \int \csc ^5(a+b x) \sec ^5(a+b x) \, dx=-\frac {3 \, \log \left ({\left | \sin \left (b x + a\right )^{2} - 1 \right |}\right )}{b} + \frac {6 \, \log \left ({\left | \sin \left (b x + a\right ) \right |}\right )}{b} - \frac {12 \, \sin \left (b x + a\right )^{6} - 18 \, \sin \left (b x + a\right )^{4} + 4 \, \sin \left (b x + a\right )^{2} + 1}{4 \, {\left (\sin \left (b x + a\right )^{4} - \sin \left (b x + a\right )^{2}\right )}^{2} b} \] Input:
integrate(csc(b*x+a)^5*sec(b*x+a)^5,x, algorithm="giac")
Output:
-3*log(abs(sin(b*x + a)^2 - 1))/b + 6*log(abs(sin(b*x + a)))/b - 1/4*(12*s in(b*x + a)^6 - 18*sin(b*x + a)^4 + 4*sin(b*x + a)^2 + 1)/((sin(b*x + a)^4 - sin(b*x + a)^2)^2*b)
Time = 25.44 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.93 \[ \int \csc ^5(a+b x) \sec ^5(a+b x) \, dx=\frac {2\,{\mathrm {tan}\left (a+b\,x\right )}^2}{b}+\frac {{\mathrm {tan}\left (a+b\,x\right )}^4}{4\,b}+\frac {6\,\ln \left (\mathrm {tan}\left (a+b\,x\right )\right )}{b}-\frac {{\mathrm {cot}\left (a+b\,x\right )}^4\,\left (2\,{\mathrm {tan}\left (a+b\,x\right )}^2+\frac {1}{4}\right )}{b} \] Input:
int(1/(cos(a + b*x)^5*sin(a + b*x)^5),x)
Output:
(2*tan(a + b*x)^2)/b + tan(a + b*x)^4/(4*b) + (6*log(tan(a + b*x)))/b - (c ot(a + b*x)^4*(2*tan(a + b*x)^2 + 1/4))/b
Time = 0.18 (sec) , antiderivative size = 269, normalized size of antiderivative = 3.90 \[ \int \csc ^5(a+b x) \sec ^5(a+b x) \, dx=\frac {-96 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right ) \sin \left (b x +a \right )^{8}+192 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right ) \sin \left (b x +a \right )^{6}-96 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right ) \sin \left (b x +a \right )^{4}-96 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right ) \sin \left (b x +a \right )^{8}+192 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right ) \sin \left (b x +a \right )^{6}-96 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right ) \sin \left (b x +a \right )^{4}+96 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \sin \left (b x +a \right )^{8}-192 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \sin \left (b x +a \right )^{6}+96 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \sin \left (b x +a \right )^{4}-41 \sin \left (b x +a \right )^{8}+34 \sin \left (b x +a \right )^{6}+31 \sin \left (b x +a \right )^{4}-16 \sin \left (b x +a \right )^{2}-4}{16 \sin \left (b x +a \right )^{4} b \left (\sin \left (b x +a \right )^{4}-2 \sin \left (b x +a \right )^{2}+1\right )} \] Input:
int(csc(b*x+a)^5*sec(b*x+a)^5,x)
Output:
( - 96*log(tan((a + b*x)/2) - 1)*sin(a + b*x)**8 + 192*log(tan((a + b*x)/2 ) - 1)*sin(a + b*x)**6 - 96*log(tan((a + b*x)/2) - 1)*sin(a + b*x)**4 - 96 *log(tan((a + b*x)/2) + 1)*sin(a + b*x)**8 + 192*log(tan((a + b*x)/2) + 1) *sin(a + b*x)**6 - 96*log(tan((a + b*x)/2) + 1)*sin(a + b*x)**4 + 96*log(t an((a + b*x)/2))*sin(a + b*x)**8 - 192*log(tan((a + b*x)/2))*sin(a + b*x)* *6 + 96*log(tan((a + b*x)/2))*sin(a + b*x)**4 - 41*sin(a + b*x)**8 + 34*si n(a + b*x)**6 + 31*sin(a + b*x)**4 - 16*sin(a + b*x)**2 - 4)/(16*sin(a + b *x)**4*b*(sin(a + b*x)**4 - 2*sin(a + b*x)**2 + 1))