Integrand size = 7, antiderivative size = 50 \[ \int \frac {1}{(\cos (x)+\sin (x))^6} \, dx=-\frac {\cos (x)-\sin (x)}{10 (\cos (x)+\sin (x))^5}-\frac {\cos (x)-\sin (x)}{15 (\cos (x)+\sin (x))^3}+\frac {2 \sin (x)}{15 (\cos (x)+\sin (x))} \] Output:
1/10*(-cos(x)+sin(x))/(cos(x)+sin(x))^5+1/15*(-cos(x)+sin(x))/(cos(x)+sin( x))^3+2/15*sin(x)/(cos(x)+sin(x))
Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.52 \[ \int \frac {1}{(\cos (x)+\sin (x))^6} \, dx=-\frac {5 \cos (3 x)-10 \sin (x)+\sin (5 x)}{30 (\cos (x)+\sin (x))^5} \] Input:
Integrate[(Cos[x] + Sin[x])^(-6),x]
Output:
-1/30*(5*Cos[3*x] - 10*Sin[x] + Sin[5*x])/(Cos[x] + Sin[x])^5
Time = 0.27 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {3042, 3555, 3042, 3555, 3042, 3554}
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.
\(\displaystyle \int \frac {1}{(\sin (x)+\cos (x))^6} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(\sin (x)+\cos (x))^6}dx\) |
\(\Big \downarrow \) 3555 |
\(\displaystyle \frac {2}{5} \int \frac {1}{(\cos (x)+\sin (x))^4}dx-\frac {\cos (x)-\sin (x)}{10 (\sin (x)+\cos (x))^5}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{5} \int \frac {1}{(\cos (x)+\sin (x))^4}dx-\frac {\cos (x)-\sin (x)}{10 (\sin (x)+\cos (x))^5}\) |
\(\Big \downarrow \) 3555 |
\(\displaystyle \frac {2}{5} \left (\frac {1}{3} \int \frac {1}{(\cos (x)+\sin (x))^2}dx-\frac {\cos (x)-\sin (x)}{6 (\sin (x)+\cos (x))^3}\right )-\frac {\cos (x)-\sin (x)}{10 (\sin (x)+\cos (x))^5}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{5} \left (\frac {1}{3} \int \frac {1}{(\cos (x)+\sin (x))^2}dx-\frac {\cos (x)-\sin (x)}{6 (\sin (x)+\cos (x))^3}\right )-\frac {\cos (x)-\sin (x)}{10 (\sin (x)+\cos (x))^5}\) |
\(\Big \downarrow \) 3554 |
\(\displaystyle \frac {2}{5} \left (\frac {\sin (x)}{3 (\sin (x)+\cos (x))}-\frac {\cos (x)-\sin (x)}{6 (\sin (x)+\cos (x))^3}\right )-\frac {\cos (x)-\sin (x)}{10 (\sin (x)+\cos (x))^5}\) |
Input:
Int[(Cos[x] + Sin[x])^(-6),x]
Output:
-1/10*(Cos[x] - Sin[x])/(Cos[x] + Sin[x])^5 + (2*(-1/6*(Cos[x] - Sin[x])/( Cos[x] + Sin[x])^3 + Sin[x]/(3*(Cos[x] + Sin[x]))))/5
Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-2), x _Symbol] :> Simp[Sin[c + d*x]/(a*d*(a*Cos[c + d*x] + b*Sin[c + d*x])), x] / ; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x _Symbol] :> Simp[(b*Cos[c + d*x] - a*Sin[c + d*x])*((a*Cos[c + d*x] + b*Sin [c + d*x])^(n + 1)/(d*(n + 1)*(a^2 + b^2))), x] + Simp[(n + 2)/((n + 1)*(a^ 2 + b^2)) Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1] && NeQ[n, -2]
Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.60
method | result | size |
risch | \(\frac {-\frac {2}{15}+\frac {4 \,{\mathrm e}^{4 i x}}{3}+\frac {2 i {\mathrm e}^{2 i x}}{3}}{\left ({\mathrm e}^{2 i x}+i\right )^{5}}\) | \(30\) |
default | \(-\frac {4}{5 \left (\tan \left (x \right )+1\right )^{5}}+\frac {2}{\left (\tan \left (x \right )+1\right )^{4}}-\frac {1}{\tan \left (x \right )+1}+\frac {2}{\left (\tan \left (x \right )+1\right )^{2}}-\frac {8}{3 \left (\tan \left (x \right )+1\right )^{3}}\) | \(42\) |
norman | \(\frac {-8 \tan \left (\frac {x}{2}\right )^{2}-2 \tan \left (\frac {x}{2}\right )-2 \tan \left (\frac {x}{2}\right )^{9}+8 \tan \left (\frac {x}{2}\right )^{8}-\frac {40 \tan \left (\frac {x}{2}\right )^{3}}{3}-\frac {40 \tan \left (\frac {x}{2}\right )^{7}}{3}-\frac {8 \tan \left (\frac {x}{2}\right )^{6}}{3}+\frac {8 \tan \left (\frac {x}{2}\right )^{4}}{3}+\frac {236 \tan \left (\frac {x}{2}\right )^{5}}{15}}{\left (\tan \left (\frac {x}{2}\right )^{2}-2 \tan \left (\frac {x}{2}\right )-1\right )^{5}}\) | \(89\) |
parallelrisch | \(\frac {-8 \tan \left (\frac {x}{2}\right )^{2}-2 \tan \left (\frac {x}{2}\right )-2 \tan \left (\frac {x}{2}\right )^{9}+8 \tan \left (\frac {x}{2}\right )^{8}-\frac {40 \tan \left (\frac {x}{2}\right )^{3}}{3}-\frac {40 \tan \left (\frac {x}{2}\right )^{7}}{3}-\frac {8 \tan \left (\frac {x}{2}\right )^{6}}{3}+\frac {8 \tan \left (\frac {x}{2}\right )^{4}}{3}+\frac {236 \tan \left (\frac {x}{2}\right )^{5}}{15}}{\left (\tan \left (\frac {x}{2}\right )^{2}-2 \tan \left (\frac {x}{2}\right )-1\right )^{5}}\) | \(90\) |
Input:
int(1/(cos(x)+sin(x))^6,x,method=_RETURNVERBOSE)
Output:
2/15*(-1+10*exp(4*I*x)+5*I*exp(2*I*x))/(exp(2*I*x)+I)^5
Time = 0.07 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.34 \[ \int \frac {1}{(\cos (x)+\sin (x))^6} \, dx=-\frac {8 \, \cos \left (x\right )^{5} - 20 \, \cos \left (x\right )^{3} - {\left (8 \, \cos \left (x\right )^{4} + 4 \, \cos \left (x\right )^{2} - 7\right )} \sin \left (x\right ) + 5 \, \cos \left (x\right )}{30 \, {\left (4 \, \cos \left (x\right )^{5} + {\left (4 \, \cos \left (x\right )^{4} - 8 \, \cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) - 5 \, \cos \left (x\right )\right )}} \] Input:
integrate(1/(cos(x)+sin(x))^6,x, algorithm="fricas")
Output:
-1/30*(8*cos(x)^5 - 20*cos(x)^3 - (8*cos(x)^4 + 4*cos(x)^2 - 7)*sin(x) + 5 *cos(x))/(4*cos(x)^5 + (4*cos(x)^4 - 8*cos(x)^2 - 1)*sin(x) - 5*cos(x))
Leaf count of result is larger than twice the leaf count of optimal. 838 vs. \(2 (51) = 102\).
Time = 2.35 (sec) , antiderivative size = 838, normalized size of antiderivative = 16.76 \[ \int \frac {1}{(\cos (x)+\sin (x))^6} \, dx=\text {Too large to display} \] Input:
integrate(1/(cos(x)+sin(x))**6,x)
Output:
-30*tan(x/2)**9/(15*tan(x/2)**10 - 150*tan(x/2)**9 + 525*tan(x/2)**8 - 600 *tan(x/2)**7 - 450*tan(x/2)**6 + 1020*tan(x/2)**5 + 450*tan(x/2)**4 - 600* tan(x/2)**3 - 525*tan(x/2)**2 - 150*tan(x/2) - 15) + 120*tan(x/2)**8/(15*t an(x/2)**10 - 150*tan(x/2)**9 + 525*tan(x/2)**8 - 600*tan(x/2)**7 - 450*ta n(x/2)**6 + 1020*tan(x/2)**5 + 450*tan(x/2)**4 - 600*tan(x/2)**3 - 525*tan (x/2)**2 - 150*tan(x/2) - 15) - 200*tan(x/2)**7/(15*tan(x/2)**10 - 150*tan (x/2)**9 + 525*tan(x/2)**8 - 600*tan(x/2)**7 - 450*tan(x/2)**6 + 1020*tan( x/2)**5 + 450*tan(x/2)**4 - 600*tan(x/2)**3 - 525*tan(x/2)**2 - 150*tan(x/ 2) - 15) - 40*tan(x/2)**6/(15*tan(x/2)**10 - 150*tan(x/2)**9 + 525*tan(x/2 )**8 - 600*tan(x/2)**7 - 450*tan(x/2)**6 + 1020*tan(x/2)**5 + 450*tan(x/2) **4 - 600*tan(x/2)**3 - 525*tan(x/2)**2 - 150*tan(x/2) - 15) + 236*tan(x/2 )**5/(15*tan(x/2)**10 - 150*tan(x/2)**9 + 525*tan(x/2)**8 - 600*tan(x/2)** 7 - 450*tan(x/2)**6 + 1020*tan(x/2)**5 + 450*tan(x/2)**4 - 600*tan(x/2)**3 - 525*tan(x/2)**2 - 150*tan(x/2) - 15) + 40*tan(x/2)**4/(15*tan(x/2)**10 - 150*tan(x/2)**9 + 525*tan(x/2)**8 - 600*tan(x/2)**7 - 450*tan(x/2)**6 + 1020*tan(x/2)**5 + 450*tan(x/2)**4 - 600*tan(x/2)**3 - 525*tan(x/2)**2 - 1 50*tan(x/2) - 15) - 200*tan(x/2)**3/(15*tan(x/2)**10 - 150*tan(x/2)**9 + 5 25*tan(x/2)**8 - 600*tan(x/2)**7 - 450*tan(x/2)**6 + 1020*tan(x/2)**5 + 45 0*tan(x/2)**4 - 600*tan(x/2)**3 - 525*tan(x/2)**2 - 150*tan(x/2) - 15) - 1 20*tan(x/2)**2/(15*tan(x/2)**10 - 150*tan(x/2)**9 + 525*tan(x/2)**8 - 6...
Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.12 \[ \int \frac {1}{(\cos (x)+\sin (x))^6} \, dx=-\frac {15 \, \tan \left (x\right )^{4} + 30 \, \tan \left (x\right )^{3} + 40 \, \tan \left (x\right )^{2} + 20 \, \tan \left (x\right ) + 7}{15 \, {\left (\tan \left (x\right )^{5} + 5 \, \tan \left (x\right )^{4} + 10 \, \tan \left (x\right )^{3} + 10 \, \tan \left (x\right )^{2} + 5 \, \tan \left (x\right ) + 1\right )}} \] Input:
integrate(1/(cos(x)+sin(x))^6,x, algorithm="maxima")
Output:
-1/15*(15*tan(x)^4 + 30*tan(x)^3 + 40*tan(x)^2 + 20*tan(x) + 7)/(tan(x)^5 + 5*tan(x)^4 + 10*tan(x)^3 + 10*tan(x)^2 + 5*tan(x) + 1)
Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.64 \[ \int \frac {1}{(\cos (x)+\sin (x))^6} \, dx=-\frac {15 \, \tan \left (x\right )^{4} + 30 \, \tan \left (x\right )^{3} + 40 \, \tan \left (x\right )^{2} + 20 \, \tan \left (x\right ) + 7}{15 \, {\left (\tan \left (x\right ) + 1\right )}^{5}} \] Input:
integrate(1/(cos(x)+sin(x))^6,x, algorithm="giac")
Output:
-1/15*(15*tan(x)^4 + 30*tan(x)^3 + 40*tan(x)^2 + 20*tan(x) + 7)/(tan(x) + 1)^5
Time = 0.16 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.76 \[ \int \frac {1}{(\cos (x)+\sin (x))^6} \, dx=\frac {2\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (15\,{\mathrm {tan}\left (\frac {x}{2}\right )}^8-60\,{\mathrm {tan}\left (\frac {x}{2}\right )}^7+100\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+20\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5-118\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4-20\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+100\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+60\,\mathrm {tan}\left (\frac {x}{2}\right )+15\right )}{15\,{\left (-{\mathrm {tan}\left (\frac {x}{2}\right )}^2+2\,\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^5} \] Input:
int(1/(cos(x) + sin(x))^6,x)
Output:
(2*tan(x/2)*(60*tan(x/2) + 100*tan(x/2)^2 - 20*tan(x/2)^3 - 118*tan(x/2)^4 + 20*tan(x/2)^5 + 100*tan(x/2)^6 - 60*tan(x/2)^7 + 15*tan(x/2)^8 + 15))/( 15*(2*tan(x/2) - tan(x/2)^2 + 1)^5)
Time = 0.15 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.34 \[ \int \frac {1}{(\cos (x)+\sin (x))^6} \, dx=\frac {-2 \cos \left (x \right ) \sin \left (x \right )^{4}-6 \cos \left (x \right ) \sin \left (x \right )^{2}+3 \cos \left (x \right )+6 \sin \left (x \right )^{5}-10 \sin \left (x \right )^{3}}{60 \cos \left (x \right ) \sin \left (x \right )^{4}-120 \cos \left (x \right ) \sin \left (x \right )^{2}-15 \cos \left (x \right )+60 \sin \left (x \right )^{5}-75 \sin \left (x \right )} \] Input:
int(1/(cos(x)+sin(x))^6,x)
Output:
( - 2*cos(x)*sin(x)**4 - 6*cos(x)*sin(x)**2 + 3*cos(x) + 6*sin(x)**5 - 10* sin(x)**3)/(15*(4*cos(x)*sin(x)**4 - 8*cos(x)*sin(x)**2 - cos(x) + 4*sin(x )**5 - 5*sin(x)))