Integrand size = 10, antiderivative size = 33 \[ \int x \cos ^2(x) \cot ^2(x) \, dx=-\frac {3 x^2}{4}-\frac {\cos ^2(x)}{4}-x \cot (x)+\log (\sin (x))-\frac {1}{2} x \cos (x) \sin (x) \] Output:
-3/4*x^2-1/4*cos(x)^2-x*cot(x)+ln(sin(x))-1/2*x*cos(x)*sin(x)
Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int x \cos ^2(x) \cot ^2(x) \, dx=-\frac {3 x^2}{4}-\frac {1}{8} \cos (2 x)-x \cot (x)+\log (\sin (x))-\frac {1}{4} x \sin (2 x) \] Input:
Integrate[x*Cos[x]^2*Cot[x]^2,x]
Output:
(-3*x^2)/4 - Cos[2*x]/8 - x*Cot[x] + Log[Sin[x]] - (x*Sin[2*x])/4
Time = 0.31 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4908, 3042, 3791, 15, 4203, 15, 25, 3042, 25, 3956}
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.
| \(\displaystyle \int x \cos ^2(x) \cot ^2(x) \, dx\) |
| \(\Big \downarrow \) 4908 |
| \(\displaystyle \int x \cot ^2(x)dx-\int x \cos ^2(x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int x \tan \left (x+\frac {\pi }{2}\right )^2dx-\int x \sin \left (x+\frac {\pi }{2}\right )^2dx\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle -\frac {\int xdx}{2}+\int x \tan \left (x+\frac {\pi }{2}\right )^2dx-\frac {1}{4} \cos ^2(x)-\frac {1}{2} x \sin (x) \cos (x)\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \int x \tan \left (x+\frac {\pi }{2}\right )^2dx-\frac {x^2}{4}-\frac {\cos ^2(x)}{4}-\frac {1}{2} x \sin (x) \cos (x)\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle -\int xdx-\int -\cot (x)dx-\frac {x^2}{4}-\frac {\cos ^2(x)}{4}-x \cot (x)-\frac {1}{2} x \sin (x) \cos (x)\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\int -\cot (x)dx-\frac {3 x^2}{4}-\frac {\cos ^2(x)}{4}-x \cot (x)-\frac {1}{2} x \sin (x) \cos (x)\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \cot (x)dx-\frac {3 x^2}{4}-\frac {\cos ^2(x)}{4}-x \cot (x)-\frac {1}{2} x \sin (x) \cos (x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\tan \left (x+\frac {\pi }{2}\right )dx-\frac {3 x^2}{4}-\frac {\cos ^2(x)}{4}-x \cot (x)-\frac {1}{2} x \sin (x) \cos (x)\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \tan \left (x+\frac {\pi }{2}\right )dx-\frac {3 x^2}{4}-\frac {\cos ^2(x)}{4}-x \cot (x)-\frac {1}{2} x \sin (x) \cos (x)\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle -\frac {3 x^2}{4}-\frac {\cos ^2(x)}{4}-x \cot (x)+\log (\sin (x))-\frac {1}{2} x \sin (x) \cos (x)\) |
Input:
Int[x*Cos[x]^2*Cot[x]^2,x]
Output:
(-3*x^2)/4 - Cos[x]^2/4 - x*Cot[x] + Log[Sin[x]] - (x*Cos[x]*Sin[x])/2
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x ]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n) Int[(c + d* x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symb ol] :> Simp[b*(c + d*x)^m*((b*Tan[e + f*x])^(n - 1)/(f*(n - 1))), x] + (-Si mp[b*d*(m/(f*(n - 1))) Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1), x] , x] - Simp[b^2 Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; Free Q[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 0]
Int[Cos[(a_.) + (b_.)*(x_)]^(n_.)*Cot[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d _.)*(x_))^(m_.), x_Symbol] :> -Int[(c + d*x)^m*Cos[a + b*x]^n*Cot[a + b*x]^ (p - 2), x] + Int[(c + d*x)^m*Cos[a + b*x]^(n - 2)*Cot[a + b*x]^p, x] /; Fr eeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(27)=54\).
Time = 0.49 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.70
| method | result | size |
| parallelrisch | \(-\frac {3}{8}-\ln \left (\sec \left (\frac {x}{2}\right )^{2}\right )+\ln \left (\tan \left (\frac {x}{2}\right )\right )+\frac {x \left (-3-2 \cos \left (x \right )+\cos \left (2 x \right )\right ) \cot \left (\frac {x}{2}\right )}{4}+\frac {\sec \left (\frac {x}{2}\right ) \csc \left (\frac {x}{2}\right ) x}{2}-\frac {3 x^{2}}{4}-\frac {\cos \left (2 x \right )}{8}\) | \(56\) |
| risch | \(-\frac {3 x^{2}}{4}+\frac {i \left (i+2 x \right ) {\mathrm e}^{2 i x}}{16}-\frac {i \left (-i+2 x \right ) {\mathrm e}^{-2 i x}}{16}-2 i x -\frac {2 i x}{{\mathrm e}^{2 i x}-1}+\ln \left ({\mathrm e}^{2 i x}-1\right )\) | \(60\) |
| norman | \(\frac {\tan \left (\frac {x}{2}\right )^{3}-\frac {x}{2}-\frac {3 x \tan \left (\frac {x}{2}\right )^{2}}{2}+\frac {3 x \tan \left (\frac {x}{2}\right )^{4}}{2}+\frac {x \tan \left (\frac {x}{2}\right )^{6}}{2}-\frac {3 x^{2} \tan \left (\frac {x}{2}\right )}{4}-\frac {3 x^{2} \tan \left (\frac {x}{2}\right )^{3}}{2}-\frac {3 \tan \left (\frac {x}{2}\right )^{5} x^{2}}{4}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{2} \tan \left (\frac {x}{2}\right )}-\ln \left (1+\tan \left (\frac {x}{2}\right )^{2}\right )+\ln \left (\tan \left (\frac {x}{2}\right )\right )\) | \(103\) |
Input:
int(x*cos(x)^4/sin(x)^2,x,method=_RETURNVERBOSE)
Output:
-3/8-ln(sec(1/2*x)^2)+ln(tan(1/2*x))+1/4*x*(-3-2*cos(x)+cos(2*x))*cot(1/2* x)+1/2*sec(1/2*x)*csc(1/2*x)*x-3/4*x^2-1/8*cos(2*x)
Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.36 \[ \int x \cos ^2(x) \cot ^2(x) \, dx=\frac {4 \, x \cos \left (x\right )^{3} - 12 \, x \cos \left (x\right ) - {\left (6 \, x^{2} + 2 \, \cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) + 8 \, \log \left (\frac {1}{2} \, \sin \left (x\right )\right ) \sin \left (x\right )}{8 \, \sin \left (x\right )} \] Input:
integrate(x*cos(x)^4/sin(x)^2,x, algorithm="fricas")
Output:
1/8*(4*x*cos(x)^3 - 12*x*cos(x) - (6*x^2 + 2*cos(x)^2 - 1)*sin(x) + 8*log( 1/2*sin(x))*sin(x))/sin(x)
Leaf count of result is larger than twice the leaf count of optimal. 507 vs. \(2 (32) = 64\).
Time = 0.56 (sec) , antiderivative size = 507, normalized size of antiderivative = 15.36 \[ \int x \cos ^2(x) \cot ^2(x) \, dx =\text {Too large to display} \] Input:
integrate(x*cos(x)**4/sin(x)**2,x)
Output:
-3*x**2*tan(x/2)**5/(4*tan(x/2)**5 + 8*tan(x/2)**3 + 4*tan(x/2)) - 6*x**2* tan(x/2)**3/(4*tan(x/2)**5 + 8*tan(x/2)**3 + 4*tan(x/2)) - 3*x**2*tan(x/2) /(4*tan(x/2)**5 + 8*tan(x/2)**3 + 4*tan(x/2)) + 2*x*tan(x/2)**6/(4*tan(x/2 )**5 + 8*tan(x/2)**3 + 4*tan(x/2)) + 6*x*tan(x/2)**4/(4*tan(x/2)**5 + 8*ta n(x/2)**3 + 4*tan(x/2)) - 6*x*tan(x/2)**2/(4*tan(x/2)**5 + 8*tan(x/2)**3 + 4*tan(x/2)) - 2*x/(4*tan(x/2)**5 + 8*tan(x/2)**3 + 4*tan(x/2)) - 4*log(ta n(x/2)**2 + 1)*tan(x/2)**5/(4*tan(x/2)**5 + 8*tan(x/2)**3 + 4*tan(x/2)) - 8*log(tan(x/2)**2 + 1)*tan(x/2)**3/(4*tan(x/2)**5 + 8*tan(x/2)**3 + 4*tan( x/2)) - 4*log(tan(x/2)**2 + 1)*tan(x/2)/(4*tan(x/2)**5 + 8*tan(x/2)**3 + 4 *tan(x/2)) + 4*log(tan(x/2))*tan(x/2)**5/(4*tan(x/2)**5 + 8*tan(x/2)**3 + 4*tan(x/2)) + 8*log(tan(x/2))*tan(x/2)**3/(4*tan(x/2)**5 + 8*tan(x/2)**3 + 4*tan(x/2)) + 4*log(tan(x/2))*tan(x/2)/(4*tan(x/2)**5 + 8*tan(x/2)**3 + 4 *tan(x/2)) + 4*tan(x/2)**3/(4*tan(x/2)**5 + 8*tan(x/2)**3 + 4*tan(x/2))
Exception generated. \[ \int x \cos ^2(x) \cot ^2(x) \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x*cos(x)^4/sin(x)^2,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (27) = 54\).
Time = 0.14 (sec) , antiderivative size = 206, normalized size of antiderivative = 6.24 \[ \int x \cos ^2(x) \cot ^2(x) \, dx=-\frac {6 \, x^{2} \tan \left (\frac {1}{2} \, x\right )^{5} - 4 \, x \tan \left (\frac {1}{2} \, x\right )^{6} - 4 \, \log \left (\frac {16 \, \tan \left (\frac {1}{2} \, x\right )^{2}}{\tan \left (\frac {1}{2} \, x\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, x\right )^{5} + 12 \, x^{2} \tan \left (\frac {1}{2} \, x\right )^{3} - 12 \, x \tan \left (\frac {1}{2} \, x\right )^{4} + \tan \left (\frac {1}{2} \, x\right )^{5} - 8 \, \log \left (\frac {16 \, \tan \left (\frac {1}{2} \, x\right )^{2}}{\tan \left (\frac {1}{2} \, x\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, x\right )^{3} + 6 \, x^{2} \tan \left (\frac {1}{2} \, x\right ) + 12 \, x \tan \left (\frac {1}{2} \, x\right )^{2} - 6 \, \tan \left (\frac {1}{2} \, x\right )^{3} - 4 \, \log \left (\frac {16 \, \tan \left (\frac {1}{2} \, x\right )^{2}}{\tan \left (\frac {1}{2} \, x\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, x\right ) + 4 \, x + \tan \left (\frac {1}{2} \, x\right )}{8 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{5} + 2 \, \tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )\right )}} \] Input:
integrate(x*cos(x)^4/sin(x)^2,x, algorithm="giac")
Output:
-1/8*(6*x^2*tan(1/2*x)^5 - 4*x*tan(1/2*x)^6 - 4*log(16*tan(1/2*x)^2/(tan(1 /2*x)^4 + 2*tan(1/2*x)^2 + 1))*tan(1/2*x)^5 + 12*x^2*tan(1/2*x)^3 - 12*x*t an(1/2*x)^4 + tan(1/2*x)^5 - 8*log(16*tan(1/2*x)^2/(tan(1/2*x)^4 + 2*tan(1 /2*x)^2 + 1))*tan(1/2*x)^3 + 6*x^2*tan(1/2*x) + 12*x*tan(1/2*x)^2 - 6*tan( 1/2*x)^3 - 4*log(16*tan(1/2*x)^2/(tan(1/2*x)^4 + 2*tan(1/2*x)^2 + 1))*tan( 1/2*x) + 4*x + tan(1/2*x))/(tan(1/2*x)^5 + 2*tan(1/2*x)^3 + tan(1/2*x))
Time = 0.37 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.70 \[ \int x \cos ^2(x) \cot ^2(x) \, dx=\ln \left ({\mathrm {e}}^{x\,2{}\mathrm {i}}-1\right )-{\mathrm {e}}^{-x\,2{}\mathrm {i}}\,\left (\frac {1}{16}+\frac {x\,1{}\mathrm {i}}{8}\right )+{\mathrm {e}}^{x\,2{}\mathrm {i}}\,\left (-\frac {1}{16}+\frac {x\,1{}\mathrm {i}}{8}\right )-\frac {3\,x^2}{4}-x\,2{}\mathrm {i}-\frac {x\,2{}\mathrm {i}}{{\mathrm {e}}^{x\,2{}\mathrm {i}}-1} \] Input:
int((x*cos(x)^4)/sin(x)^2,x)
Output:
log(exp(x*2i) - 1) - x*2i - exp(-x*2i)*((x*1i)/8 + 1/16) + exp(x*2i)*((x*1 i)/8 - 1/16) - (x*2i)/(exp(x*2i) - 1) - (3*x^2)/4
Time = 0.16 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.76 \[ \int x \cos ^2(x) \cot ^2(x) \, dx=\frac {-2 \cos \left (x \right ) \sin \left (x \right )^{2} x -4 \cos \left (x \right ) x -4 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2}+1\right ) \sin \left (x \right )+4 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )+\sin \left (x \right )^{3}-3 \sin \left (x \right ) x^{2}-2 \sin \left (x \right )}{4 \sin \left (x \right )} \] Input:
int(x*cos(x)^4/sin(x)^2,x)
Output:
( - 2*cos(x)*sin(x)**2*x - 4*cos(x)*x - 4*log(tan(x/2)**2 + 1)*sin(x) + 4* log(tan(x/2))*sin(x) + sin(x)**3 - 3*sin(x)*x**2 - 2*sin(x))/(4*sin(x))