Integrand size = 13, antiderivative size = 81 \[ \int \frac {\csc ^6(x)}{a+b \cot (x)} \, dx=\frac {a \left (a^2+2 b^2\right ) \cot (x)}{b^4}-\frac {\left (a^2+2 b^2\right ) \cot ^2(x)}{2 b^3}+\frac {a \cot ^3(x)}{3 b^2}-\frac {\cot ^4(x)}{4 b}-\frac {\left (a^2+b^2\right )^2 \log (a+b \cot (x))}{b^5} \] Output:
a*(a^2+2*b^2)*cot(x)/b^4-1/2*(a^2+2*b^2)*cot(x)^2/b^3+1/3*a*cot(x)^3/b^2-1 /4*cot(x)^4/b-(a^2+b^2)^2*ln(a+b*cot(x))/b^5
Time = 1.06 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.05 \[ \int \frac {\csc ^6(x)}{a+b \cot (x)} \, dx=\frac {-6 b^2 \left (a^2+b^2\right ) \csc ^2(x)-3 b^4 \csc ^4(x)+4 a b \cot (x) \left (3 a^2+5 b^2+b^2 \csc ^2(x)\right )+12 \left (a^2+b^2\right )^2 (\log (\sin (x))-\log (b \cos (x)+a \sin (x)))}{12 b^5} \] Input:
Integrate[Csc[x]^6/(a + b*Cot[x]),x]
Output:
(-6*b^2*(a^2 + b^2)*Csc[x]^2 - 3*b^4*Csc[x]^4 + 4*a*b*Cot[x]*(3*a^2 + 5*b^ 2 + b^2*Csc[x]^2) + 12*(a^2 + b^2)^2*(Log[Sin[x]] - Log[b*Cos[x] + a*Sin[x ]]))/(12*b^5)
Time = 0.29 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 3987, 27, 476, 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.
\(\displaystyle \int \frac {\csc ^6(x)}{a+b \cot (x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sec \left (x-\frac {\pi }{2}\right )^6}{a-b \tan \left (x-\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 3987 |
\(\displaystyle -\frac {\int \frac {\left (\cot ^2(x) b^2+b^2\right )^2}{b^4 (a+b \cot (x))}d(b \cot (x))}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {\left (\cot ^2(x) b^2+b^2\right )^2}{a+b \cot (x)}d(b \cot (x))}{b^5}\) |
\(\Big \downarrow \) 476 |
\(\displaystyle -\frac {\int \left (b^3 \cot ^3(x)-a b^2 \cot ^2(x)+b \left (a^2+2 b^2\right ) \cot (x)-a \left (a^2+2 b^2\right )+\frac {\left (a^2+b^2\right )^2}{a+b \cot (x)}\right )d(b \cot (x))}{b^5}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\frac {1}{2} b^2 \left (a^2+2 b^2\right ) \cot ^2(x)-a b \left (a^2+2 b^2\right ) \cot (x)+\left (a^2+b^2\right )^2 \log (a+b \cot (x))-\frac {1}{3} a b^3 \cot ^3(x)+\frac {1}{4} b^4 \cot ^4(x)}{b^5}\) |
Input:
Int[Csc[x]^6/(a + b*Cot[x]),x]
Output:
-((-(a*b*(a^2 + 2*b^2)*Cot[x]) + (b^2*(a^2 + 2*b^2)*Cot[x]^2)/2 - (a*b^3*C ot[x]^3)/3 + (b^4*Cot[x]^4)/4 + (a^2 + b^2)^2*Log[a + b*Cot[x]])/b^5)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_ ), x_Symbol] :> Simp[1/(b*f) Subst[Int[(a + x)^n*(1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && NeQ[a^2 + b^2, 0] && IntegerQ[m/2]
Time = 2.79 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.31
method | result | size |
default | \(-\frac {\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \ln \left (\tan \left (x \right ) a +b \right )}{b^{5}}-\frac {1}{4 b \tan \left (x \right )^{4}}-\frac {a^{2}+2 b^{2}}{2 b^{3} \tan \left (x \right )^{2}}+\frac {\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \ln \left (\tan \left (x \right )\right )}{b^{5}}+\frac {a}{3 b^{2} \tan \left (x \right )^{3}}+\frac {\left (a^{2}+2 b^{2}\right ) a}{b^{4} \tan \left (x \right )}\) | \(106\) |
risch | \(\frac {2 i a^{3} {\mathrm e}^{6 i x}+2 i a \,b^{2} {\mathrm e}^{6 i x}+2 a^{2} b \,{\mathrm e}^{6 i x}+2 b^{3} {\mathrm e}^{6 i x}-6 i a^{3} {\mathrm e}^{4 i x}-10 i a \,b^{2} {\mathrm e}^{4 i x}-4 a^{2} b \,{\mathrm e}^{4 i x}-8 b^{3} {\mathrm e}^{4 i x}+6 i a^{3} {\mathrm e}^{2 i x}+\frac {34 i a \,b^{2} {\mathrm e}^{2 i x}}{3}+2 a^{2} b \,{\mathrm e}^{2 i x}+2 b^{3} {\mathrm e}^{2 i x}-2 i a^{3}-\frac {10 i a \,b^{2}}{3}}{b^{4} \left ({\mathrm e}^{2 i x}-1\right )^{4}}+\frac {\ln \left ({\mathrm e}^{2 i x}-1\right ) a^{4}}{b^{5}}+\frac {2 \ln \left ({\mathrm e}^{2 i x}-1\right ) a^{2}}{b^{3}}+\frac {\ln \left ({\mathrm e}^{2 i x}-1\right )}{b}-\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {i b -a}{i b +a}\right ) a^{4}}{b^{5}}-\frac {2 \ln \left ({\mathrm e}^{2 i x}+\frac {i b -a}{i b +a}\right ) a^{2}}{b^{3}}-\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {i b -a}{i b +a}\right )}{b}\) | \(298\) |
Input:
int(csc(x)^6/(a+b*cot(x)),x,method=_RETURNVERBOSE)
Output:
-(a^4+2*a^2*b^2+b^4)/b^5*ln(tan(x)*a+b)-1/4/b/tan(x)^4-1/2*(a^2+2*b^2)/b^3 /tan(x)^2+(a^4+2*a^2*b^2+b^4)/b^5*ln(tan(x))+1/3/b^2*a/tan(x)^3+(a^2+2*b^2 )/b^4*a/tan(x)
Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (75) = 150\).
Time = 0.10 (sec) , antiderivative size = 248, normalized size of antiderivative = 3.06 \[ \int \frac {\csc ^6(x)}{a+b \cot (x)} \, dx=-\frac {6 \, a^{2} b^{2} + 9 \, b^{4} - 6 \, {\left (a^{2} b^{2} + b^{4}\right )} \cos \left (x\right )^{2} + 6 \, {\left ({\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (x\right )^{4} + a^{4} + 2 \, a^{2} b^{2} + b^{4} - 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (x\right )^{2}\right )} \log \left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) - {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + a^{2}\right ) - 6 \, {\left ({\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (x\right )^{4} + a^{4} + 2 \, a^{2} b^{2} + b^{4} - 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (x\right )^{2}\right )} \log \left (-\frac {1}{4} \, \cos \left (x\right )^{2} + \frac {1}{4}\right ) + 4 \, {\left ({\left (3 \, a^{3} b + 5 \, a b^{3}\right )} \cos \left (x\right )^{3} - 3 \, {\left (a^{3} b + 2 \, a b^{3}\right )} \cos \left (x\right )\right )} \sin \left (x\right )}{12 \, {\left (b^{5} \cos \left (x\right )^{4} - 2 \, b^{5} \cos \left (x\right )^{2} + b^{5}\right )}} \] Input:
integrate(csc(x)^6/(a+b*cot(x)),x, algorithm="fricas")
Output:
-1/12*(6*a^2*b^2 + 9*b^4 - 6*(a^2*b^2 + b^4)*cos(x)^2 + 6*((a^4 + 2*a^2*b^ 2 + b^4)*cos(x)^4 + a^4 + 2*a^2*b^2 + b^4 - 2*(a^4 + 2*a^2*b^2 + b^4)*cos( x)^2)*log(2*a*b*cos(x)*sin(x) - (a^2 - b^2)*cos(x)^2 + a^2) - 6*((a^4 + 2* a^2*b^2 + b^4)*cos(x)^4 + a^4 + 2*a^2*b^2 + b^4 - 2*(a^4 + 2*a^2*b^2 + b^4 )*cos(x)^2)*log(-1/4*cos(x)^2 + 1/4) + 4*((3*a^3*b + 5*a*b^3)*cos(x)^3 - 3 *(a^3*b + 2*a*b^3)*cos(x))*sin(x))/(b^5*cos(x)^4 - 2*b^5*cos(x)^2 + b^5)
Timed out. \[ \int \frac {\csc ^6(x)}{a+b \cot (x)} \, dx=\text {Timed out} \] Input:
integrate(csc(x)**6/(a+b*cot(x)),x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.31 \[ \int \frac {\csc ^6(x)}{a+b \cot (x)} \, dx=-\frac {{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (a \tan \left (x\right ) + b\right )}{b^{5}} + \frac {{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (x\right )\right )}{b^{5}} + \frac {4 \, a b^{2} \tan \left (x\right ) + 12 \, {\left (a^{3} + 2 \, a b^{2}\right )} \tan \left (x\right )^{3} - 3 \, b^{3} - 6 \, {\left (a^{2} b + 2 \, b^{3}\right )} \tan \left (x\right )^{2}}{12 \, b^{4} \tan \left (x\right )^{4}} \] Input:
integrate(csc(x)^6/(a+b*cot(x)),x, algorithm="maxima")
Output:
-(a^4 + 2*a^2*b^2 + b^4)*log(a*tan(x) + b)/b^5 + (a^4 + 2*a^2*b^2 + b^4)*l og(tan(x))/b^5 + 1/12*(4*a*b^2*tan(x) + 12*(a^3 + 2*a*b^2)*tan(x)^3 - 3*b^ 3 - 6*(a^2*b + 2*b^3)*tan(x)^2)/(b^4*tan(x)^4)
Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (75) = 150\).
Time = 0.14 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.86 \[ \int \frac {\csc ^6(x)}{a+b \cot (x)} \, dx=\frac {{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \log \left ({\left | \tan \left (x\right ) \right |}\right )}{b^{5}} - \frac {{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \log \left ({\left | a \tan \left (x\right ) + b \right |}\right )}{a b^{5}} - \frac {25 \, a^{4} \tan \left (x\right )^{4} + 50 \, a^{2} b^{2} \tan \left (x\right )^{4} + 25 \, b^{4} \tan \left (x\right )^{4} - 12 \, a^{3} b \tan \left (x\right )^{3} - 24 \, a b^{3} \tan \left (x\right )^{3} + 6 \, a^{2} b^{2} \tan \left (x\right )^{2} + 12 \, b^{4} \tan \left (x\right )^{2} - 4 \, a b^{3} \tan \left (x\right ) + 3 \, b^{4}}{12 \, b^{5} \tan \left (x\right )^{4}} \] Input:
integrate(csc(x)^6/(a+b*cot(x)),x, algorithm="giac")
Output:
(a^4 + 2*a^2*b^2 + b^4)*log(abs(tan(x)))/b^5 - (a^5 + 2*a^3*b^2 + a*b^4)*l og(abs(a*tan(x) + b))/(a*b^5) - 1/12*(25*a^4*tan(x)^4 + 50*a^2*b^2*tan(x)^ 4 + 25*b^4*tan(x)^4 - 12*a^3*b*tan(x)^3 - 24*a*b^3*tan(x)^3 + 6*a^2*b^2*ta n(x)^2 + 12*b^4*tan(x)^2 - 4*a*b^3*tan(x) + 3*b^4)/(b^5*tan(x)^4)
Time = 9.50 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.36 \[ \int \frac {\csc ^6(x)}{a+b \cot (x)} \, dx=-\frac {\frac {1}{4\,b}-\frac {a\,\mathrm {tan}\left (x\right )}{3\,b^2}+\frac {{\mathrm {tan}\left (x\right )}^2\,\left (a^2+2\,b^2\right )}{2\,b^3}-\frac {a\,{\mathrm {tan}\left (x\right )}^3\,\left (a^2+2\,b^2\right )}{b^4}}{{\mathrm {tan}\left (x\right )}^4}-\frac {2\,\mathrm {atanh}\left (\frac {\left (b+2\,a\,\mathrm {tan}\left (x\right )\right )\,{\left (a^2+b^2\right )}^2}{b\,\left (a^4+2\,a^2\,b^2+b^4\right )}\right )\,{\left (a^2+b^2\right )}^2}{b^5} \] Input:
int(1/(sin(x)^6*(a + b*cot(x))),x)
Output:
- (1/(4*b) - (a*tan(x))/(3*b^2) + (tan(x)^2*(a^2 + 2*b^2))/(2*b^3) - (a*ta n(x)^3*(a^2 + 2*b^2))/b^4)/tan(x)^4 - (2*atanh(((b + 2*a*tan(x))*(a^2 + b^ 2)^2)/(b*(a^4 + b^4 + 2*a^2*b^2)))*(a^2 + b^2)^2)/b^5
Time = 0.19 (sec) , antiderivative size = 226, normalized size of antiderivative = 2.79 \[ \int \frac {\csc ^6(x)}{a+b \cot (x)} \, dx=\frac {96 \cos \left (x \right ) \sin \left (x \right )^{3} a^{3} b +160 \cos \left (x \right ) \sin \left (x \right )^{3} a \,b^{3}+32 \cos \left (x \right ) \sin \left (x \right ) a \,b^{3}-96 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2} b -2 \tan \left (\frac {x}{2}\right ) a -b \right ) \sin \left (x \right )^{4} a^{4}-192 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2} b -2 \tan \left (\frac {x}{2}\right ) a -b \right ) \sin \left (x \right )^{4} a^{2} b^{2}-96 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2} b -2 \tan \left (\frac {x}{2}\right ) a -b \right ) \sin \left (x \right )^{4} b^{4}+96 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{4} a^{4}+192 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{4} a^{2} b^{2}+96 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{4} b^{4}+24 \sin \left (x \right )^{4} a^{2} b^{2}+33 \sin \left (x \right )^{4} b^{4}-48 \sin \left (x \right )^{2} a^{2} b^{2}-48 \sin \left (x \right )^{2} b^{4}-24 b^{4}}{96 \sin \left (x \right )^{4} b^{5}} \] Input:
int(csc(x)^6/(a+b*cot(x)),x)
Output:
(96*cos(x)*sin(x)**3*a**3*b + 160*cos(x)*sin(x)**3*a*b**3 + 32*cos(x)*sin( x)*a*b**3 - 96*log(tan(x/2)**2*b - 2*tan(x/2)*a - b)*sin(x)**4*a**4 - 192* log(tan(x/2)**2*b - 2*tan(x/2)*a - b)*sin(x)**4*a**2*b**2 - 96*log(tan(x/2 )**2*b - 2*tan(x/2)*a - b)*sin(x)**4*b**4 + 96*log(tan(x/2))*sin(x)**4*a** 4 + 192*log(tan(x/2))*sin(x)**4*a**2*b**2 + 96*log(tan(x/2))*sin(x)**4*b** 4 + 24*sin(x)**4*a**2*b**2 + 33*sin(x)**4*b**4 - 48*sin(x)**2*a**2*b**2 - 48*sin(x)**2*b**4 - 24*b**4)/(96*sin(x)**4*b**5)