Integrand size = 13, antiderivative size = 101 \[ \int \frac {\csc ^5(x)}{a+b \cot (x)} \, dx=\frac {a \text {arctanh}(\cos (x))}{2 b^2}+\frac {a \left (a^2+b^2\right ) \text {arctanh}(\cos (x))}{b^4}+\frac {\left (a^2+b^2\right )^{3/2} \text {arctanh}\left (\frac {(b-a \cot (x)) \sin (x)}{\sqrt {a^2+b^2}}\right )}{b^4}-\frac {\left (a^2+b^2\right ) \csc (x)}{b^3}+\frac {a \cot (x) \csc (x)}{2 b^2}-\frac {\csc ^3(x)}{3 b} \] Output:
1/2*a*arctanh(cos(x))/b^2+a*(a^2+b^2)*arctanh(cos(x))/b^4+(a^2+b^2)^(3/2)* arctanh((b-a*cot(x))*sin(x)/(a^2+b^2)^(1/2))/b^4-(a^2+b^2)*csc(x)/b^3+1/2* a*cot(x)*csc(x)/b^2-1/3*csc(x)^3/b
Time = 0.99 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.96 \[ \int \frac {\csc ^5(x)}{a+b \cot (x)} \, dx=-\frac {-96 \left (a^2+b^2\right )^{3/2} \text {arctanh}\left (\frac {-a+b \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )+4 b \left (6 a^2+7 b^2\right ) \cot \left (\frac {x}{2}\right )-6 a b^2 \csc ^2\left (\frac {x}{2}\right )-48 a^3 \log \left (\cos \left (\frac {x}{2}\right )\right )-72 a b^2 \log \left (\cos \left (\frac {x}{2}\right )\right )+48 a^3 \log \left (\sin \left (\frac {x}{2}\right )\right )+72 a b^2 \log \left (\sin \left (\frac {x}{2}\right )\right )+6 a b^2 \sec ^2\left (\frac {x}{2}\right )+16 b^3 \csc ^3(x) \sin ^4\left (\frac {x}{2}\right )+b^3 \csc ^4\left (\frac {x}{2}\right ) \sin (x)+24 a^2 b \tan \left (\frac {x}{2}\right )+28 b^3 \tan \left (\frac {x}{2}\right )}{48 b^4} \] Input:
Integrate[Csc[x]^5/(a + b*Cot[x]),x]
Output:
-1/48*(-96*(a^2 + b^2)^(3/2)*ArcTanh[(-a + b*Tan[x/2])/Sqrt[a^2 + b^2]] + 4*b*(6*a^2 + 7*b^2)*Cot[x/2] - 6*a*b^2*Csc[x/2]^2 - 48*a^3*Log[Cos[x/2]] - 72*a*b^2*Log[Cos[x/2]] + 48*a^3*Log[Sin[x/2]] + 72*a*b^2*Log[Sin[x/2]] + 6*a*b^2*Sec[x/2]^2 + 16*b^3*Csc[x]^3*Sin[x/2]^4 + b^3*Csc[x/2]^4*Sin[x] + 24*a^2*b*Tan[x/2] + 28*b^3*Tan[x/2])/b^4
Time = 0.90 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.98, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.077, Rules used = {3042, 3989, 3042, 3967, 3042, 3989, 3042, 3967, 3042, 3988, 219, 4255, 3042, 4257}
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.
| \(\displaystyle \int \frac {\csc ^5(x)}{a+b \cot (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sec \left (x-\frac {\pi }{2}\right )^5}{a-b \tan \left (x-\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3989 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {\csc ^3(x)}{a+b \cot (x)}dx}{b^2}-\frac {\int (a-b \cot (x)) \csc ^3(x)dx}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {\sec \left (x-\frac {\pi }{2}\right )^3}{a-b \tan \left (x-\frac {\pi }{2}\right )}dx}{b^2}-\frac {\int \sec \left (x-\frac {\pi }{2}\right )^3 \left (a+b \tan \left (x-\frac {\pi }{2}\right )\right )dx}{b^2}\) |
| \(\Big \downarrow \) 3967 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {\sec \left (x-\frac {\pi }{2}\right )^3}{a-b \tan \left (x-\frac {\pi }{2}\right )}dx}{b^2}-\frac {a \int \csc ^3(x)dx+\frac {1}{3} b \csc ^3(x)}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {\sec \left (x-\frac {\pi }{2}\right )^3}{a-b \tan \left (x-\frac {\pi }{2}\right )}dx}{b^2}-\frac {a \int \csc (x)^3dx+\frac {1}{3} b \csc ^3(x)}{b^2}\) |
| \(\Big \downarrow \) 3989 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\left (a^2+b^2\right ) \int \frac {\csc (x)}{a+b \cot (x)}dx}{b^2}-\frac {\int (a-b \cot (x)) \csc (x)dx}{b^2}\right )}{b^2}-\frac {a \int \csc (x)^3dx+\frac {1}{3} b \csc ^3(x)}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\left (a^2+b^2\right ) \int \frac {\sec \left (x-\frac {\pi }{2}\right )}{a-b \tan \left (x-\frac {\pi }{2}\right )}dx}{b^2}-\frac {\int \sec \left (x-\frac {\pi }{2}\right ) \left (a+b \tan \left (x-\frac {\pi }{2}\right )\right )dx}{b^2}\right )}{b^2}-\frac {a \int \csc (x)^3dx+\frac {1}{3} b \csc ^3(x)}{b^2}\) |
| \(\Big \downarrow \) 3967 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\left (a^2+b^2\right ) \int \frac {\sec \left (x-\frac {\pi }{2}\right )}{a-b \tan \left (x-\frac {\pi }{2}\right )}dx}{b^2}-\frac {a \int \csc (x)dx+b \csc (x)}{b^2}\right )}{b^2}-\frac {a \int \csc (x)^3dx+\frac {1}{3} b \csc ^3(x)}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\left (a^2+b^2\right ) \int \frac {\sec \left (x-\frac {\pi }{2}\right )}{a-b \tan \left (x-\frac {\pi }{2}\right )}dx}{b^2}-\frac {a \int \csc (x)dx+b \csc (x)}{b^2}\right )}{b^2}-\frac {a \int \csc (x)^3dx+\frac {1}{3} b \csc ^3(x)}{b^2}\) |
| \(\Big \downarrow \) 3988 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (-\frac {\left (a^2+b^2\right ) \int \frac {1}{a^2+b^2-(b-a \cot (x))^2 \sin ^2(x)}d(-((b-a \cot (x)) \sin (x)))}{b^2}-\frac {a \int \csc (x)dx+b \csc (x)}{b^2}\right )}{b^2}-\frac {a \int \csc (x)^3dx+\frac {1}{3} b \csc ^3(x)}{b^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {\sin (x) (b-a \cot (x))}{\sqrt {a^2+b^2}}\right )}{b^2}-\frac {a \int \csc (x)dx+b \csc (x)}{b^2}\right )}{b^2}-\frac {a \int \csc (x)^3dx+\frac {1}{3} b \csc ^3(x)}{b^2}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {\sin (x) (b-a \cot (x))}{\sqrt {a^2+b^2}}\right )}{b^2}-\frac {a \int \csc (x)dx+b \csc (x)}{b^2}\right )}{b^2}-\frac {a \left (\frac {\int \csc (x)dx}{2}-\frac {1}{2} \cot (x) \csc (x)\right )+\frac {1}{3} b \csc ^3(x)}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {\sin (x) (b-a \cot (x))}{\sqrt {a^2+b^2}}\right )}{b^2}-\frac {a \int \csc (x)dx+b \csc (x)}{b^2}\right )}{b^2}-\frac {a \left (\frac {\int \csc (x)dx}{2}-\frac {1}{2} \cot (x) \csc (x)\right )+\frac {1}{3} b \csc ^3(x)}{b^2}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {\sin (x) (b-a \cot (x))}{\sqrt {a^2+b^2}}\right )}{b^2}-\frac {b \csc (x)-a \text {arctanh}(\cos (x))}{b^2}\right )}{b^2}-\frac {a \left (-\frac {1}{2} \text {arctanh}(\cos (x))-\frac {1}{2} \cot (x) \csc (x)\right )+\frac {1}{3} b \csc ^3(x)}{b^2}\) |
Input:
Int[Csc[x]^5/(a + b*Cot[x]),x]
Output:
((a^2 + b^2)*((Sqrt[a^2 + b^2]*ArcTanh[((b - a*Cot[x])*Sin[x])/Sqrt[a^2 + b^2]])/b^2 - (-(a*ArcTanh[Cos[x]]) + b*Csc[x])/b^2))/b^2 - ((b*Csc[x]^3)/3 + a*(-1/2*ArcTanh[Cos[x]] - (Cot[x]*Csc[x])/2))/b^2
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( x_)]), x_Symbol] :> Simp[b*((d*Sec[e + f*x])^m/(f*m)), x] + Simp[a Int[(d *Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2*m] || NeQ[a^2 + b^2, 0])
Int[sec[(e_.) + (f_.)*(x_)]/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbo l] :> Simp[-f^(-1) Subst[Int[1/(a^2 + b^2 - x^2), x], x, (b - a*Tan[e + f *x])/Sec[e + f*x]], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 + b^2, 0]
Int[sec[(e_.) + (f_.)*(x_)]^(m_)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_ Symbol] :> Simp[-(b^2)^(-1) Int[Sec[e + f*x]^(m - 2)*(a - b*Tan[e + f*x]) , x], x] + Simp[(a^2 + b^2)/b^2 Int[Sec[e + f*x]^(m - 2)/(a + b*Tan[e + f *x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 + b^2, 0] && IGtQ[(m - 1) /2, 0]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 1.58 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.67
| method | result | size |
| default | \(-\frac {\frac {\tan \left (\frac {x}{2}\right )^{3} b^{2}}{3}+a b \tan \left (\frac {x}{2}\right )^{2}+4 a^{2} \tan \left (\frac {x}{2}\right )+5 b^{2} \tan \left (\frac {x}{2}\right )}{8 b^{3}}-\frac {1}{24 b \tan \left (\frac {x}{2}\right )^{3}}-\frac {4 a^{2}+5 b^{2}}{8 b^{3} \tan \left (\frac {x}{2}\right )}+\frac {a}{8 b^{2} \tan \left (\frac {x}{2}\right )^{2}}-\frac {a \left (2 a^{2}+3 b^{2}\right ) \ln \left (\tan \left (\frac {x}{2}\right )\right )}{2 b^{4}}+\frac {\left (-16 a^{4}-32 a^{2} b^{2}-16 b^{4}\right ) \operatorname {arctanh}\left (\frac {-2 b \tan \left (\frac {x}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{8 b^{4} \sqrt {a^{2}+b^{2}}}\) | \(169\) |
| risch | \(-\frac {i {\mathrm e}^{i x} \left (-3 i a b \,{\mathrm e}^{4 i x}+6 a^{2} {\mathrm e}^{4 i x}+6 b^{2} {\mathrm e}^{4 i x}-12 a^{2} {\mathrm e}^{2 i x}-20 b^{2} {\mathrm e}^{2 i x}+3 i a b +6 a^{2}+6 b^{2}\right )}{3 b^{3} \left ({\mathrm e}^{2 i x}-1\right )^{3}}-\frac {a^{3} \ln \left ({\mathrm e}^{i x}-1\right )}{b^{4}}-\frac {3 a \ln \left ({\mathrm e}^{i x}-1\right )}{2 b^{2}}+\frac {i \sqrt {-a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i x}-\frac {\left (i a +b \right ) \sqrt {-a^{2}-b^{2}}}{a^{2}+b^{2}}\right ) a^{2}}{b^{4}}+\frac {i \sqrt {-a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i x}-\frac {\left (i a +b \right ) \sqrt {-a^{2}-b^{2}}}{a^{2}+b^{2}}\right )}{b^{2}}-\frac {i \sqrt {-a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i x}+\frac {\left (i a +b \right ) \sqrt {-a^{2}-b^{2}}}{a^{2}+b^{2}}\right ) a^{2}}{b^{4}}-\frac {i \sqrt {-a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i x}+\frac {\left (i a +b \right ) \sqrt {-a^{2}-b^{2}}}{a^{2}+b^{2}}\right )}{b^{2}}+\frac {a^{3} \ln \left ({\mathrm e}^{i x}+1\right )}{b^{4}}+\frac {3 a \ln \left ({\mathrm e}^{i x}+1\right )}{2 b^{2}}\) | \(387\) |
Input:
int(csc(x)^5/(a+b*cot(x)),x,method=_RETURNVERBOSE)
Output:
-1/8/b^3*(1/3*tan(1/2*x)^3*b^2+a*b*tan(1/2*x)^2+4*a^2*tan(1/2*x)+5*b^2*tan (1/2*x))-1/24/b/tan(1/2*x)^3-1/8*(4*a^2+5*b^2)/b^3/tan(1/2*x)+1/8*a/b^2/ta n(1/2*x)^2-1/2/b^4*a*(2*a^2+3*b^2)*ln(tan(1/2*x))+1/8*(-16*a^4-32*a^2*b^2- 16*b^4)/b^4/(a^2+b^2)^(1/2)*arctanh(1/2*(-2*b*tan(1/2*x)+2*a)/(a^2+b^2)^(1 /2))
Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (93) = 186\).
Time = 0.24 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.61 \[ \int \frac {\csc ^5(x)}{a+b \cot (x)} \, dx=-\frac {6 \, a b^{2} \cos \left (x\right ) \sin \left (x\right ) - 6 \, {\left ({\left (a^{2} + b^{2}\right )} \cos \left (x\right )^{2} - a^{2} - b^{2}\right )} \sqrt {a^{2} + b^{2}} \log \left (-\frac {2 \, a b \cos \left (x\right ) \sin \left (x\right ) - {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - a^{2} - 2 \, b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (a \cos \left (x\right ) - b \sin \left (x\right )\right )}}{2 \, a b \cos \left (x\right ) \sin \left (x\right ) - {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + a^{2}}\right ) \sin \left (x\right ) - 12 \, a^{2} b - 16 \, b^{3} + 12 \, {\left (a^{2} b + b^{3}\right )} \cos \left (x\right )^{2} + 3 \, {\left (2 \, a^{3} + 3 \, a b^{2} - {\left (2 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (x\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) \sin \left (x\right ) - 3 \, {\left (2 \, a^{3} + 3 \, a b^{2} - {\left (2 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (x\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) \sin \left (x\right )}{12 \, {\left (b^{4} \cos \left (x\right )^{2} - b^{4}\right )} \sin \left (x\right )} \] Input:
integrate(csc(x)^5/(a+b*cot(x)),x, algorithm="fricas")
Output:
-1/12*(6*a*b^2*cos(x)*sin(x) - 6*((a^2 + b^2)*cos(x)^2 - a^2 - b^2)*sqrt(a ^2 + b^2)*log(-(2*a*b*cos(x)*sin(x) - (a^2 - b^2)*cos(x)^2 - a^2 - 2*b^2 + 2*sqrt(a^2 + b^2)*(a*cos(x) - b*sin(x)))/(2*a*b*cos(x)*sin(x) - (a^2 - b^ 2)*cos(x)^2 + a^2))*sin(x) - 12*a^2*b - 16*b^3 + 12*(a^2*b + b^3)*cos(x)^2 + 3*(2*a^3 + 3*a*b^2 - (2*a^3 + 3*a*b^2)*cos(x)^2)*log(1/2*cos(x) + 1/2)* sin(x) - 3*(2*a^3 + 3*a*b^2 - (2*a^3 + 3*a*b^2)*cos(x)^2)*log(-1/2*cos(x) + 1/2)*sin(x))/((b^4*cos(x)^2 - b^4)*sin(x))
Timed out. \[ \int \frac {\csc ^5(x)}{a+b \cot (x)} \, dx=\text {Timed out} \] Input:
integrate(csc(x)**5/(a+b*cot(x)),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (93) = 186\).
Time = 0.11 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.14 \[ \int \frac {\csc ^5(x)}{a+b \cot (x)} \, dx=-\frac {\frac {3 \, a b \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {b^{2} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {3 \, {\left (4 \, a^{2} + 5 \, b^{2}\right )} \sin \left (x\right )}{\cos \left (x\right ) + 1}}{24 \, b^{3}} - \frac {{\left (2 \, a^{3} + 3 \, a b^{2}\right )} \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{2 \, b^{4}} - \frac {{\left (b^{2} - \frac {3 \, a b \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {3 \, {\left (4 \, a^{2} + 5 \, b^{2}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (x\right ) + 1\right )}^{3}}{24 \, b^{3} \sin \left (x\right )^{3}} - \frac {{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (\frac {a - \frac {b \sin \left (x\right )}{\cos \left (x\right ) + 1} + \sqrt {a^{2} + b^{2}}}{a - \frac {b \sin \left (x\right )}{\cos \left (x\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} b^{4}} \] Input:
integrate(csc(x)^5/(a+b*cot(x)),x, algorithm="maxima")
Output:
-1/24*(3*a*b*sin(x)^2/(cos(x) + 1)^2 + b^2*sin(x)^3/(cos(x) + 1)^3 + 3*(4* a^2 + 5*b^2)*sin(x)/(cos(x) + 1))/b^3 - 1/2*(2*a^3 + 3*a*b^2)*log(sin(x)/( cos(x) + 1))/b^4 - 1/24*(b^2 - 3*a*b*sin(x)/(cos(x) + 1) + 3*(4*a^2 + 5*b^ 2)*sin(x)^2/(cos(x) + 1)^2)*(cos(x) + 1)^3/(b^3*sin(x)^3) - (a^4 + 2*a^2*b ^2 + b^4)*log((a - b*sin(x)/(cos(x) + 1) + sqrt(a^2 + b^2))/(a - b*sin(x)/ (cos(x) + 1) - sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*b^4)
Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (93) = 186\).
Time = 0.14 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.18 \[ \int \frac {\csc ^5(x)}{a+b \cot (x)} \, dx=-\frac {b^{2} \tan \left (\frac {1}{2} \, x\right )^{3} + 3 \, a b \tan \left (\frac {1}{2} \, x\right )^{2} + 12 \, a^{2} \tan \left (\frac {1}{2} \, x\right ) + 15 \, b^{2} \tan \left (\frac {1}{2} \, x\right )}{24 \, b^{3}} - \frac {{\left (2 \, a^{3} + 3 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{2 \, b^{4}} - \frac {{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (\frac {{\left | 2 \, b \tan \left (\frac {1}{2} \, x\right ) - 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b \tan \left (\frac {1}{2} \, x\right ) - 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} b^{4}} + \frac {44 \, a^{3} \tan \left (\frac {1}{2} \, x\right )^{3} + 66 \, a b^{2} \tan \left (\frac {1}{2} \, x\right )^{3} - 12 \, a^{2} b \tan \left (\frac {1}{2} \, x\right )^{2} - 15 \, b^{3} \tan \left (\frac {1}{2} \, x\right )^{2} + 3 \, a b^{2} \tan \left (\frac {1}{2} \, x\right ) - b^{3}}{24 \, b^{4} \tan \left (\frac {1}{2} \, x\right )^{3}} \] Input:
integrate(csc(x)^5/(a+b*cot(x)),x, algorithm="giac")
Output:
-1/24*(b^2*tan(1/2*x)^3 + 3*a*b*tan(1/2*x)^2 + 12*a^2*tan(1/2*x) + 15*b^2* tan(1/2*x))/b^3 - 1/2*(2*a^3 + 3*a*b^2)*log(abs(tan(1/2*x)))/b^4 - (a^4 + 2*a^2*b^2 + b^4)*log(abs(2*b*tan(1/2*x) - 2*a - 2*sqrt(a^2 + b^2))/abs(2*b *tan(1/2*x) - 2*a + 2*sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*b^4) + 1/24*(44*a ^3*tan(1/2*x)^3 + 66*a*b^2*tan(1/2*x)^3 - 12*a^2*b*tan(1/2*x)^2 - 15*b^3*t an(1/2*x)^2 + 3*a*b^2*tan(1/2*x) - b^3)/(b^4*tan(1/2*x)^3)
Time = 9.61 (sec) , antiderivative size = 674, normalized size of antiderivative = 6.67 \[ \int \frac {\csc ^5(x)}{a+b \cot (x)} \, dx =\text {Too large to display} \] Input:
int(1/(sin(x)^5*(a + b*cot(x))),x)
Output:
- tan(x/2)*(5/(8*b) + a^2/(2*b^3)) - tan(x/2)^3/(24*b) - (a*tan(x/2)^2)/(8 *b^2) - (tan(x/2)^2*(4*a^2 + 5*b^2) + b^2/3 - a*b*tan(x/2))/(8*b^3*tan(x/2 )^3) - (atan(((((a^2 + b^2)^3)^(1/2)*((2*b^8 + 7*a^2*b^6 + 4*a^4*b^4)/b^6 + (tan(x/2)*(7*a*b^6 + 16*a^3*b^4 + 8*a^5*b^2))/b^5 + (((a^2 + b^2)^3)^(1/ 2)*(2*a*b^2 + (tan(x/2)*(6*b^8 + 8*a^2*b^6))/b^5))/b^4)*1i)/b^4 + (((a^2 + b^2)^3)^(1/2)*((2*b^8 + 7*a^2*b^6 + 4*a^4*b^4)/b^6 + (tan(x/2)*(7*a*b^6 + 16*a^3*b^4 + 8*a^5*b^2))/b^5 - (((a^2 + b^2)^3)^(1/2)*(2*a*b^2 + (tan(x/2 )*(6*b^8 + 8*a^2*b^6))/b^5))/b^4)*1i)/b^4)/((2*(3*a*b^6 + 2*a^7 + 8*a^3*b^ 4 + 7*a^5*b^2))/b^6 + (((a^2 + b^2)^3)^(1/2)*((2*b^8 + 7*a^2*b^6 + 4*a^4*b ^4)/b^6 + (tan(x/2)*(7*a*b^6 + 16*a^3*b^4 + 8*a^5*b^2))/b^5 + (((a^2 + b^2 )^3)^(1/2)*(2*a*b^2 + (tan(x/2)*(6*b^8 + 8*a^2*b^6))/b^5))/b^4))/b^4 - ((( a^2 + b^2)^3)^(1/2)*((2*b^8 + 7*a^2*b^6 + 4*a^4*b^4)/b^6 + (tan(x/2)*(7*a* b^6 + 16*a^3*b^4 + 8*a^5*b^2))/b^5 - (((a^2 + b^2)^3)^(1/2)*(2*a*b^2 + (ta n(x/2)*(6*b^8 + 8*a^2*b^6))/b^5))/b^4))/b^4 - (2*tan(x/2)*(2*a^6 + 4*b^6 + 10*a^2*b^4 + 8*a^4*b^2))/b^5))*((a^2 + b^2)^3)^(1/2)*2i)/b^4 - (log(tan(x /2))*((3*a*b^2)/2 + a^3))/b^4
Time = 0.17 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.55 \[ \int \frac {\csc ^5(x)}{a+b \cot (x)} \, dx=\frac {-12 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {x}{2}\right ) b i -a i}{\sqrt {a^{2}+b^{2}}}\right ) \sin \left (x \right )^{3} a^{2} i -12 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {x}{2}\right ) b i -a i}{\sqrt {a^{2}+b^{2}}}\right ) \sin \left (x \right )^{3} b^{2} i +3 \cos \left (x \right ) \sin \left (x \right ) a \,b^{2}-6 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{3} a^{3}-9 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{3} a \,b^{2}-6 \sin \left (x \right )^{2} a^{2} b -6 \sin \left (x \right )^{2} b^{3}-2 b^{3}}{6 \sin \left (x \right )^{3} b^{4}} \] Input:
int(csc(x)^5/(a+b*cot(x)),x)
Output:
( - 12*sqrt(a**2 + b**2)*atan((tan(x/2)*b*i - a*i)/sqrt(a**2 + b**2))*sin( x)**3*a**2*i - 12*sqrt(a**2 + b**2)*atan((tan(x/2)*b*i - a*i)/sqrt(a**2 + b**2))*sin(x)**3*b**2*i + 3*cos(x)*sin(x)*a*b**2 - 6*log(tan(x/2))*sin(x)* *3*a**3 - 9*log(tan(x/2))*sin(x)**3*a*b**2 - 6*sin(x)**2*a**2*b - 6*sin(x) **2*b**3 - 2*b**3)/(6*sin(x)**3*b**4)