Integrand size = 33, antiderivative size = 976 \[ \int \cot ^5(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=-\frac {\sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \arctan \left (\frac {b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )-b \sqrt {a^2+b^2-2 a c+c^2} \cot (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}-\frac {b \text {arctanh}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{2 \sqrt {c} e}+\frac {b \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{16 c^{5/2} e}-\frac {b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c \cot (d+e x)}{2 \sqrt {c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{256 c^{9/2} e}+\frac {\sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \text {arctanh}\left (\frac {b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )+b \sqrt {a^2+b^2-2 a c+c^2} \cot (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}-\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{e}-\frac {b (b+2 c \cot (d+e x)) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{8 c^2 e}+\frac {b \left (7 b^2-12 a c\right ) (b+2 c \cot (d+e x)) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{128 c^4 e}+\frac {\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}}{3 c e}-\frac {\cot ^2(d+e x) \left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}}{5 c e}-\frac {\left (35 b^2-32 a c-42 b c \cot (d+e x)\right ) \left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}}{240 c^3 e} \]
1/16*b*(-4*a*c+b^2)*arctanh(1/2*(b+2*c*cot(e*x+d))/c^(1/2)/(a+b*cot(e*x+d) +c*cot(e*x+d)^2)^(1/2))/c^(5/2)/e-1/256*b*(-12*a*c+7*b^2)*(-4*a*c+b^2)*arc tanh(1/2*(b+2*c*cot(e*x+d))/c^(1/2)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)) /c^(9/2)/e+1/3*(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/2)/c/e-1/5*cot(e*x+d)^2* (a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/2)/c/e-1/240*(35*b^2-32*a*c-42*b*c*cot( e*x+d))*(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/2)/c^3/e-1/2*b*arctanh(1/2*(b+2 *c*cot(e*x+d))/c^(1/2)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2))/e/c^(1/2)-(a +b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)/e-1/8*b*(b+2*c*cot(e*x+d))*(a+b*cot(e* x+d)+c*cot(e*x+d)^2)^(1/2)/c^2/e+1/128*b*(-12*a*c+7*b^2)*(b+2*c*cot(e*x+d) )*(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)/c^4/e+1/2*arctanh(1/2*(b^2+b*cot(e *x+d)*(a^2-2*a*c+b^2+c^2)^(1/2)+(a-c)*(a-c+(a^2-2*a*c+b^2+c^2)^(1/2)))/(a^ 2-2*a*c+b^2+c^2)^(1/4)*2^(1/2)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)/(a^2+ b^2+c*(c-(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c-(a^2-2*a*c+b^2+c^2)^(1/2)))^(1/ 2))*(a^2+b^2+c*(c-(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c-(a^2-2*a*c+b^2+c^2)^(1 /2)))^(1/2)/(a^2-2*a*c+b^2+c^2)^(1/4)/e*2^(1/2)-1/2*arctan(1/2*(b^2+(a-c)* (a-c-(a^2-2*a*c+b^2+c^2)^(1/2))-b*cot(e*x+d)*(a^2-2*a*c+b^2+c^2)^(1/2))/(a ^2-2*a*c+b^2+c^2)^(1/4)*2^(1/2)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)/(a^2 +b^2+c*(c+(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c+(a^2-2*a*c+b^2+c^2)^(1/2)))^(1 /2))*(a^2+b^2+c*(c+(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c+(a^2-2*a*c+b^2+c^2)^( 1/2)))^(1/2)/(a^2-2*a*c+b^2+c^2)^(1/4)/e*2^(1/2)
Result contains complex when optimal does not.
Time = 6.71 (sec) , antiderivative size = 1214, normalized size of antiderivative = 1.24 \[ \int \cot ^5(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=-\frac {i \sqrt {a+i b-c} \arctan \left (\frac {i b-2 c+(2 i a-b) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right ) \tan (d+e x) \sqrt {\cot ^2(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )}}{2 e \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}-\frac {i \sqrt {a-i b-c} \arctan \left (\frac {i b+2 c+(2 i a+b) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right ) \tan (d+e x) \sqrt {\cot ^2(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )}}{2 e \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}-\frac {\sqrt {a} \text {arctanh}\left (\frac {b+2 a \tan (d+e x)}{2 \sqrt {a} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right ) \tan (d+e x) \sqrt {\cot ^2(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )}}{e \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}+\frac {\tan (d+e x) \sqrt {\cot ^2(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )} \left (2 \sqrt {a} \text {arctanh}\left (\frac {b+2 a \tan (d+e x)}{2 \sqrt {a} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )-\frac {b \text {arctanh}\left (\frac {2 c+b \tan (d+e x)}{2 \sqrt {c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )}{\sqrt {c}}-2 \cot (d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}\right )}{2 e \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}+\frac {\tan (d+e x) \sqrt {\cot ^2(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )} \left (\frac {16 \cot ^3(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )^{3/2}}{c}+\frac {3 b \left (\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 c+b \tan (d+e x)}{2 \sqrt {c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )}{c^{3/2}}-\frac {2 \cot ^2(d+e x) (2 c+b \tan (d+e x)) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}{c}\right )}{c}\right )}{48 e \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}+\frac {\tan (d+e x) \sqrt {\cot ^2(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )} \left (-\frac {\cot ^5(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )^{3/2}}{5 c}-\frac {-\frac {7 b \cot ^4(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )^{3/2}}{8 c}-\frac {\frac {\left (-35 b^2+32 a c\right ) \cot ^3(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )^{3/2}}{12 c}-\frac {\left (-7 a b c+\frac {1}{4} b \left (35 b^2-32 a c\right )\right ) \left (\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 c+b \tan (d+e x)}{2 \sqrt {c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )}{8 c^{3/2}}+\frac {\cot ^2(d+e x) (-2 c-b \tan (d+e x)) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}{4 c}\right )}{2 c}}{4 c}}{5 c}\right )}{e \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}} \]
((-1/2*I)*Sqrt[a + I*b - c]*ArcTan[(I*b - 2*c + ((2*I)*a - b)*Tan[d + e*x] )/(2*Sqrt[a + I*b - c]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])]*Tan[d + e*x]*Sqrt[Cot[d + e*x]^2*(c + b*Tan[d + e*x] + a*Tan[d + e*x]^2)])/(e*S qrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2]) - ((I/2)*Sqrt[a - I*b - c]*Arc Tan[(I*b + 2*c + ((2*I)*a + b)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])]*Tan[d + e*x]*Sqrt[Cot[d + e*x]^2*(c + b*Tan[d + e*x] + a*Tan[d + e*x]^2)])/(e*Sqrt[c + b*Tan[d + e*x] + a*Tan[ d + e*x]^2]) - (Sqrt[a]*ArcTanh[(b + 2*a*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])]*Tan[d + e*x]*Sqrt[Cot[d + e*x]^2*(c + b*Tan[d + e*x] + a*Tan[d + e*x]^2)])/(e*Sqrt[c + b*Tan[d + e*x] + a*Tan[ d + e*x]^2]) + (Tan[d + e*x]*Sqrt[Cot[d + e*x]^2*(c + b*Tan[d + e*x] + a*T an[d + e*x]^2)]*(2*Sqrt[a]*ArcTanh[(b + 2*a*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[ c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])] - (b*ArcTanh[(2*c + b*Tan[d + e*x ])/(2*Sqrt[c]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])])/Sqrt[c] - 2*C ot[d + e*x]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2]))/(2*e*Sqrt[c + b* Tan[d + e*x] + a*Tan[d + e*x]^2]) + (Tan[d + e*x]*Sqrt[Cot[d + e*x]^2*(c + b*Tan[d + e*x] + a*Tan[d + e*x]^2)]*((16*Cot[d + e*x]^3*(c + b*Tan[d + e* x] + a*Tan[d + e*x]^2)^(3/2))/c + (3*b*(((b^2 - 4*a*c)*ArcTanh[(2*c + b*Ta n[d + e*x])/(2*Sqrt[c]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])])/c^(3 /2) - (2*Cot[d + e*x]^2*(2*c + b*Tan[d + e*x])*Sqrt[c + b*Tan[d + e*x] ...
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.
| \(\displaystyle \int \cot ^5(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cot (d+e x)^5 \sqrt {a+b \cot (d+e x)+c \cot (d+e x)^2}dx\) |
| \(\Big \downarrow \) 4184 |
| \(\displaystyle -\frac {\int \frac {\cot ^5(d+e x) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}{\cot ^2(d+e x)+1}d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {\int \left (\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot ^3(d+e x)+\frac {\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot (d+e x)}{\cot ^2(d+e x)+1}-\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot (d+e x)\right )d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\int \frac {\cot ^5(d+e x) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}{\cot ^2(d+e x)+1}d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {\int \left (\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot ^3(d+e x)+\frac {\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot (d+e x)}{\cot ^2(d+e x)+1}-\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot (d+e x)\right )d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\int \frac {\cot ^5(d+e x) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}{\cot ^2(d+e x)+1}d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {\int \left (\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot ^3(d+e x)+\frac {\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot (d+e x)}{\cot ^2(d+e x)+1}-\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot (d+e x)\right )d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\int \frac {\cot ^5(d+e x) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}{\cot ^2(d+e x)+1}d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {\int \left (\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot ^3(d+e x)+\frac {\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot (d+e x)}{\cot ^2(d+e x)+1}-\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot (d+e x)\right )d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\int \frac {\cot ^5(d+e x) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}{\cot ^2(d+e x)+1}d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {\int \left (\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot ^3(d+e x)+\frac {\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot (d+e x)}{\cot ^2(d+e x)+1}-\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot (d+e x)\right )d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\int \frac {\cot ^5(d+e x) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}{\cot ^2(d+e x)+1}d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {\int \left (\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot ^3(d+e x)+\frac {\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot (d+e x)}{\cot ^2(d+e x)+1}-\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot (d+e x)\right )d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\int \frac {\cot ^5(d+e x) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}{\cot ^2(d+e x)+1}d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {\int \left (\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot ^3(d+e x)+\frac {\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot (d+e x)}{\cot ^2(d+e x)+1}-\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot (d+e x)\right )d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\int \frac {\cot ^5(d+e x) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}{\cot ^2(d+e x)+1}d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {\int \left (\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot ^3(d+e x)+\frac {\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot (d+e x)}{\cot ^2(d+e x)+1}-\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot (d+e x)\right )d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\int \frac {\cot ^5(d+e x) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}{\cot ^2(d+e x)+1}d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {\int \left (\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot ^3(d+e x)+\frac {\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot (d+e x)}{\cot ^2(d+e x)+1}-\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot (d+e x)\right )d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\int \frac {\cot ^5(d+e x) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}{\cot ^2(d+e x)+1}d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {\int \left (\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot ^3(d+e x)+\frac {\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot (d+e x)}{\cot ^2(d+e x)+1}-\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot (d+e x)\right )d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\int \frac {\cot ^5(d+e x) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}{\cot ^2(d+e x)+1}d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {\int \left (\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot ^3(d+e x)+\frac {\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot (d+e x)}{\cot ^2(d+e x)+1}-\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot (d+e x)\right )d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\int \frac {\cot ^5(d+e x) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}{\cot ^2(d+e x)+1}d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {\int \left (\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot ^3(d+e x)+\frac {\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot (d+e x)}{\cot ^2(d+e x)+1}-\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot (d+e x)\right )d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\int \frac {\cot ^5(d+e x) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}{\cot ^2(d+e x)+1}d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {\int \left (\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot ^3(d+e x)+\frac {\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot (d+e x)}{\cot ^2(d+e x)+1}-\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot (d+e x)\right )d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\int \frac {\cot ^5(d+e x) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}{\cot ^2(d+e x)+1}d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {\int \left (\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot ^3(d+e x)+\frac {\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot (d+e x)}{\cot ^2(d+e x)+1}-\sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \cot (d+e x)\right )d\cot (d+e x)}{e}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\int \frac {\cot ^5(d+e x) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}{\cot ^2(d+e x)+1}d\cot (d+e x)}{e}\) |
3.1.6.3.1 Defintions of rubi rules used
Int[cot[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*(cot[(d_.) + (e_.)*(x_)]*( f_.))^(n_.) + (c_.)*(cot[(d_.) + (e_.)*(x_)]*(f_.))^(n2_.))^(p_), x_Symbol] :> Simp[-f/e Subst[Int[(x/f)^m*((a + b*x^n + c*x^(2*n))^p/(f^2 + x^2)), x], x, f*Cot[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[ n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 6.86 (sec) , antiderivative size = 17768513, normalized size of antiderivative = 18205.44
\[\text {output too large to display}\]
Leaf count of result is larger than twice the leaf count of optimal. 6335 vs. \(2 (877) = 1754\).
Time = 3.02 (sec) , antiderivative size = 12721, normalized size of antiderivative = 13.03 \[ \int \cot ^5(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\text {Too large to display} \]
\[ \int \cot ^5(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\int \sqrt {a + b \cot {\left (d + e x \right )} + c \cot ^{2}{\left (d + e x \right )}} \cot ^{5}{\left (d + e x \right )}\, dx \]
\[ \int \cot ^5(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\int { \sqrt {c \cot \left (e x + d\right )^{2} + b \cot \left (e x + d\right ) + a} \cot \left (e x + d\right )^{5} \,d x } \]
\[ \int \cot ^5(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\int { \sqrt {c \cot \left (e x + d\right )^{2} + b \cot \left (e x + d\right ) + a} \cot \left (e x + d\right )^{5} \,d x } \]
Timed out. \[ \int \cot ^5(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx=\text {Hanged} \]