Integrand size = 25, antiderivative size = 270 \[ \int \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^3 \, dx=\frac {(a+b) \left (a^2-4 a b+b^2\right ) \sqrt {e} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \sqrt {e} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \sqrt {e} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}+\sqrt {e} \cot (c+d x)}\right )}{\sqrt {2} d}-\frac {2 b \left (3 a^2-b^2\right ) \sqrt {e \cot (c+d x)}}{d}-\frac {8 a b^2 (e \cot (c+d x))^{3/2}}{5 d e}-\frac {2 b^2 (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))}{5 d e} \] Output:
1/2*(a+b)*(a^2-4*a*b+b^2)*e^(1/2)*arctan(1-2^(1/2)*(e*cot(d*x+c))^(1/2)/e^ (1/2))*2^(1/2)/d-1/2*(a+b)*(a^2-4*a*b+b^2)*e^(1/2)*arctan(1+2^(1/2)*(e*cot (d*x+c))^(1/2)/e^(1/2))*2^(1/2)/d+1/2*(a-b)*(a^2+4*a*b+b^2)*e^(1/2)*arctan h(2^(1/2)*(e*cot(d*x+c))^(1/2)/(e^(1/2)+e^(1/2)*cot(d*x+c)))*2^(1/2)/d-2*b *(3*a^2-b^2)*(e*cot(d*x+c))^(1/2)/d-8/5*a*b^2*(e*cot(d*x+c))^(3/2)/d/e-2/5 *b^2*(e*cot(d*x+c))^(3/2)*(a+b*cot(d*x+c))/d/e
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.53 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.91 \[ \int \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^3 \, dx=-\frac {\sqrt {e \cot (c+d x)} \left (2 a b^2 \cot ^{\frac {3}{2}}(c+d x)+\frac {2}{5} b^3 \cot ^{\frac {5}{2}}(c+d x)+\frac {2}{3} a \left (a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )-\frac {1}{4} b \left (-3 a^2+b^2\right ) \left (2 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+8 \sqrt {\cot (c+d x)}+\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )\right )}{d \sqrt {\cot (c+d x)}} \] Input:
Integrate[Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x])^3,x]
Output:
-((Sqrt[e*Cot[c + d*x]]*(2*a*b^2*Cot[c + d*x]^(3/2) + (2*b^3*Cot[c + d*x]^ (5/2))/5 + (2*a*(a^2 - 3*b^2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2])/3 - (b*(-3*a^2 + b^2)*(2*Sqrt[2]*ArcTan[1 - Sqrt[2 ]*Sqrt[Cot[c + d*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 8*Sqrt[Cot[c + d*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/4) )/(d*Sqrt[Cot[c + d*x]]))
Time = 1.06 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.17, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.720, Rules used = {3042, 4049, 27, 3042, 4113, 3042, 4011, 3042, 4017, 27, 1482, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )^3dx\) |
| \(\Big \downarrow \) 4049 |
| \(\displaystyle -\frac {2 \int -\frac {1}{2} \sqrt {e \cot (c+d x)} \left (12 a b^2 e \cot ^2(c+d x)+5 b \left (3 a^2-b^2\right ) e \cot (c+d x)+a \left (5 a^2-3 b^2\right ) e\right )dx}{5 e}-\frac {2 b^2 (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))}{5 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \sqrt {e \cot (c+d x)} \left (12 a b^2 e \cot ^2(c+d x)+5 b \left (3 a^2-b^2\right ) e \cot (c+d x)+a \left (5 a^2-3 b^2\right ) e\right )dx}{5 e}-\frac {2 b^2 (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))}{5 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (12 a b^2 e \tan \left (c+d x+\frac {\pi }{2}\right )^2-5 b \left (3 a^2-b^2\right ) e \tan \left (c+d x+\frac {\pi }{2}\right )+a \left (5 a^2-3 b^2\right ) e\right )dx}{5 e}-\frac {2 b^2 (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))}{5 d e}\) |
| \(\Big \downarrow \) 4113 |
| \(\displaystyle \frac {\int \sqrt {e \cot (c+d x)} \left (5 a \left (a^2-3 b^2\right ) e+5 b \left (3 a^2-b^2\right ) \cot (c+d x) e\right )dx-\frac {8 a b^2 (e \cot (c+d x))^{3/2}}{d}}{5 e}-\frac {2 b^2 (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))}{5 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (5 a \left (a^2-3 b^2\right ) e-5 b \left (3 a^2-b^2\right ) e \tan \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {8 a b^2 (e \cot (c+d x))^{3/2}}{d}}{5 e}-\frac {2 b^2 (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))}{5 d e}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {\int \frac {5 a \left (a^2-3 b^2\right ) e^2 \cot (c+d x)-5 b \left (3 a^2-b^2\right ) e^2}{\sqrt {e \cot (c+d x)}}dx-\frac {10 b e \left (3 a^2-b^2\right ) \sqrt {e \cot (c+d x)}}{d}-\frac {8 a b^2 (e \cot (c+d x))^{3/2}}{d}}{5 e}-\frac {2 b^2 (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))}{5 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {-5 b \left (3 a^2-b^2\right ) e^2-5 a \left (a^2-3 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right ) e^2}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {10 b e \left (3 a^2-b^2\right ) \sqrt {e \cot (c+d x)}}{d}-\frac {8 a b^2 (e \cot (c+d x))^{3/2}}{d}}{5 e}-\frac {2 b^2 (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))}{5 d e}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {\frac {2 \int \frac {5 e^2 \left (b \left (3 a^2-b^2\right ) e-a \left (a^2-3 b^2\right ) e \cot (c+d x)\right )}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d}-\frac {10 b e \left (3 a^2-b^2\right ) \sqrt {e \cot (c+d x)}}{d}-\frac {8 a b^2 (e \cot (c+d x))^{3/2}}{d}}{5 e}-\frac {2 b^2 (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))}{5 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {10 e^2 \int \frac {b \left (3 a^2-b^2\right ) e-a \left (a^2-3 b^2\right ) e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d}-\frac {10 b e \left (3 a^2-b^2\right ) \sqrt {e \cot (c+d x)}}{d}-\frac {8 a b^2 (e \cot (c+d x))^{3/2}}{d}}{5 e}-\frac {2 b^2 (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))}{5 d e}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {\frac {10 e^2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {\cot (c+d x) e+e}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d}-\frac {10 b e \left (3 a^2-b^2\right ) \sqrt {e \cot (c+d x)}}{d}-\frac {8 a b^2 (e \cot (c+d x))^{3/2}}{d}}{5 e}-\frac {2 b^2 (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))}{5 d e}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {10 e^2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}\right )\right )}{d}-\frac {10 b e \left (3 a^2-b^2\right ) \sqrt {e \cot (c+d x)}}{d}-\frac {8 a b^2 (e \cot (c+d x))^{3/2}}{d}}{5 e}-\frac {2 b^2 (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))}{5 d e}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {10 e^2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {10 b e \left (3 a^2-b^2\right ) \sqrt {e \cot (c+d x)}}{d}-\frac {8 a b^2 (e \cot (c+d x))^{3/2}}{d}}{5 e}-\frac {2 b^2 (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))}{5 d e}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {10 e^2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {10 b e \left (3 a^2-b^2\right ) \sqrt {e \cot (c+d x)}}{d}-\frac {8 a b^2 (e \cot (c+d x))^{3/2}}{d}}{5 e}-\frac {2 b^2 (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))}{5 d e}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {10 e^2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {10 b e \left (3 a^2-b^2\right ) \sqrt {e \cot (c+d x)}}{d}-\frac {8 a b^2 (e \cot (c+d x))^{3/2}}{d}}{5 e}-\frac {2 b^2 (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))}{5 d e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {10 e^2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {10 b e \left (3 a^2-b^2\right ) \sqrt {e \cot (c+d x)}}{d}-\frac {8 a b^2 (e \cot (c+d x))^{3/2}}{d}}{5 e}-\frac {2 b^2 (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))}{5 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {10 e^2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {10 b e \left (3 a^2-b^2\right ) \sqrt {e \cot (c+d x)}}{d}-\frac {8 a b^2 (e \cot (c+d x))^{3/2}}{d}}{5 e}-\frac {2 b^2 (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))}{5 d e}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {10 e^2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}-\frac {10 b e \left (3 a^2-b^2\right ) \sqrt {e \cot (c+d x)}}{d}-\frac {8 a b^2 (e \cot (c+d x))^{3/2}}{d}}{5 e}-\frac {2 b^2 (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))}{5 d e}\) |
Input:
Int[Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x])^3,x]
Output:
(-2*b^2*(e*Cot[c + d*x])^(3/2)*(a + b*Cot[c + d*x]))/(5*d*e) + ((-10*b*(3* a^2 - b^2)*e*Sqrt[e*Cot[c + d*x]])/d - (8*a*b^2*(e*Cot[c + d*x])^(3/2))/d + (10*e^2*(-1/2*((a + b)*(a^2 - 4*a*b + b^2)*(-(ArcTan[1 - (Sqrt[2]*Sqrt[e *Cot[c + d*x]])/Sqrt[e]]/(Sqrt[2]*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*Sqrt[e*C ot[c + d*x]])/Sqrt[e]]/(Sqrt[2]*Sqrt[e]))) + ((a - b)*(a^2 + 4*a*b + b^2)* (-1/2*Log[e + e*Cot[c + d*x] - Sqrt[2]*Sqrt[e]*Sqrt[e*Cot[c + d*x]]]/(Sqrt [2]*Sqrt[e]) + Log[e + e*Cot[c + d*x] + Sqrt[2]*Sqrt[e]*Sqrt[e*Cot[c + d*x ]]]/(2*Sqrt[2]*Sqrt[e])))/2))/d)/(5*e)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a*c, 2]}, Simp[(d*q + a*e)/(2*a*c) Int[(q + c*x^2)/(a + c*x^4), x], x] + Simp[(d*q - a*e)/(2*a*c) Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a , c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- a)*c]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Int [(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x], x] , x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ )]], x_Symbol] :> Simp[2/f Subst[Int[(b*c + d*x^2)/(b^2 + x^4), x], x, Sq rt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] & & NeQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Simp[1/(d*(m + n - 1)) Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[ e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2 , 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || I ntegerQ[m]) && !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])) )
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Si mp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && !LeQ[m, -1]
Time = 0.30 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.37
| method | result | size |
| derivativedivides | \(-\frac {2 \left (\frac {b^{3} \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+a e \,b^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}+3 a^{2} b \,e^{2} \sqrt {e \cot \left (d x +c \right )}-b^{3} e^{2} \sqrt {e \cot \left (d x +c \right )}+e^{3} \left (\frac {\left (-3 a^{2} b e +b^{3} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (a^{3}-3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d \,e^{2}}\) | \(371\) |
| default | \(-\frac {2 \left (\frac {b^{3} \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+a e \,b^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}+3 a^{2} b \,e^{2} \sqrt {e \cot \left (d x +c \right )}-b^{3} e^{2} \sqrt {e \cot \left (d x +c \right )}+e^{3} \left (\frac {\left (-3 a^{2} b e +b^{3} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (a^{3}-3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d \,e^{2}}\) | \(371\) |
| parts | \(-\frac {a^{3} e \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d \left (e^{2}\right )^{\frac {1}{4}}}-\frac {2 b^{3} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-e^{2} \sqrt {e \cot \left (d x +c \right )}+\frac {e^{2} \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8}\right )}{d \,e^{2}}+\frac {3 a^{2} b \left (-2 \sqrt {e \cot \left (d x +c \right )}+\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4}\right )}{d}-\frac {6 a \,b^{2} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-\frac {e^{2} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d e}\) | \(625\) |
Input:
int((e*cot(d*x+c))^(1/2)*(a+b*cot(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
-2/d/e^2*(1/5*b^3*(e*cot(d*x+c))^(5/2)+a*e*b^2*(e*cot(d*x+c))^(3/2)+3*a^2* b*e^2*(e*cot(d*x+c))^(1/2)-b^3*e^2*(e*cot(d*x+c))^(1/2)+e^3*(1/8*(-3*a^2*b *e+b^3*e)*(e^2)^(1/4)/e^2*2^(1/2)*(ln((e*cot(d*x+c)+(e^2)^(1/4)*(e*cot(d*x +c))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^( 1/2)*2^(1/2)+(e^2)^(1/2)))+2*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/ 2)+1)-2*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1))+1/8*(a^3-3*a* b^2)/(e^2)^(1/4)*2^(1/2)*(ln((e*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2 )*2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1 /2)+(e^2)^(1/2)))+2*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)-2*a rctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1))))
Leaf count of result is larger than twice the leaf count of optimal. 1643 vs. \(2 (230) = 460\).
Time = 0.14 (sec) , antiderivative size = 1643, normalized size of antiderivative = 6.09 \[ \int \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^3 \, dx=\text {Too large to display} \] Input:
integrate((e*cot(d*x+c))^(1/2)*(a+b*cot(d*x+c))^3,x, algorithm="fricas")
Output:
1/10*(5*(d*cos(2*d*x + 2*c) - d)*sqrt((d^2*sqrt(-(a^12 - 30*a^10*b^2 + 255 *a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)*e^2/d^4) + 2*(3 *a^5*b - 10*a^3*b^3 + 3*a*b^5)*e)/d^2)*log(-(a^12 - 12*a^10*b^2 - 27*a^8*b ^4 + 27*a^4*b^8 + 12*a^2*b^10 - b^12)*e*sqrt((e*cos(2*d*x + 2*c) + e)/sin( 2*d*x + 2*c)) + ((a^3 - 3*a*b^2)*d^3*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b ^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)*e^2/d^4) - (3*a^8*b - 46*a^6*b^3 + 60*a^4*b^5 - 18*a^2*b^7 + b^9)*d*e)*sqrt((d^2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12 )*e^2/d^4) + 2*(3*a^5*b - 10*a^3*b^3 + 3*a*b^5)*e)/d^2)) - 5*(d*cos(2*d*x + 2*c) - d)*sqrt((d^2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^ 6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)*e^2/d^4) + 2*(3*a^5*b - 10*a^3*b^3 + 3*a*b^5)*e)/d^2)*log(-(a^12 - 12*a^10*b^2 - 27*a^8*b^4 + 27*a^4*b^8 + 12* a^2*b^10 - b^12)*e*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) - ((a^3 - 3*a*b^2)*d^3*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 25 5*a^4*b^8 - 30*a^2*b^10 + b^12)*e^2/d^4) - (3*a^8*b - 46*a^6*b^3 + 60*a^4* b^5 - 18*a^2*b^7 + b^9)*d*e)*sqrt((d^2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8 *b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)*e^2/d^4) + 2*(3*a^5 *b - 10*a^3*b^3 + 3*a*b^5)*e)/d^2)) - 5*(d*cos(2*d*x + 2*c) - d)*sqrt(-(d^ 2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30 *a^2*b^10 + b^12)*e^2/d^4) - 2*(3*a^5*b - 10*a^3*b^3 + 3*a*b^5)*e)/d^2)...
\[ \int \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^3 \, dx=\int \sqrt {e \cot {\left (c + d x \right )}} \left (a + b \cot {\left (c + d x \right )}\right )^{3}\, dx \] Input:
integrate((e*cot(d*x+c))**(1/2)*(a+b*cot(d*x+c))**3,x)
Output:
Integral(sqrt(e*cot(c + d*x))*(a + b*cot(c + d*x))**3, x)
Exception generated. \[ \int \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^3 \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*cot(d*x+c))^(1/2)*(a+b*cot(d*x+c))^3,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^3 \, dx=\int { {\left (b \cot \left (d x + c\right ) + a\right )}^{3} \sqrt {e \cot \left (d x + c\right )} \,d x } \] Input:
integrate((e*cot(d*x+c))^(1/2)*(a+b*cot(d*x+c))^3,x, algorithm="giac")
Output:
integrate((b*cot(d*x + c) + a)^3*sqrt(e*cot(d*x + c)), x)
Time = 11.29 (sec) , antiderivative size = 2071, normalized size of antiderivative = 7.67 \[ \int \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^3 \, dx=\text {Too large to display} \] Input:
int((e*cot(c + d*x))^(1/2)*(a + b*cot(c + d*x))^3,x)
Output:
(e*cot(c + d*x))^(1/2)*((2*b^3)/d - (6*a^2*b)/d) + atan((((16*(e*cot(c + d *x))^(1/2)*(a^6*e^4 - b^6*e^4 + 15*a^2*b^4*e^4 - 15*a^4*b^2*e^4))/d^2 - (8 *(4*b^3*d^2*e^4 - 12*a^2*b*d^2*e^4)*((b^6*e*1i - a^6*e*1i - a^2*b^4*e*15i - 20*a^3*b^3*e + a^4*b^2*e*15i + 6*a*b^5*e + 6*a^5*b*e)/(4*d^2))^(1/2))/d^ 3)*((b^6*e*1i - a^6*e*1i - a^2*b^4*e*15i - 20*a^3*b^3*e + a^4*b^2*e*15i + 6*a*b^5*e + 6*a^5*b*e)/(4*d^2))^(1/2)*1i + ((16*(e*cot(c + d*x))^(1/2)*(a^ 6*e^4 - b^6*e^4 + 15*a^2*b^4*e^4 - 15*a^4*b^2*e^4))/d^2 + (8*(4*b^3*d^2*e^ 4 - 12*a^2*b*d^2*e^4)*((b^6*e*1i - a^6*e*1i - a^2*b^4*e*15i - 20*a^3*b^3*e + a^4*b^2*e*15i + 6*a*b^5*e + 6*a^5*b*e)/(4*d^2))^(1/2))/d^3)*((b^6*e*1i - a^6*e*1i - a^2*b^4*e*15i - 20*a^3*b^3*e + a^4*b^2*e*15i + 6*a*b^5*e + 6* a^5*b*e)/(4*d^2))^(1/2)*1i)/(((16*(e*cot(c + d*x))^(1/2)*(a^6*e^4 - b^6*e^ 4 + 15*a^2*b^4*e^4 - 15*a^4*b^2*e^4))/d^2 - (8*(4*b^3*d^2*e^4 - 12*a^2*b*d ^2*e^4)*((b^6*e*1i - a^6*e*1i - a^2*b^4*e*15i - 20*a^3*b^3*e + a^4*b^2*e*1 5i + 6*a*b^5*e + 6*a^5*b*e)/(4*d^2))^(1/2))/d^3)*((b^6*e*1i - a^6*e*1i - a ^2*b^4*e*15i - 20*a^3*b^3*e + a^4*b^2*e*15i + 6*a*b^5*e + 6*a^5*b*e)/(4*d^ 2))^(1/2) - ((16*(e*cot(c + d*x))^(1/2)*(a^6*e^4 - b^6*e^4 + 15*a^2*b^4*e^ 4 - 15*a^4*b^2*e^4))/d^2 + (8*(4*b^3*d^2*e^4 - 12*a^2*b*d^2*e^4)*((b^6*e*1 i - a^6*e*1i - a^2*b^4*e*15i - 20*a^3*b^3*e + a^4*b^2*e*15i + 6*a*b^5*e + 6*a^5*b*e)/(4*d^2))^(1/2))/d^3)*((b^6*e*1i - a^6*e*1i - a^2*b^4*e*15i - 20 *a^3*b^3*e + a^4*b^2*e*15i + 6*a*b^5*e + 6*a^5*b*e)/(4*d^2))^(1/2) + (1...
\[ \int \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^3 \, dx=\frac {\sqrt {e}\, \left (-2 \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{2} b^{3}-30 \sqrt {\cot \left (d x +c \right )}\, a^{2} b +10 \sqrt {\cot \left (d x +c \right )}\, b^{3}-15 \left (\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )}d x \right ) a^{2} b d +5 \left (\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )}d x \right ) b^{3} d +5 \left (\int \sqrt {\cot \left (d x +c \right )}d x \right ) a^{3} d +15 \left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{2}d x \right ) a \,b^{2} d \right )}{5 d} \] Input:
int((e*cot(d*x+c))^(1/2)*(a+b*cot(d*x+c))^3,x)
Output:
(sqrt(e)*( - 2*sqrt(cot(c + d*x))*cot(c + d*x)**2*b**3 - 30*sqrt(cot(c + d *x))*a**2*b + 10*sqrt(cot(c + d*x))*b**3 - 15*int(sqrt(cot(c + d*x))/cot(c + d*x),x)*a**2*b*d + 5*int(sqrt(cot(c + d*x))/cot(c + d*x),x)*b**3*d + 5* int(sqrt(cot(c + d*x)),x)*a**3*d + 15*int(sqrt(cot(c + d*x))*cot(c + d*x)* *2,x)*a*b**2*d))/(5*d)