Integrand size = 23, antiderivative size = 262 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d}-\frac {263 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{512 \sqrt {2} a^{5/2} d}+\frac {199 a^2}{288 d (a+a \sec (c+d x))^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}+\frac {135 a}{448 d (a+a \sec (c+d x))^{7/2}}+\frac {7}{640 d (a+a \sec (c+d x))^{5/2}}-\frac {83}{256 a d (a+a \sec (c+d x))^{3/2}}-\frac {761}{512 a^2 d \sqrt {a+a \sec (c+d x)}} \] Output:
2*arctanh((a+a*sec(d*x+c))^(1/2)/a^(1/2))/a^(5/2)/d-263/1024*arctanh(1/2*( a+a*sec(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*2^(1/2)/a^(5/2)/d+199/288*a^2/d/(a+ a*sec(d*x+c))^(9/2)-1/4*a^2/d/(1-sec(d*x+c))^2/(a+a*sec(d*x+c))^(9/2)-21/1 6*a^2/d/(1-sec(d*x+c))/(a+a*sec(d*x+c))^(9/2)+135/448*a/d/(a+a*sec(d*x+c)) ^(7/2)+7/640/d/(a+a*sec(d*x+c))^(5/2)-83/256/a/d/(a+a*sec(d*x+c))^(3/2)-76 1/512/a^2/d/(a+a*sec(d*x+c))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.21 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.38 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {\cot ^4(c+d x) \left (-450+263 \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},1,-\frac {7}{2},\frac {1}{2} (1+\sec (c+d x))\right ) (-1+\sec (c+d x))^2-64 \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},1,-\frac {7}{2},1+\sec (c+d x)\right ) (-1+\sec (c+d x))^2+378 \sec (c+d x)\right )}{288 d (a (1+\sec (c+d x)))^{5/2}} \] Input:
Integrate[Cot[c + d*x]^5/(a + a*Sec[c + d*x])^(5/2),x]
Output:
(Cot[c + d*x]^4*(-450 + 263*Hypergeometric2F1[-9/2, 1, -7/2, (1 + Sec[c + d*x])/2]*(-1 + Sec[c + d*x])^2 - 64*Hypergeometric2F1[-9/2, 1, -7/2, 1 + S ec[c + d*x]]*(-1 + Sec[c + d*x])^2 + 378*Sec[c + d*x]))/(288*d*(a*(1 + Sec [c + d*x]))^(5/2))
Time = 0.46 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.11, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 25, 4368, 25, 27, 114, 27, 168, 27, 169, 27, 169, 27, 169, 27, 169, 27, 169, 27, 174, 73, 219, 220}
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.
| \(\displaystyle \int \frac {\cot ^5(c+d x)}{(a \sec (c+d x)+a)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\cot \left (c+d x+\frac {\pi }{2}\right )^5 \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\cot \left (\frac {1}{2} (2 c+\pi )+d x\right )^5 \left (\csc \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4368 |
| \(\displaystyle \frac {a^6 \int -\frac {\cos (c+d x)}{a^3 (1-\sec (c+d x))^3 (\sec (c+d x) a+a)^{11/2}}d\sec (c+d x)}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a^6 \int \frac {\cos (c+d x)}{a^3 (1-\sec (c+d x))^3 (\sec (c+d x) a+a)^{11/2}}d\sec (c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \int \frac {\cos (c+d x)}{(1-\sec (c+d x))^3 (\sec (c+d x) a+a)^{11/2}}d\sec (c+d x)}{d}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{9/2}}-\frac {\int -\frac {a \cos (c+d x) (13 \sec (c+d x)+8)}{2 (1-\sec (c+d x))^2 (\sec (c+d x) a+a)^{11/2}}d\sec (c+d x)}{4 a}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \int \frac {\cos (c+d x) (13 \sec (c+d x)+8)}{(1-\sec (c+d x))^2 (\sec (c+d x) a+a)^{11/2}}d\sec (c+d x)+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{9/2}}\right )}{d}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {21}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{9/2}}-\frac {\int -\frac {a \cos (c+d x) (231 \sec (c+d x)+32)}{2 (1-\sec (c+d x)) (\sec (c+d x) a+a)^{11/2}}d\sec (c+d x)}{2 a}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{9/2}}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \int \frac {\cos (c+d x) (231 \sec (c+d x)+32)}{(1-\sec (c+d x)) (\sec (c+d x) a+a)^{11/2}}d\sec (c+d x)+\frac {21}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{9/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{9/2}}\right )}{d}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\int \frac {9 a \cos (c+d x) (199 \sec (c+d x)+64)}{2 (1-\sec (c+d x)) (\sec (c+d x) a+a)^{9/2}}d\sec (c+d x)}{9 a^2}-\frac {199}{9 a (a \sec (c+d x)+a)^{9/2}}\right )+\frac {21}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{9/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{9/2}}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\int \frac {\cos (c+d x) (199 \sec (c+d x)+64)}{(1-\sec (c+d x)) (\sec (c+d x) a+a)^{9/2}}d\sec (c+d x)}{2 a}-\frac {199}{9 a (a \sec (c+d x)+a)^{9/2}}\right )+\frac {21}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{9/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{9/2}}\right )}{d}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\int \frac {7 a \cos (c+d x) (135 \sec (c+d x)+128)}{2 (1-\sec (c+d x)) (\sec (c+d x) a+a)^{7/2}}d\sec (c+d x)}{7 a^2}-\frac {135}{7 a (a \sec (c+d x)+a)^{7/2}}}{2 a}-\frac {199}{9 a (a \sec (c+d x)+a)^{9/2}}\right )+\frac {21}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{9/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{9/2}}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\int \frac {\cos (c+d x) (135 \sec (c+d x)+128)}{(1-\sec (c+d x)) (\sec (c+d x) a+a)^{7/2}}d\sec (c+d x)}{2 a}-\frac {135}{7 a (a \sec (c+d x)+a)^{7/2}}}{2 a}-\frac {199}{9 a (a \sec (c+d x)+a)^{9/2}}\right )+\frac {21}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{9/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{9/2}}\right )}{d}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\frac {\int \frac {5 a \cos (c+d x) (7 \sec (c+d x)+256)}{2 (1-\sec (c+d x)) (\sec (c+d x) a+a)^{5/2}}d\sec (c+d x)}{5 a^2}-\frac {7}{5 a (a \sec (c+d x)+a)^{5/2}}}{2 a}-\frac {135}{7 a (a \sec (c+d x)+a)^{7/2}}}{2 a}-\frac {199}{9 a (a \sec (c+d x)+a)^{9/2}}\right )+\frac {21}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{9/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{9/2}}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\frac {\int \frac {\cos (c+d x) (7 \sec (c+d x)+256)}{(1-\sec (c+d x)) (\sec (c+d x) a+a)^{5/2}}d\sec (c+d x)}{2 a}-\frac {7}{5 a (a \sec (c+d x)+a)^{5/2}}}{2 a}-\frac {135}{7 a (a \sec (c+d x)+a)^{7/2}}}{2 a}-\frac {199}{9 a (a \sec (c+d x)+a)^{9/2}}\right )+\frac {21}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{9/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{9/2}}\right )}{d}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\frac {\frac {\int \frac {3 a \cos (c+d x) (512-249 \sec (c+d x))}{2 (1-\sec (c+d x)) (\sec (c+d x) a+a)^{3/2}}d\sec (c+d x)}{3 a^2}+\frac {83}{a (a \sec (c+d x)+a)^{3/2}}}{2 a}-\frac {7}{5 a (a \sec (c+d x)+a)^{5/2}}}{2 a}-\frac {135}{7 a (a \sec (c+d x)+a)^{7/2}}}{2 a}-\frac {199}{9 a (a \sec (c+d x)+a)^{9/2}}\right )+\frac {21}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{9/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{9/2}}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\frac {\frac {\int \frac {\cos (c+d x) (512-249 \sec (c+d x))}{(1-\sec (c+d x)) (\sec (c+d x) a+a)^{3/2}}d\sec (c+d x)}{2 a}+\frac {83}{a (a \sec (c+d x)+a)^{3/2}}}{2 a}-\frac {7}{5 a (a \sec (c+d x)+a)^{5/2}}}{2 a}-\frac {135}{7 a (a \sec (c+d x)+a)^{7/2}}}{2 a}-\frac {199}{9 a (a \sec (c+d x)+a)^{9/2}}\right )+\frac {21}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{9/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{9/2}}\right )}{d}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\frac {\frac {\frac {\int \frac {a \cos (c+d x) (1024-761 \sec (c+d x))}{2 (1-\sec (c+d x)) \sqrt {\sec (c+d x) a+a}}d\sec (c+d x)}{a^2}+\frac {761}{a \sqrt {a \sec (c+d x)+a}}}{2 a}+\frac {83}{a (a \sec (c+d x)+a)^{3/2}}}{2 a}-\frac {7}{5 a (a \sec (c+d x)+a)^{5/2}}}{2 a}-\frac {135}{7 a (a \sec (c+d x)+a)^{7/2}}}{2 a}-\frac {199}{9 a (a \sec (c+d x)+a)^{9/2}}\right )+\frac {21}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{9/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{9/2}}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\frac {\frac {\frac {\int \frac {\cos (c+d x) (1024-761 \sec (c+d x))}{(1-\sec (c+d x)) \sqrt {\sec (c+d x) a+a}}d\sec (c+d x)}{2 a}+\frac {761}{a \sqrt {a \sec (c+d x)+a}}}{2 a}+\frac {83}{a (a \sec (c+d x)+a)^{3/2}}}{2 a}-\frac {7}{5 a (a \sec (c+d x)+a)^{5/2}}}{2 a}-\frac {135}{7 a (a \sec (c+d x)+a)^{7/2}}}{2 a}-\frac {199}{9 a (a \sec (c+d x)+a)^{9/2}}\right )+\frac {21}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{9/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{9/2}}\right )}{d}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\frac {\frac {\frac {263 \int \frac {1}{(1-\sec (c+d x)) \sqrt {\sec (c+d x) a+a}}d\sec (c+d x)+1024 \int \frac {\cos (c+d x)}{\sqrt {\sec (c+d x) a+a}}d\sec (c+d x)}{2 a}+\frac {761}{a \sqrt {a \sec (c+d x)+a}}}{2 a}+\frac {83}{a (a \sec (c+d x)+a)^{3/2}}}{2 a}-\frac {7}{5 a (a \sec (c+d x)+a)^{5/2}}}{2 a}-\frac {135}{7 a (a \sec (c+d x)+a)^{7/2}}}{2 a}-\frac {199}{9 a (a \sec (c+d x)+a)^{9/2}}\right )+\frac {21}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{9/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{9/2}}\right )}{d}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\frac {\frac {\frac {\frac {526 \int \frac {1}{2-\frac {\sec (c+d x) a+a}{a}}d\sqrt {\sec (c+d x) a+a}}{a}+\frac {2048 \int \frac {1}{\frac {\sec (c+d x) a+a}{a}-1}d\sqrt {\sec (c+d x) a+a}}{a}}{2 a}+\frac {761}{a \sqrt {a \sec (c+d x)+a}}}{2 a}+\frac {83}{a (a \sec (c+d x)+a)^{3/2}}}{2 a}-\frac {7}{5 a (a \sec (c+d x)+a)^{5/2}}}{2 a}-\frac {135}{7 a (a \sec (c+d x)+a)^{7/2}}}{2 a}-\frac {199}{9 a (a \sec (c+d x)+a)^{9/2}}\right )+\frac {21}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{9/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{9/2}}\right )}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\frac {\frac {\frac {\frac {2048 \int \frac {1}{\frac {\sec (c+d x) a+a}{a}-1}d\sqrt {\sec (c+d x) a+a}}{a}+\frac {263 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {a}}}{2 a}+\frac {761}{a \sqrt {a \sec (c+d x)+a}}}{2 a}+\frac {83}{a (a \sec (c+d x)+a)^{3/2}}}{2 a}-\frac {7}{5 a (a \sec (c+d x)+a)^{5/2}}}{2 a}-\frac {135}{7 a (a \sec (c+d x)+a)^{7/2}}}{2 a}-\frac {199}{9 a (a \sec (c+d x)+a)^{9/2}}\right )+\frac {21}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{9/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{9/2}}\right )}{d}\) |
| \(\Big \downarrow \) 220 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\frac {\frac {\frac {\frac {263 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {a}}-\frac {2048 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{\sqrt {a}}}{2 a}+\frac {761}{a \sqrt {a \sec (c+d x)+a}}}{2 a}+\frac {83}{a (a \sec (c+d x)+a)^{3/2}}}{2 a}-\frac {7}{5 a (a \sec (c+d x)+a)^{5/2}}}{2 a}-\frac {135}{7 a (a \sec (c+d x)+a)^{7/2}}}{2 a}-\frac {199}{9 a (a \sec (c+d x)+a)^{9/2}}\right )+\frac {21}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{9/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{9/2}}\right )}{d}\) |
Input:
Int[Cot[c + d*x]^5/(a + a*Sec[c + d*x])^(5/2),x]
Output:
-((a^3*(1/(4*a*(1 - Sec[c + d*x])^2*(a + a*Sec[c + d*x])^(9/2)) + (21/(2*a *(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(9/2)) + (-199/(9*a*(a + a*Sec[c + d*x])^(9/2)) + (-135/(7*a*(a + a*Sec[c + d*x])^(7/2)) + (-7/(5*a*(a + a* Sec[c + d*x])^(5/2)) + (83/(a*(a + a*Sec[c + d*x])^(3/2)) + (((-2048*ArcTa nh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/Sqrt[a] + (263*Sqrt[2]*ArcTanh[Sqrt[ a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/Sqrt[a])/(2*a) + 761/(a*Sqrt[a + a *Sec[c + d*x]]))/(2*a))/(2*a))/(2*a))/(2*a))/4)/8))/d)
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _), x_Symbol] :> Simp[-(d*b^(m - 1))^(-1) Subst[Int[(-a + b*x)^((m - 1)/2 )*((a + b*x)^((m - 1)/2 + n)/x), x], x, Csc[c + d*x]], x] /; FreeQ[{a, b, c , d, n}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && !IntegerQ[n]
Leaf count of result is larger than twice the leaf count of optimal. \(551\) vs. \(2(217)=434\).
Time = 1.04 (sec) , antiderivative size = 552, normalized size of antiderivative = 2.11
| method | result | size |
| default | \(-\frac {\sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (\left (26756150 \cos \left (d x +c \right )^{9}+47383570 \cos \left (d x +c \right )^{8}-20121800 \cos \left (d x +c \right )^{7}-157684280 \cos \left (d x +c \right )^{6}-201315980 \cos \left (d x +c \right )^{5}-58355492 \cos \left (d x +c \right )^{4}+99588632 \cos \left (d x +c \right )^{3}+114632232 \cos \left (d x +c \right )^{2}+48966918 \cos \left (d x +c \right )+7897890\right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cot \left (d x +c \right ) \csc \left (d x +c \right )^{3}+\left (46126080 \cos \left (d x +c \right )^{6}+276756480 \cos \left (d x +c \right )^{5}+691891200 \cos \left (d x +c \right )^{4}+922521600 \cos \left (d x +c \right )^{3}+691891200 \cos \left (d x +c \right )^{2}+276756480 \cos \left (d x +c \right )+46126080\right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{2}\right )+\left (11846835 \cos \left (d x +c \right )^{6}+71081010 \cos \left (d x +c \right )^{5}+177702525 \cos \left (d x +c \right )^{4}+236936700 \cos \left (d x +c \right )^{3}+177702525 \cos \left (d x +c \right )^{2}+71081010 \cos \left (d x +c \right )+11846835\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )+\left (162821786 \cos \left (d x +c \right )^{9}+538358018 \cos \left (d x +c \right )^{8}+638522576 \cos \left (d x +c \right )^{7}-281374720 \cos \left (d x +c \right )^{6}-1354120300 \cos \left (d x +c \right )^{5}-1027763308 \cos \left (d x +c \right )^{4}+191533056 \cos \left (d x +c \right )^{3}+689921232 \cos \left (d x +c \right )^{2}+373543170 \cos \left (d x +c \right )+68558490\right ) \cot \left (d x +c \right ) \csc \left (d x +c \right )^{3}\right )}{46126080 d \,a^{3} \left (1+\cos \left (d x +c \right )\right ) \left (1+\cos \left (d x +c \right )^{5}+5 \cos \left (d x +c \right )^{4}+10 \cos \left (d x +c \right )^{3}+10 \cos \left (d x +c \right )^{2}+5 \cos \left (d x +c \right )\right )}\) | \(552\) |
Input:
int(cot(d*x+c)^5/(a+a*sec(d*x+c))^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/46126080/d/a^3*(a*(1+sec(d*x+c)))^(1/2)/(1+cos(d*x+c))/(1+cos(d*x+c)^5+ 5*cos(d*x+c)^4+10*cos(d*x+c)^3+10*cos(d*x+c)^2+5*cos(d*x+c))*((26756150*co s(d*x+c)^9+47383570*cos(d*x+c)^8-20121800*cos(d*x+c)^7-157684280*cos(d*x+c )^6-201315980*cos(d*x+c)^5-58355492*cos(d*x+c)^4+99588632*cos(d*x+c)^3+114 632232*cos(d*x+c)^2+48966918*cos(d*x+c)+7897890)*2^(1/2)*(-2*cos(d*x+c)/(1 +cos(d*x+c)))^(1/2)*(-cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cot(d*x+c)*csc(d*x+ c)^3+(46126080*cos(d*x+c)^6+276756480*cos(d*x+c)^5+691891200*cos(d*x+c)^4+ 922521600*cos(d*x+c)^3+691891200*cos(d*x+c)^2+276756480*cos(d*x+c)+4612608 0)*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*2^(1/2)*(-2*cos (d*x+c)/(1+cos(d*x+c)))^(1/2))+(11846835*cos(d*x+c)^6+71081010*cos(d*x+c)^ 5+177702525*cos(d*x+c)^4+236936700*cos(d*x+c)^3+177702525*cos(d*x+c)^2+710 81010*cos(d*x+c)+11846835)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2 *2^(1/2)/(-cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+(162821786*cos(d*x+c)^9+53835 8018*cos(d*x+c)^8+638522576*cos(d*x+c)^7-281374720*cos(d*x+c)^6-1354120300 *cos(d*x+c)^5-1027763308*cos(d*x+c)^4+191533056*cos(d*x+c)^3+689921232*cos (d*x+c)^2+373543170*cos(d*x+c)+68558490)*cot(d*x+c)*csc(d*x+c)^3)
Leaf count of result is larger than twice the leaf count of optimal. 438 vs. \(2 (213) = 426\).
Time = 0.21 (sec) , antiderivative size = 905, normalized size of antiderivative = 3.45 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate(cot(d*x+c)^5/(a+a*sec(d*x+c))^(5/2),x, algorithm="fricas")
Output:
[1/645120*(82845*sqrt(2)*(cos(d*x + c)^7 + 3*cos(d*x + c)^6 + cos(d*x + c) ^5 - 5*cos(d*x + c)^4 - 5*cos(d*x + c)^3 + cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*sqrt(a)*log(-(2*sqrt(2)*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) - 3*a*cos(d*x + c) - a)/(cos(d*x + c) - 1)) + 322560*(cos (d*x + c)^7 + 3*cos(d*x + c)^6 + cos(d*x + c)^5 - 5*cos(d*x + c)^4 - 5*cos (d*x + c)^3 + cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*sqrt(a)*log(-8*a*cos(d* x + c)^2 - 4*(2*cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c ) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) - 4*(382201*cos(d*x + c)^7 + 591520*cos(d*x + c)^6 - 403607*cos(d*x + c)^5 - 1112040*cos(d*x + c)^4 - 1 89189*cos(d*x + c)^3 + 531720*cos(d*x + c)^2 + 239715*cos(d*x + c))*sqrt(( a*cos(d*x + c) + a)/cos(d*x + c)))/(a^3*d*cos(d*x + c)^7 + 3*a^3*d*cos(d*x + c)^6 + a^3*d*cos(d*x + c)^5 - 5*a^3*d*cos(d*x + c)^4 - 5*a^3*d*cos(d*x + c)^3 + a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d), 1/322560*(8 2845*sqrt(2)*(cos(d*x + c)^7 + 3*cos(d*x + c)^6 + cos(d*x + c)^5 - 5*cos(d *x + c)^4 - 5*cos(d*x + c)^3 + cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*sqrt(- a)*arctan(sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)) - 322560*(cos(d*x + c)^7 + 3*cos(d*x + c)^6 + cos(d*x + c)^5 - 5*cos(d*x + c)^4 - 5*cos(d*x + c)^3 + cos(d*x + c)^2 + 3* cos(d*x + c) + 1)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos (d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a)) - 2*(382201*cos(d*x + c...
\[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\int \frac {\cot ^{5}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(cot(d*x+c)**5/(a+a*sec(d*x+c))**(5/2),x)
Output:
Integral(cot(c + d*x)**5/(a*(sec(c + d*x) + 1))**(5/2), x)
Timed out. \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)^5/(a+a*sec(d*x+c))^(5/2),x, algorithm="maxima")
Output:
Timed out
Time = 0.99 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.39 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {\frac {82845 \, \sqrt {2} \arctan \left (\frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {645120 \, \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {315 \, {\left (33 \, \sqrt {2} {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} - 31 \, \sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a\right )}}{a^{4} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}} - \frac {8 \, \sqrt {2} {\left (35 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{4} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{56} - 225 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{3} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{57} + 1008 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{58} + 4410 \, {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} a^{59} + 31185 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{60}\right )}}{a^{63} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}}{322560 \, d} \] Input:
integrate(cot(d*x+c)^5/(a+a*sec(d*x+c))^(5/2),x, algorithm="giac")
Output:
1/322560*(82845*sqrt(2)*arctan(sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)/sqrt(-a ))/(sqrt(-a)*a^2*sgn(cos(d*x + c))) - 645120*arctan(1/2*sqrt(2)*sqrt(-a*ta n(1/2*d*x + 1/2*c)^2 + a)/sqrt(-a))/(sqrt(-a)*a^2*sgn(cos(d*x + c))) - 315 *(33*sqrt(2)*(-a*tan(1/2*d*x + 1/2*c)^2 + a)^(3/2) - 31*sqrt(2)*sqrt(-a*ta n(1/2*d*x + 1/2*c)^2 + a)*a)/(a^4*sgn(cos(d*x + c))*tan(1/2*d*x + 1/2*c)^4 ) - 8*sqrt(2)*(35*(a*tan(1/2*d*x + 1/2*c)^2 - a)^4*sqrt(-a*tan(1/2*d*x + 1 /2*c)^2 + a)*a^56 - 225*(a*tan(1/2*d*x + 1/2*c)^2 - a)^3*sqrt(-a*tan(1/2*d *x + 1/2*c)^2 + a)*a^57 + 1008*(a*tan(1/2*d*x + 1/2*c)^2 - a)^2*sqrt(-a*ta n(1/2*d*x + 1/2*c)^2 + a)*a^58 + 4410*(-a*tan(1/2*d*x + 1/2*c)^2 + a)^(3/2 )*a^59 + 31185*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)*a^60)/(a^63*sgn(cos(d*x + c))))/d
Timed out. \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^5}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \] Input:
int(cot(c + d*x)^5/(a + a/cos(c + d*x))^(5/2),x)
Output:
int(cot(c + d*x)^5/(a + a/cos(c + d*x))^(5/2), x)
\[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sec \left (d x +c \right )+1}\, \cot \left (d x +c \right )^{5}}{\sec \left (d x +c \right )^{3}+3 \sec \left (d x +c \right )^{2}+3 \sec \left (d x +c \right )+1}d x \right )}{a^{3}} \] Input:
int(cot(d*x+c)^5/(a+a*sec(d*x+c))^(5/2),x)
Output:
(sqrt(a)*int((sqrt(sec(c + d*x) + 1)*cot(c + d*x)**5)/(sec(c + d*x)**3 + 3 *sec(c + d*x)**2 + 3*sec(c + d*x) + 1),x))/a**3