Integrand size = 23, antiderivative size = 238 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {203 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{256 \sqrt {2} a^{3/2} d}+\frac {139 a^2}{224 d (a+a \sec (c+d x))^{7/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {15 a}{64 d (a+a \sec (c+d x))^{5/2}}-\frac {53}{384 d (a+a \sec (c+d x))^{3/2}}-\frac {309}{256 a d \sqrt {a+a \sec (c+d x)}} \] Output:
2*arctanh((a+a*sec(d*x+c))^(1/2)/a^(1/2))/a^(3/2)/d-203/512*arctanh(1/2*(a +a*sec(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*2^(1/2)/a^(3/2)/d+139/224*a^2/d/(a+a *sec(d*x+c))^(7/2)-1/4*a^2/d/(1-sec(d*x+c))^2/(a+a*sec(d*x+c))^(7/2)-19/16 *a^2/d/(1-sec(d*x+c))/(a+a*sec(d*x+c))^(7/2)+15/64*a/d/(a+a*sec(d*x+c))^(5 /2)-53/384/d/(a+a*sec(d*x+c))^(3/2)-309/256/a/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 = 0.19 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.42 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {\cot ^4(c+d x) \left (-322+203 \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},1,-\frac {5}{2},\frac {1}{2} (1+\sec (c+d x))\right ) (-1+\sec (c+d x))^2-64 \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},1,-\frac {5}{2},1+\sec (c+d x)\right ) (-1+\sec (c+d x))^2+266 \sec (c+d x)\right )}{224 d (a (1+\sec (c+d x)))^{3/2}} \] Input:
Integrate[Cot[c + d*x]^5/(a + a*Sec[c + d*x])^(3/2),x]
Output:
(Cot[c + d*x]^4*(-322 + 203*Hypergeometric2F1[-7/2, 1, -5/2, (1 + Sec[c + d*x])/2]*(-1 + Sec[c + d*x])^2 - 64*Hypergeometric2F1[-7/2, 1, -5/2, 1 + S ec[c + d*x]]*(-1 + Sec[c + d*x])^2 + 266*Sec[c + d*x]))/(224*d*(a*(1 + Sec [c + d*x]))^(3/2))
Time = 0.43 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.11, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.913, Rules used = {3042, 25, 4368, 25, 27, 114, 27, 168, 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.
| \(\displaystyle \int \frac {\cot ^5(c+d x)}{(a \sec (c+d x)+a)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\cot \left (c+d x+\frac {\pi }{2}\right )^5 \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\cot \left (\frac {1}{2} (2 c+\pi )+d x\right )^5 \left (\csc \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4368 |
| \(\displaystyle \frac {a^6 \int -\frac {\cos (c+d x)}{a^3 (1-\sec (c+d x))^3 (\sec (c+d x) a+a)^{9/2}}d\sec (c+d x)}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a^6 \int \frac {\cos (c+d x)}{a^3 (1-\sec (c+d x))^3 (\sec (c+d x) a+a)^{9/2}}d\sec (c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \int \frac {\cos (c+d x)}{(1-\sec (c+d x))^3 (\sec (c+d x) a+a)^{9/2}}d\sec (c+d x)}{d}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{7/2}}-\frac {\int -\frac {a \cos (c+d x) (11 \sec (c+d x)+8)}{2 (1-\sec (c+d x))^2 (\sec (c+d x) a+a)^{9/2}}d\sec (c+d x)}{4 a}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \int \frac {\cos (c+d x) (11 \sec (c+d x)+8)}{(1-\sec (c+d x))^2 (\sec (c+d x) a+a)^{9/2}}d\sec (c+d x)+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{7/2}}\right )}{d}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {19}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}-\frac {\int -\frac {a \cos (c+d x) (171 \sec (c+d x)+32)}{2 (1-\sec (c+d x)) (\sec (c+d x) a+a)^{9/2}}d\sec (c+d x)}{2 a}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{7/2}}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \int \frac {\cos (c+d x) (171 \sec (c+d x)+32)}{(1-\sec (c+d x)) (\sec (c+d x) a+a)^{9/2}}d\sec (c+d x)+\frac {19}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{7/2}}\right )}{d}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\int \frac {7 a \cos (c+d x) (139 \sec (c+d x)+64)}{2 (1-\sec (c+d x)) (\sec (c+d x) a+a)^{7/2}}d\sec (c+d x)}{7 a^2}-\frac {139}{7 a (a \sec (c+d x)+a)^{7/2}}\right )+\frac {19}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{7/2}}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\int \frac {\cos (c+d x) (139 \sec (c+d x)+64)}{(1-\sec (c+d x)) (\sec (c+d x) a+a)^{7/2}}d\sec (c+d x)}{2 a}-\frac {139}{7 a (a \sec (c+d x)+a)^{7/2}}\right )+\frac {19}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{7/2}}\right )}{d}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\int \frac {5 a \cos (c+d x) (75 \sec (c+d x)+128)}{2 (1-\sec (c+d x)) (\sec (c+d x) a+a)^{5/2}}d\sec (c+d x)}{5 a^2}-\frac {15}{a (a \sec (c+d x)+a)^{5/2}}}{2 a}-\frac {139}{7 a (a \sec (c+d x)+a)^{7/2}}\right )+\frac {19}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{7/2}}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\int \frac {\cos (c+d x) (75 \sec (c+d x)+128)}{(1-\sec (c+d x)) (\sec (c+d x) a+a)^{5/2}}d\sec (c+d x)}{2 a}-\frac {15}{a (a \sec (c+d x)+a)^{5/2}}}{2 a}-\frac {139}{7 a (a \sec (c+d x)+a)^{7/2}}\right )+\frac {19}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{7/2}}\right )}{d}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\frac {\int \frac {3 a \cos (c+d x) (256-53 \sec (c+d x))}{2 (1-\sec (c+d x)) (\sec (c+d x) a+a)^{3/2}}d\sec (c+d x)}{3 a^2}+\frac {53}{3 a (a \sec (c+d x)+a)^{3/2}}}{2 a}-\frac {15}{a (a \sec (c+d x)+a)^{5/2}}}{2 a}-\frac {139}{7 a (a \sec (c+d x)+a)^{7/2}}\right )+\frac {19}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{7/2}}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\frac {\int \frac {\cos (c+d x) (256-53 \sec (c+d x))}{(1-\sec (c+d x)) (\sec (c+d x) a+a)^{3/2}}d\sec (c+d x)}{2 a}+\frac {53}{3 a (a \sec (c+d x)+a)^{3/2}}}{2 a}-\frac {15}{a (a \sec (c+d x)+a)^{5/2}}}{2 a}-\frac {139}{7 a (a \sec (c+d x)+a)^{7/2}}\right )+\frac {19}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{7/2}}\right )}{d}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\frac {\frac {\int \frac {a \cos (c+d x) (512-309 \sec (c+d x))}{2 (1-\sec (c+d x)) \sqrt {\sec (c+d x) a+a}}d\sec (c+d x)}{a^2}+\frac {309}{a \sqrt {a \sec (c+d x)+a}}}{2 a}+\frac {53}{3 a (a \sec (c+d x)+a)^{3/2}}}{2 a}-\frac {15}{a (a \sec (c+d x)+a)^{5/2}}}{2 a}-\frac {139}{7 a (a \sec (c+d x)+a)^{7/2}}\right )+\frac {19}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{7/2}}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\frac {\frac {\int \frac {\cos (c+d x) (512-309 \sec (c+d x))}{(1-\sec (c+d x)) \sqrt {\sec (c+d x) a+a}}d\sec (c+d x)}{2 a}+\frac {309}{a \sqrt {a \sec (c+d x)+a}}}{2 a}+\frac {53}{3 a (a \sec (c+d x)+a)^{3/2}}}{2 a}-\frac {15}{a (a \sec (c+d x)+a)^{5/2}}}{2 a}-\frac {139}{7 a (a \sec (c+d x)+a)^{7/2}}\right )+\frac {19}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{7/2}}\right )}{d}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\frac {\frac {203 \int \frac {1}{(1-\sec (c+d x)) \sqrt {\sec (c+d x) a+a}}d\sec (c+d x)+512 \int \frac {\cos (c+d x)}{\sqrt {\sec (c+d x) a+a}}d\sec (c+d x)}{2 a}+\frac {309}{a \sqrt {a \sec (c+d x)+a}}}{2 a}+\frac {53}{3 a (a \sec (c+d x)+a)^{3/2}}}{2 a}-\frac {15}{a (a \sec (c+d x)+a)^{5/2}}}{2 a}-\frac {139}{7 a (a \sec (c+d x)+a)^{7/2}}\right )+\frac {19}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{7/2}}\right )}{d}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\frac {\frac {\frac {406 \int \frac {1}{2-\frac {\sec (c+d x) a+a}{a}}d\sqrt {\sec (c+d x) a+a}}{a}+\frac {1024 \int \frac {1}{\frac {\sec (c+d x) a+a}{a}-1}d\sqrt {\sec (c+d x) a+a}}{a}}{2 a}+\frac {309}{a \sqrt {a \sec (c+d x)+a}}}{2 a}+\frac {53}{3 a (a \sec (c+d x)+a)^{3/2}}}{2 a}-\frac {15}{a (a \sec (c+d x)+a)^{5/2}}}{2 a}-\frac {139}{7 a (a \sec (c+d x)+a)^{7/2}}\right )+\frac {19}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{7/2}}\right )}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\frac {\frac {\frac {1024 \int \frac {1}{\frac {\sec (c+d x) a+a}{a}-1}d\sqrt {\sec (c+d x) a+a}}{a}+\frac {203 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {a}}}{2 a}+\frac {309}{a \sqrt {a \sec (c+d x)+a}}}{2 a}+\frac {53}{3 a (a \sec (c+d x)+a)^{3/2}}}{2 a}-\frac {15}{a (a \sec (c+d x)+a)^{5/2}}}{2 a}-\frac {139}{7 a (a \sec (c+d x)+a)^{7/2}}\right )+\frac {19}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{7/2}}\right )}{d}\) |
| \(\Big \downarrow \) 220 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\frac {\frac {\frac {203 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {a}}-\frac {1024 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{\sqrt {a}}}{2 a}+\frac {309}{a \sqrt {a \sec (c+d x)+a}}}{2 a}+\frac {53}{3 a (a \sec (c+d x)+a)^{3/2}}}{2 a}-\frac {15}{a (a \sec (c+d x)+a)^{5/2}}}{2 a}-\frac {139}{7 a (a \sec (c+d x)+a)^{7/2}}\right )+\frac {19}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{7/2}}\right )}{d}\) |
Input:
Int[Cot[c + d*x]^5/(a + a*Sec[c + d*x])^(3/2),x]
Output:
-((a^3*(1/(4*a*(1 - Sec[c + d*x])^2*(a + a*Sec[c + d*x])^(7/2)) + (19/(2*a *(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(7/2)) + (-139/(7*a*(a + a*Sec[c + d*x])^(7/2)) + (-15/(a*(a + a*Sec[c + d*x])^(5/2)) + (53/(3*a*(a + a*Sec [c + d*x])^(3/2)) + (((-1024*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/Sq rt[a] + (203*Sqrt[2]*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/ Sqrt[a])/(2*a) + 309/(a*Sqrt[a + a*Sec[c + d*x]]))/(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. \(501\) vs. \(2(197)=394\).
Time = 0.94 (sec) , antiderivative size = 502, normalized size of antiderivative = 2.11
| method | result | size |
| default | \(-\frac {\sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (\left (13718558 \cos \left (d x +c \right )^{8}+15407084 \cos \left (d x +c \right )^{7}-34527892 \cos \left (d x +c \right )^{6}-90257860 \cos \left (d x +c \right )^{5}-65130520 \cos \left (d x +c \right )^{4}+20088068 \cos \left (d x +c \right )^{3}+56780724 \cos \left (d x +c \right )^{2}+31699668 \cos \left (d x +c \right )+6096090\right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-\frac {2 \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 (23063040 \cos \left (d x +c \right )^{5}+115315200 \cos \left (d x +c \right )^{4}+230630400 \cos \left (d x +c \right )^{3}+230630400 \cos \left (d x +c \right )^{2}+115315200 \cos \left (d x +c \right )+23063040\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{2}\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+\left (9144135 \cos \left (d x +c \right )^{5}+45720675 \cos \left (d x +c \right )^{4}+91441350 \cos \left (d x +c \right )^{3}+91441350 \cos \left (d x +c \right )^{2}+45720675 \cos \left (d x +c \right )+9144135\right ) \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+\left (71894386 \cos \left (d x +c \right )^{8}+171227592 \cos \left (d x +c \right )^{7}+93427144 \cos \left (d x +c \right )^{6}-234114504 \cos \left (d x +c \right )^{5}-352765556 \cos \left (d x +c \right )^{4}-70936008 \cos \left (d x +c \right )^{3}+166702536 \cos \left (d x +c \right )^{2}+126726600 \cos \left (d x +c \right )+27837810\right ) \cot \left (d x +c \right ) \csc \left (d x +c \right )^{3}\right )}{23063040 d \,a^{2} \left (1+\cos \left (d x +c \right )\right ) \left (1+\cos \left (d x +c \right )^{4}+4 \cos \left (d x +c \right )^{3}+6 \cos \left (d x +c \right )^{2}+4 \cos \left (d x +c \right )\right )}\) | \(502\) |
Input:
int(cot(d*x+c)^5/(a+a*sec(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/23063040/d/a^2*(a*(1+sec(d*x+c)))^(1/2)/(1+cos(d*x+c))/(1+cos(d*x+c)^4+ 4*cos(d*x+c)^3+6*cos(d*x+c)^2+4*cos(d*x+c))*((13718558*cos(d*x+c)^8+154070 84*cos(d*x+c)^7-34527892*cos(d*x+c)^6-90257860*cos(d*x+c)^5-65130520*cos(d *x+c)^4+20088068*cos(d*x+c)^3+56780724*cos(d*x+c)^2+31699668*cos(d*x+c)+60 96090)*2^(1/2)*(-cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(-2*cos(d*x+c)/(1+cos(d* x+c)))^(1/2)*cot(d*x+c)*csc(d*x+c)^3+(23063040*cos(d*x+c)^5+115315200*cos( d*x+c)^4+230630400*cos(d*x+c)^3+230630400*cos(d*x+c)^2+115315200*cos(d*x+c )+23063040)*2^(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2 ))*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+(9144135*cos(d*x+c)^5+45720675*cos (d*x+c)^4+91441350*cos(d*x+c)^3+91441350*cos(d*x+c)^2+45720675*cos(d*x+c)+ 9144135)*arctan(1/2*2^(1/2)/(-cos(d*x+c)/(1+cos(d*x+c)))^(1/2))*(-2*cos(d* x+c)/(1+cos(d*x+c)))^(1/2)+(71894386*cos(d*x+c)^8+171227592*cos(d*x+c)^7+9 3427144*cos(d*x+c)^6-234114504*cos(d*x+c)^5-352765556*cos(d*x+c)^4-7093600 8*cos(d*x+c)^3+166702536*cos(d*x+c)^2+126726600*cos(d*x+c)+27837810)*cot(d *x+c)*csc(d*x+c)^3)
Leaf count of result is larger than twice the leaf count of optimal. 404 vs. \(2 (193) = 386\).
Time = 0.21 (sec) , antiderivative size = 837, normalized size of antiderivative = 3.52 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate(cot(d*x+c)^5/(a+a*sec(d*x+c))^(3/2),x, algorithm="fricas")
Output:
[1/21504*(4263*sqrt(2)*(cos(d*x + c)^6 + 2*cos(d*x + c)^5 - cos(d*x + c)^4 - 4*cos(d*x + c)^3 - cos(d*x + c)^2 + 2*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)) + 10752*(cos(d*x + c)^6 + 2*cos(d* x + c)^5 - cos(d*x + c)^4 - 4*cos(d*x + c)^3 - cos(d*x + c)^2 + 2*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*(10363*cos(d*x + c)^6 + 8037*cos(d*x + c)^5 - 16538*cos(d*x + c)^4 - 14238*cos(d*x + c)^3 + 7231*cos(d*x + c)^2 + 6489*cos(d*x + c))*sqrt((a *cos(d*x + c) + a)/cos(d*x + c)))/(a^2*d*cos(d*x + c)^6 + 2*a^2*d*cos(d*x + c)^5 - a^2*d*cos(d*x + c)^4 - 4*a^2*d*cos(d*x + c)^3 - a^2*d*cos(d*x + c )^2 + 2*a^2*d*cos(d*x + c) + a^2*d), 1/10752*(4263*sqrt(2)*(cos(d*x + c)^6 + 2*cos(d*x + c)^5 - cos(d*x + c)^4 - 4*cos(d*x + c)^3 - cos(d*x + c)^2 + 2*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)) - 10752*(cos(d*x + c)^6 + 2*cos(d*x + c)^5 - cos(d*x + c)^4 - 4*cos(d*x + c)^3 - cos(d*x + c) ^2 + 2*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*(10363*cos(d*x + c)^6 + 8037*cos(d*x + c)^5 - 16538*cos(d*x + c)^4 - 14238*cos(d*x + c)^3 + 7231*cos(d*x + c)^2 + 6489*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/c...
\[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {\cot ^{5}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(cot(d*x+c)**5/(a+a*sec(d*x+c))**(3/2),x)
Output:
Integral(cot(c + d*x)**5/(a*(sec(c + d*x) + 1))**(3/2), x)
\[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int { \frac {\cot \left (d x + c\right )^{5}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(cot(d*x+c)^5/(a+a*sec(d*x+c))^(3/2),x, algorithm="maxima")
Output:
integrate(cot(d*x + c)^5/(a*sec(d*x + c) + a)^(3/2), x)
Time = 0.67 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.35 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {\frac {4263 \, \sqrt {2} \arctan \left (\frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {21504 \, \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {21 \, {\left (29 \, \sqrt {2} {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} - 27 \, \sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a\right )}}{a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}} + \frac {8 \, \sqrt {2} {\left (3 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{3} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{30} - 21 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{31} - 112 \, {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} a^{32} - 882 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{33}\right )}}{a^{35} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}}{10752 \, d} \] Input:
integrate(cot(d*x+c)^5/(a+a*sec(d*x+c))^(3/2),x, algorithm="giac")
Output:
1/10752*(4263*sqrt(2)*arctan(sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)/sqrt(-a)) /(sqrt(-a)*a*sgn(cos(d*x + c))) - 21504*arctan(1/2*sqrt(2)*sqrt(-a*tan(1/2 *d*x + 1/2*c)^2 + a)/sqrt(-a))/(sqrt(-a)*a*sgn(cos(d*x + c))) - 21*(29*sqr t(2)*(-a*tan(1/2*d*x + 1/2*c)^2 + a)^(3/2) - 27*sqrt(2)*sqrt(-a*tan(1/2*d* x + 1/2*c)^2 + a)*a)/(a^3*sgn(cos(d*x + c))*tan(1/2*d*x + 1/2*c)^4) + 8*sq rt(2)*(3*(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^30 - 21*(a*tan(1/2*d*x + 1/2*c)^2 - a)^2*sqrt(-a*tan(1/2*d*x + 1/2*c )^2 + a)*a^31 - 112*(-a*tan(1/2*d*x + 1/2*c)^2 + a)^(3/2)*a^32 - 882*sqrt( -a*tan(1/2*d*x + 1/2*c)^2 + a)*a^33)/(a^35*sgn(cos(d*x + c))))/d
Timed out. \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^5}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \] Input:
int(cot(c + d*x)^5/(a + a/cos(c + d*x))^(3/2),x)
Output:
int(cot(c + d*x)^5/(a + a/cos(c + d*x))^(3/2), x)
\[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sec \left (d x +c \right )+1}\, \cot \left (d x +c \right )^{5}}{\sec \left (d x +c \right )^{2}+2 \sec \left (d x +c \right )+1}d x \right )}{a^{2}} \] Input:
int(cot(d*x+c)^5/(a+a*sec(d*x+c))^(3/2),x)
Output:
(sqrt(a)*int((sqrt(sec(c + d*x) + 1)*cot(c + d*x)**5)/(sec(c + d*x)**2 + 2 *sec(c + d*x) + 1),x))/a**2