Integrand size = 23, antiderivative size = 214 \[ \int \frac {\cot ^5(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {151 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{128 \sqrt {2} \sqrt {a} d}+\frac {87 a^2}{160 d (a+a \sec (c+d x))^{5/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{5/2}}-\frac {17 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{5/2}}+\frac {23 a}{192 d (a+a \sec (c+d x))^{3/2}}-\frac {105}{128 d \sqrt {a+a \sec (c+d x)}} \] Output:
2*arctanh((a+a*sec(d*x+c))^(1/2)/a^(1/2))/a^(1/2)/d-151/256*arctanh(1/2*(a +a*sec(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*2^(1/2)/a^(1/2)/d+87/160*a^2/d/(a+a* sec(d*x+c))^(5/2)-1/4*a^2/d/(1-sec(d*x+c))^2/(a+a*sec(d*x+c))^(5/2)-17/16* a^2/d/(1-sec(d*x+c))/(a+a*sec(d*x+c))^(5/2)+23/192*a/d/(a+a*sec(d*x+c))^(3 /2)-105/128/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 = 0.18 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.48 \[ \int \frac {\cot ^5(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\frac {\cot ^4(c+d x) \left (-2 \left (105+32 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},1+\sec (c+d x)\right ) (-1+\sec (c+d x))^2-85 \sec (c+d x)\right )+151 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},\frac {1}{2} (1+\sec (c+d x))\right ) (-1+\sec (c+d x))^2\right )}{160 d \sqrt {a (1+\sec (c+d x))}} \] Input:
Integrate[Cot[c + d*x]^5/Sqrt[a + a*Sec[c + d*x]],x]
Output:
(Cot[c + d*x]^4*(-2*(105 + 32*Hypergeometric2F1[-5/2, 1, -3/2, 1 + Sec[c + d*x]]*(-1 + Sec[c + d*x])^2 - 85*Sec[c + d*x]) + 151*Hypergeometric2F1[-5 /2, 1, -3/2, (1 + Sec[c + d*x])/2]*(-1 + Sec[c + d*x])^2))/(160*d*Sqrt[a*( 1 + Sec[c + d*x])])
Time = 0.41 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.10, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.826, Rules used = {3042, 25, 4368, 25, 27, 114, 27, 168, 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.
| \(\displaystyle \int \frac {\cot ^5(c+d x)}{\sqrt {a \sec (c+d x)+a}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\cot \left (c+d x+\frac {\pi }{2}\right )^5 \sqrt {a \csc \left (c+d x+\frac {\pi }{2}\right )+a}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\cot \left (\frac {1}{2} (2 c+\pi )+d x\right )^5 \sqrt {\csc \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a}}dx\) |
| \(\Big \downarrow \) 4368 |
| \(\displaystyle \frac {a^6 \int -\frac {\cos (c+d x)}{a^3 (1-\sec (c+d x))^3 (\sec (c+d x) a+a)^{7/2}}d\sec (c+d x)}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a^6 \int \frac {\cos (c+d x)}{a^3 (1-\sec (c+d x))^3 (\sec (c+d x) a+a)^{7/2}}d\sec (c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \int \frac {\cos (c+d x)}{(1-\sec (c+d x))^3 (\sec (c+d x) a+a)^{7/2}}d\sec (c+d x)}{d}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{5/2}}-\frac {\int -\frac {a \cos (c+d x) (9 \sec (c+d x)+8)}{2 (1-\sec (c+d x))^2 (\sec (c+d x) a+a)^{7/2}}d\sec (c+d x)}{4 a}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \int \frac {\cos (c+d x) (9 \sec (c+d x)+8)}{(1-\sec (c+d x))^2 (\sec (c+d x) a+a)^{7/2}}d\sec (c+d x)+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {17}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{5/2}}-\frac {\int -\frac {a \cos (c+d x) (119 \sec (c+d x)+32)}{2 (1-\sec (c+d x)) (\sec (c+d x) a+a)^{7/2}}d\sec (c+d x)}{2 a}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \int \frac {\cos (c+d x) (119 \sec (c+d x)+32)}{(1-\sec (c+d x)) (\sec (c+d x) a+a)^{7/2}}d\sec (c+d x)+\frac {17}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{5/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\int \frac {5 a \cos (c+d x) (87 \sec (c+d x)+64)}{2 (1-\sec (c+d x)) (\sec (c+d x) a+a)^{5/2}}d\sec (c+d x)}{5 a^2}-\frac {87}{5 a (a \sec (c+d x)+a)^{5/2}}\right )+\frac {17}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{5/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\int \frac {\cos (c+d x) (87 \sec (c+d x)+64)}{(1-\sec (c+d x)) (\sec (c+d x) a+a)^{5/2}}d\sec (c+d x)}{2 a}-\frac {87}{5 a (a \sec (c+d x)+a)^{5/2}}\right )+\frac {17}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{5/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\int \frac {3 a \cos (c+d x) (23 \sec (c+d x)+128)}{2 (1-\sec (c+d x)) (\sec (c+d x) a+a)^{3/2}}d\sec (c+d x)}{3 a^2}-\frac {23}{3 a (a \sec (c+d x)+a)^{3/2}}}{2 a}-\frac {87}{5 a (a \sec (c+d x)+a)^{5/2}}\right )+\frac {17}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{5/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\int \frac {\cos (c+d x) (23 \sec (c+d x)+128)}{(1-\sec (c+d x)) (\sec (c+d x) a+a)^{3/2}}d\sec (c+d x)}{2 a}-\frac {23}{3 a (a \sec (c+d x)+a)^{3/2}}}{2 a}-\frac {87}{5 a (a \sec (c+d x)+a)^{5/2}}\right )+\frac {17}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{5/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\frac {\int \frac {a \cos (c+d x) (256-105 \sec (c+d x))}{2 (1-\sec (c+d x)) \sqrt {\sec (c+d x) a+a}}d\sec (c+d x)}{a^2}+\frac {105}{a \sqrt {a \sec (c+d x)+a}}}{2 a}-\frac {23}{3 a (a \sec (c+d x)+a)^{3/2}}}{2 a}-\frac {87}{5 a (a \sec (c+d x)+a)^{5/2}}\right )+\frac {17}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{5/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\frac {\int \frac {\cos (c+d x) (256-105 \sec (c+d x))}{(1-\sec (c+d x)) \sqrt {\sec (c+d x) a+a}}d\sec (c+d x)}{2 a}+\frac {105}{a \sqrt {a \sec (c+d x)+a}}}{2 a}-\frac {23}{3 a (a \sec (c+d x)+a)^{3/2}}}{2 a}-\frac {87}{5 a (a \sec (c+d x)+a)^{5/2}}\right )+\frac {17}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{5/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\frac {151 \int \frac {1}{(1-\sec (c+d x)) \sqrt {\sec (c+d x) a+a}}d\sec (c+d x)+256 \int \frac {\cos (c+d x)}{\sqrt {\sec (c+d x) a+a}}d\sec (c+d x)}{2 a}+\frac {105}{a \sqrt {a \sec (c+d x)+a}}}{2 a}-\frac {23}{3 a (a \sec (c+d x)+a)^{3/2}}}{2 a}-\frac {87}{5 a (a \sec (c+d x)+a)^{5/2}}\right )+\frac {17}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{5/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\frac {\frac {302 \int \frac {1}{2-\frac {\sec (c+d x) a+a}{a}}d\sqrt {\sec (c+d x) a+a}}{a}+\frac {512 \int \frac {1}{\frac {\sec (c+d x) a+a}{a}-1}d\sqrt {\sec (c+d x) a+a}}{a}}{2 a}+\frac {105}{a \sqrt {a \sec (c+d x)+a}}}{2 a}-\frac {23}{3 a (a \sec (c+d x)+a)^{3/2}}}{2 a}-\frac {87}{5 a (a \sec (c+d x)+a)^{5/2}}\right )+\frac {17}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{5/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\frac {\frac {512 \int \frac {1}{\frac {\sec (c+d x) a+a}{a}-1}d\sqrt {\sec (c+d x) a+a}}{a}+\frac {151 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {a}}}{2 a}+\frac {105}{a \sqrt {a \sec (c+d x)+a}}}{2 a}-\frac {23}{3 a (a \sec (c+d x)+a)^{3/2}}}{2 a}-\frac {87}{5 a (a \sec (c+d x)+a)^{5/2}}\right )+\frac {17}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{5/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{5/2}}\right )}{d}\) |
| \(\Big \downarrow \) 220 |
| \(\displaystyle -\frac {a^3 \left (\frac {1}{8} \left (\frac {1}{4} \left (\frac {\frac {\frac {\frac {151 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {a}}-\frac {512 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{\sqrt {a}}}{2 a}+\frac {105}{a \sqrt {a \sec (c+d x)+a}}}{2 a}-\frac {23}{3 a (a \sec (c+d x)+a)^{3/2}}}{2 a}-\frac {87}{5 a (a \sec (c+d x)+a)^{5/2}}\right )+\frac {17}{2 a (1-\sec (c+d x)) (a \sec (c+d x)+a)^{5/2}}\right )+\frac {1}{4 a (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{5/2}}\right )}{d}\) |
Input:
Int[Cot[c + d*x]^5/Sqrt[a + a*Sec[c + d*x]],x]
Output:
-((a^3*(1/(4*a*(1 - Sec[c + d*x])^2*(a + a*Sec[c + d*x])^(5/2)) + (17/(2*a *(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(5/2)) + (-87/(5*a*(a + a*Sec[c + d*x])^(5/2)) + (-23/(3*a*(a + a*Sec[c + d*x])^(3/2)) + (((-512*ArcTanh[Sq rt[a + a*Sec[c + d*x]]/Sqrt[a]])/Sqrt[a] + (151*Sqrt[2]*ArcTanh[Sqrt[a + a *Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/Sqrt[a])/(2*a) + 105/(a*Sqrt[a + a*Sec[ c + d*x]]))/(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. \(451\) vs. \(2(177)=354\).
Time = 0.91 (sec) , antiderivative size = 452, normalized size of antiderivative = 2.11
| method | result | size |
| default | \(-\frac {\sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (\left (491182 \cos \left (d x +c \right )^{7}+72414 \cos \left (d x +c \right )^{6}-2208722 \cos \left (d x +c \right )^{5}-3092178 \cos \left (d x +c \right )^{4}-634502 \cos \left (d x +c \right )^{3}+1783914 \cos \left (d x +c \right )^{2}+1465002 \cos \left (d x +c \right )+348810\right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cot \left (d x +c \right ) \csc \left (d x +c \right )^{3}+\left (523215 \cos \left (d x +c \right )^{4}+2092860 \cos \left (d x +c \right )^{3}+3139290 \cos \left (d x +c \right )^{2}+2092860 \cos \left (d x +c \right )+523215\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )+\left (887040 \cos \left (d x +c \right )^{4}+3548160 \cos \left (d x +c \right )^{3}+5322240 \cos \left (d x +c \right )^{2}+3548160 \cos \left (d x +c \right )+887040\right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{2}\right )+\left (2285666 \cos \left (d x +c \right )^{7}+3200806 \cos \left (d x +c \right )^{6}-1116062 \cos \left (d x +c \right )^{5}-6516202 \cos \left (d x +c \right )^{4}-3621354 \cos \left (d x +c \right )^{3}+2265186 \cos \left (d x +c \right )^{2}+2774310 \cos \left (d x +c \right )+727650\right ) \cot \left (d x +c \right ) \csc \left (d x +c \right )^{3}\right )}{887040 d a \left (1+\cos \left (d x +c \right )\right ) \left (\cos \left (d x +c \right )^{3}+3 \cos \left (d x +c \right )^{2}+3 \cos \left (d x +c \right )+1\right )}\) | \(452\) |
Input:
int(cot(d*x+c)^5/(a+a*sec(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/887040/d/a*(a*(1+sec(d*x+c)))^(1/2)/(1+cos(d*x+c))/(cos(d*x+c)^3+3*cos( d*x+c)^2+3*cos(d*x+c)+1)*((491182*cos(d*x+c)^7+72414*cos(d*x+c)^6-2208722* cos(d*x+c)^5-3092178*cos(d*x+c)^4-634502*cos(d*x+c)^3+1783914*cos(d*x+c)^2 +1465002*cos(d*x+c)+348810)*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+(523215*cos(d*x+ c)^4+2092860*cos(d*x+c)^3+3139290*cos(d*x+c)^2+2092860*cos(d*x+c)+523215)* (-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*2^(1/2)/(-cos(d*x+c)/(1+co s(d*x+c)))^(1/2))+(887040*cos(d*x+c)^4+3548160*cos(d*x+c)^3+5322240*cos(d* x+c)^2+3548160*cos(d*x+c)+887040)*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))+(2285666*cos (d*x+c)^7+3200806*cos(d*x+c)^6-1116062*cos(d*x+c)^5-6516202*cos(d*x+c)^4-3 621354*cos(d*x+c)^3+2265186*cos(d*x+c)^2+2774310*cos(d*x+c)+727650)*cot(d* x+c)*csc(d*x+c)^3)
Time = 0.17 (sec) , antiderivative size = 705, normalized size of antiderivative = 3.29 \[ \int \frac {\cot ^5(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx =\text {Too large to display} \] Input:
integrate(cot(d*x+c)^5/(a+a*sec(d*x+c))^(1/2),x, algorithm="fricas")
Output:
[1/7680*(2265*sqrt(2)*(cos(d*x + c)^5 + cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 2*cos(d*x + c)^2 + cos(d*x + c) + 1)*sqrt(a)*log(-(2*sqrt(2)*sqrt(a)*sqr t((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)) + 3840*(cos(d*x + c)^5 + cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 2*cos(d*x + c)^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*(2821*cos(d*x + c)^5 + 278*cos(d *x + c)^4 - 3964*cos(d*x + c)^3 - 230*cos(d*x + c)^2 + 1575*cos(d*x + c))* sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a*d*cos(d*x + c)^5 + a*d*cos(d*x + c)^4 - 2*a*d*cos(d*x + c)^3 - 2*a*d*cos(d*x + c)^2 + a*d*cos(d*x + c) + a*d), 1/3840*(2265*sqrt(2)*(cos(d*x + c)^5 + cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 2*cos(d*x + c)^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)) - 3840*(cos(d*x + c)^5 + cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 2*cos(d* x + c)^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*(2821*cos(d *x + c)^5 + 278*cos(d*x + c)^4 - 3964*cos(d*x + c)^3 - 230*cos(d*x + c)^2 + 1575*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a*d*cos(d*x + c)^5 + a*d*cos(d*x + c)^4 - 2*a*d*cos(d*x + c)^3 - 2*a*d*cos(d*x + c)^2 + a*d*cos(d*x + c) + a*d)]
\[ \int \frac {\cot ^5(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\int \frac {\cot ^{5}{\left (c + d x \right )}}{\sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )}}\, dx \] Input:
integrate(cot(d*x+c)**5/(a+a*sec(d*x+c))**(1/2),x)
Output:
Integral(cot(c + d*x)**5/sqrt(a*(sec(c + d*x) + 1)), x)
\[ \int \frac {\cot ^5(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\int { \frac {\cot \left (d x + c\right )^{5}}{\sqrt {a \sec \left (d x + c\right ) + a}} \,d x } \] Input:
integrate(cot(d*x+c)^5/(a+a*sec(d*x+c))^(1/2),x, algorithm="maxima")
Output:
integrate(cot(d*x + c)^5/sqrt(a*sec(d*x + c) + a), x)
Time = 0.59 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.13 \[ \int \frac {\cot ^5(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=-\frac {\sqrt {2} {\left (\frac {3840 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a}} - \frac {2265 \, \arctan \left (\frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + \frac {15 \, {\left (25 \, {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} - 23 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a\right )}}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}} + \frac {8 \, {\left (3 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{12} + 25 \, {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} a^{13} + 240 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{14}\right )}}{a^{15}}\right )}}{3840 \, d \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} \] Input:
integrate(cot(d*x+c)^5/(a+a*sec(d*x+c))^(1/2),x, algorithm="giac")
Output:
-1/3840*sqrt(2)*(3840*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-a*tan(1/2*d*x + 1/2 *c)^2 + a)/sqrt(-a))/sqrt(-a) - 2265*arctan(sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)/sqrt(-a))/sqrt(-a) + 15*(25*(-a*tan(1/2*d*x + 1/2*c)^2 + a)^(3/2) - 23*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)*a)/(a^2*tan(1/2*d*x + 1/2*c)^4) + 8 *(3*(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 ^12 + 25*(-a*tan(1/2*d*x + 1/2*c)^2 + a)^(3/2)*a^13 + 240*sqrt(-a*tan(1/2* d*x + 1/2*c)^2 + a)*a^14)/a^15)/(d*sgn(cos(d*x + c)))
Timed out. \[ \int \frac {\cot ^5(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^5}{\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}}} \,d x \] Input:
int(cot(c + d*x)^5/(a + a/cos(c + d*x))^(1/2),x)
Output:
int(cot(c + d*x)^5/(a + a/cos(c + d*x))^(1/2), x)
\[ \int \frac {\cot ^5(c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sec \left (d x +c \right )+1}\, \cot \left (d x +c \right )^{5}}{\sec \left (d x +c \right )+1}d x \right )}{a} \] Input:
int(cot(d*x+c)^5/(a+a*sec(d*x+c))^(1/2),x)
Output:
(sqrt(a)*int((sqrt(sec(c + d*x) + 1)*cot(c + d*x)**5)/(sec(c + d*x) + 1),x ))/a