\(\int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx\) [206]

Optimal result

Integrand size = 23, antiderivative size = 439 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a^{5/2} d}+\frac {74461 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{32768 \sqrt {2} a^{5/2} d}+\frac {8925 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{32768 a^3 d}-\frac {41693 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{49152 a^4 d}+\frac {58077 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{40960 a^5 d}-\frac {9467 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{8192 a^5 d}-\frac {2473 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{12288 a^5 d}-\frac {155 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{3072 a^5 d}-\frac {7 \cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{512 a^5 d}-\frac {\cos ^5(c+d x) \cot ^5(c+d x) \sec ^{10}\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{320 a^5 d} \]

[Out]

-2*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/a^(5/2)/d-41693/49152*cot(d*x+c)^3*(a+a*sec(d*x+c))^(3/2)
/a^4/d+58077/40960*cot(d*x+c)^5*(a+a*sec(d*x+c))^(5/2)/a^5/d-9467/8192*cos(d*x+c)*cot(d*x+c)^5*sec(1/2*d*x+1/2
*c)^2*(a+a*sec(d*x+c))^(5/2)/a^5/d-2473/12288*cos(d*x+c)^2*cot(d*x+c)^5*sec(1/2*d*x+1/2*c)^4*(a+a*sec(d*x+c))^
(5/2)/a^5/d-155/3072*cos(d*x+c)^3*cot(d*x+c)^5*sec(1/2*d*x+1/2*c)^6*(a+a*sec(d*x+c))^(5/2)/a^5/d-7/512*cos(d*x
+c)^4*cot(d*x+c)^5*sec(1/2*d*x+1/2*c)^8*(a+a*sec(d*x+c))^(5/2)/a^5/d-1/320*cos(d*x+c)^5*cot(d*x+c)^5*sec(1/2*d
*x+1/2*c)^10*(a+a*sec(d*x+c))^(5/2)/a^5/d+74461/65536*arctan(1/2*a^(1/2)*tan(d*x+c)*2^(1/2)/(a+a*sec(d*x+c))^(
1/2))*2^(1/2)/a^(5/2)/d+8925/32768*cot(d*x+c)*(a+a*sec(d*x+c))^(1/2)/a^3/d

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3972, 483, 593, 597, 536, 209} \[ \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{a^{5/2} d}+\frac {74461 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{32768 \sqrt {2} a^{5/2} d}+\frac {58077 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{40960 a^5 d}-\frac {\cos ^5(c+d x) \cot ^5(c+d x) \sec ^{10}\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{320 a^5 d}-\frac {7 \cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{512 a^5 d}-\frac {155 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{3072 a^5 d}-\frac {2473 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{12288 a^5 d}-\frac {9467 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{8192 a^5 d}-\frac {41693 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{49152 a^4 d}+\frac {8925 \cot (c+d x) \sqrt {a \sec (c+d x)+a}}{32768 a^3 d} \]

[In]

Int[Cot[c + d*x]^6/(a + a*Sec[c + d*x])^(5/2),x]

[Out]

(-2*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) + (74461*ArcTan[(Sqrt[a]*Tan[c + d*x]
)/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(32768*Sqrt[2]*a^(5/2)*d) + (8925*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]]
)/(32768*a^3*d) - (41693*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(49152*a^4*d) + (58077*Cot[c + d*x]^5*(a +
a*Sec[c + d*x])^(5/2))/(40960*a^5*d) - (9467*Cos[c + d*x]*Cot[c + d*x]^5*Sec[(c + d*x)/2]^2*(a + a*Sec[c + d*
x])^(5/2))/(8192*a^5*d) - (2473*Cos[c + d*x]^2*Cot[c + d*x]^5*Sec[(c + d*x)/2]^4*(a + a*Sec[c + d*x])^(5/2))/(
12288*a^5*d) - (155*Cos[c + d*x]^3*Cot[c + d*x]^5*Sec[(c + d*x)/2]^6*(a + a*Sec[c + d*x])^(5/2))/(3072*a^5*d)
- (7*Cos[c + d*x]^4*Cot[c + d*x]^5*Sec[(c + d*x)/2]^8*(a + a*Sec[c + d*x])^(5/2))/(512*a^5*d) - (Cos[c + d*x]^
5*Cot[c + d*x]^5*Sec[(c + d*x)/2]^10*(a + a*Sec[c + d*x])^(5/2))/(320*a^5*d)

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 483

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*(e*
x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*e*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a*d)
*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*(b*c - a*d)*(p + 1) + d*b*(m + n
*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ
[p, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 593

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*g*n*(b*c - a*d)*(p +
1))), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)
*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 597

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rule 3972

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.), x_Symbol] :> Dist[-2*(a^(m/2 +
n + 1/2)/d), Subst[Int[x^m*((2 + a*x^2)^(m/2 + n - 1/2)/(1 + a*x^2)), x], x, Cot[c + d*x]/Sqrt[a + b*Csc[c +
d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m/2] && IntegerQ[n - 1/2]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \text {Subst}\left (\int \frac {1}{x^6 \left (1+a x^2\right ) \left (2+a x^2\right )^6} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a^5 d} \\ & = -\frac {\cos ^5(c+d x) \cot ^5(c+d x) \sec ^{10}\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{320 a^5 d}-\frac {\text {Subst}\left (\int \frac {5 a-15 a^2 x^2}{x^6 \left (1+a x^2\right ) \left (2+a x^2\right )^5} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{10 a^6 d} \\ & = -\frac {7 \cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{512 a^5 d}-\frac {\cos ^5(c+d x) \cot ^5(c+d x) \sec ^{10}\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{320 a^5 d}-\frac {\text {Subst}\left (\int \frac {-135 a^2-455 a^3 x^2}{x^6 \left (1+a x^2\right ) \left (2+a x^2\right )^4} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{160 a^7 d} \\ & = -\frac {155 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{3072 a^5 d}-\frac {7 \cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{512 a^5 d}-\frac {\cos ^5(c+d x) \cot ^5(c+d x) \sec ^{10}\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{320 a^5 d}-\frac {\text {Subst}\left (\int \frac {-4685 a^3-8525 a^4 x^2}{x^6 \left (1+a x^2\right ) \left (2+a x^2\right )^3} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{1920 a^8 d} \\ & = -\frac {2473 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{12288 a^5 d}-\frac {155 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{3072 a^5 d}-\frac {7 \cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{512 a^5 d}-\frac {\cos ^5(c+d x) \cot ^5(c+d x) \sec ^{10}\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{320 a^5 d}-\frac {\text {Subst}\left (\int \frac {-80565 a^4-111285 a^5 x^2}{x^6 \left (1+a x^2\right ) \left (2+a x^2\right )^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{15360 a^9 d} \\ & = -\frac {9467 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{8192 a^5 d}-\frac {2473 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{12288 a^5 d}-\frac {155 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{3072 a^5 d}-\frac {7 \cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{512 a^5 d}-\frac {\cos ^5(c+d x) \cot ^5(c+d x) \sec ^{10}\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{320 a^5 d}-\frac {\text {Subst}\left (\int \frac {-871155 a^5-994035 a^6 x^2}{x^6 \left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{61440 a^{10} d} \\ & = \frac {58077 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{40960 a^5 d}-\frac {9467 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{8192 a^5 d}-\frac {2473 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{12288 a^5 d}-\frac {155 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{3072 a^5 d}-\frac {7 \cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{512 a^5 d}-\frac {\cos ^5(c+d x) \cot ^5(c+d x) \sec ^{10}\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{320 a^5 d}+\frac {\text {Subst}\left (\int \frac {-3126975 a^6-4355775 a^7 x^2}{x^4 \left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{614400 a^{10} d} \\ & = -\frac {41693 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{49152 a^4 d}+\frac {58077 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{40960 a^5 d}-\frac {9467 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{8192 a^5 d}-\frac {2473 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{12288 a^5 d}-\frac {155 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{3072 a^5 d}-\frac {7 \cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{512 a^5 d}-\frac {\cos ^5(c+d x) \cot ^5(c+d x) \sec ^{10}\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{320 a^5 d}-\frac {\text {Subst}\left (\int \frac {-2008125 a^7-9380925 a^8 x^2}{x^2 \left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{3686400 a^{10} d} \\ & = \frac {8925 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{32768 a^3 d}-\frac {41693 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{49152 a^4 d}+\frac {58077 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{40960 a^5 d}-\frac {9467 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{8192 a^5 d}-\frac {2473 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{12288 a^5 d}-\frac {155 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{3072 a^5 d}-\frac {7 \cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{512 a^5 d}-\frac {\cos ^5(c+d x) \cot ^5(c+d x) \sec ^{10}\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{320 a^5 d}+\frac {\text {Subst}\left (\int \frac {12737475 a^8-2008125 a^9 x^2}{\left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{7372800 a^{10} d} \\ & = \frac {8925 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{32768 a^3 d}-\frac {41693 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{49152 a^4 d}+\frac {58077 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{40960 a^5 d}-\frac {9467 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{8192 a^5 d}-\frac {2473 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{12288 a^5 d}-\frac {155 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{3072 a^5 d}-\frac {7 \cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{512 a^5 d}-\frac {\cos ^5(c+d x) \cot ^5(c+d x) \sec ^{10}\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{320 a^5 d}+\frac {2 \text {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a^2 d}-\frac {74461 \text {Subst}\left (\int \frac {1}{2+a x^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{32768 a^2 d} \\ & = -\frac {2 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a^{5/2} d}+\frac {74461 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{32768 \sqrt {2} a^{5/2} d}+\frac {8925 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{32768 a^3 d}-\frac {41693 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{49152 a^4 d}+\frac {58077 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{40960 a^5 d}-\frac {9467 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{8192 a^5 d}-\frac {2473 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{12288 a^5 d}-\frac {155 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{3072 a^5 d}-\frac {7 \cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{512 a^5 d}-\frac {\cos ^5(c+d x) \cot ^5(c+d x) \sec ^{10}\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{320 a^5 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.53 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.67 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {\cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \left (-\frac {\sqrt {\frac {1}{2+2 \cos (c+d x)}} (3364685+2115266 \cos (c+d x)+3550428 \cos (2 (c+d x))+1005782 \cos (3 (c+d x))+714844 \cos (4 (c+d x))-1338430 \cos (5 (c+d x))+1168164 \cos (6 (c+d x))+1363110 \cos (7 (c+d x))+639063 \cos (8 (c+d x))) \csc ^5\left (\frac {1}{2} (c+d x)\right ) \sec ^9\left (\frac {1}{2} (c+d x)\right ) \sqrt {\sec (c+d x)}}{8192}+1116915 \arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\frac {1}{1+\sec (c+d x)}} \sqrt {1+\sec (c+d x)}-983040 \sqrt {2} \arctan \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\frac {1}{1+\sec (c+d x)}}}\right ) \sqrt {\frac {1}{1+\sec (c+d x)}} \sqrt {1+\sec (c+d x)}\right )}{122880 d \sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )} (a (1+\sec (c+d x)))^{5/2}} \]

[In]

Integrate[Cot[c + d*x]^6/(a + a*Sec[c + d*x])^(5/2),x]

[Out]

(Cos[(c + d*x)/2]^4*Sec[c + d*x]^(5/2)*(-1/8192*(Sqrt[(2 + 2*Cos[c + d*x])^(-1)]*(3364685 + 2115266*Cos[c + d*
x] + 3550428*Cos[2*(c + d*x)] + 1005782*Cos[3*(c + d*x)] + 714844*Cos[4*(c + d*x)] - 1338430*Cos[5*(c + d*x)]
+ 1168164*Cos[6*(c + d*x)] + 1363110*Cos[7*(c + d*x)] + 639063*Cos[8*(c + d*x)])*Csc[(c + d*x)/2]^5*Sec[(c + d
*x)/2]^9*Sqrt[Sec[c + d*x]]) + 1116915*ArcSin[Tan[(c + d*x)/2]]*Sqrt[(1 + Sec[c + d*x])^(-1)]*Sqrt[1 + Sec[c +
d*x]] - 983040*Sqrt[2]*ArcTan[Tan[(c + d*x)/2]/Sqrt[(1 + Sec[c + d*x])^(-1)]]*Sqrt[(1 + Sec[c + d*x])^(-1)]*S
qrt[1 + Sec[c + d*x]]))/(122880*d*Sqrt[Sec[(c + d*x)/2]^2]*(a*(1 + Sec[c + d*x]))^(5/2))

Maple [A] (verified)

Time = 1.74 (sec) , antiderivative size = 746, normalized size of antiderivative = 1.70

method	result	size
default	\(\frac {\sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (1116915 \cos \left (d x +c \right )^{3} \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {2}\, \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )+\sqrt {\cot \left (d x +c \right )^{2}-2 \cot \left (d x +c \right ) \csc \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}\right )+3350745 \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )+\sqrt {\cot \left (d x +c \right )^{2}-2 \cot \left (d x +c \right ) \csc \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}\right ) \cos \left (d x +c \right )^{2}-1966080 \cos \left (d x +c \right )^{3} \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+3350745 \sqrt {2}\, \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )+\sqrt {\cot \left (d x +c \right )^{2}-2 \cot \left (d x +c \right ) \csc \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )-5898240 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )^{2}+1116915 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )+\sqrt {\cot \left (d x +c \right )^{2}-2 \cot \left (d x +c \right ) \csc \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}\right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-1278126 \cos \left (d x +c \right )^{3} \cot \left (d x +c \right )^{5}-5898240 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )-1363110 \cos \left (d x +c \right )^{2} \cot \left (d x +c \right )^{5}-1966080 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+1972170 \cos \left (d x +c \right ) \cot \left (d x +c \right )^{5}+2720050 \cot \left (d x +c \right )^{5}-810890 \cot \left (d x +c \right )^{4} \csc \left (d x +c \right )-1673842 \cot \left (d x +c \right )^{3} \csc \left (d x +c \right )^{2}-30610 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )^{3}+267750 \cot \left (d x +c \right ) \csc \left (d x +c \right )^{4}\right )}{983040 d \,a^{3} \left (\cos \left (d x +c \right )+1\right )^{3}}\)	\(746\)

[In]

int(cot(d*x+c)^6/(a+a*sec(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/983040/d/a^3*(a*(1+sec(d*x+c)))^(1/2)/(cos(d*x+c)+1)^3*(1116915*cos(d*x+c)^3*(-cos(d*x+c)/(cos(d*x+c)+1))^(1
/2)*2^(1/2)*ln(csc(d*x+c)-cot(d*x+c)+(cot(d*x+c)^2-2*cot(d*x+c)*csc(d*x+c)+csc(d*x+c)^2-1)^(1/2))+3350745*2^(1
/2)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln(csc(d*x+c)-cot(d*x+c)+(cot(d*x+c)^2-2*cot(d*x+c)*csc(d*x+c)+csc(d*x+
c)^2-1)^(1/2))*cos(d*x+c)^2-1966080*cos(d*x+c)^3*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(sin(d*x+c)/(cos(d*
x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+3350745*2^(1/2)*ln(csc(d*x+c)-cot(d*x+c)+(cot(d*x+c)^2-2*cot(d*x+c
)*csc(d*x+c)+csc(d*x+c)^2-1)^(1/2))*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)-5898240*(-cos(d*x+c)/(cos(d*
x+c)+1))^(1/2)*arctanh(sin(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^2+1116915*ln(c
sc(d*x+c)-cot(d*x+c)+(cot(d*x+c)^2-2*cot(d*x+c)*csc(d*x+c)+csc(d*x+c)^2-1)^(1/2))*2^(1/2)*(-cos(d*x+c)/(cos(d*
x+c)+1))^(1/2)-1278126*cos(d*x+c)^3*cot(d*x+c)^5-5898240*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(sin(d*x+c)
/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)-1363110*cos(d*x+c)^2*cot(d*x+c)^5-1966080*(-cos
(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(sin(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+1972170*co
s(d*x+c)*cot(d*x+c)^5+2720050*cot(d*x+c)^5-810890*cot(d*x+c)^4*csc(d*x+c)-1673842*cot(d*x+c)^3*csc(d*x+c)^2-30
610*cot(d*x+c)^2*csc(d*x+c)^3+267750*cot(d*x+c)*csc(d*x+c)^4)

Fricas [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 1023, normalized size of antiderivative = 2.33 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

[-1/1966080*(1116915*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)*sin(d*x + c) + 3*a*cos(d*x + c)^2 + 2*a*cos(d*x + c) - a)/(cos(d*x + c)^2 + 2*cos(d*x
+ c) + 1))*sin(d*x + c) + 983040*(cos(d*x + c)^7 + 3*cos(d*x + c)^6 + cos(d*x + c)^5 - 5*cos(d*x + c)^4 - 5*co
s(d*x + c)^3 + cos(d*x + c)^2 + 3*cos(d*x + c) + 1)*sqrt(-a)*log(-(8*a*cos(d*x + c)^3 - 4*(2*cos(d*x + c)^2 -
cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c) + a)/(cos(d*x +
c) + 1))*sin(d*x + c) + 4*(639063*cos(d*x + c)^8 + 681555*cos(d*x + c)^7 - 986085*cos(d*x + c)^6 - 1360025*co
s(d*x + c)^5 + 405445*cos(d*x + c)^4 + 836921*cos(d*x + c)^3 + 15305*cos(d*x + c)^2 - 133875*cos(d*x + c))*sqr
t((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)*sin(d*
x + c)), -1/983040*(1116915*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*cos(d*x + c) + a)/cos(d*
x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))*sin(d*x + c) + 983040*(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)*sin(d*x + c)/(2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a))*
sin(d*x + c) + 2*(639063*cos(d*x + c)^8 + 681555*cos(d*x + c)^7 - 986085*cos(d*x + c)^6 - 1360025*cos(d*x + c)
^5 + 405445*cos(d*x + c)^4 + 836921*cos(d*x + c)^3 + 15305*cos(d*x + c)^2 - 133875*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*c
os(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)*sin(d*x + c))]

Sympy [F]

\[ \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\int \frac {\cot ^{6}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(cot(d*x+c)**6/(a+a*sec(d*x+c))**(5/2),x)

[Out]

Integral(cot(c + d*x)**6/(a*(sec(c + d*x) + 1))**(5/2), x)

Maxima [F(-1)]

Timed out. \[ \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

Timed out

Giac [A] (verification not implemented)

none

Time = 1.62 (sec) , antiderivative size = 412, normalized size of antiderivative = 0.94 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {{\left (2 \, {\left (4 \, {\left (6 \, {\left (\frac {8 \, \sqrt {2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {91 \, \sqrt {2}}{a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {3043 \, \sqrt {2}}{a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \frac {47185 \, \sqrt {2}}{a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {349965 \, \sqrt {2}}{a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {1024 \, \sqrt {2} {\left (345 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{8} - 1230 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{6} a + 1760 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{4} a^{2} - 1150 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} a^{3} + 299 \, a^{4}\right )}}{{\left ({\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} - a\right )}^{5} \sqrt {-a} a \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}}{983040 \, d} \]

[In]

integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^(5/2),x, algorithm="giac")

[Out]

1/983040*((2*(4*(6*(8*sqrt(2)*tan(1/2*d*x + 1/2*c)^2/(a^3*sgn(cos(d*x + c))) - 91*sqrt(2)/(a^3*sgn(cos(d*x + c
))))*tan(1/2*d*x + 1/2*c)^2 + 3043*sqrt(2)/(a^3*sgn(cos(d*x + c))))*tan(1/2*d*x + 1/2*c)^2 - 47185*sqrt(2)/(a^
3*sgn(cos(d*x + c))))*tan(1/2*d*x + 1/2*c)^2 + 349965*sqrt(2)/(a^3*sgn(cos(d*x + c))))*sqrt(-a*tan(1/2*d*x + 1
/2*c)^2 + a)*tan(1/2*d*x + 1/2*c) - 1024*sqrt(2)*(345*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1
/2*c)^2 + a))^8 - 1230*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^6*a + 1760*(sqrt(
-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^4*a^2 - 1150*(sqrt(-a)*tan(1/2*d*x + 1/2*c) -
sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2*a^3 + 299*a^4)/(((sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x +
1/2*c)^2 + a))^2 - a)^5*sqrt(-a)*a*sgn(cos(d*x + c))))/d

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^6}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]

[In]

int(cot(c + d*x)^6/(a + a/cos(c + d*x))^(5/2),x)

[Out]

int(cot(c + d*x)^6/(a + a/cos(c + d*x))^(5/2), x)