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

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.420103, antiderivative size = 439, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3887, 472, 579, 583, 522, 203} \[ \frac{58077 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{40960 a^5 d}-\frac{41693 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{49152 a^4 d}+\frac{8925 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{32768 a^3 d}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{a^{5/2} d}+\frac{74461 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{32768 \sqrt{2} a^{5/2} d}-\frac{\cos ^5(c+d x) \cot ^5(c+d x) \sec ^{10}\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{320 a^5 d}-\frac{7 \cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{512 a^5 d}-\frac{155 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{3072 a^5 d}-\frac{2473 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{12288 a^5 d}-\frac{9467 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{8192 a^5 d} \]

Antiderivative was successfully verified.

[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 3887

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]

Rule 472

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 579

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 583

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 522

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 203

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

Rubi steps

\begin{align*} \int \frac{\cot ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx &=-\frac{2 \operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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 \operatorname{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 \operatorname{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 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{a^{5/2} d}+\frac{74461 \tan ^{-1}\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 [C]  time = 23.7097, size = 5698, normalized size = 12.98 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [B]  time = 0.444, size = 1412, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/983040/d/a^3*(a*(cos(d*x+c)+1)/cos(d*x+c))^(1/2)*(cos(d*x+c)+1)^2*(-1+cos(d*x+c))^5*(-983040*cos(d*x+c)^7*si
n(d*x+c)*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)
*sin(d*x+c)/cos(d*x+c))-1116915*cos(d*x+c)^7*sin(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln(-(-(-2*cos(d*x
+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))-2949120*cos(d*x+c)^6*sin(d*x+c)*2^(1/2)*(-2*cos
(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))-
3350745*cos(d*x+c)^6*sin(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln(-(-(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2
)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))-983040*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)^5*sin(d*x+c)*2^(
1/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))+1278126*cos(d*x+c)^8-1116
915*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)^5*sin(d*x+c)*ln(-(-(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*si
n(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))+4915200*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*2^(1/2)*cos(d*x+c)^4*sin(d*x+c
)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))+1363110*cos(d*x+c)^7+5584575
*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)^4*sin(d*x+c)*ln(-(-(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d
*x+c)+cos(d*x+c)-1)/sin(d*x+c))+4915200*2^(1/2)*cos(d*x+c)^3*sin(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a
rctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))-1972170*cos(d*x+c)^6+5584575*co
s(d*x+c)^3*sin(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln(-(-(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+
c)+cos(d*x+c)-1)/sin(d*x+c))-983040*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arcta
nh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))-2720050*cos(d*x+c)^5-1116915*cos(d*
x+c)^2*sin(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln(-(-(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+c
os(d*x+c)-1)/sin(d*x+c))-2949120*2^(1/2)*cos(d*x+c)*sin(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(1/
2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))+810890*cos(d*x+c)^4-3350745*cos(d*x+c)*s
in(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln(-(-(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+cos(d*x+c
)-1)/sin(d*x+c))-983040*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*2^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d
*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*sin(d*x+c)+1673842*cos(d*x+c)^3-1116915*sin(d*x+c)*ln(-(-(-2*cos(d*x+c)
/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+30610*cos(d*x
+c)^2-267750*cos(d*x+c))/sin(d*x+c)^15

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 9.6942, size = 613, normalized size = 1.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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(tan(1/2*d*x + 1/2*c)^2 - 1)) - 91*sqrt(2)/(a^3*
sgn(tan(1/2*d*x + 1/2*c)^2 - 1)))*tan(1/2*d*x + 1/2*c)^2 + 3043*sqrt(2)/(a^3*sgn(tan(1/2*d*x + 1/2*c)^2 - 1)))
*tan(1/2*d*x + 1/2*c)^2 - 47185*sqrt(2)/(a^3*sgn(tan(1/2*d*x + 1/2*c)^2 - 1)))*tan(1/2*d*x + 1/2*c)^2 + 349965
*sqrt(2)/(a^3*sgn(tan(1/2*d*x + 1/2*c)^2 - 1)))*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)*tan(1/2*d*x + 1/2*c) - 102
4*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
(tan(1/2*d*x + 1/2*c)^2 - 1)))/d