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

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

[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]  time = 0.42, antiderivative size = 439, normalized size of antiderivative = 1.00, 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 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])

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 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 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 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]

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*}

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Mathematica [C]  time = 24.27, size = 5688, normalized size = 12.96 \[ \text {Result too large to show} \]

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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fricas [A]  time = 0.81, size = 1023, normalized size = 2.33 \[ \text {result too large to display} \]

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]

[-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))]

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giac [A]  time = 3.67, size = 454, normalized size = 1.03 \[ -\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 (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} - \frac {91 \, \sqrt {2}}{a^{3} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {3043 \, \sqrt {2}}{a^{3} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \frac {47185 \, \sqrt {2}}{a^{3} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {349965 \, \sqrt {2}}{a^{3} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\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 (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}}{983040 \, d} \]

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

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maple [B]  time = 1.62, size = 1412, normalized size = 3.22 \[ \text {result too large to display} \]

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*(1+cos(d*x+c))/cos(d*x+c))^(1/2)*(1+cos(d*x+c))^2*(-1+cos(d*x+c))^5*(-983040*cos(d*x+c)^7*sin(d*
x+c)*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/
cos(d*x+c)*2^(1/2))-1116915*cos(d*x+c)^7*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/
(1+cos(d*x+c)))^(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)/(1+cos(d*x+c)))^(1/2)*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))-3350
745*cos(d*x+c)^6*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*si
n(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))-983040*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)^5*sin(d*x+c)*2^(1/2)
*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))+1278126*cos(d*x+c)^8-1116915*
(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)^5*sin(d*x+c)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*
x+c)+cos(d*x+c)-1)/sin(d*x+c))+4915200*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)^4*sin(d*x+c)*2^(1/2)*ar
ctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))+1363110*cos(d*x+c)^7+5584575*(-2
*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)^4*sin(d*x+c)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c
)+cos(d*x+c)-1)/sin(d*x+c))+4915200*cos(d*x+c)^3*sin(d*x+c)*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arcta
nh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))-1972170*cos(d*x+c)^6+5584575*cos(d*
x+c)^3*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+c
os(d*x+c)-1)/sin(d*x+c))-983040*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctanh(1/2*(-2*cos(d*x+c)/(1+cos
(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))*cos(d*x+c)^2*sin(d*x+c)-2720050*cos(d*x+c)^5-1116915*(-2*cos(d*
x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))*cos
(d*x+c)^2*sin(d*x+c)-2949120*cos(d*x+c)*sin(d*x+c)*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctanh(1/2*(-
2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))+810890*cos(d*x+c)^4-3350745*cos(d*x+c)*sin(d
*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*ln(-(-(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)
/sin(d*x+c))-983040*2^(1/2)*sin(d*x+c)*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*
2^(1/2))*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+1673842*cos(d*x+c)^3-1116915*sin(d*x+c)*ln(-(-(-2*cos(d*x+c)/(1+
cos(d*x+c)))^(1/2)*sin(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+30610*cos(d*x+c)^
2-267750*cos(d*x+c))/sin(d*x+c)^15/a^3

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {cot}\left (c+d\,x\right )}^6}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{6}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]

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]

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

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