\(\int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx\) [205]

Optimal result

Integrand size = 23, antiderivative size = 355 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a^{5/2} d}-\frac {9683 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{4096 \sqrt {2} a^{5/2} d}-\frac {1491 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{4096 a^3 d}+\frac {5587 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{6144 a^4 d}-\frac {1527 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{2048 a^4 d}-\frac {145 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{1024 a^4 d}-\frac {9 \cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{256 a^4 d}-\frac {\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^4 d} \]

[Out]

2*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/a^(5/2)/d+5587/6144*cot(d*x+c)^3*(a+a*sec(d*x+c))^(3/2)/a^
4/d-1527/2048*cos(d*x+c)*cot(d*x+c)^3*sec(1/2*d*x+1/2*c)^2*(a+a*sec(d*x+c))^(3/2)/a^4/d-145/1024*cos(d*x+c)^2*
cot(d*x+c)^3*sec(1/2*d*x+1/2*c)^4*(a+a*sec(d*x+c))^(3/2)/a^4/d-9/256*cos(d*x+c)^3*cot(d*x+c)^3*sec(1/2*d*x+1/2
*c)^6*(a+a*sec(d*x+c))^(3/2)/a^4/d-1/128*cos(d*x+c)^4*cot(d*x+c)^3*sec(1/2*d*x+1/2*c)^8*(a+a*sec(d*x+c))^(3/2)
/a^4/d-9683/8192*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-1491/4096*cot
(d*x+c)*(a+a*sec(d*x+c))^(1/2)/a^3/d

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3972, 483, 593, 597, 536, 209} \[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{a^{5/2} d}-\frac {9683 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{4096 \sqrt {2} a^{5/2} d}+\frac {5587 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{6144 a^4 d}-\frac {\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{128 a^4 d}-\frac {9 \cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{256 a^4 d}-\frac {145 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{1024 a^4 d}-\frac {1527 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{2048 a^4 d}-\frac {1491 \cot (c+d x) \sqrt {a \sec (c+d x)+a}}{4096 a^3 d} \]

[In]

Int[Cot[c + d*x]^4/(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) - (9683*ArcTan[(Sqrt[a]*Tan[c + d*x])/
(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(4096*Sqrt[2]*a^(5/2)*d) - (1491*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(
4096*a^3*d) + (5587*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(6144*a^4*d) - (1527*Cos[c + d*x]*Cot[c + d*x]^
3*Sec[(c + d*x)/2]^2*(a + a*Sec[c + d*x])^(3/2))/(2048*a^4*d) - (145*Cos[c + d*x]^2*Cot[c + d*x]^3*Sec[(c + d*
x)/2]^4*(a + a*Sec[c + d*x])^(3/2))/(1024*a^4*d) - (9*Cos[c + d*x]^3*Cot[c + d*x]^3*Sec[(c + d*x)/2]^6*(a + a*
Sec[c + d*x])^(3/2))/(256*a^4*d) - (Cos[c + d*x]^4*Cot[c + d*x]^3*Sec[(c + d*x)/2]^8*(a + a*Sec[c + d*x])^(3/2
))/(128*a^4*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^4 \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 )}{a^4 d} \\ & = -\frac {\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^4 d}-\frac {\text {Subst}\left (\int \frac {5 a-11 a^2 x^2}{x^4 \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 )}{8 a^5 d} \\ & = -\frac {9 \cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{256 a^4 d}-\frac {\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^4 d}-\frac {\text {Subst}\left (\int \frac {-51 a^2-243 a^3 x^2}{x^4 \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 )}{96 a^6 d} \\ & = -\frac {145 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{1024 a^4 d}-\frac {9 \cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{256 a^4 d}-\frac {\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^4 d}-\frac {\text {Subst}\left (\int \frac {-1509 a^3-3045 a^4 x^2}{x^4 \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 )}{768 a^7 d} \\ & = -\frac {1527 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{2048 a^4 d}-\frac {145 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{1024 a^4 d}-\frac {9 \cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{256 a^4 d}-\frac {\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^4 d}-\frac {\text {Subst}\left (\int \frac {-16761 a^4-22905 a^5 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 )}{3072 a^8 d} \\ & = \frac {5587 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{6144 a^4 d}-\frac {1527 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{2048 a^4 d}-\frac {145 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{1024 a^4 d}-\frac {9 \cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{256 a^4 d}-\frac {\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^4 d}+\frac {\text {Subst}\left (\int \frac {-13419 a^5-50283 a^6 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 )}{18432 a^8 d} \\ & = -\frac {1491 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{4096 a^3 d}+\frac {5587 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{6144 a^4 d}-\frac {1527 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{2048 a^4 d}-\frac {145 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{1024 a^4 d}-\frac {9 \cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{256 a^4 d}-\frac {\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^4 d}-\frac {\text {Subst}\left (\int \frac {60309 a^6-13419 a^7 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 )}{36864 a^8 d} \\ & = -\frac {1491 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{4096 a^3 d}+\frac {5587 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{6144 a^4 d}-\frac {1527 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{2048 a^4 d}-\frac {145 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{1024 a^4 d}-\frac {9 \cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{256 a^4 d}-\frac {\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^4 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 {9683 \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 )}{4096 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 {9683 \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{4096 \sqrt {2} a^{5/2} d}-\frac {1491 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{4096 a^3 d}+\frac {5587 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{6144 a^4 d}-\frac {1527 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{2048 a^4 d}-\frac {145 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{1024 a^4 d}-\frac {9 \cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{256 a^4 d}-\frac {\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^4 d} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 3.44 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.81 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {\cos ^5\left (\frac {1}{2} (c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \left (\left (-29258+3200 \csc ^2\left (\frac {1}{2} (c+d x)\right )-128 \csc ^4\left (\frac {1}{2} (c+d x)\right )+18225 \sec ^2\left (\frac {1}{2} (c+d x)\right )-4470 \sec ^4\left (\frac {1}{2} (c+d x)\right )+696 \sec ^6\left (\frac {1}{2} (c+d x)\right )-48 \sec ^8\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\sec (c+d x)}+49152 \arctan \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\frac {1}{1+\sec (c+d x)}}}\right ) \cot \left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {\sec (c+d x)}{(1+\sec (c+d x))^2}} \sqrt {1+\sec (c+d x)}-29049 \arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ) \cot \left (\frac {1}{2} (c+d x)\right ) \sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {1}{1+\sec (c+d x)}} \sqrt {1+\sec (c+d x)}\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{3072 d (a (1+\sec (c+d x)))^{5/2}} \]

[In]

Integrate[Cot[c + d*x]^4/(a + a*Sec[c + d*x])^(5/2),x]

[Out]

(Cos[(c + d*x)/2]^5*Sec[c + d*x]^(5/2)*((-29258 + 3200*Csc[(c + d*x)/2]^2 - 128*Csc[(c + d*x)/2]^4 + 18225*Sec
[(c + d*x)/2]^2 - 4470*Sec[(c + d*x)/2]^4 + 696*Sec[(c + d*x)/2]^6 - 48*Sec[(c + d*x)/2]^8)*Sqrt[Sec[c + d*x]]
+ 49152*ArcTan[Tan[(c + d*x)/2]/Sqrt[(1 + Sec[c + d*x])^(-1)]]*Cot[(c + d*x)/2]*Sqrt[Sec[c + d*x]/(1 + Sec[c
+ d*x])^2]*Sqrt[1 + Sec[c + d*x]] - 29049*ArcSin[Tan[(c + d*x)/2]]*Cot[(c + d*x)/2]*Sqrt[Sec[(c + d*x)/2]^2]*S
qrt[(1 + Sec[c + d*x])^(-1)]*Sqrt[1 + Sec[c + d*x]])*Sin[(c + d*x)/2])/(3072*d*(a*(1 + Sec[c + d*x]))^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(709\) vs. \(2(310)=620\).

Time = 1.93 (sec) , antiderivative size = 710, normalized size of antiderivative = 2.00

method	result	size
default	\(-\frac {\sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (29049 \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 )-49152 \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 )+87147 \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}-147456 \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}+87147 \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 )-147456 \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 )+29049 \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}}+29258 \cos \left (d x +c \right )^{3} \cot \left (d x +c \right )^{3}-49152 \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 )+28466 \cot \left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}-28116 \cos \left (d x +c \right ) \cot \left (d x +c \right )^{3}-34852 \cot \left (d x +c \right )^{3}+4490 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )+8946 \cot \left (d x +c \right ) \csc \left (d x +c \right )^{2}\right )}{24576 d \,a^{3} \left (\cos \left (d x +c \right )+1\right )^{3}}\)	\(710\)

[In]

int(cot(d*x+c)^4/(a+a*sec(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/24576/d/a^3*(a*(1+sec(d*x+c)))^(1/2)/(cos(d*x+c)+1)^3*(29049*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))-49152*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))+
87147*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)^2-147456*(-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+87147*2^(1/2)*ln(csc(d*x+c)-cot(d*x+c)+(cot(d*x+c)^2-2*co
t(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)-147456*(-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)+29049*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))*2^(1/2)*(-cos(d*x+c)/(cos(
d*x+c)+1))^(1/2)+29258*cos(d*x+c)^3*cot(d*x+c)^3-49152*(-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))+28466*cot(d*x+c)^3*cos(d*x+c)^2-28116*cos(d*x+c)*cot(d*x+c)^
3-34852*cot(d*x+c)^3+4490*cot(d*x+c)^2*csc(d*x+c)+8946*cot(d*x+c)*csc(d*x+c)^2)

Fricas [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 868, normalized size of antiderivative = 2.45 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate(cot(d*x+c)^4/(a+a*sec(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

[-1/49152*(29049*sqrt(2)*(cos(d*x + c)^5 + 3*cos(d*x + c)^4 + 2*cos(d*x + c)^3 - 2*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) + 24576*(cos(
d*x + c)^5 + 3*cos(d*x + c)^4 + 2*cos(d*x + c)^3 - 2*cos(d*x + c)^2 - 3*cos(d*x + c) - 1)*sqrt(-a)*log(-(8*a*c
os(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*(14629*cos(d*x + c)^6 + 14233*cos(d*x + c)^5
- 14058*cos(d*x + c)^4 - 17426*cos(d*x + c)^3 + 2245*cos(d*x + c)^2 + 4473*cos(d*x + c))*sqrt((a*cos(d*x + c)
+ a)/cos(d*x + c)))/((a^3*d*cos(d*x + c)^5 + 3*a^3*d*cos(d*x + c)^4 + 2*a^3*d*cos(d*x + c)^3 - 2*a^3*d*cos(d*x
+ c)^2 - 3*a^3*d*cos(d*x + c) - a^3*d)*sin(d*x + c)), 1/24576*(29049*sqrt(2)*(cos(d*x + c)^5 + 3*cos(d*x + c)
^4 + 2*cos(d*x + c)^3 - 2*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) + 24576*(cos(d*x + c)^5 + 3*cos(d*x + c)^4 +
2*cos(d*x + c)^3 - 2*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*(14629*cos
(d*x + c)^6 + 14233*cos(d*x + c)^5 - 14058*cos(d*x + c)^4 - 17426*cos(d*x + c)^3 + 2245*cos(d*x + c)^2 + 4473*
cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((a^3*d*cos(d*x + c)^5 + 3*a^3*d*cos(d*x + c)^4 + 2*a^3
*d*cos(d*x + c)^3 - 2*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]

\[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\int \frac {\cot ^{4}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(cot(d*x+c)**4/(a+a*sec(d*x+c))**(5/2),x)

[Out]

Integral(cot(c + d*x)**4/(a*(sec(c + d*x) + 1))**(5/2), x)

Maxima [F]

\[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\int { \frac {\cot \left (d x + c\right )^{4}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(cot(d*x+c)^4/(a+a*sec(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate(cot(d*x + c)^4/(a*sec(d*x + c) + a)^(5/2), x)

Giac [A] (verification not implemented)

none

Time = 1.39 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.83 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {3 \, {\left (2 \, {\left (4 \, {\left (\frac {2 \, \sqrt {2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {19 \, \sqrt {2}}{a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {369 \, \sqrt {2}}{a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \frac {2989 \, \sqrt {2}}{a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {512 \, \sqrt {2} {\left (12 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{4} - 21 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} a + 11 \, a^{2}\right )}}{{\left ({\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} - a\right )}^{3} \sqrt {-a} a \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}}{24576 \, d} \]

[In]

integrate(cot(d*x+c)^4/(a+a*sec(d*x+c))^(5/2),x, algorithm="giac")

[Out]

1/24576*(3*(2*(4*(2*sqrt(2)*tan(1/2*d*x + 1/2*c)^2/(a^3*sgn(cos(d*x + c))) - 19*sqrt(2)/(a^3*sgn(cos(d*x + c))
))*tan(1/2*d*x + 1/2*c)^2 + 369*sqrt(2)/(a^3*sgn(cos(d*x + c))))*tan(1/2*d*x + 1/2*c)^2 - 2989*sqrt(2)/(a^3*sg
n(cos(d*x + c))))*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)*tan(1/2*d*x + 1/2*c) + 512*sqrt(2)*(12*(sqrt(-a)*tan(1/2
*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^4 - 21*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d
*x + 1/2*c)^2 + a))^2*a + 11*a^2)/(((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*sqrt(-a)*a*sgn(cos(d*x + c))))/d

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^4}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]

[In]

int(cot(c + d*x)^4/(a + a/cos(c + d*x))^(5/2),x)

[Out]

int(cot(c + d*x)^4/(a + a/cos(c + d*x))^(5/2), x)