\(\int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx\) [188]

Optimal result

Integrand size = 23, antiderivative size = 238 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {203 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{256 \sqrt {2} a^{3/2} d}+\frac {139 a^2}{224 d (a+a \sec (c+d x))^{7/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {15 a}{64 d (a+a \sec (c+d x))^{5/2}}-\frac {53}{384 d (a+a \sec (c+d x))^{3/2}}-\frac {309}{256 a d \sqrt {a+a \sec (c+d x)}} \]

[Out]

2*arctanh((a+a*sec(d*x+c))^(1/2)/a^(1/2))/a^(3/2)/d+139/224*a^2/d/(a+a*sec(d*x+c))^(7/2)-1/4*a^2/d/(1-sec(d*x+
c))^2/(a+a*sec(d*x+c))^(7/2)-19/16*a^2/d/(1-sec(d*x+c))/(a+a*sec(d*x+c))^(7/2)+15/64*a/d/(a+a*sec(d*x+c))^(5/2
)-53/384/d/(a+a*sec(d*x+c))^(3/2)-203/512*arctanh(1/2*(a+a*sec(d*x+c))^(1/2)*2^(1/2)/a^(1/2))/a^(3/2)/d*2^(1/2
)-309/256/a/d/(a+a*sec(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3965, 105, 156, 157, 162, 65, 213} \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {203 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{256 \sqrt {2} a^{3/2} d}+\frac {139 a^2}{224 d (a \sec (c+d x)+a)^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{7/2}}+\frac {15 a}{64 d (a \sec (c+d x)+a)^{5/2}}-\frac {53}{384 d (a \sec (c+d x)+a)^{3/2}}-\frac {309}{256 a d \sqrt {a \sec (c+d x)+a}} \]

[In]

Int[Cot[c + d*x]^5/(a + a*Sec[c + d*x])^(3/2),x]

[Out]

(2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) - (203*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqr
t[a])])/(256*Sqrt[2]*a^(3/2)*d) + (139*a^2)/(224*d*(a + a*Sec[c + d*x])^(7/2)) - a^2/(4*d*(1 - Sec[c + d*x])^2
*(a + a*Sec[c + d*x])^(7/2)) - (19*a^2)/(16*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(7/2)) + (15*a)/(64*d*(a
+ a*Sec[c + d*x])^(5/2)) - 53/(384*d*(a + a*Sec[c + d*x])^(3/2)) - 309/(256*a*d*Sqrt[a + a*Sec[c + d*x]])

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]

Rule 157

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 213

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

Rule 3965

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

Rubi steps \begin{align*} \text {integral}& = \frac {a^6 \text {Subst}\left (\int \frac {1}{x (-a+a x)^3 (a+a x)^{9/2}} \, dx,x,\sec (c+d x)\right )}{d} \\ & = -\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{7/2}}-\frac {a^3 \text {Subst}\left (\int \frac {4 a^2+\frac {11 a^2 x}{2}}{x (-a+a x)^2 (a+a x)^{9/2}} \, dx,x,\sec (c+d x)\right )}{4 d} \\ & = -\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {\text {Subst}\left (\int \frac {8 a^4+\frac {171 a^4 x}{4}}{x (-a+a x) (a+a x)^{9/2}} \, dx,x,\sec (c+d x)\right )}{8 d} \\ & = \frac {139 a^2}{224 d (a+a \sec (c+d x))^{7/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}-\frac {\text {Subst}\left (\int \frac {-56 a^6-\frac {973 a^6 x}{8}}{x (-a+a x) (a+a x)^{7/2}} \, dx,x,\sec (c+d x)\right )}{56 a^3 d} \\ & = \frac {139 a^2}{224 d (a+a \sec (c+d x))^{7/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {15 a}{64 d (a+a \sec (c+d x))^{5/2}}+\frac {\text {Subst}\left (\int \frac {280 a^8+\frac {2625 a^8 x}{16}}{x (-a+a x) (a+a x)^{5/2}} \, dx,x,\sec (c+d x)\right )}{280 a^6 d} \\ & = \frac {139 a^2}{224 d (a+a \sec (c+d x))^{7/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {15 a}{64 d (a+a \sec (c+d x))^{5/2}}-\frac {53}{384 d (a+a \sec (c+d x))^{3/2}}-\frac {\text {Subst}\left (\int \frac {-840 a^{10}+\frac {5565 a^{10} x}{32}}{x (-a+a x) (a+a x)^{3/2}} \, dx,x,\sec (c+d x)\right )}{840 a^9 d} \\ & = \frac {139 a^2}{224 d (a+a \sec (c+d x))^{7/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {15 a}{64 d (a+a \sec (c+d x))^{5/2}}-\frac {53}{384 d (a+a \sec (c+d x))^{3/2}}-\frac {309}{256 a d \sqrt {a+a \sec (c+d x)}}+\frac {\text {Subst}\left (\int \frac {840 a^{12}-\frac {32445 a^{12} x}{64}}{x (-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{840 a^{12} d} \\ & = \frac {139 a^2}{224 d (a+a \sec (c+d x))^{7/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {15 a}{64 d (a+a \sec (c+d x))^{5/2}}-\frac {53}{384 d (a+a \sec (c+d x))^{3/2}}-\frac {309}{256 a d \sqrt {a+a \sec (c+d x)}}+\frac {203 \text {Subst}\left (\int \frac {1}{(-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{512 d}-\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{a d} \\ & = \frac {139 a^2}{224 d (a+a \sec (c+d x))^{7/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {15 a}{64 d (a+a \sec (c+d x))^{5/2}}-\frac {53}{384 d (a+a \sec (c+d x))^{3/2}}-\frac {309}{256 a d \sqrt {a+a \sec (c+d x)}}-\frac {2 \text {Subst}\left (\int \frac {1}{-1+\frac {x^2}{a}} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{a^2 d}+\frac {203 \text {Subst}\left (\int \frac {1}{-2 a+x^2} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{256 a d} \\ & = \frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {203 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{256 \sqrt {2} a^{3/2} d}+\frac {139 a^2}{224 d (a+a \sec (c+d x))^{7/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {15 a}{64 d (a+a \sec (c+d x))^{5/2}}-\frac {53}{384 d (a+a \sec (c+d x))^{3/2}}-\frac {309}{256 a d \sqrt {a+a \sec (c+d x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.21 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.42 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {\cot ^4(c+d x) \left (-322+203 \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},1,-\frac {5}{2},\frac {1}{2} (1+\sec (c+d x))\right ) (-1+\sec (c+d x))^2-64 \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},1,-\frac {5}{2},1+\sec (c+d x)\right ) (-1+\sec (c+d x))^2+266 \sec (c+d x)\right )}{224 d (a (1+\sec (c+d x)))^{3/2}} \]

[In]

Integrate[Cot[c + d*x]^5/(a + a*Sec[c + d*x])^(3/2),x]

[Out]

(Cot[c + d*x]^4*(-322 + 203*Hypergeometric2F1[-7/2, 1, -5/2, (1 + Sec[c + d*x])/2]*(-1 + Sec[c + d*x])^2 - 64*
Hypergeometric2F1[-7/2, 1, -5/2, 1 + Sec[c + d*x]]*(-1 + Sec[c + d*x])^2 + 266*Sec[c + d*x]))/(224*d*(a*(1 + S
ec[c + d*x]))^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(435\) vs. \(2(197)=394\).

Time = 1.82 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.83

method	result	size
default	\(-\frac {\sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (4263 \cos \left (d x +c \right )^{2} \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+8526 \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+21504 \cos \left (d x +c \right )^{2} \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+4263 \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+43008 \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \cos \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+21504 \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+20726 \cos \left (d x +c \right )^{2} \cot \left (d x +c \right )^{4}+16074 \cos \left (d x +c \right ) \cot \left (d x +c \right )^{4}-33076 \cot \left (d x +c \right )^{4}-28476 \cot \left (d x +c \right )^{3} \csc \left (d x +c \right )+14462 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )^{2}+12978 \csc \left (d x +c \right )^{3} \cot \left (d x +c \right )\right )}{10752 d \,a^{2} \left (\cos \left (d x +c \right )+1\right )^{2}}\)	\(436\)

[In]

int(cot(d*x+c)^5/(a+a*sec(d*x+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/10752/d/a^2*(a*(1+sec(d*x+c)))^(1/2)/(cos(d*x+c)+1)^2*(4263*cos(d*x+c)^2*arctan(1/2*2^(1/2)/(-cos(d*x+c)/(c
os(d*x+c)+1))^(1/2))*2^(1/2)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+8526*arctan(1/2*2^(1/2)/(-cos(d*x+c)/(cos(d*x+
c)+1))^(1/2))*cos(d*x+c)*2^(1/2)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+21504*cos(d*x+c)^2*arctan((-cos(d*x+c)/(co
s(d*x+c)+1))^(1/2))*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+4263*2^(1/2)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(
1/2*2^(1/2)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+43008*arctan((-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)*(-
cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+21504*arctan((-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*(-cos(d*x+c)/(cos(d*x+c)+1))
^(1/2)+20726*cos(d*x+c)^2*cot(d*x+c)^4+16074*cos(d*x+c)*cot(d*x+c)^4-33076*cot(d*x+c)^4-28476*cot(d*x+c)^3*csc
(d*x+c)+14462*cot(d*x+c)^2*csc(d*x+c)^2+12978*csc(d*x+c)^3*cot(d*x+c))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 404 vs. \(2 (193) = 386\).

Time = 0.40 (sec) , antiderivative size = 837, normalized size of antiderivative = 3.52 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\left [\frac {4263 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \log \left (-\frac {2 \, \sqrt {2} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) - 3 \, a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) - 1}\right ) + 10752 \, {\left (\cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \log \left (-8 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} - 8 \, a \cos \left (d x + c\right ) - a\right ) - 4 \, {\left (10363 \, \cos \left (d x + c\right )^{6} + 8037 \, \cos \left (d x + c\right )^{5} - 16538 \, \cos \left (d x + c\right )^{4} - 14238 \, \cos \left (d x + c\right )^{3} + 7231 \, \cos \left (d x + c\right )^{2} + 6489 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{21504 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} + 2 \, a^{2} d \cos \left (d x + c\right )^{5} - a^{2} d \cos \left (d x + c\right )^{4} - 4 \, a^{2} d \cos \left (d x + c\right )^{3} - a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}}, \frac {4263 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {2} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{a \cos \left (d x + c\right ) + a}\right ) - 10752 \, {\left (\cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, a \cos \left (d x + c\right ) + a}\right ) - 2 \, {\left (10363 \, \cos \left (d x + c\right )^{6} + 8037 \, \cos \left (d x + c\right )^{5} - 16538 \, \cos \left (d x + c\right )^{4} - 14238 \, \cos \left (d x + c\right )^{3} + 7231 \, \cos \left (d x + c\right )^{2} + 6489 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{10752 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} + 2 \, a^{2} d \cos \left (d x + c\right )^{5} - a^{2} d \cos \left (d x + c\right )^{4} - 4 \, a^{2} d \cos \left (d x + c\right )^{3} - a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}}\right ] \]

[In]

integrate(cot(d*x+c)^5/(a+a*sec(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

[1/21504*(4263*sqrt(2)*(cos(d*x + c)^6 + 2*cos(d*x + c)^5 - cos(d*x + c)^4 - 4*cos(d*x + c)^3 - cos(d*x + c)^2
+ 2*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) -
3*a*cos(d*x + c) - a)/(cos(d*x + c) - 1)) + 10752*(cos(d*x + c)^6 + 2*cos(d*x + c)^5 - cos(d*x + c)^4 - 4*cos(
d*x + c)^3 - cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(a)*log(-8*a*cos(d*x + c)^2 - 4*(2*cos(d*x + c)^2 + cos(
d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) - 4*(10363*cos(d*x + c)^6 +
8037*cos(d*x + c)^5 - 16538*cos(d*x + c)^4 - 14238*cos(d*x + c)^3 + 7231*cos(d*x + c)^2 + 6489*cos(d*x + c))*s
qrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a^2*d*cos(d*x + c)^6 + 2*a^2*d*cos(d*x + c)^5 - a^2*d*cos(d*x + c)^4
- 4*a^2*d*cos(d*x + c)^3 - a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d), 1/10752*(4263*sqrt(2)*(cos(d*
x + c)^6 + 2*cos(d*x + c)^5 - cos(d*x + c)^4 - 4*cos(d*x + c)^3 - cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*sqrt(-a
)*arctan(sqrt(2)*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)) - 10752*(
cos(d*x + c)^6 + 2*cos(d*x + c)^5 - cos(d*x + c)^4 - 4*cos(d*x + c)^3 - cos(d*x + c)^2 + 2*cos(d*x + c) + 1)*s
qrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + a)) - 2*(10
363*cos(d*x + c)^6 + 8037*cos(d*x + c)^5 - 16538*cos(d*x + c)^4 - 14238*cos(d*x + c)^3 + 7231*cos(d*x + c)^2 +
6489*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(a^2*d*cos(d*x + c)^6 + 2*a^2*d*cos(d*x + c)^5 -
a^2*d*cos(d*x + c)^4 - 4*a^2*d*cos(d*x + c)^3 - a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d)]

Sympy [F]

\[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {\cot ^{5}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(cot(d*x+c)**5/(a+a*sec(d*x+c))**(3/2),x)

[Out]

Integral(cot(c + d*x)**5/(a*(sec(c + d*x) + 1))**(3/2), x)

Maxima [F]

\[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int { \frac {\cot \left (d x + c\right )^{5}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(cot(d*x+c)^5/(a+a*sec(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate(cot(d*x + c)^5/(a*sec(d*x + c) + a)^(3/2), x)

Giac [A] (verification not implemented)

none

Time = 1.01 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.35 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\frac {\frac {4263 \, \sqrt {2} \arctan \left (\frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {21504 \, \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {21 \, {\left (29 \, \sqrt {2} {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} - 27 \, \sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a\right )}}{a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}} + \frac {8 \, \sqrt {2} {\left (3 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{3} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{30} - 21 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{31} - 112 \, {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} a^{32} - 882 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{33}\right )}}{a^{35} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}}{10752 \, d} \]

[In]

integrate(cot(d*x+c)^5/(a+a*sec(d*x+c))^(3/2),x, algorithm="giac")

[Out]

1/10752*(4263*sqrt(2)*arctan(sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)/sqrt(-a))/(sqrt(-a)*a*sgn(cos(d*x + c))) - 21
504*arctan(1/2*sqrt(2)*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)/sqrt(-a))/(sqrt(-a)*a*sgn(cos(d*x + c))) - 21*(29*s
qrt(2)*(-a*tan(1/2*d*x + 1/2*c)^2 + a)^(3/2) - 27*sqrt(2)*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)*a)/(a^3*sgn(cos(
d*x + c))*tan(1/2*d*x + 1/2*c)^4) + 8*sqrt(2)*(3*(a*tan(1/2*d*x + 1/2*c)^2 - a)^3*sqrt(-a*tan(1/2*d*x + 1/2*c)
^2 + a)*a^30 - 21*(a*tan(1/2*d*x + 1/2*c)^2 - a)^2*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)*a^31 - 112*(-a*tan(1/2*
d*x + 1/2*c)^2 + a)^(3/2)*a^32 - 882*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)*a^33)/(a^35*sgn(cos(d*x + c))))/d

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^5}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]

[In]

int(cot(c + d*x)^5/(a + a/cos(c + d*x))^(3/2),x)

[Out]

int(cot(c + d*x)^5/(a + a/cos(c + d*x))^(3/2), x)