\(\int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx\) [200]

Optimal result

Integrand size = 23, antiderivative size = 262 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d}-\frac {263 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{512 \sqrt {2} a^{5/2} d}+\frac {199 a^2}{288 d (a+a \sec (c+d x))^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}+\frac {135 a}{448 d (a+a \sec (c+d x))^{7/2}}+\frac {7}{640 d (a+a \sec (c+d x))^{5/2}}-\frac {83}{256 a d (a+a \sec (c+d x))^{3/2}}-\frac {761}{512 a^2 d \sqrt {a+a \sec (c+d x)}} \]

[Out]

2*arctanh((a+a*sec(d*x+c))^(1/2)/a^(1/2))/a^(5/2)/d+199/288*a^2/d/(a+a*sec(d*x+c))^(9/2)-1/4*a^2/d/(1-sec(d*x+
c))^2/(a+a*sec(d*x+c))^(9/2)-21/16*a^2/d/(1-sec(d*x+c))/(a+a*sec(d*x+c))^(9/2)+135/448*a/d/(a+a*sec(d*x+c))^(7
/2)+7/640/d/(a+a*sec(d*x+c))^(5/2)-83/256/a/d/(a+a*sec(d*x+c))^(3/2)-263/1024*arctanh(1/2*(a+a*sec(d*x+c))^(1/
2)*2^(1/2)/a^(1/2))/a^(5/2)/d*2^(1/2)-761/512/a^2/d/(a+a*sec(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3965, 105, 156, 157, 162, 65, 213} \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{a^{5/2} d}-\frac {263 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{512 \sqrt {2} a^{5/2} d}+\frac {199 a^2}{288 d (a \sec (c+d x)+a)^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{9/2}}-\frac {761}{512 a^2 d \sqrt {a \sec (c+d x)+a}}+\frac {135 a}{448 d (a \sec (c+d x)+a)^{7/2}}+\frac {7}{640 d (a \sec (c+d x)+a)^{5/2}}-\frac {83}{256 a d (a \sec (c+d x)+a)^{3/2}} \]

[In]

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

[Out]

(2*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/(a^(5/2)*d) - (263*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqr
t[a])])/(512*Sqrt[2]*a^(5/2)*d) + (199*a^2)/(288*d*(a + a*Sec[c + d*x])^(9/2)) - a^2/(4*d*(1 - Sec[c + d*x])^2
*(a + a*Sec[c + d*x])^(9/2)) - (21*a^2)/(16*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(9/2)) + (135*a)/(448*d*
(a + a*Sec[c + d*x])^(7/2)) + 7/(640*d*(a + a*Sec[c + d*x])^(5/2)) - 83/(256*a*d*(a + a*Sec[c + d*x])^(3/2)) -
761/(512*a^2*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)^{11/2}} \, 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))^{9/2}}-\frac {a^3 \text {Subst}\left (\int \frac {4 a^2+\frac {13 a^2 x}{2}}{x (-a+a x)^2 (a+a x)^{11/2}} \, 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))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}+\frac {\text {Subst}\left (\int \frac {8 a^4+\frac {231 a^4 x}{4}}{x (-a+a x) (a+a x)^{11/2}} \, dx,x,\sec (c+d x)\right )}{8 d} \\ & = \frac {199 a^2}{288 d (a+a \sec (c+d x))^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}-\frac {\text {Subst}\left (\int \frac {-72 a^6-\frac {1791 a^6 x}{8}}{x (-a+a x) (a+a x)^{9/2}} \, dx,x,\sec (c+d x)\right )}{72 a^3 d} \\ & = \frac {199 a^2}{288 d (a+a \sec (c+d x))^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}+\frac {135 a}{448 d (a+a \sec (c+d x))^{7/2}}+\frac {\text {Subst}\left (\int \frac {504 a^8+\frac {8505 a^8 x}{16}}{x (-a+a x) (a+a x)^{7/2}} \, dx,x,\sec (c+d x)\right )}{504 a^6 d} \\ & = \frac {199 a^2}{288 d (a+a \sec (c+d x))^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}+\frac {135 a}{448 d (a+a \sec (c+d x))^{7/2}}+\frac {7}{640 d (a+a \sec (c+d x))^{5/2}}-\frac {\text {Subst}\left (\int \frac {-2520 a^{10}-\frac {2205 a^{10} x}{32}}{x (-a+a x) (a+a x)^{5/2}} \, dx,x,\sec (c+d x)\right )}{2520 a^9 d} \\ & = \frac {199 a^2}{288 d (a+a \sec (c+d x))^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}+\frac {135 a}{448 d (a+a \sec (c+d x))^{7/2}}+\frac {7}{640 d (a+a \sec (c+d x))^{5/2}}-\frac {83}{256 a d (a+a \sec (c+d x))^{3/2}}+\frac {\text {Subst}\left (\int \frac {7560 a^{12}-\frac {235305 a^{12} x}{64}}{x (-a+a x) (a+a x)^{3/2}} \, dx,x,\sec (c+d x)\right )}{7560 a^{12} d} \\ & = \frac {199 a^2}{288 d (a+a \sec (c+d x))^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}+\frac {135 a}{448 d (a+a \sec (c+d x))^{7/2}}+\frac {7}{640 d (a+a \sec (c+d x))^{5/2}}-\frac {83}{256 a d (a+a \sec (c+d x))^{3/2}}-\frac {761}{512 a^2 d \sqrt {a+a \sec (c+d x)}}-\frac {\text {Subst}\left (\int \frac {-7560 a^{14}+\frac {719145 a^{14} x}{128}}{x (-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{7560 a^{15} d} \\ & = \frac {199 a^2}{288 d (a+a \sec (c+d x))^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}+\frac {135 a}{448 d (a+a \sec (c+d x))^{7/2}}+\frac {7}{640 d (a+a \sec (c+d x))^{5/2}}-\frac {83}{256 a d (a+a \sec (c+d x))^{3/2}}-\frac {761}{512 a^2 d \sqrt {a+a \sec (c+d x)}}-\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{a^2 d}+\frac {263 \text {Subst}\left (\int \frac {1}{(-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{1024 a d} \\ & = \frac {199 a^2}{288 d (a+a \sec (c+d x))^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}+\frac {135 a}{448 d (a+a \sec (c+d x))^{7/2}}+\frac {7}{640 d (a+a \sec (c+d x))^{5/2}}-\frac {83}{256 a d (a+a \sec (c+d x))^{3/2}}-\frac {761}{512 a^2 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^3 d}+\frac {263 \text {Subst}\left (\int \frac {1}{-2 a+x^2} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{512 a^2 d} \\ & = \frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d}-\frac {263 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{512 \sqrt {2} a^{5/2} d}+\frac {199 a^2}{288 d (a+a \sec (c+d x))^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}+\frac {135 a}{448 d (a+a \sec (c+d x))^{7/2}}+\frac {7}{640 d (a+a \sec (c+d x))^{5/2}}-\frac {83}{256 a d (a+a \sec (c+d x))^{3/2}}-\frac {761}{512 a^2 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 = 0.35 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.38 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {\cot ^4(c+d x) \left (-450+263 \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},1,-\frac {7}{2},\frac {1}{2} (1+\sec (c+d x))\right ) (-1+\sec (c+d x))^2-64 \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},1,-\frac {7}{2},1+\sec (c+d x)\right ) (-1+\sec (c+d x))^2+378 \sec (c+d x)\right )}{288 d (a (1+\sec (c+d x)))^{5/2}} \]

[In]

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

[Out]

(Cot[c + d*x]^4*(-450 + 263*Hypergeometric2F1[-9/2, 1, -7/2, (1 + Sec[c + d*x])/2]*(-1 + Sec[c + d*x])^2 - 64*
Hypergeometric2F1[-9/2, 1, -7/2, 1 + Sec[c + d*x]]*(-1 + Sec[c + d*x])^2 + 378*Sec[c + d*x]))/(288*d*(a*(1 + S
ec[c + d*x]))^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(563\) vs. \(2(217)=434\).

Time = 1.68 (sec) , antiderivative size = 564, normalized size of antiderivative = 2.15

method	result	size
default	\(-\frac {\sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (82845 \cos \left (d x +c \right )^{3} \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 )+645120 \cos \left (d x +c \right )^{3} \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right )+248535 \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}}+1935360 \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}}+248535 \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}}+1935360 \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}}+82845 \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 )+764402 \cos \left (d x +c \right )^{3} \cot \left (d x +c \right )^{4}+645120 \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}}+1183040 \cos \left (d x +c \right )^{2} \cot \left (d x +c \right )^{4}-807214 \cos \left (d x +c \right ) \cot \left (d x +c \right )^{4}-2224080 \cot \left (d x +c \right )^{4}-378378 \cot \left (d x +c \right )^{3} \csc \left (d x +c \right )+1063440 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )^{2}+479430 \csc \left (d x +c \right )^{3} \cot \left (d x +c \right )\right )}{322560 d \,a^{3} \left (\cos \left (d x +c \right )+1\right )^{3}}\)	\(564\)

[In]

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

[Out]

-1/322560/d/a^3*(a*(1+sec(d*x+c)))^(1/2)/(cos(d*x+c)+1)^3*(82845*cos(d*x+c)^3*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))+645120*cos(d*x+c)^3*(-cos(d*x+c)/(cos(d*x+c)
+1))^(1/2)*arctan((-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+248535*cos(d*x+c)^2*arctan(1/2*2^(1/2)/(-cos(d*x+c)/(cos
(d*x+c)+1))^(1/2))*2^(1/2)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+1935360*cos(d*x+c)^2*arctan((-cos(d*x+c)/(cos(d*
x+c)+1))^(1/2))*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+248535*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)+1935360*arctan((-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*co
s(d*x+c)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+82845*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))+764402*cos(d*x+c)^3*cot(d*x+c)^4+645120*arctan((-cos(d*x+c)/(cos(d*x+c)+
1))^(1/2))*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+1183040*cos(d*x+c)^2*cot(d*x+c)^4-807214*cos(d*x+c)*cot(d*x+c)^4
-2224080*cot(d*x+c)^4-378378*cot(d*x+c)^3*csc(d*x+c)+1063440*cot(d*x+c)^2*csc(d*x+c)^2+479430*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. 438 vs. \(2 (213) = 426\).

Time = 0.39 (sec) , antiderivative size = 905, normalized size of antiderivative = 3.45 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/645120*(82845*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) - 3*a*cos(d*x + c) - a)/(cos(d*x + c) - 1)) + 322560*(cos(d*x + c)^7 + 3*cos(d*x + c)^6 + c
os(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(-8*a*co
s(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*(382201*cos(d*x + c)^7 + 591520*cos(d*x + c)^6 - 403607*cos(d*x + c)^5 - 1112040*cos(d*x + c)
^4 - 189189*cos(d*x + c)^3 + 531720*cos(d*x + c)^2 + 239715*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*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), 1/322560*(82845*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)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)
) - 322560*(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)/
(2*a*cos(d*x + c) + a)) - 2*(382201*cos(d*x + c)^7 + 591520*cos(d*x + c)^6 - 403607*cos(d*x + c)^5 - 1112040*c
os(d*x + c)^4 - 189189*cos(d*x + c)^3 + 531720*cos(d*x + c)^2 + 239715*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*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)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-1)]

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

[In]

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

[Out]

Timed out

Giac [A] (verification not implemented)

none

Time = 1.27 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.39 \[ \int \frac {\cot ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {\frac {82845 \, \sqrt {2} \arctan \left (\frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {645120 \, \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {315 \, {\left (33 \, \sqrt {2} {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} - 31 \, \sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a\right )}}{a^{4} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}} - \frac {8 \, \sqrt {2} {\left (35 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{4} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{56} - 225 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{3} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{57} + 1008 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{58} + 4410 \, {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} a^{59} + 31185 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{60}\right )}}{a^{63} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}}{322560 \, d} \]

[In]

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

[Out]

1/322560*(82845*sqrt(2)*arctan(sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)/sqrt(-a))/(sqrt(-a)*a^2*sgn(cos(d*x + c)))
- 645120*arctan(1/2*sqrt(2)*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)/sqrt(-a))/(sqrt(-a)*a^2*sgn(cos(d*x + c))) - 3
15*(33*sqrt(2)*(-a*tan(1/2*d*x + 1/2*c)^2 + a)^(3/2) - 31*sqrt(2)*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)*a)/(a^4*
sgn(cos(d*x + c))*tan(1/2*d*x + 1/2*c)^4) - 8*sqrt(2)*(35*(a*tan(1/2*d*x + 1/2*c)^2 - a)^4*sqrt(-a*tan(1/2*d*x
+ 1/2*c)^2 + a)*a^56 - 225*(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^57 + 1008*(
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^58 + 4410*(-a*tan(1/2*d*x + 1/2*c)^2 + a
)^(3/2)*a^59 + 31185*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)*a^60)/(a^63*sgn(cos(d*x + c))))/d

Mupad [F(-1)]

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

[In]

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

[Out]

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