\(\int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx\) [199]

Optimal result

Integrand size = 23, antiderivative size = 200 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {13 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{32 \sqrt {2} a^{5/2} d}-\frac {5 a}{28 d (a+a \sec (c+d x))^{7/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {3}{40 d (a+a \sec (c+d x))^{5/2}}+\frac {19}{48 a d (a+a \sec (c+d x))^{3/2}}+\frac {51}{32 a^2 d \sqrt {a+a \sec (c+d x)}} \]

[Out]

-2*arctanh((a+a*sec(d*x+c))^(1/2)/a^(1/2))/a^(5/2)/d-5/28*a/d/(a+a*sec(d*x+c))^(7/2)+1/2*a/d/(1-sec(d*x+c))/(a
+a*sec(d*x+c))^(7/2)+3/40/d/(a+a*sec(d*x+c))^(5/2)+19/48/a/d/(a+a*sec(d*x+c))^(3/2)+13/64*arctanh(1/2*(a+a*sec
(d*x+c))^(1/2)*2^(1/2)/a^(1/2))/a^(5/2)/d*2^(1/2)+51/32/a^2/d/(a+a*sec(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3965, 105, 157, 162, 65, 213} \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {13 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{32 \sqrt {2} a^{5/2} d}+\frac {51}{32 a^2 d \sqrt {a \sec (c+d x)+a}}-\frac {5 a}{28 d (a \sec (c+d x)+a)^{7/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}+\frac {3}{40 d (a \sec (c+d x)+a)^{5/2}}+\frac {19}{48 a d (a \sec (c+d x)+a)^{3/2}} \]

[In]

Int[Cot[c + d*x]^3/(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) + (13*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2]*Sqr
t[a])])/(32*Sqrt[2]*a^(5/2)*d) - (5*a)/(28*d*(a + a*Sec[c + d*x])^(7/2)) + a/(2*d*(1 - Sec[c + d*x])*(a + a*Se
c[c + d*x])^(7/2)) + 3/(40*d*(a + a*Sec[c + d*x])^(5/2)) + 19/(48*a*d*(a + a*Sec[c + d*x])^(3/2)) + 51/(32*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 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^4 \text {Subst}\left (\int \frac {1}{x (-a+a x)^2 (a+a x)^{9/2}} \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}-\frac {a \text {Subst}\left (\int \frac {2 a^2+\frac {9 a^2 x}{2}}{x (-a+a x) (a+a x)^{9/2}} \, dx,x,\sec (c+d x)\right )}{2 d} \\ & = -\frac {5 a}{28 d (a+a \sec (c+d x))^{7/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {\text {Subst}\left (\int \frac {-14 a^4-\frac {35 a^4 x}{4}}{x (-a+a x) (a+a x)^{7/2}} \, dx,x,\sec (c+d x)\right )}{14 a^2 d} \\ & = -\frac {5 a}{28 d (a+a \sec (c+d x))^{7/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {3}{40 d (a+a \sec (c+d x))^{5/2}}-\frac {\text {Subst}\left (\int \frac {70 a^6-\frac {105 a^6 x}{8}}{x (-a+a x) (a+a x)^{5/2}} \, dx,x,\sec (c+d x)\right )}{70 a^5 d} \\ & = -\frac {5 a}{28 d (a+a \sec (c+d x))^{7/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {3}{40 d (a+a \sec (c+d x))^{5/2}}+\frac {19}{48 a d (a+a \sec (c+d x))^{3/2}}+\frac {\text {Subst}\left (\int \frac {-210 a^8+\frac {1995 a^8 x}{16}}{x (-a+a x) (a+a x)^{3/2}} \, dx,x,\sec (c+d x)\right )}{210 a^8 d} \\ & = -\frac {5 a}{28 d (a+a \sec (c+d x))^{7/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {3}{40 d (a+a \sec (c+d x))^{5/2}}+\frac {19}{48 a d (a+a \sec (c+d x))^{3/2}}+\frac {51}{32 a^2 d \sqrt {a+a \sec (c+d x)}}-\frac {\text {Subst}\left (\int \frac {210 a^{10}-\frac {5355 a^{10} x}{32}}{x (-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{210 a^{11} d} \\ & = -\frac {5 a}{28 d (a+a \sec (c+d x))^{7/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {3}{40 d (a+a \sec (c+d x))^{5/2}}+\frac {19}{48 a d (a+a \sec (c+d x))^{3/2}}+\frac {51}{32 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 {13 \text {Subst}\left (\int \frac {1}{(-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{64 a d} \\ & = -\frac {5 a}{28 d (a+a \sec (c+d x))^{7/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {3}{40 d (a+a \sec (c+d x))^{5/2}}+\frac {19}{48 a d (a+a \sec (c+d x))^{3/2}}+\frac {51}{32 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 {13 \text {Subst}\left (\int \frac {1}{-2 a+x^2} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{32 a^2 d} \\ & = -\frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {13 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{32 \sqrt {2} a^{5/2} d}-\frac {5 a}{28 d (a+a \sec (c+d x))^{7/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {3}{40 d (a+a \sec (c+d x))^{5/2}}+\frac {19}{48 a d (a+a \sec (c+d x))^{3/2}}+\frac {51}{32 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 = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.45 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\frac {a \left (-14-13 \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},1,-\frac {5}{2},\frac {1}{2} (1+\sec (c+d x))\right ) (-1+\sec (c+d x))+8 \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},1,-\frac {5}{2},1+\sec (c+d x)\right ) (-1+\sec (c+d x))\right )}{28 d (-1+\sec (c+d x)) (a (1+\sec (c+d x)))^{7/2}} \]

[In]

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

[Out]

(a*(-14 - 13*Hypergeometric2F1[-7/2, 1, -5/2, (1 + Sec[c + d*x])/2]*(-1 + Sec[c + d*x]) + 8*Hypergeometric2F1[
-7/2, 1, -5/2, 1 + Sec[c + d*x]]*(-1 + Sec[c + d*x])))/(28*d*(-1 + Sec[c + d*x])*(a*(1 + Sec[c + d*x]))^(7/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(527\) vs. \(2(163)=326\).

Time = 1.76 (sec) , antiderivative size = 528, normalized size of antiderivative = 2.64

method	result	size
default	\(\frac {\sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (1365 \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 )+13440 \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 )+4095 \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}}+40320 \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}}+4095 \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}}+40320 \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}}+1365 \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 )-16034 \cos \left (d x +c \right )^{3} \cot \left (d x +c \right )^{2}+13440 \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}}-25280 \cos \left (d x +c \right )^{2} \cot \left (d x +c \right )^{2}+3164 \cos \left (d x +c \right ) \cot \left (d x +c \right )^{2}+24080 \cot \left (d x +c \right )^{2}+10710 \cot \left (d x +c \right ) \csc \left (d x +c \right )\right )}{6720 d \,a^{3} \left (\cos \left (d x +c \right )+1\right )^{3}}\)	\(528\)

[In]

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

[Out]

1/6720/d/a^3*(a*(1+sec(d*x+c)))^(1/2)/(cos(d*x+c)+1)^3*(1365*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))+13440*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))+4095*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)+40320*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)+4095*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)+40320*arctan((-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)*(-co
s(d*x+c)/(cos(d*x+c)+1))^(1/2)+1365*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))-16034*cos(d*x+c)^3*cot(d*x+c)^2+13440*arctan((-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*(-cos
(d*x+c)/(cos(d*x+c)+1))^(1/2)-25280*cos(d*x+c)^2*cot(d*x+c)^2+3164*cos(d*x+c)*cot(d*x+c)^2+24080*cot(d*x+c)^2+
10710*cot(d*x+c)*csc(d*x+c))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 361 vs. \(2 (161) = 322\).

Time = 0.38 (sec) , antiderivative size = 748, normalized size of antiderivative = 3.74 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=\left [\frac {1365 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{5} + 3 \, \cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} - 3 \, \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 ) + 6720 \, {\left (\cos \left (d x + c\right )^{5} + 3 \, \cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} - 3 \, \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 (8017 \, \cos \left (d x + c\right )^{5} + 12640 \, \cos \left (d x + c\right )^{4} - 1582 \, \cos \left (d x + c\right )^{3} - 12040 \, \cos \left (d x + c\right )^{2} - 5355 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{13440 \, {\left (a^{3} d \cos \left (d x + c\right )^{5} + 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 2 \, a^{3} d \cos \left (d x + c\right )^{3} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} - 3 \, a^{3} d \cos \left (d x + c\right ) - a^{3} d\right )}}, -\frac {1365 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{5} + 3 \, \cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} - 3 \, \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 ) - 6720 \, {\left (\cos \left (d x + c\right )^{5} + 3 \, \cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} - 3 \, \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 (8017 \, \cos \left (d x + c\right )^{5} + 12640 \, \cos \left (d x + c\right )^{4} - 1582 \, \cos \left (d x + c\right )^{3} - 12040 \, \cos \left (d x + c\right )^{2} - 5355 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{6720 \, {\left (a^{3} d \cos \left (d x + c\right )^{5} + 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 2 \, a^{3} d \cos \left (d x + c\right )^{3} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} - 3 \, a^{3} d \cos \left (d x + c\right ) - a^{3} d\right )}}\right ] \]

[In]

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

[Out]

[1/13440*(1365*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) + 3*a*cos(d*x + c)
+ a)/(cos(d*x + c) - 1)) + 6720*(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*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*(8017*cos(d*x + c)^5 + 12640*cos(d*x + c)^4 - 1582*cos(d
*x + c)^3 - 12040*cos(d*x + c)^2 - 5355*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), -1/6720*(1365*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)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x +
c) + a)) - 6720*(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)/(2*a*cos(d*x + c) + a)) - 2*
(8017*cos(d*x + c)^5 + 12640*cos(d*x + c)^4 - 1582*cos(d*x + c)^3 - 12040*cos(d*x + c)^2 - 5355*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)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 1.08 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.48 \[ \int \frac {\cot ^3(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx=-\frac {\frac {1365 \, \sqrt {2} \arctan \left (\frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {13440 \, \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} + \frac {105 \, \sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} + \frac {2 \, \sqrt {2} {\left (15 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{3} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{36} - 84 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{37} - 385 \, {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} a^{38} - 2730 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{39}\right )}}{a^{42} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}}{6720 \, d} \]

[In]

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

[Out]

-1/6720*(1365*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))) -
13440*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))) + 105*
sqrt(2)*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)/(a^3*sgn(cos(d*x + c))*tan(1/2*d*x + 1/2*c)^2) + 2*sqrt(2)*(15*(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^36 - 84*(a*tan(1/2*d*x + 1/2*c)^2 - a)^2*s
qrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)*a^37 - 385*(-a*tan(1/2*d*x + 1/2*c)^2 + a)^(3/2)*a^38 - 2730*sqrt(-a*tan(1/
2*d*x + 1/2*c)^2 + a)*a^39)/(a^42*sgn(cos(d*x + c))))/d

Mupad [F(-1)]

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

[In]

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

[Out]

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