\(\int \cot ^5(c+d x) \sqrt {a+a \sec (c+d x)} \, dx\) [140]

Optimal result

Integrand size = 23, antiderivative size = 193 \[ \int \cot ^5(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{d}-\frac {107 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{64 \sqrt {2} d}+\frac {43 a^2}{96 d (a+a \sec (c+d x))^{3/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{3/2}}-\frac {15 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{3/2}}-\frac {21 a}{64 d \sqrt {a+a \sec (c+d x)}} \]

[Out]

43/96*a^2/d/(a+a*sec(d*x+c))^(3/2)-1/4*a^2/d/(1-sec(d*x+c))^2/(a+a*sec(d*x+c))^(3/2)-15/16*a^2/d/(1-sec(d*x+c)
)/(a+a*sec(d*x+c))^(3/2)+2*arctanh((a+a*sec(d*x+c))^(1/2)/a^(1/2))*a^(1/2)/d-107/128*arctanh(1/2*(a+a*sec(d*x+
c))^(1/2)*2^(1/2)/a^(1/2))*2^(1/2)*a^(1/2)/d-21/64*a/d/(a+a*sec(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3965, 105, 156, 157, 162, 65, 213} \[ \int \cot ^5(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\frac {43 a^2}{96 d (a \sec (c+d x)+a)^{3/2}}-\frac {15 a^2}{16 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{3/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{3/2}}+\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{d}-\frac {107 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{64 \sqrt {2} d}-\frac {21 a}{64 d \sqrt {a \sec (c+d x)+a}} \]

[In]

Int[Cot[c + d*x]^5*Sqrt[a + a*Sec[c + d*x]],x]

[Out]

(2*Sqrt[a]*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d - (107*Sqrt[a]*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[
2]*Sqrt[a])])/(64*Sqrt[2]*d) + (43*a^2)/(96*d*(a + a*Sec[c + d*x])^(3/2)) - a^2/(4*d*(1 - Sec[c + d*x])^2*(a +
a*Sec[c + d*x])^(3/2)) - (15*a^2)/(16*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(3/2)) - (21*a)/(64*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)^{5/2}} \, 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))^{3/2}}-\frac {a^3 \text {Subst}\left (\int \frac {4 a^2+\frac {7 a^2 x}{2}}{x (-a+a x)^2 (a+a x)^{5/2}} \, 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))^{3/2}}-\frac {15 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{3/2}}+\frac {\text {Subst}\left (\int \frac {8 a^4+\frac {75 a^4 x}{4}}{x (-a+a x) (a+a x)^{5/2}} \, dx,x,\sec (c+d x)\right )}{8 d} \\ & = \frac {43 a^2}{96 d (a+a \sec (c+d x))^{3/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{3/2}}-\frac {15 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{3/2}}-\frac {\text {Subst}\left (\int \frac {-24 a^6-\frac {129 a^6 x}{8}}{x (-a+a x) (a+a x)^{3/2}} \, dx,x,\sec (c+d x)\right )}{24 a^3 d} \\ & = \frac {43 a^2}{96 d (a+a \sec (c+d x))^{3/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{3/2}}-\frac {15 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{3/2}}-\frac {21 a}{64 d \sqrt {a+a \sec (c+d x)}}+\frac {\text {Subst}\left (\int \frac {24 a^8-\frac {63 a^8 x}{16}}{x (-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{24 a^6 d} \\ & = \frac {43 a^2}{96 d (a+a \sec (c+d x))^{3/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{3/2}}-\frac {15 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{3/2}}-\frac {21 a}{64 d \sqrt {a+a \sec (c+d x)}}-\frac {a \text {Subst}\left (\int \frac {1}{x \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{d}+\frac {\left (107 a^2\right ) \text {Subst}\left (\int \frac {1}{(-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{128 d} \\ & = \frac {43 a^2}{96 d (a+a \sec (c+d x))^{3/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{3/2}}-\frac {15 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{3/2}}-\frac {21 a}{64 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 )}{d}+\frac {(107 a) \text {Subst}\left (\int \frac {1}{-2 a+x^2} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{64 d} \\ & = \frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{d}-\frac {107 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{64 \sqrt {2} d}+\frac {43 a^2}{96 d (a+a \sec (c+d x))^{3/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{3/2}}-\frac {15 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{3/2}}-\frac {21 a}{64 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 = 0.59 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.53 \[ \int \cot ^5(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\frac {\cot ^4(c+d x) \left (-2 \left (57+32 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},1+\sec (c+d x)\right ) (-1+\sec (c+d x))^2-45 \sec (c+d x)\right )+107 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {1}{2} (1+\sec (c+d x))\right ) (-1+\sec (c+d x))^2\right ) \sqrt {a (1+\sec (c+d x))}}{96 d} \]

[In]

Integrate[Cot[c + d*x]^5*Sqrt[a + a*Sec[c + d*x]],x]

[Out]

(Cot[c + d*x]^4*(-2*(57 + 32*Hypergeometric2F1[-3/2, 1, -1/2, 1 + Sec[c + d*x]]*(-1 + Sec[c + d*x])^2 - 45*Sec
[c + d*x]) + 107*Hypergeometric2F1[-3/2, 1, -1/2, (1 + Sec[c + d*x])/2]*(-1 + Sec[c + d*x])^2)*Sqrt[a*(1 + Sec
[c + d*x])])/(96*d)

Maple [A] (verified)

Time = 3.04 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.90

method	result	size
default	\(-\frac {\sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (321 \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+768 \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+410 \cot \left (d x +c \right )^{4}-142 \cot \left (d x +c \right )^{3} \csc \left (d x +c \right )-298 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )^{2}+126 \csc \left (d x +c \right )^{3} \cot \left (d x +c \right )\right )}{384 d}\)	\(173\)

[In]

int(cot(d*x+c)^5*(a+a*sec(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/384/d*(a*(1+sec(d*x+c)))^(1/2)*(321*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))+768*arctan((-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+4
10*cot(d*x+c)^4-142*cot(d*x+c)^3*csc(d*x+c)-298*cot(d*x+c)^2*csc(d*x+c)^2+126*csc(d*x+c)^3*cot(d*x+c))

Fricas [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 529, normalized size of antiderivative = 2.74 \[ \int \cot ^5(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\left [\frac {384 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 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 ) + 321 \, {\left (\sqrt {2} \cos \left (d x + c\right )^{4} - 2 \, \sqrt {2} \cos \left (d x + c\right )^{2} + \sqrt {2}\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 ) - 4 \, {\left (205 \, \cos \left (d x + c\right )^{4} - 71 \, \cos \left (d x + c\right )^{3} - 149 \, \cos \left (d x + c\right )^{2} + 63 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{768 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}}, \frac {321 \, {\left (\sqrt {2} \cos \left (d x + c\right )^{4} - 2 \, \sqrt {2} \cos \left (d x + c\right )^{2} + \sqrt {2}\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 ) - 384 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 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 (205 \, \cos \left (d x + c\right )^{4} - 71 \, \cos \left (d x + c\right )^{3} - 149 \, \cos \left (d x + c\right )^{2} + 63 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{384 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}}\right ] \]

[In]

integrate(cot(d*x+c)^5*(a+a*sec(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

[1/768*(384*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*sqrt(a)*log(-8*a*cos(d*x + c)^2 - 4*(2*cos(d*x + c)^2 + co
s(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)) - 8*a*cos(d*x + c) - a) + 321*(sqrt(2)*cos(d*x + c
)^4 - 2*sqrt(2)*cos(d*x + c)^2 + sqrt(2))*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)) - 4*(205*cos(d*x + c)^4 - 71*cos(d*x + c)^3 - 149
*cos(d*x + c)^2 + 63*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^4 - 2*d*cos(d*x +
c)^2 + d), 1/384*(321*(sqrt(2)*cos(d*x + c)^4 - 2*sqrt(2)*cos(d*x + c)^2 + sqrt(2))*sqrt(-a)*arctan(sqrt(2)*sq
rt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)) - 384*(cos(d*x + c)^4 - 2*co
s(d*x + c)^2 + 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*(205*cos(d*x + c)^4 - 71*cos(d*x + c)^3 - 149*cos(d*x + c)^2 + 63*cos(d*x + c))*sqrt((a*cos(d*
x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-1)]

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

[In]

integrate(cot(d*x+c)^5*(a+a*sec(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

Timed out

Giac [F]

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

[In]

integrate(cot(d*x+c)^5*(a+a*sec(d*x+c))^(1/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

int(cot(c + d*x)^5*(a + a/cos(c + d*x))^(1/2),x)

[Out]

int(cot(c + d*x)^5*(a + a/cos(c + d*x))^(1/2), x)