\(\int \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2} \, dx\) [164]

Optimal result

Integrand size = 23, antiderivative size = 147 \[ \int \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\frac {2 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{d}-\frac {43 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{16 \sqrt {2} d}-\frac {a^2 \sqrt {a+a \sec (c+d x)}}{4 d (1-\sec (c+d x))^2}-\frac {11 a^2 \sqrt {a+a \sec (c+d x)}}{16 d (1-\sec (c+d x))} \]

[Out]

2*a^(5/2)*arctanh((a+a*sec(d*x+c))^(1/2)/a^(1/2))/d-43/32*a^(5/2)*arctanh(1/2*(a+a*sec(d*x+c))^(1/2)*2^(1/2)/a
^(1/2))*2^(1/2)/d-1/4*a^2*(a+a*sec(d*x+c))^(1/2)/d/(1-sec(d*x+c))^2-11/16*a^2*(a+a*sec(d*x+c))^(1/2)/d/(1-sec(
d*x+c))

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3965, 105, 156, 162, 65, 213} \[ \int \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\frac {2 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{d}-\frac {43 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{16 \sqrt {2} d}-\frac {11 a^2 \sqrt {a \sec (c+d x)+a}}{16 d (1-\sec (c+d x))}-\frac {a^2 \sqrt {a \sec (c+d x)+a}}{4 d (1-\sec (c+d x))^2} \]

[In]

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

[Out]

(2*a^(5/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/d - (43*a^(5/2)*ArcTanh[Sqrt[a + a*Sec[c + d*x]]/(Sqrt[2
]*Sqrt[a])])/(16*Sqrt[2]*d) - (a^2*Sqrt[a + a*Sec[c + d*x]])/(4*d*(1 - Sec[c + d*x])^2) - (11*a^2*Sqrt[a + a*S
ec[c + d*x]])/(16*d*(1 - 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 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 \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{d} \\ & = -\frac {a^2 \sqrt {a+a \sec (c+d x)}}{4 d (1-\sec (c+d x))^2}-\frac {a^3 \text {Subst}\left (\int \frac {4 a^2+\frac {3 a^2 x}{2}}{x (-a+a x)^2 \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{4 d} \\ & = -\frac {a^2 \sqrt {a+a \sec (c+d x)}}{4 d (1-\sec (c+d x))^2}-\frac {11 a^2 \sqrt {a+a \sec (c+d x)}}{16 d (1-\sec (c+d x))}+\frac {\text {Subst}\left (\int \frac {8 a^4+\frac {11 a^4 x}{4}}{x (-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{8 d} \\ & = -\frac {a^2 \sqrt {a+a \sec (c+d x)}}{4 d (1-\sec (c+d x))^2}-\frac {11 a^2 \sqrt {a+a \sec (c+d x)}}{16 d (1-\sec (c+d x))}-\frac {a^3 \text {Subst}\left (\int \frac {1}{x \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{d}+\frac {\left (43 a^4\right ) \text {Subst}\left (\int \frac {1}{(-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{32 d} \\ & = -\frac {a^2 \sqrt {a+a \sec (c+d x)}}{4 d (1-\sec (c+d x))^2}-\frac {11 a^2 \sqrt {a+a \sec (c+d x)}}{16 d (1-\sec (c+d x))}-\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{-1+\frac {x^2}{a}} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{d}+\frac {\left (43 a^3\right ) \text {Subst}\left (\int \frac {1}{-2 a+x^2} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{16 d} \\ & = \frac {2 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{d}-\frac {43 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{16 \sqrt {2} d}-\frac {a^2 \sqrt {a+a \sec (c+d x)}}{4 d (1-\sec (c+d x))^2}-\frac {11 a^2 \sqrt {a+a \sec (c+d x)}}{16 d (1-\sec (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.74 \[ \int \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\frac {(a (1+\sec (c+d x)))^{5/2} \left (64 \text {arctanh}\left (\sqrt {1+\sec (c+d x)}\right )-43 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\sec (c+d x)}}{\sqrt {2}}\right )+\frac {2 \sqrt {1+\sec (c+d x)} (-15+11 \sec (c+d x))}{(-1+\sec (c+d x))^2}\right )}{32 d (1+\sec (c+d x))^{5/2}} \]

[In]

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

[Out]

((a*(1 + Sec[c + d*x]))^(5/2)*(64*ArcTanh[Sqrt[1 + Sec[c + d*x]]] - 43*Sqrt[2]*ArcTanh[Sqrt[1 + Sec[c + d*x]]/
Sqrt[2]] + (2*Sqrt[1 + Sec[c + d*x]]*(-15 + 11*Sec[c + d*x]))/(-1 + Sec[c + d*x])^2))/(32*d*(1 + Sec[c + d*x])
^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(271\) vs. \(2(122)=244\).

Time = 88.52 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.85

method	result	size
default	\(\frac {a^{2} \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (-43 \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-64 \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \cos \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+43 \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+64 \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+30 \cos \left (d x +c \right ) \cot \left (d x +c \right )^{2}+8 \cot \left (d x +c \right )^{2}-22 \cot \left (d x +c \right ) \csc \left (d x +c \right )\right )}{32 d \left (\cos \left (d x +c \right )-1\right )}\)	\(272\)

[In]

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

[Out]

1/32/d*a^2*(a*(1+sec(d*x+c)))^(1/2)/(cos(d*x+c)-1)*(-43*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)-64*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)+43*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))+64*arctan((-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+3
0*cos(d*x+c)*cot(d*x+c)^2+8*cot(d*x+c)^2-22*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. 247 vs. \(2 (118) = 236\).

Time = 0.33 (sec) , antiderivative size = 503, normalized size of antiderivative = 3.42 \[ \int \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\left [\frac {64 \, {\left (a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \cos \left (d x + c\right ) + a^{2}\right )} \sqrt {a} \log \left (-2 \, a \cos \left (d x + c\right ) - 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\right ) + 43 \, {\left (\sqrt {2} a^{2} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} a^{2} \cos \left (d x + c\right ) + \sqrt {2} a^{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 (15 \, a^{2} \cos \left (d x + c\right )^{2} - 11 \, a^{2} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{64 \, {\left (d \cos \left (d x + c\right )^{2} - 2 \, d \cos \left (d x + c\right ) + d\right )}}, \frac {43 \, {\left (\sqrt {2} a^{2} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} a^{2} \cos \left (d x + c\right ) + \sqrt {2} a^{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 ) - 64 \, {\left (a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \cos \left (d x + c\right ) + a^{2}\right )} \sqrt {-a} \arctan \left (\frac {\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 ) - 2 \, {\left (15 \, a^{2} \cos \left (d x + c\right )^{2} - 11 \, a^{2} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{32 \, {\left (d \cos \left (d x + c\right )^{2} - 2 \, d \cos \left (d x + c\right ) + d\right )}}\right ] \]

[In]

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

[Out]

[1/64*(64*(a^2*cos(d*x + c)^2 - 2*a^2*cos(d*x + c) + a^2)*sqrt(a)*log(-2*a*cos(d*x + c) - 2*sqrt(a)*sqrt((a*co
s(d*x + c) + a)/cos(d*x + c))*cos(d*x + c) - a) + 43*(sqrt(2)*a^2*cos(d*x + c)^2 - 2*sqrt(2)*a^2*cos(d*x + c)
+ sqrt(2)*a^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*(15*a^2*cos(d*x + c)^2 - 11*a^2*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/
cos(d*x + c)))/(d*cos(d*x + c)^2 - 2*d*cos(d*x + c) + d), 1/32*(43*(sqrt(2)*a^2*cos(d*x + c)^2 - 2*sqrt(2)*a^2
*cos(d*x + c) + sqrt(2)*a^2)*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)) - 64*(a^2*cos(d*x + c)^2 - 2*a^2*cos(d*x + c) + a^2)*sqrt(-a)*arctan(sqrt(-a)*sqrt(
(a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)) - 2*(15*a^2*cos(d*x + c)^2 - 11*a^2*cos(
d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^2 - 2*d*cos(d*x + c) + d)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-1)]

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

[In]

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

[Out]

Timed out

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[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)