\(\int \cot ^6(c+d x) (a+a \sec (c+d x))^{5/2} \, dx\) [170]

Optimal result

Integrand size = 23, antiderivative size = 176 \[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=-\frac {2 a^{5/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}+\frac {a^{5/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{4 \sqrt {2} d}-\frac {7 a^2 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{4 d}+\frac {a \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{2 d}-\frac {\cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{5 d} \]

[Out]

-2*a^(5/2)*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/d+1/2*a*cot(d*x+c)^3*(a+a*sec(d*x+c))^(3/2)/d-1/5
*cot(d*x+c)^5*(a+a*sec(d*x+c))^(5/2)/d+1/8*a^(5/2)*arctan(1/2*a^(1/2)*tan(d*x+c)*2^(1/2)/(a+a*sec(d*x+c))^(1/2
))/d*2^(1/2)-7/4*a^2*cot(d*x+c)*(a+a*sec(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3972, 491, 597, 536, 209} \[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=-\frac {2 a^{5/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}+\frac {a^{5/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{4 \sqrt {2} d}-\frac {7 a^2 \cot (c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}-\frac {\cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d}+\frac {a \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{2 d} \]

[In]

Int[Cot[c + d*x]^6*(a + a*Sec[c + d*x])^(5/2),x]

[Out]

(-2*a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a^(5/2)*ArcTan[(Sqrt[a]*Tan[c + d*x]
)/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(4*Sqrt[2]*d) - (7*a^2*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(4*d) + (
a*Cot[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(2*d) - (Cot[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2))/(5*d)

Rule 209

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

Rule 491

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*e*(m + 1))), x] - Dist[1/(a*c*e^n*(m + 1)), Int[(e*x)^(m +
n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[(b*c + a*d)*(m + n + 1) + n*(b*c*p + a*d*q) + b*d*(m + n*(p + q + 2) + 1)*
x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntBino
mialQ[a, b, c, d, e, m, n, p, q, x]

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 597

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

Rule 3972

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

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \text {Subst}\left (\int \frac {1}{x^6 \left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d} \\ & = -\frac {\cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{5 d}-\frac {\text {Subst}\left (\int \frac {-15 a-5 a^2 x^2}{x^4 \left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{5 d} \\ & = \frac {a \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{2 d}-\frac {\cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{5 d}+\frac {\text {Subst}\left (\int \frac {-105 a^2-45 a^3 x^2}{x^2 \left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{30 d} \\ & = -\frac {7 a^2 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{4 d}+\frac {a \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{2 d}-\frac {\cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{5 d}-\frac {\text {Subst}\left (\int \frac {-225 a^3-105 a^4 x^2}{\left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{60 d} \\ & = -\frac {7 a^2 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{4 d}+\frac {a \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{2 d}-\frac {\cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{5 d}-\frac {a^3 \text {Subst}\left (\int \frac {1}{2+a x^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{4 d}+\frac {\left (2 a^3\right ) \text {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d} \\ & = -\frac {2 a^{5/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}+\frac {a^{5/2} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{4 \sqrt {2} d}-\frac {7 a^2 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{4 d}+\frac {a \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{2 d}-\frac {\cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{5 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.40 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.15 \[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\frac {\sec ^5\left (\frac {1}{2} (c+d x)\right ) (a (1+\sec (c+d x)))^{5/2} \left (\frac {1}{40} \cos ^3(c+d x) (160 \cos (c+d x)-7 (17+7 \cos (2 (c+d x)))) \csc ^5\left (\frac {1}{2} (c+d x)\right )+\frac {\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )-8 \sqrt {2} \arctan \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}}}\right )\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sec \left (\frac {1}{2} (c+d x)\right ) \sqrt {1+\sec (c+d x)}}{\sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )} \sec ^{\frac {5}{2}}(c+d x)}\right )}{32 d} \]

[In]

Integrate[Cot[c + d*x]^6*(a + a*Sec[c + d*x])^(5/2),x]

[Out]

(Sec[(c + d*x)/2]^5*(a*(1 + Sec[c + d*x]))^(5/2)*((Cos[c + d*x]^3*(160*Cos[c + d*x] - 7*(17 + 7*Cos[2*(c + d*x
)]))*Csc[(c + d*x)/2]^5)/40 + ((ArcSin[Tan[(c + d*x)/2]] - 8*Sqrt[2]*ArcTan[Tan[(c + d*x)/2]/Sqrt[Cos[c + d*x]
/(1 + Cos[c + d*x])]])*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[1 + Sec[c + d*x]])/(Sqrt[Se
c[(c + d*x)/2]^2]*Sec[c + d*x]^(5/2))))/(32*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(371\) vs. \(2(147)=294\).

Time = 194.44 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.11

method	result	size
default	\(\frac {a^{2} \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (5 \sqrt {2}\, \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )+\sqrt {\cot \left (d x +c \right )^{2}-2 \cot \left (d x +c \right ) \csc \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )-80 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )-5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )+\sqrt {\cot \left (d x +c \right )^{2}-2 \cot \left (d x +c \right ) \csc \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}\right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+80 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )+98 \cos \left (d x +c \right ) \cot \left (d x +c \right )^{3}-62 \cot \left (d x +c \right )^{3}-90 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )+70 \cot \left (d x +c \right ) \csc \left (d x +c \right )^{2}\right )}{40 d \left (\cos \left (d x +c \right )-1\right )}\)	\(372\)

[In]

int(cot(d*x+c)^6*(a+a*sec(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/40/d*a^2*(a*(1+sec(d*x+c)))^(1/2)/(cos(d*x+c)-1)*(5*2^(1/2)*ln(csc(d*x+c)-cot(d*x+c)+(cot(d*x+c)^2-2*cot(d*x
+c)*csc(d*x+c)+csc(d*x+c)^2-1)^(1/2))*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)-80*(-cos(d*x+c)/(cos(d*x+c
)+1))^(1/2)*arctanh(sin(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)-5*ln(csc(d*x+c)-c
ot(d*x+c)+(cot(d*x+c)^2-2*cot(d*x+c)*csc(d*x+c)+csc(d*x+c)^2-1)^(1/2))*2^(1/2)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1
/2)+80*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(sin(d*x+c)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)
)+98*cos(d*x+c)*cot(d*x+c)^3-62*cot(d*x+c)^3-90*cot(d*x+c)^2*csc(d*x+c)+70*cot(d*x+c)*csc(d*x+c)^2)

Fricas [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 650, normalized size of antiderivative = 3.69 \[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\left [\frac {5 \, {\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 ) \sin \left (d x + c\right ) - 3 \, a \cos \left (d x + c\right )^{2} - 2 \, a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) \sin \left (d x + c\right ) + 40 \, {\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 (-\frac {8 \, a \cos \left (d x + c\right )^{3} + 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 )}} \sin \left (d x + c\right ) - 7 \, a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right ) + 1}\right ) \sin \left (d x + c\right ) - 4 \, {\left (49 \, a^{2} \cos \left (d x + c\right )^{3} - 80 \, a^{2} \cos \left (d x + c\right )^{2} + 35 \, 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 )}}}{80 \, {\left (d \cos \left (d x + c\right )^{2} - 2 \, d \cos \left (d x + c\right ) + d\right )} \sin \left (d x + c\right )}, -\frac {40 \, {\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 {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 ) \sin \left (d x + c\right )}{2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - a}\right ) \sin \left (d x + c\right ) + 5 \, {\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 {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) + 2 \, {\left (49 \, a^{2} \cos \left (d x + c\right )^{3} - 80 \, a^{2} \cos \left (d x + c\right )^{2} + 35 \, 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 )}}}{40 \, {\left (d \cos \left (d x + c\right )^{2} - 2 \, d \cos \left (d x + c\right ) + d\right )} \sin \left (d x + c\right )}\right ] \]

[In]

integrate(cot(d*x+c)^6*(a+a*sec(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

[1/80*(5*(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)*sin(d*x + c) - 3*a*cos(d*x + c)^2 - 2*a*cos(d*x + c)
+ a)/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1))*sin(d*x + c) + 40*(a^2*cos(d*x + c)^2 - 2*a^2*cos(d*x + c) + a^2)
*sqrt(-a)*log(-(8*a*cos(d*x + c)^3 + 4*(2*cos(d*x + c)^2 - cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/co
s(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c) + a)/(cos(d*x + c) + 1))*sin(d*x + c) - 4*(49*a^2*cos(d*x + c)^3 -
80*a^2*cos(d*x + c)^2 + 35*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)*sin(d*x + c)), -1/40*(40*(a^2*cos(d*x + c)^2 - 2*a^2*cos(d*x + c) + a^2)*sqrt(a)*arctan(2*
sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c)/(2*a*cos(d*x + c)^2 + a*cos(d*x + c)
- a))*sin(d*x + c) + 5*(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*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))*sin(d*x + c) + 2*(49*a^2
*cos(d*x + c)^3 - 80*a^2*cos(d*x + c)^2 + 35*a^2*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((d*co
s(d*x + c)^2 - 2*d*cos(d*x + c) + d)*sin(d*x + c))]

Sympy [F(-1)]

Timed out. \[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\text {Timed out} \]

[In]

integrate(cot(d*x+c)**6*(a+a*sec(d*x+c))**(5/2),x)

[Out]

Timed out

Maxima [F(-1)]

Timed out. \[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\text {Timed out} \]

[In]

integrate(cot(d*x+c)^6*(a+a*sec(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

Timed out

Giac [F]

\[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cot \left (d x + c\right )^{6} \,d x } \]

[In]

integrate(cot(d*x+c)^6*(a+a*sec(d*x+c))^(5/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \cot ^6(c+d x) (a+a \sec (c+d x))^{5/2} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^6\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \]

[In]

int(cot(c + d*x)^6*(a + a/cos(c + d*x))^(5/2),x)

[Out]

int(cot(c + d*x)^6*(a + a/cos(c + d*x))^(5/2), x)