\(\int \cot ^4(c+d x) \sqrt {a+a \sec (c+d x)} \, dx\) [145]

Optimal result

Integrand size = 23, antiderivative size = 196 \[ \int \cot ^4(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}-\frac {9 \sqrt {a} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{8 \sqrt {2} d}+\frac {7 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{8 d}+\frac {\cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{12 a d}-\frac {\cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{4 a d} \]

[Out]

1/12*cot(d*x+c)^3*(a+a*sec(d*x+c))^(3/2)/a/d-1/4*cos(d*x+c)*cot(d*x+c)^3*sec(1/2*d*x+1/2*c)^2*(a+a*sec(d*x+c))
^(3/2)/a/d+2*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))*a^(1/2)/d-9/16*arctan(1/2*a^(1/2)*tan(d*x+c)*2^
(1/2)/(a+a*sec(d*x+c))^(1/2))*a^(1/2)/d*2^(1/2)+7/8*cot(d*x+c)*(a+a*sec(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 196, 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, 483, 597, 536, 209} \[ \int \cot ^4(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}-\frac {9 \sqrt {a} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{8 \sqrt {2} d}+\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{12 a d}+\frac {7 \cot (c+d x) \sqrt {a \sec (c+d x)+a}}{8 d}-\frac {\cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{4 a d} \]

[In]

Int[Cot[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]],x]

[Out]

(2*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d - (9*Sqrt[a]*ArcTan[(Sqrt[a]*Tan[c + d*x
])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(8*Sqrt[2]*d) + (7*Cot[c + d*x]*Sqrt[a + a*Sec[c + d*x]])/(8*d) + (Cot
[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2))/(12*a*d) - (Cos[c + d*x]*Cot[c + d*x]^3*Sec[(c + d*x)/2]^2*(a + a*Sec[
c + d*x])^(3/2))/(4*a*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 483

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*(e*
x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*e*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a*d)
*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*(b*c - a*d)*(p + 1) + d*b*(m + n
*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ
[p, -1] && IntBinomialQ[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^4 \left (1+a x^2\right ) \left (2+a x^2\right )^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a d} \\ & = -\frac {\cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{4 a d}-\frac {\text {Subst}\left (\int \frac {-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 )}{2 a^2 d} \\ & = \frac {\cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{12 a d}-\frac {\cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{4 a d}+\frac {\text {Subst}\left (\int \frac {21 a^2-3 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 )}{12 a^2 d} \\ & = \frac {7 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{8 d}+\frac {\cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{12 a d}-\frac {\cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{4 a d}-\frac {\text {Subst}\left (\int \frac {69 a^3+21 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 )}{24 a^2 d} \\ & = \frac {7 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{8 d}+\frac {\cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{12 a d}-\frac {\cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{4 a d}+\frac {(9 a) \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 )}{8 d}-\frac {(2 a) \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 \sqrt {a} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}-\frac {9 \sqrt {a} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{8 \sqrt {2} d}+\frac {7 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{8 d}+\frac {\cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{12 a d}-\frac {\cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{4 a d} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 7.21 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.18 \[ \int \cot ^4(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\frac {\left (-9 \arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )+16 \sqrt {2} \arctan \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}}}\right )\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {1+\sec (c+d x)} \sqrt {a (1+\sec (c+d x))}}{16 d \sqrt {\sec (c+d x)}}+\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\sec (c+d x))} \left (\frac {2}{3} \csc \left (\frac {1}{2} (c+d x)\right )-\frac {1}{24} \csc ^3\left (\frac {1}{2} (c+d x)\right )-\frac {31}{24} \sin \left (\frac {1}{2} (c+d x)\right )+\frac {1}{16} \sec \left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{d} \]

[In]

Integrate[Cot[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]],x]

[Out]

((-9*ArcSin[Tan[(c + d*x)/2]] + 16*Sqrt[2]*ArcTan[Tan[(c + d*x)/2]/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]])*Sqr
t[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[1 + Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])
/(16*d*Sqrt[Sec[c + d*x]]) + (Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*((2*Csc[(c + d*x)/2])/3 - Csc[(c + d
*x)/2]^3/24 - (31*Sin[(c + d*x)/2])/24 + (Sec[(c + d*x)/2]*Tan[(c + d*x)/2])/16))/d

Maple [A] (verified)

Time = 2.78 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00

method	result	size
default	\(\frac {\sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (-27 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )+\sqrt {\cot \left (d x +c \right )^{2}-2 \cot \left (d x +c \right ) \csc \left (d x +c \right )+\csc \left (d x +c \right )^{2}-1}\right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+96 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )-62 \cot \left (d x +c \right )^{3}+4 \cot \left (d x +c \right )^{2} \csc \left (d x +c \right )+42 \cot \left (d x +c \right ) \csc \left (d x +c \right )^{2}\right )}{48 d}\)	\(196\)

[In]

int(cot(d*x+c)^4*(a+a*sec(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/48/d*(a*(1+sec(d*x+c)))^(1/2)*(-27*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))*2^(1/2)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+96*(-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))-62*cot(d*x+c)^3+4*cot(d*x+c)^2*csc(d*x+c)+42*cot(d*x+c)*
csc(d*x+c)^2)

Fricas [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 547, normalized size of antiderivative = 2.79 \[ \int \cot ^4(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\left [\frac {27 \, {\left (\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 ) \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 ) + 48 \, {\left (\cos \left (d x + c\right )^{2} - 1\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 (31 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} - 21 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{96 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )}, \frac {48 \, {\left (\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 ) \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 ) + 27 \, {\left (\sqrt {2} \cos \left (d x + c\right )^{2} - \sqrt {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 (31 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} - 21 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{48 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )}\right ] \]

[In]

integrate(cot(d*x+c)^4*(a+a*sec(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

[1/96*(27*(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)*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) + 48*(cos(d*x + c)^2 - 1)*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)/cos(d*x + c))*sin(d*x + c) - 7*a*cos(d*x + c) + a)/(cos(d*x + c) +
1))*sin(d*x + c) + 4*(31*cos(d*x + c)^3 - 2*cos(d*x + c)^2 - 21*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d
*x + c)))/((d*cos(d*x + c)^2 - d)*sin(d*x + c)), 1/48*(48*(cos(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)*sin(d*x + c)/(2*a*cos(d*x + c)^2 + a*cos(d*x + c) - a))*sin(d*x
+ c) + 27*(sqrt(2)*cos(d*x + c)^2 - sqrt(2))*sqrt(a)*arctan(sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*c
os(d*x + c)/(sqrt(a)*sin(d*x + c)))*sin(d*x + c) + 2*(31*cos(d*x + c)^3 - 2*cos(d*x + c)^2 - 21*cos(d*x + c))*
sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/((d*cos(d*x + c)^2 - d)*sin(d*x + c))]

Sympy [F]

\[ \int \cot ^4(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\int \sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )} \cot ^{4}{\left (c + d x \right )}\, dx \]

[In]

integrate(cot(d*x+c)**4*(a+a*sec(d*x+c))**(1/2),x)

[Out]

Integral(sqrt(a*(sec(c + d*x) + 1))*cot(c + d*x)**4, x)

Maxima [F]

\[ \int \cot ^4(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\int { \sqrt {a \sec \left (d x + c\right ) + a} \cot \left (d x + c\right )^{4} \,d x } \]

[In]

integrate(cot(d*x+c)^4*(a+a*sec(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a*sec(d*x + c) + a)*cot(d*x + c)^4, x)

Giac [F]

\[ \int \cot ^4(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\int { \sqrt {a \sec \left (d x + c\right ) + a} \cot \left (d x + c\right )^{4} \,d x } \]

[In]

integrate(cot(d*x+c)^4*(a+a*sec(d*x+c))^(1/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^4\,\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}} \,d x \]

[In]

int(cot(c + d*x)^4*(a + a/cos(c + d*x))^(1/2),x)

[Out]

int(cot(c + d*x)^4*(a + a/cos(c + d*x))^(1/2), x)