\(\int \cot ^3(c+d x) \sqrt {a+a \sec (c+d x)} \, dx\) [139]

Optimal result

Integrand size = 23, antiderivative size = 131 \[ \int \cot ^3(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {7 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{4 \sqrt {2} d}+\frac {a}{4 d \sqrt {a+a \sec (c+d x)}}+\frac {a}{2 d (1-\sec (c+d x)) \sqrt {a+a \sec (c+d x)}} \] Output:

-2*a^(1/2)*arctanh((a+a*sec(d*x+c))^(1/2)/a^(1/2))/d+7/8*2^(1/2)*a^(1/2)*a 
rctanh(1/2*(a+a*sec(d*x+c))^(1/2)*2^(1/2)/a^(1/2))/d+1/4*a/d/(a+a*sec(d*x+ 
c))^(1/2)+1/2*a/d/(1-sec(d*x+c))/(a+a*sec(d*x+c))^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.19 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.66 \[ \int \cot ^3(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\frac {\cot ^2(c+d x) \left (-2-7 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {1}{2} (1+\sec (c+d x))\right ) (-1+\sec (c+d x))+8 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1+\sec (c+d x)\right ) (-1+\sec (c+d x))\right ) \sqrt {a (1+\sec (c+d x))}}{4 d} \] Input:

Integrate[Cot[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]],x]
 

Output:

(Cot[c + d*x]^2*(-2 - 7*Hypergeometric2F1[-1/2, 1, 1/2, (1 + Sec[c + d*x]) 
/2]*(-1 + Sec[c + d*x]) + 8*Hypergeometric2F1[-1/2, 1, 1/2, 1 + Sec[c + d* 
x]]*(-1 + Sec[c + d*x]))*Sqrt[a*(1 + Sec[c + d*x])])/(4*d)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.05, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {3042, 25, 4368, 27, 114, 27, 169, 27, 174, 73, 219, 220}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \cot ^3(c+d x) \sqrt {a \sec (c+d x)+a} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -\frac {\sqrt {a \csc \left (c+d x+\frac {\pi }{2}\right )+a}}{\cot \left (c+d x+\frac {\pi }{2}\right )^3}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \frac {\sqrt {\csc \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a}}{\cot \left (\frac {1}{2} (2 c+\pi )+d x\right )^3}dx\)

\(\Big \downarrow \) 4368

\(\displaystyle \frac {a^4 \int \frac {\cos (c+d x)}{a^2 (1-\sec (c+d x))^2 (\sec (c+d x) a+a)^{3/2}}d\sec (c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a^2 \int \frac {\cos (c+d x)}{(1-\sec (c+d x))^2 (\sec (c+d x) a+a)^{3/2}}d\sec (c+d x)}{d}\)

\(\Big \downarrow \) 114

\(\displaystyle \frac {a^2 \left (\frac {1}{2 a (1-\sec (c+d x)) \sqrt {a \sec (c+d x)+a}}-\frac {\int -\frac {a \cos (c+d x) (3 \sec (c+d x)+4)}{2 (1-\sec (c+d x)) (\sec (c+d x) a+a)^{3/2}}d\sec (c+d x)}{2 a}\right )}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a^2 \left (\frac {1}{4} \int \frac {\cos (c+d x) (3 \sec (c+d x)+4)}{(1-\sec (c+d x)) (\sec (c+d x) a+a)^{3/2}}d\sec (c+d x)+\frac {1}{2 a (1-\sec (c+d x)) \sqrt {a \sec (c+d x)+a}}\right )}{d}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {a^2 \left (\frac {1}{4} \left (\frac {\int \frac {a \cos (c+d x) (8-\sec (c+d x))}{2 (1-\sec (c+d x)) \sqrt {\sec (c+d x) a+a}}d\sec (c+d x)}{a^2}+\frac {1}{a \sqrt {a \sec (c+d x)+a}}\right )+\frac {1}{2 a (1-\sec (c+d x)) \sqrt {a \sec (c+d x)+a}}\right )}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a^2 \left (\frac {1}{4} \left (\frac {\int \frac {\cos (c+d x) (8-\sec (c+d x))}{(1-\sec (c+d x)) \sqrt {\sec (c+d x) a+a}}d\sec (c+d x)}{2 a}+\frac {1}{a \sqrt {a \sec (c+d x)+a}}\right )+\frac {1}{2 a (1-\sec (c+d x)) \sqrt {a \sec (c+d x)+a}}\right )}{d}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {a^2 \left (\frac {1}{4} \left (\frac {7 \int \frac {1}{(1-\sec (c+d x)) \sqrt {\sec (c+d x) a+a}}d\sec (c+d x)+8 \int \frac {\cos (c+d x)}{\sqrt {\sec (c+d x) a+a}}d\sec (c+d x)}{2 a}+\frac {1}{a \sqrt {a \sec (c+d x)+a}}\right )+\frac {1}{2 a (1-\sec (c+d x)) \sqrt {a \sec (c+d x)+a}}\right )}{d}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {a^2 \left (\frac {1}{4} \left (\frac {\frac {14 \int \frac {1}{2-\frac {\sec (c+d x) a+a}{a}}d\sqrt {\sec (c+d x) a+a}}{a}+\frac {16 \int \frac {1}{\frac {\sec (c+d x) a+a}{a}-1}d\sqrt {\sec (c+d x) a+a}}{a}}{2 a}+\frac {1}{a \sqrt {a \sec (c+d x)+a}}\right )+\frac {1}{2 a (1-\sec (c+d x)) \sqrt {a \sec (c+d x)+a}}\right )}{d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {a^2 \left (\frac {1}{4} \left (\frac {\frac {16 \int \frac {1}{\frac {\sec (c+d x) a+a}{a}-1}d\sqrt {\sec (c+d x) a+a}}{a}+\frac {7 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {a}}}{2 a}+\frac {1}{a \sqrt {a \sec (c+d x)+a}}\right )+\frac {1}{2 a (1-\sec (c+d x)) \sqrt {a \sec (c+d x)+a}}\right )}{d}\)

\(\Big \downarrow \) 220

\(\displaystyle \frac {a^2 \left (\frac {1}{4} \left (\frac {\frac {7 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {a}}-\frac {16 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{\sqrt {a}}}{2 a}+\frac {1}{a \sqrt {a \sec (c+d x)+a}}\right )+\frac {1}{2 a (1-\sec (c+d x)) \sqrt {a \sec (c+d x)+a}}\right )}{d}\)

Input:

Int[Cot[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]],x]
 

Output:

(a^2*(1/(2*a*(1 - Sec[c + d*x])*Sqrt[a + a*Sec[c + d*x]]) + (((-16*ArcTanh 
[Sqrt[a + a*Sec[c + d*x]]/Sqrt[a]])/Sqrt[a] + (7*Sqrt[2]*ArcTanh[Sqrt[a + 
a*Sec[c + d*x]]/(Sqrt[2]*Sqrt[a])])/Sqrt[a])/(2*a) + 1/(a*Sqrt[a + a*Sec[c 
 + d*x]]))/4))/d
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> 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] + Simp[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 169

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> 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] + S 
imp[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 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(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 219

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

rule 220

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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4368

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n 
_), x_Symbol] :> Simp[-(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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(294\) vs. \(2(106)=212\).

Time = 0.69 (sec) , antiderivative size = 295, normalized size of antiderivative = 2.25

method	result	size
default	\(-\frac {\sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (\left (34 \cos \left (d x +c \right )^{3}-50 \cos \left (d x +c \right )^{2}-154 \cos \left (d x +c \right )-70\right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cot \left (d x +c \right ) \csc \left (d x +c \right )+\left (-120 \cos \left (d x +c \right )^{2}-240 \cos \left (d x +c \right )-120\right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{2}\right )+\left (-105 \cos \left (d x +c \right )^{2}-210 \cos \left (d x +c \right )-105\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )+\left (158 \cos \left (d x +c \right )^{3}-18 \cos \left (d x +c \right )^{2}-110 \cos \left (d x +c \right )-30\right ) \cot \left (d x +c \right ) \csc \left (d x +c \right )\right )}{120 d \left (1+\cos \left (d x +c \right )\right )^{2}}\)	\(295\)

Input:

int(cot(d*x+c)^3*(a+a*sec(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

-1/120/d*(a*(1+sec(d*x+c)))^(1/2)/(1+cos(d*x+c))^2*((34*cos(d*x+c)^3-50*co 
s(d*x+c)^2-154*cos(d*x+c)-70)*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2) 
*(-cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cot(d*x+c)*csc(d*x+c)+(-120*cos(d*x+c) 
^2-240*cos(d*x+c)-120)*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan 
(1/2*2^(1/2)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+(-105*cos(d*x+c)^2-210* 
cos(d*x+c)-105)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*arctan(1/2*2^(1/2)/(- 
cos(d*x+c)/(1+cos(d*x+c)))^(1/2))+(158*cos(d*x+c)^3-18*cos(d*x+c)^2-110*co 
s(d*x+c)-30)*cot(d*x+c)*csc(d*x+c))
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 417, normalized size of antiderivative = 3.18 \[ \int \cot ^3(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\left [\frac {7 \, \sqrt {\frac {1}{2}} {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sqrt {a} \log \left (\frac {4 \, \sqrt {\frac {1}{2}} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) + 3 \, a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right ) - 1}\right ) + 4 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sqrt {a} \log \left (-8 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} - 8 \, a \cos \left (d x + c\right ) - a\right ) + 2 \, {\left (3 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{8 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}}, -\frac {7 \, \sqrt {\frac {1}{2}} {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{a \cos \left (d x + c\right ) + a}\right ) - 4 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, a \cos \left (d x + c\right ) + a}\right ) - {\left (3 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{4 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}}\right ] \] Input:

integrate(cot(d*x+c)^3*(a+a*sec(d*x+c))^(1/2),x, algorithm="fricas")
 

Output:

[1/8*(7*sqrt(1/2)*(cos(d*x + c)^2 - 1)*sqrt(a)*log((4*sqrt(1/2)*sqrt(a)*sq 
rt((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*(cos(d*x + c)^2 - 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) + 2*(3*cos(d*x + c)^2 - cos(d*x + 
c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^2 - d), -1/4* 
(7*sqrt(1/2)*(cos(d*x + c)^2 - 1)*sqrt(-a)*arctan(2*sqrt(1/2)*sqrt(-a)*sqr 
t((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(a*cos(d*x + c) + a)) - 
4*(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)/(2*a*cos(d*x + c) + a)) - (3*cos(d*x + c)^2 - 
 cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c)))/(d*cos(d*x + c)^2 
- d)]
 

Sympy [F]

\[ \int \cot ^3(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\int \sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )} \cot ^{3}{\left (c + d x \right )}\, dx \] Input:

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

Output:

Integral(sqrt(a*(sec(c + d*x) + 1))*cot(c + d*x)**3, x)
 

Maxima [F]

\[ \int \cot ^3(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\int { \sqrt {a \sec \left (d x + c\right ) + a} \cot \left (d x + c\right )^{3} \,d x } \] Input:

integrate(cot(d*x+c)^3*(a+a*sec(d*x+c))^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(a*sec(d*x + c) + a)*cot(d*x + c)^3, x)
 

Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.07 \[ \int \cot ^3(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\frac {\sqrt {2} {\left (\frac {8 \, \sqrt {2} a \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a}} - \frac {7 \, a \arctan \left (\frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + 2 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} - \frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}\right )} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}{8 \, d} \] Input:

integrate(cot(d*x+c)^3*(a+a*sec(d*x+c))^(1/2),x, algorithm="giac")
 

Output:

1/8*sqrt(2)*(8*sqrt(2)*a*arctan(1/2*sqrt(2)*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 
 + a)/sqrt(-a))/sqrt(-a) - 7*a*arctan(sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)/ 
sqrt(-a))/sqrt(-a) + 2*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a) - sqrt(-a*tan(1 
/2*d*x + 1/2*c)^2 + a)/tan(1/2*d*x + 1/2*c)^2)*sgn(cos(d*x + c))/d
 

Mupad [F(-1)]

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

int(cot(c + d*x)^3*(a + a/cos(c + d*x))^(1/2),x)
 

Output:

int(cot(c + d*x)^3*(a + a/cos(c + d*x))^(1/2), x)
 

Reduce [F]

\[ \int \cot ^3(c+d x) \sqrt {a+a \sec (c+d x)} \, dx=\sqrt {a}\, \left (\int \sqrt {\sec \left (d x +c \right )+1}\, \cot \left (d x +c \right )^{3}d x \right ) \] Input:

int(cot(d*x+c)^3*(a+a*sec(d*x+c))^(1/2),x)
 

Output:

sqrt(a)*int(sqrt(sec(c + d*x) + 1)*cot(c + d*x)**3,x)