\(\int \frac {\cot ^3(e+f x)}{(a+b \sec ^2(e+f x))^{5/2}} \, dx\) [432]

Optimal result

Integrand size = 25, antiderivative size = 200 \[ \int \frac {\cot ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \sec ^2(e+f x)}}{\sqrt {a}}\right )}{a^{5/2} f}+\frac {(2 a+7 b) \text {arctanh}\left (\frac {\sqrt {a+b \sec ^2(e+f x)}}{\sqrt {a+b}}\right )}{2 (a+b)^{7/2} f}-\frac {(3 a-2 b) b}{6 a (a+b)^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cot ^2(e+f x)}{2 (a+b) f \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {b \left (a^2-6 a b-2 b^2\right )}{2 a^2 (a+b)^3 f \sqrt {a+b \sec ^2(e+f x)}} \] Output:

-arctanh((a+b*sec(f*x+e)^2)^(1/2)/a^(1/2))/a^(5/2)/f+1/2*(2*a+7*b)*arctanh 
((a+b*sec(f*x+e)^2)^(1/2)/(a+b)^(1/2))/(a+b)^(7/2)/f-1/6*(3*a-2*b)*b/a/(a+ 
b)^2/f/(a+b*sec(f*x+e)^2)^(3/2)-1/2*cot(f*x+e)^2/(a+b)/f/(a+b*sec(f*x+e)^2 
)^(3/2)-1/2*b*(a^2-6*a*b-2*b^2)/a^2/(a+b)^3/f/(a+b*sec(f*x+e)^2)^(1/2)
 

Mathematica [F]

\[ \int \frac {\cot ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\cot ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx \] Input:

Integrate[Cot[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(5/2),x]
 

Output:

Integrate[Cot[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(5/2), x]
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.14, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 4627, 354, 114, 27, 169, 27, 169, 27, 174, 73, 221}

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.

\(\displaystyle \int \frac {\cot ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\tan (e+f x)^3 \left (a+b \sec (e+f x)^2\right )^{5/2}}dx\)

\(\Big \downarrow \) 4627

\(\displaystyle \frac {\int \frac {\cos (e+f x)}{\left (1-\sec ^2(e+f x)\right )^2 \left (b \sec ^2(e+f x)+a\right )^{5/2}}d\sec (e+f x)}{f}\)

\(\Big \downarrow \) 354

\(\displaystyle \frac {\int \frac {\cos (e+f x)}{\left (1-\sec ^2(e+f x)\right )^2 \left (b \sec ^2(e+f x)+a\right )^{5/2}}d\sec ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 114

\(\displaystyle \frac {\frac {1}{(a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\int -\frac {\cos (e+f x) \left (5 b \sec ^2(e+f x)+2 a+2 b\right )}{2 \left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a\right )^{5/2}}d\sec ^2(e+f x)}{a+b}}{2 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\cos (e+f x) \left (5 b \sec ^2(e+f x)+2 (a+b)\right )}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a\right )^{5/2}}d\sec ^2(e+f x)}{2 (a+b)}+\frac {1}{(a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 f}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {\frac {\frac {2 \int \frac {3 \cos (e+f x) \left (2 (a+b)^2+(3 a-2 b) b \sec ^2(e+f x)\right )}{2 \left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a\right )^{3/2}}d\sec ^2(e+f x)}{3 a (a+b)}+\frac {2 b \left (\frac {2}{a}-\frac {5}{a+b}\right )}{3 \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 (a+b)}+\frac {1}{(a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {\cos (e+f x) \left (2 (a+b)^2+(3 a-2 b) b \sec ^2(e+f x)\right )}{\left (1-\sec ^2(e+f x)\right ) \left (b \sec ^2(e+f x)+a\right )^{3/2}}d\sec ^2(e+f x)}{a (a+b)}+\frac {2 b \left (\frac {2}{a}-\frac {5}{a+b}\right )}{3 \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 (a+b)}+\frac {1}{(a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 f}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {\frac {\frac {\frac {2 \int \frac {\cos (e+f x) \left (2 (a+b)^3+b \left (a^2-6 b a-2 b^2\right ) \sec ^2(e+f x)\right )}{2 \left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a}}d\sec ^2(e+f x)}{a (a+b)}-\frac {2 b \left (a^2-6 a b-2 b^2\right )}{a (a+b) \sqrt {a+b \sec ^2(e+f x)}}}{a (a+b)}+\frac {2 b \left (\frac {2}{a}-\frac {5}{a+b}\right )}{3 \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 (a+b)}+\frac {1}{(a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\frac {\int \frac {\cos (e+f x) \left (2 (a+b)^3+b \left (a^2-6 b a-2 b^2\right ) \sec ^2(e+f x)\right )}{\left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a}}d\sec ^2(e+f x)}{a (a+b)}-\frac {2 b \left (a^2-6 a b-2 b^2\right )}{a (a+b) \sqrt {a+b \sec ^2(e+f x)}}}{a (a+b)}+\frac {2 b \left (\frac {2}{a}-\frac {5}{a+b}\right )}{3 \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 (a+b)}+\frac {1}{(a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 f}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {\frac {\frac {\frac {a^2 (2 a+7 b) \int \frac {1}{\left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a}}d\sec ^2(e+f x)+2 (a+b)^3 \int \frac {\cos (e+f x)}{\sqrt {b \sec ^2(e+f x)+a}}d\sec ^2(e+f x)}{a (a+b)}-\frac {2 b \left (a^2-6 a b-2 b^2\right )}{a (a+b) \sqrt {a+b \sec ^2(e+f x)}}}{a (a+b)}+\frac {2 b \left (\frac {2}{a}-\frac {5}{a+b}\right )}{3 \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 (a+b)}+\frac {1}{(a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 f}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {\frac {\frac {\frac {2 a^2 (2 a+7 b) \int \frac {1}{\frac {a+b}{b}-\frac {\sec ^4(e+f x)}{b}}d\sqrt {b \sec ^2(e+f x)+a}}{b}+\frac {4 (a+b)^3 \int \frac {1}{\frac {\sec ^4(e+f x)}{b}-\frac {a}{b}}d\sqrt {b \sec ^2(e+f x)+a}}{b}}{a (a+b)}-\frac {2 b \left (a^2-6 a b-2 b^2\right )}{a (a+b) \sqrt {a+b \sec ^2(e+f x)}}}{a (a+b)}+\frac {2 b \left (\frac {2}{a}-\frac {5}{a+b}\right )}{3 \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 (a+b)}+\frac {1}{(a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {\frac {\frac {2 a^2 (2 a+7 b) \text {arctanh}\left (\frac {\sqrt {a+b \sec ^2(e+f x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b}}-\frac {4 (a+b)^3 \text {arctanh}\left (\frac {\sqrt {a+b \sec ^2(e+f x)}}{\sqrt {a}}\right )}{\sqrt {a}}}{a (a+b)}-\frac {2 b \left (a^2-6 a b-2 b^2\right )}{a (a+b) \sqrt {a+b \sec ^2(e+f x)}}}{a (a+b)}+\frac {2 b \left (\frac {2}{a}-\frac {5}{a+b}\right )}{3 \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 (a+b)}+\frac {1}{(a+b) \left (1-\sec ^2(e+f x)\right ) \left (a+b \sec ^2(e+f x)\right )^{3/2}}}{2 f}\)

Input:

Int[Cot[e + f*x]^3/(a + b*Sec[e + f*x]^2)^(5/2),x]
 

Output:

(1/((a + b)*(1 - Sec[e + f*x]^2)*(a + b*Sec[e + f*x]^2)^(3/2)) + ((2*b*(2/ 
a - 5/(a + b)))/(3*(a + b*Sec[e + f*x]^2)^(3/2)) + (((-4*(a + b)^3*ArcTanh 
[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a]])/Sqrt[a] + (2*a^2*(2*a + 7*b)*ArcTanh 
[Sqrt[a + b*Sec[e + f*x]^2]/Sqrt[a + b]])/Sqrt[a + b])/(a*(a + b)) - (2*b* 
(a^2 - 6*a*b - 2*b^2))/(a*(a + b)*Sqrt[a + b*Sec[e + f*x]^2]))/(a*(a + b)) 
)/(2*(a + b)))/(2*f)
 

Defintions of rubi rules used

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 221

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

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

rule 3042

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

rule 4627

Int[((a_) + (b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*tan[(e_.) + ( 
f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = FreeFactors[Sec[e + f*x], x]}, Si 
mp[1/f   Subst[Int[(-1 + ff^2*x^2)^((m - 1)/2)*((a + b*(c*ff*x)^n)^p/x), x] 
, x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[( 
m - 1)/2] && (GtQ[m, 0] || EqQ[n, 2] || EqQ[n, 4] || IGtQ[p, 0] || Integers 
Q[2*n, p])
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(12013\) vs. \(2(174)=348\).

Time = 8.63 (sec) , antiderivative size = 12014, normalized size of antiderivative = 60.07

method	result	size
default	\(\text {Expression too large to display}\)	\(12014\)

Input:

int(cot(f*x+e)^3/(a+b*sec(f*x+e)^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 830 vs. \(2 (174) = 348\).

Time = 4.48 (sec) , antiderivative size = 3507, normalized size of antiderivative = 17.54 \[ \int \frac {\cot ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(cot(f*x+e)^3/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {\cot ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\cot ^{3}{\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(cot(f*x+e)**3/(a+b*sec(f*x+e)**2)**(5/2),x)
 

Output:

Integral(cot(e + f*x)**3/(a + b*sec(e + f*x)**2)**(5/2), x)
                                                                                    
                                                                                    
 

Maxima [F(-1)]

Timed out. \[ \int \frac {\cot ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(cot(f*x+e)^3/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2160 vs. \(2 (174) = 348\).

Time = 2.14 (sec) , antiderivative size = 2160, normalized size of antiderivative = 10.80 \[ \int \frac {\cot ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(cot(f*x+e)^3/(a+b*sec(f*x+e)^2)^(5/2),x, algorithm="giac")
 

Output:

1/24*(((((3*(a^20*b^2*sgn(cos(f*x + e)) + 10*a^19*b^3*sgn(cos(f*x + e)) + 
45*a^18*b^4*sgn(cos(f*x + e)) + 120*a^17*b^5*sgn(cos(f*x + e)) + 210*a^16* 
b^6*sgn(cos(f*x + e)) + 252*a^15*b^7*sgn(cos(f*x + e)) + 210*a^14*b^8*sgn( 
cos(f*x + e)) + 120*a^13*b^9*sgn(cos(f*x + e)) + 45*a^12*b^10*sgn(cos(f*x 
+ e)) + 10*a^11*b^11*sgn(cos(f*x + e)) + a^10*b^12*sgn(cos(f*x + e)))*tan( 
1/2*f*x + 1/2*e)^2/(a^21*b^2 + 11*a^20*b^3 + 55*a^19*b^4 + 165*a^18*b^5 + 
330*a^17*b^6 + 462*a^16*b^7 + 462*a^15*b^8 + 330*a^14*b^9 + 165*a^13*b^10 
+ 55*a^12*b^11 + 11*a^11*b^12 + a^10*b^13) - 4*(3*a^20*b^2*sgn(cos(f*x + e 
)) + 24*a^19*b^3*sgn(cos(f*x + e)) + 101*a^18*b^4*sgn(cos(f*x + e)) + 330* 
a^17*b^5*sgn(cos(f*x + e)) + 900*a^16*b^6*sgn(cos(f*x + e)) + 1896*a^15*b^ 
7*sgn(cos(f*x + e)) + 2898*a^14*b^8*sgn(cos(f*x + e)) + 3132*a^13*b^9*sgn( 
cos(f*x + e)) + 2355*a^12*b^10*sgn(cos(f*x + e)) + 1200*a^11*b^11*sgn(cos( 
f*x + e)) + 393*a^10*b^12*sgn(cos(f*x + e)) + 74*a^9*b^13*sgn(cos(f*x + e) 
) + 6*a^8*b^14*sgn(cos(f*x + e)))/(a^21*b^2 + 11*a^20*b^3 + 55*a^19*b^4 + 
165*a^18*b^5 + 330*a^17*b^6 + 462*a^16*b^7 + 462*a^15*b^8 + 330*a^14*b^9 + 
 165*a^13*b^10 + 55*a^12*b^11 + 11*a^11*b^12 + a^10*b^13))*tan(1/2*f*x + 1 
/2*e)^2 + 6*(3*a^20*b^2*sgn(cos(f*x + e)) + 22*a^19*b^3*sgn(cos(f*x + e)) 
+ 111*a^18*b^4*sgn(cos(f*x + e)) + 460*a^17*b^5*sgn(cos(f*x + e)) + 1330*a 
^16*b^6*sgn(cos(f*x + e)) + 2516*a^15*b^7*sgn(cos(f*x + e)) + 3094*a^14*b^ 
8*sgn(cos(f*x + e)) + 2432*a^13*b^9*sgn(cos(f*x + e)) + 1135*a^12*b^10*...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {{\mathrm {cot}\left (e+f\,x\right )}^3}{{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{5/2}} \,d x \] Input:

int(cot(e + f*x)^3/(a + b/cos(e + f*x)^2)^(5/2),x)
 

Output:

int(cot(e + f*x)^3/(a + b/cos(e + f*x)^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\cot ^3(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\sqrt {\sec \left (f x +e \right )^{2} b +a}\, \cot \left (f x +e \right )^{3}}{\sec \left (f x +e \right )^{6} b^{3}+3 \sec \left (f x +e \right )^{4} a \,b^{2}+3 \sec \left (f x +e \right )^{2} a^{2} b +a^{3}}d x \] Input:

int(cot(f*x+e)^3/(a+b*sec(f*x+e)^2)^(5/2),x)
 

Output:

int((sqrt(sec(e + f*x)**2*b + a)*cot(e + f*x)**3)/(sec(e + f*x)**6*b**3 + 
3*sec(e + f*x)**4*a*b**2 + 3*sec(e + f*x)**2*a**2*b + a**3),x)