\(\int \frac {\sec ^6(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx\) [121]

Optimal result

Integrand size = 28, antiderivative size = 262 \[ \int \frac {\sec ^6(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=-\frac {3 a \text {arctanh}(\sin (c+d x))}{8 b^2 d}-\frac {a \left (a^2+b^2\right ) \text {arctanh}(\sin (c+d x))}{2 b^4 d}-\frac {a \left (a^2+b^2\right )^2 \text {arctanh}(\sin (c+d x))}{b^6 d}-\frac {\left (a^2+b^2\right )^{5/2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^6 d}+\frac {\left (a^2+b^2\right )^2 \sec (c+d x)}{b^5 d}+\frac {\left (a^2+b^2\right ) \sec ^3(c+d x)}{3 b^3 d}+\frac {\sec ^5(c+d x)}{5 b d}-\frac {3 a \sec (c+d x) \tan (c+d x)}{8 b^2 d}-\frac {a \left (a^2+b^2\right ) \sec (c+d x) \tan (c+d x)}{2 b^4 d}-\frac {a \sec ^3(c+d x) \tan (c+d x)}{4 b^2 d} \] Output:

-3/8*a*arctanh(sin(d*x+c))/b^2/d-1/2*a*(a^2+b^2)*arctanh(sin(d*x+c))/b^4/d 
-a*(a^2+b^2)^2*arctanh(sin(d*x+c))/b^6/d-(a^2+b^2)^(5/2)*arctanh((b*cos(d* 
x+c)-a*sin(d*x+c))/(a^2+b^2)^(1/2))/b^6/d+(a^2+b^2)^2*sec(d*x+c)/b^5/d+1/3 
*(a^2+b^2)*sec(d*x+c)^3/b^3/d+1/5*sec(d*x+c)^5/b/d-3/8*a*sec(d*x+c)*tan(d* 
x+c)/b^2/d-1/2*a*(a^2+b^2)*sec(d*x+c)*tan(d*x+c)/b^4/d-1/4*a*sec(d*x+c)^3* 
tan(d*x+c)/b^2/d
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(661\) vs. \(2(262)=524\).

Time = 4.01 (sec) , antiderivative size = 661, normalized size of antiderivative = 2.52 \[ \int \frac {\sec ^6(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx =\text {Too large to display} \] Input:

Integrate[Sec[c + d*x]^6/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]
 

Output:

(Sec[c + d*x]*(240*a^4*b + 520*a^2*b^3 + 298*b^5 + 480*(a^2 + b^2)^(5/2)*A 
rcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]] + 30*a*(8*a^4 + 20*a^2*b 
^2 + 15*b^4)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 30*a*(8*a^4 + 20*a 
^2*b^2 + 15*b^4)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (3*b^4*(-5*a + 
 2*b))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4 + (b^2*(-60*a^3 + 20*a^2*b 
- 105*a*b^2 + 29*b^3))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (12*b^5*S 
in[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5 + (2*b^3*(20*a^2 
+ 29*b^2)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (2*b 
*(120*a^4 + 260*a^2*b^2 + 149*b^4)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - S 
in[(c + d*x)/2]) - (12*b^5*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + 
d*x)/2])^5 + (3*b^4*(5*a + 2*b))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 - 
 (2*b^3*(20*a^2 + 29*b^2)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d 
*x)/2])^3 + (b^2*(60*a^3 + 20*a^2*b + 105*a*b^2 + 29*b^3))/(Cos[(c + d*x)/ 
2] + Sin[(c + d*x)/2])^2 - (2*b*(120*a^4 + 260*a^2*b^2 + 149*b^4)*Sin[(c + 
 d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))*(a*Cos[c + d*x] + b*Sin[c 
 + d*x]))/(240*b^6*d*(a + b*Tan[c + d*x]))
 

Rubi [A] (verified)

Time = 1.18 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.94, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3583, 3042, 3583, 3042, 3583, 3042, 3553, 219, 4255, 3042, 4255, 3042, 4257}

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.

\(\displaystyle \int \frac {\sec ^6(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\cos (c+d x)^6 (a \cos (c+d x)+b \sin (c+d x))}dx\)

\(\Big \downarrow \) 3583

\(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {\sec ^4(c+d x)}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}-\frac {a \int \sec ^5(c+d x)dx}{b^2}+\frac {\sec ^5(c+d x)}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {1}{\cos (c+d x)^4 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^5dx}{b^2}+\frac {\sec ^5(c+d x)}{5 b d}\)

\(\Big \downarrow \) 3583

\(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\left (a^2+b^2\right ) \int \frac {\sec ^2(c+d x)}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}-\frac {a \int \sec ^3(c+d x)dx}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}\right )}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^5dx}{b^2}+\frac {\sec ^5(c+d x)}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\left (a^2+b^2\right ) \int \frac {1}{\cos (c+d x)^2 (a \cos (c+d x)+b \sin (c+d x))}dx}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}\right )}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^5dx}{b^2}+\frac {\sec ^5(c+d x)}{5 b d}\)

\(\Big \downarrow \) 3583

\(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\left (a^2+b^2\right ) \left (\frac {\left (a^2+b^2\right ) \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}-\frac {a \int \sec (c+d x)dx}{b^2}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}\right )}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^5dx}{b^2}+\frac {\sec ^5(c+d x)}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\left (a^2+b^2\right ) \left (\frac {\left (a^2+b^2\right ) \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)}dx}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}\right )}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^5dx}{b^2}+\frac {\sec ^5(c+d x)}{5 b d}\)

\(\Big \downarrow \) 3553

\(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\left (a^2+b^2\right ) \left (-\frac {\left (a^2+b^2\right ) \int \frac {1}{a^2+b^2-(b \cos (c+d x)-a \sin (c+d x))^2}d(b \cos (c+d x)-a \sin (c+d x))}{b^2 d}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}\right )}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^5dx}{b^2}+\frac {\sec ^5(c+d x)}{5 b d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\left (a^2+b^2\right ) \left (-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}\right )}{b^2}-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )^5dx}{b^2}+\frac {\sec ^5(c+d x)}{5 b d}\)

\(\Big \downarrow \) 4255

\(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\left (a^2+b^2\right ) \left (-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}\right )}{b^2}-\frac {a \left (\frac {3}{4} \int \sec ^3(c+d x)dx+\frac {\tan (c+d x) \sec ^3(c+d x)}{4 d}\right )}{b^2}+\frac {\sec ^5(c+d x)}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\left (a^2+b^2\right ) \left (-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}\right )}{b^2}-\frac {a \left (\frac {3}{4} \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx+\frac {\tan (c+d x) \sec ^3(c+d x)}{4 d}\right )}{b^2}+\frac {\sec ^5(c+d x)}{5 b d}\)

\(\Big \downarrow \) 4255

\(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\left (a^2+b^2\right ) \left (-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}\right )}{b^2}-\frac {a \left (\frac {3}{4} \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {\tan (c+d x) \sec ^3(c+d x)}{4 d}\right )}{b^2}+\frac {\sec ^5(c+d x)}{5 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\left (a^2+b^2\right ) \left (-\frac {a \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}\right )}{b^2}-\frac {a \left (\frac {3}{4} \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {\tan (c+d x) \sec ^3(c+d x)}{4 d}\right )}{b^2}+\frac {\sec ^5(c+d x)}{5 b d}\)

\(\Big \downarrow \) 4257

\(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\left (a^2+b^2\right ) \left (-\frac {\sqrt {a^2+b^2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {a \text {arctanh}(\sin (c+d x))}{b^2 d}+\frac {\sec (c+d x)}{b d}\right )}{b^2}-\frac {a \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )}{b^2}+\frac {\sec ^3(c+d x)}{3 b d}\right )}{b^2}-\frac {a \left (\frac {3}{4} \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {\tan (c+d x) \sec ^3(c+d x)}{4 d}\right )}{b^2}+\frac {\sec ^5(c+d x)}{5 b d}\)

Input:

Int[Sec[c + d*x]^6/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]
 

Output:

Sec[c + d*x]^5/(5*b*d) - (a*((Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (3*(Arc 
Tanh[Sin[c + d*x]]/(2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*d)))/4))/b^2 + ( 
(a^2 + b^2)*(Sec[c + d*x]^3/(3*b*d) + ((a^2 + b^2)*(-((a*ArcTanh[Sin[c + d 
*x]])/(b^2*d)) - (Sqrt[a^2 + b^2]*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x] 
)/Sqrt[a^2 + b^2]])/(b^2*d) + Sec[c + d*x]/(b*d)))/b^2 - (a*(ArcTanh[Sin[c 
 + d*x]]/(2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*d)))/b^2))/b^2
 

Defintions of rubi rules used

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 3042

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

rule 3553

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x 
_Symbol] :> Simp[-d^(-1)   Subst[Int[1/(a^2 + b^2 - x^2), x], x, b*Cos[c + 
d*x] - a*Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
 

rule 3583

Int[cos[(c_.) + (d_.)*(x_)]^(m_)/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin 
[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-Cos[c + d*x]^(m + 1)/(b*d*(m + 1) 
), x] + (-Simp[a/b^2   Int[Cos[c + d*x]^(m + 1), x], x] + Simp[(a^2 + b^2)/ 
b^2   Int[Cos[c + d*x]^(m + 2)/(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x]) / 
; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]
 

rule 4255

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) 
  Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] 
&& IntegerQ[2*n]
 

rule 4257

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] 
 /; FreeQ[{c, d}, x]
 

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.83

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (-a^{6}-3 a^{4} b^{2}-3 a^{2} b^{4}-b^{6}\right ) \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{6} \sqrt {a^{2}+b^{2}}}-\frac {1}{5 b \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}-\frac {a +2 b}{4 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {4 a^{2}+6 a b +13 b^{2}}{12 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {4 a^{3}+4 a^{2} b +11 a \,b^{2}+9 b^{3}}{8 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {8 a^{4}+4 a^{3} b +20 a^{2} b^{2}+9 b^{3} a +15 b^{4}}{8 b^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a \left (8 a^{4}+20 a^{2} b^{2}+15 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 b^{6}}+\frac {1}{5 b \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {-a +2 b}{4 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {-4 a^{2}+6 a b -13 b^{2}}{12 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-4 a^{3}+4 a^{2} b -11 a \,b^{2}+9 b^{3}}{8 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-8 a^{4}+4 a^{3} b -20 a^{2} b^{2}+9 b^{3} a -15 b^{4}}{8 b^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {a \left (8 a^{4}+20 a^{2} b^{2}+15 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 b^{6}}}{d}\)	\(479\)
default	\(\frac {-\frac {2 \left (-a^{6}-3 a^{4} b^{2}-3 a^{2} b^{4}-b^{6}\right ) \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{6} \sqrt {a^{2}+b^{2}}}-\frac {1}{5 b \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}-\frac {a +2 b}{4 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {4 a^{2}+6 a b +13 b^{2}}{12 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {4 a^{3}+4 a^{2} b +11 a \,b^{2}+9 b^{3}}{8 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {8 a^{4}+4 a^{3} b +20 a^{2} b^{2}+9 b^{3} a +15 b^{4}}{8 b^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a \left (8 a^{4}+20 a^{2} b^{2}+15 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 b^{6}}+\frac {1}{5 b \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {-a +2 b}{4 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {-4 a^{2}+6 a b -13 b^{2}}{12 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-4 a^{3}+4 a^{2} b -11 a \,b^{2}+9 b^{3}}{8 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-8 a^{4}+4 a^{3} b -20 a^{2} b^{2}+9 b^{3} a -15 b^{4}}{8 b^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {a \left (8 a^{4}+20 a^{2} b^{2}+15 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 b^{6}}}{d}\)	\(479\)
risch	\(\frac {{\mathrm e}^{i \left (d x +c \right )} \left (105 i a \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-60 i a^{3} b +120 a^{4} {\mathrm e}^{8 i \left (d x +c \right )}+240 a^{2} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+120 b^{4} {\mathrm e}^{8 i \left (d x +c \right )}+120 i a^{3} b \,{\mathrm e}^{6 i \left (d x +c \right )}-105 i b^{3} a +480 a^{4} {\mathrm e}^{6 i \left (d x +c \right )}+1120 a^{2} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+640 b^{4} {\mathrm e}^{6 i \left (d x +c \right )}+720 a^{4} {\mathrm e}^{4 i \left (d x +c \right )}+1760 a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+1424 b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+60 i a^{3} b \,{\mathrm e}^{8 i \left (d x +c \right )}-120 i a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}+480 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+1120 a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+640 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-330 i a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+330 i a \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+120 a^{4}+240 a^{2} b^{2}+120 b^{4}\right )}{60 d \,b^{5} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}+\frac {a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d \,b^{6}}+\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 b^{4} d}+\frac {15 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 b^{2} d}-\frac {a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \,b^{6}}-\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 b^{4} d}-\frac {15 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 b^{2} d}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} \left (i a -b \right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}\right )}{d \,b^{6}}-\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} \left (i a -b \right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}\right )}{d \,b^{6}}\)	\(660\)

Input:

int(sec(d*x+c)^6/(a*cos(d*x+c)+b*sin(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

1/d*(-2/b^6*(-a^6-3*a^4*b^2-3*a^2*b^4-b^6)/(a^2+b^2)^(1/2)*arctanh(1/2*(2* 
a*tan(1/2*d*x+1/2*c)-2*b)/(a^2+b^2)^(1/2))-1/5/b/(tan(1/2*d*x+1/2*c)-1)^5- 
1/4*(a+2*b)/b^2/(tan(1/2*d*x+1/2*c)-1)^4-1/12*(4*a^2+6*a*b+13*b^2)/b^3/(ta 
n(1/2*d*x+1/2*c)-1)^3-1/8*(4*a^3+4*a^2*b+11*a*b^2+9*b^3)/b^4/(tan(1/2*d*x+ 
1/2*c)-1)^2-1/8*(8*a^4+4*a^3*b+20*a^2*b^2+9*a*b^3+15*b^4)/b^5/(tan(1/2*d*x 
+1/2*c)-1)+1/8*a*(8*a^4+20*a^2*b^2+15*b^4)/b^6*ln(tan(1/2*d*x+1/2*c)-1)+1/ 
5/b/(tan(1/2*d*x+1/2*c)+1)^5-1/4*(-a+2*b)/b^2/(tan(1/2*d*x+1/2*c)+1)^4-1/1 
2*(-4*a^2+6*a*b-13*b^2)/b^3/(tan(1/2*d*x+1/2*c)+1)^3-1/8*(-4*a^3+4*a^2*b-1 
1*a*b^2+9*b^3)/b^4/(tan(1/2*d*x+1/2*c)+1)^2-1/8*(-8*a^4+4*a^3*b-20*a^2*b^2 
+9*a*b^3-15*b^4)/b^5/(tan(1/2*d*x+1/2*c)+1)-1/8*a*(8*a^4+20*a^2*b^2+15*b^4 
)/b^6*ln(tan(1/2*d*x+1/2*c)+1))
 

Fricas [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.32 \[ \int \frac {\sec ^6(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {120 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}} \cos \left (d x + c\right )^{5} \log \left (-\frac {2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right ) - 15 \, {\left (8 \, a^{5} + 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, {\left (8 \, a^{5} + 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 48 \, b^{5} + 240 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} + 80 \, {\left (a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2} - 30 \, {\left (2 \, a b^{4} \cos \left (d x + c\right ) + {\left (4 \, a^{3} b^{2} + 7 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{240 \, b^{6} d \cos \left (d x + c\right )^{5}} \] Input:

integrate(sec(d*x+c)^6/(a*cos(d*x+c)+b*sin(d*x+c)),x, algorithm="fricas")
 

Output:

1/240*(120*(a^4 + 2*a^2*b^2 + b^4)*sqrt(a^2 + b^2)*cos(d*x + c)^5*log(-(2* 
a*b*cos(d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 - 2*a^2 - b^2 + 
 2*sqrt(a^2 + b^2)*(b*cos(d*x + c) - a*sin(d*x + c)))/(2*a*b*cos(d*x + c)* 
sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 + b^2)) - 15*(8*a^5 + 20*a^3*b^2 
 + 15*a*b^4)*cos(d*x + c)^5*log(sin(d*x + c) + 1) + 15*(8*a^5 + 20*a^3*b^2 
 + 15*a*b^4)*cos(d*x + c)^5*log(-sin(d*x + c) + 1) + 48*b^5 + 240*(a^4*b + 
 2*a^2*b^3 + b^5)*cos(d*x + c)^4 + 80*(a^2*b^3 + b^5)*cos(d*x + c)^2 - 30* 
(2*a*b^4*cos(d*x + c) + (4*a^3*b^2 + 7*a*b^4)*cos(d*x + c)^3)*sin(d*x + c) 
)/(b^6*d*cos(d*x + c)^5)
 

Sympy [F]

\[ \int \frac {\sec ^6(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\int \frac {\sec ^{6}{\left (c + d x \right )}}{a \cos {\left (c + d x \right )} + b \sin {\left (c + d x \right )}}\, dx \] Input:

integrate(sec(d*x+c)**6/(a*cos(d*x+c)+b*sin(d*x+c)),x)
 

Output:

Integral(sec(c + d*x)**6/(a*cos(c + d*x) + b*sin(c + d*x)), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 625 vs. \(2 (244) = 488\).

Time = 0.13 (sec) , antiderivative size = 625, normalized size of antiderivative = 2.39 \[ \int \frac {\sec ^6(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx =\text {Too large to display} \] Input:

integrate(sec(d*x+c)^6/(a*cos(d*x+c)+b*sin(d*x+c)),x, algorithm="maxima")
 

Output:

1/120*(2*(120*a^4 + 280*a^2*b^2 + 184*b^4 - 15*(4*a^3*b + 9*a*b^3)*sin(d*x 
 + c)/(cos(d*x + c) + 1) - 80*(6*a^4 + 13*a^2*b^2 + 7*b^4)*sin(d*x + c)^2/ 
(cos(d*x + c) + 1)^2 + 30*(4*a^3*b + 5*a*b^3)*sin(d*x + c)^3/(cos(d*x + c) 
 + 1)^3 + 80*(9*a^4 + 20*a^2*b^2 + 14*b^4)*sin(d*x + c)^4/(cos(d*x + c) + 
1)^4 - 240*(2*a^4 + 5*a^2*b^2 + 3*b^4)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 
 - 30*(4*a^3*b + 5*a*b^3)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 120*(a^4 + 
 3*a^2*b^2 + 3*b^4)*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 15*(4*a^3*b + 9* 
a*b^3)*sin(d*x + c)^9/(cos(d*x + c) + 1)^9)/(b^5 - 5*b^5*sin(d*x + c)^2/(c 
os(d*x + c) + 1)^2 + 10*b^5*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 10*b^5*s 
in(d*x + c)^6/(cos(d*x + c) + 1)^6 + 5*b^5*sin(d*x + c)^8/(cos(d*x + c) + 
1)^8 - b^5*sin(d*x + c)^10/(cos(d*x + c) + 1)^10) - 15*(8*a^5 + 20*a^3*b^2 
 + 15*a*b^4)*log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/b^6 + 15*(8*a^5 + 20 
*a^3*b^2 + 15*a*b^4)*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/b^6 - 120*(a 
^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*log((b - a*sin(d*x + c)/(cos(d*x + c) + 
1) + sqrt(a^2 + b^2))/(b - a*sin(d*x + c)/(cos(d*x + c) + 1) - sqrt(a^2 + 
b^2)))/(sqrt(a^2 + b^2)*b^6))/d
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 554 vs. \(2 (244) = 488\).

Time = 0.19 (sec) , antiderivative size = 554, normalized size of antiderivative = 2.11 \[ \int \frac {\sec ^6(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx =\text {Too large to display} \] Input:

integrate(sec(d*x+c)^6/(a*cos(d*x+c)+b*sin(d*x+c)),x, algorithm="giac")
 

Output:

-1/120*(15*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*log(abs(tan(1/2*d*x + 1/2*c) + 
1))/b^6 - 15*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*log(abs(tan(1/2*d*x + 1/2*c) 
- 1))/b^6 + 120*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*log(abs(2*a*tan(1/2*d* 
x + 1/2*c) - 2*b - 2*sqrt(a^2 + b^2))/abs(2*a*tan(1/2*d*x + 1/2*c) - 2*b + 
 2*sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*b^6) + 2*(60*a^3*b*tan(1/2*d*x + 1/2 
*c)^9 + 135*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 120*a^4*tan(1/2*d*x + 1/2*c)^8 
+ 360*a^2*b^2*tan(1/2*d*x + 1/2*c)^8 + 360*b^4*tan(1/2*d*x + 1/2*c)^8 - 12 
0*a^3*b*tan(1/2*d*x + 1/2*c)^7 - 150*a*b^3*tan(1/2*d*x + 1/2*c)^7 - 480*a^ 
4*tan(1/2*d*x + 1/2*c)^6 - 1200*a^2*b^2*tan(1/2*d*x + 1/2*c)^6 - 720*b^4*t 
an(1/2*d*x + 1/2*c)^6 + 720*a^4*tan(1/2*d*x + 1/2*c)^4 + 1600*a^2*b^2*tan( 
1/2*d*x + 1/2*c)^4 + 1120*b^4*tan(1/2*d*x + 1/2*c)^4 + 120*a^3*b*tan(1/2*d 
*x + 1/2*c)^3 + 150*a*b^3*tan(1/2*d*x + 1/2*c)^3 - 480*a^4*tan(1/2*d*x + 1 
/2*c)^2 - 1040*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 - 560*b^4*tan(1/2*d*x + 1/2* 
c)^2 - 60*a^3*b*tan(1/2*d*x + 1/2*c) - 135*a*b^3*tan(1/2*d*x + 1/2*c) + 12 
0*a^4 + 280*a^2*b^2 + 184*b^4)/((tan(1/2*d*x + 1/2*c)^2 - 1)^5*b^5))/d
 

Mupad [B] (verification not implemented)

Time = 19.22 (sec) , antiderivative size = 2979, normalized size of antiderivative = 11.37 \[ \int \frac {\sec ^6(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\text {Too large to display} \] Input:

int(1/(cos(c + d*x)^6*(a*cos(c + d*x) + b*sin(c + d*x))),x)
 

Output:

(atan(((((a^2 + b^2)^5)^(1/2)*(((225*a^4*b^13)/2 + 300*a^6*b^11 + 320*a^8* 
b^9 + 160*a^10*b^7 + 32*a^12*b^5)/b^14 + (tan(c/2 + (d*x)/2)*(64*a*b^17 + 
834*a^3*b^15 + 2385*a^5*b^13 + 3160*a^7*b^11 + 2240*a^9*b^9 + 832*a^11*b^7 
 + 128*a^13*b^5))/(2*b^15) - (((a^2 + b^2)^5)^(1/2)*((28*a^2*b^16 + 44*a^4 
*b^14 + 16*a^6*b^12)/b^14 - (tan(c/2 + (d*x)/2)*(128*a*b^18 + 384*a^3*b^16 
 + 384*a^5*b^14 + 128*a^7*b^12))/(2*b^15) + (((a^2 + b^2)^5)^(1/2)*(32*a^2 
*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^19 + 128*a^3*b^17))/(2*b^15)))/b^6))/b 
^6)*1i)/b^6 + (((a^2 + b^2)^5)^(1/2)*(((225*a^4*b^13)/2 + 300*a^6*b^11 + 3 
20*a^8*b^9 + 160*a^10*b^7 + 32*a^12*b^5)/b^14 + (tan(c/2 + (d*x)/2)*(64*a* 
b^17 + 834*a^3*b^15 + 2385*a^5*b^13 + 3160*a^7*b^11 + 2240*a^9*b^9 + 832*a 
^11*b^7 + 128*a^13*b^5))/(2*b^15) - (((a^2 + b^2)^5)^(1/2)*((tan(c/2 + (d* 
x)/2)*(128*a*b^18 + 384*a^3*b^16 + 384*a^5*b^14 + 128*a^7*b^12))/(2*b^15) 
- (28*a^2*b^16 + 44*a^4*b^14 + 16*a^6*b^12)/b^14 + (((a^2 + b^2)^5)^(1/2)* 
(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^19 + 128*a^3*b^17))/(2*b^15)))/ 
b^6))/b^6)*1i)/b^6)/((32*a^16 + 120*a^2*b^14 + 655*a^4*b^12 + 1549*a^6*b^1 
0 + 2069*a^8*b^8 + 1695*a^10*b^6 + 856*a^12*b^4 + 248*a^14*b^2)/b^14 + ((( 
a^2 + b^2)^5)^(1/2)*(((225*a^4*b^13)/2 + 300*a^6*b^11 + 320*a^8*b^9 + 160* 
a^10*b^7 + 32*a^12*b^5)/b^14 + (tan(c/2 + (d*x)/2)*(64*a*b^17 + 834*a^3*b^ 
15 + 2385*a^5*b^13 + 3160*a^7*b^11 + 2240*a^9*b^9 + 832*a^11*b^7 + 128*a^1 
3*b^5))/(2*b^15) - (((a^2 + b^2)^5)^(1/2)*((28*a^2*b^16 + 44*a^4*b^14 +...
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 1415, normalized size of antiderivative = 5.40 \[ \int \frac {\sec ^6(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx =\text {Too large to display} \] Input:

int(sec(d*x+c)^6/(a*cos(d*x+c)+b*sin(d*x+c)),x)
 

Output:

( - 240*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b* 
*2))*cos(c + d*x)*sin(c + d*x)**4*a**4*i - 480*sqrt(a**2 + b**2)*atan((tan 
((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2))*cos(c + d*x)*sin(c + d*x)**4*a 
**2*b**2*i - 240*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt( 
a**2 + b**2))*cos(c + d*x)*sin(c + d*x)**4*b**4*i + 480*sqrt(a**2 + b**2)* 
atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2))*cos(c + d*x)*sin(c + 
d*x)**2*a**4*i + 960*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/s 
qrt(a**2 + b**2))*cos(c + d*x)*sin(c + d*x)**2*a**2*b**2*i + 480*sqrt(a**2 
 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2))*cos(c + d*x) 
*sin(c + d*x)**2*b**4*i - 240*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i 
 - b*i)/sqrt(a**2 + b**2))*cos(c + d*x)*a**4*i - 480*sqrt(a**2 + b**2)*ata 
n((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2))*cos(c + d*x)*a**2*b**2*i 
 - 240*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b** 
2))*cos(c + d*x)*b**4*i + 120*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c 
 + d*x)**4*a**5 + 300*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)* 
*4*a**3*b**2 + 225*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4* 
a*b**4 - 240*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a**5 - 
 600*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a**3*b**2 - 45 
0*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a*b**4 + 120*cos( 
c + d*x)*log(tan((c + d*x)/2) - 1)*a**5 + 300*cos(c + d*x)*log(tan((c +...