3.2.52 \(\int (a+a \sec (c+d x))^n \sin ^4(c+d x) \, dx\) [152]

3.2.52.1 Optimal result

Integrand size = 21, antiderivative size = 230 \[ \int (a+a \sec (c+d x))^n \sin ^4(c+d x) \, dx=-\frac {\operatorname {AppellF1}\left (1-n,-\frac {1}{2},\frac {1}{2}-n,2-n,\cos (c+d x),-\cos (c+d x)\right ) (1+\cos (c+d x))^{\frac {1}{2}-n} (n-n \cos (c+d x)) \cot (c+d x) (a+a \sec (c+d x))^n}{d (1-n) \sqrt {1-\cos (c+d x)}}-\frac {\cos (c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d}+\frac {2^{\frac {1}{2}+n} \operatorname {AppellF1}\left (\frac {1}{2},-4+n,\frac {1}{2}-n,\frac {3}{2},1-\cos (c+d x),\frac {1}{2} (1-\cos (c+d x))\right ) \cos ^n(c+d x) (1+\cos (c+d x))^{-\frac {1}{2}-n} (a+a \sec (c+d x))^n \sin (c+d x)}{d} \]

output

-cos(d*x+c)*(a+a*sec(d*x+c))^n*sin(d*x+c)/d+2^(1/2+n)*AppellF1(1/2,-4+n,1/ 
2-n,3/2,1-cos(d*x+c),1/2-1/2*cos(d*x+c))*cos(d*x+c)^n*(1+cos(d*x+c))^(-1/2 
-n)*(a+a*sec(d*x+c))^n*sin(d*x+c)/d-AppellF1(1-n,1/2-n,-1/2,2-n,-cos(d*x+c 
),cos(d*x+c))*(1+cos(d*x+c))^(1/2-n)*(n-n*cos(d*x+c))*cot(d*x+c)*(a+a*sec( 
d*x+c))^n/d/(1-n)/(1-cos(d*x+c))^(1/2)
 

3.2.52.2 Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 21.75 (sec) , antiderivative size = 7069, normalized size of antiderivative = 30.73 \[ \int (a+a \sec (c+d x))^n \sin ^4(c+d x) \, dx=\text {Result too large to show} \]

input

Integrate[(a + a*Sec[c + d*x])^n*Sin[c + d*x]^4,x]
 

output

Result too large to show
 

3.2.52.3 Rubi [A] (verified)

Time = 1.81 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.38, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {3042, 4364, 3042, 3360, 3042, 3266, 3042, 3265, 3042, 3264, 150, 3525, 27, 3042, 3487, 152, 152, 150}

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.

\(\displaystyle \int \sin ^4(c+d x) (a \sec (c+d x)+a)^n \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos \left (c+d x-\frac {\pi }{2}\right )^4 \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^ndx\)

\(\Big \downarrow \) 4364

\(\displaystyle (-\cos (c+d x))^n (a (-\cos (c+d x))-a)^{-n} (a \sec (c+d x)+a)^n \int (-\cos (c+d x))^{-n} (-\cos (c+d x) a-a)^n \sin ^4(c+d x)dx\)

\(\Big \downarrow \) 3042

\(\displaystyle (-\cos (c+d x))^n (a (-\cos (c+d x))-a)^{-n} (a \sec (c+d x)+a)^n \int \cos \left (c+d x+\frac {\pi }{2}\right )^4 \left (-\sin \left (c+d x+\frac {\pi }{2}\right )\right )^{-n} \left (-\sin \left (c+d x+\frac {\pi }{2}\right ) a-a\right )^ndx\)

\(\Big \downarrow \) 3360

\(\displaystyle (-\cos (c+d x))^n (a (-\cos (c+d x))-a)^{-n} (a \sec (c+d x)+a)^n \left (\int (-\cos (c+d x))^{-n} (-\cos (c+d x) a-a)^n \left (1-2 \cos ^2(c+d x)\right )dx+\int (-\cos (c+d x))^{4-n} (-\cos (c+d x) a-a)^ndx\right )\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3266

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3265

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3264

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

\(\Big \downarrow \) 150

\(\displaystyle (-\cos (c+d x))^n (a (-\cos (c+d x))-a)^{-n} (a \sec (c+d x)+a)^n \left (\int \left (-\sin \left (c+d x+\frac {\pi }{2}\right )\right )^{-n} \left (-\sin \left (c+d x+\frac {\pi }{2}\right ) a-a\right )^n \left (1-2 \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+\frac {2^{n+\frac {1}{2}} \sin (c+d x) (-\cos (c+d x))^{-n} \cos ^n(c+d x) (\cos (c+d x)+1)^{-n-\frac {1}{2}} (a (-\cos (c+d x))-a)^n \operatorname {AppellF1}\left (\frac {1}{2},n-4,\frac {1}{2}-n,\frac {3}{2},1-\cos (c+d x),\frac {1}{2} (1-\cos (c+d x))\right )}{d}\right )\)

\(\Big \downarrow \) 3525

\(\displaystyle (-\cos (c+d x))^n (a (-\cos (c+d x))-a)^{-n} (a \sec (c+d x)+a)^n \left (\frac {\int 2 (-\cos (c+d x))^{-n} (-\cos (c+d x) a-a)^n (a n-a n \cos (c+d x))dx}{2 a}+\frac {2^{n+\frac {1}{2}} \sin (c+d x) (-\cos (c+d x))^{-n} \cos ^n(c+d x) (\cos (c+d x)+1)^{-n-\frac {1}{2}} (a (-\cos (c+d x))-a)^n \operatorname {AppellF1}\left (\frac {1}{2},n-4,\frac {1}{2}-n,\frac {3}{2},1-\cos (c+d x),\frac {1}{2} (1-\cos (c+d x))\right )}{d}+\frac {\sin (c+d x) (-\cos (c+d x))^{1-n} (a (-\cos (c+d x))-a)^n}{d}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle (-\cos (c+d x))^n (a (-\cos (c+d x))-a)^{-n} (a \sec (c+d x)+a)^n \left (\frac {\int (-\cos (c+d x))^{-n} (-\cos (c+d x) a-a)^n (a n-a n \cos (c+d x))dx}{a}+\frac {2^{n+\frac {1}{2}} \sin (c+d x) (-\cos (c+d x))^{-n} \cos ^n(c+d x) (\cos (c+d x)+1)^{-n-\frac {1}{2}} (a (-\cos (c+d x))-a)^n \operatorname {AppellF1}\left (\frac {1}{2},n-4,\frac {1}{2}-n,\frac {3}{2},1-\cos (c+d x),\frac {1}{2} (1-\cos (c+d x))\right )}{d}+\frac {\sin (c+d x) (-\cos (c+d x))^{1-n} (a (-\cos (c+d x))-a)^n}{d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle (-\cos (c+d x))^n (a (-\cos (c+d x))-a)^{-n} (a \sec (c+d x)+a)^n \left (\frac {\int \left (-\sin \left (c+d x+\frac {\pi }{2}\right )\right )^{-n} \left (-\sin \left (c+d x+\frac {\pi }{2}\right ) a-a\right )^n \left (a n-a n \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{a}+\frac {2^{n+\frac {1}{2}} \sin (c+d x) (-\cos (c+d x))^{-n} \cos ^n(c+d x) (\cos (c+d x)+1)^{-n-\frac {1}{2}} (a (-\cos (c+d x))-a)^n \operatorname {AppellF1}\left (\frac {1}{2},n-4,\frac {1}{2}-n,\frac {3}{2},1-\cos (c+d x),\frac {1}{2} (1-\cos (c+d x))\right )}{d}+\frac {\sin (c+d x) (-\cos (c+d x))^{1-n} (a (-\cos (c+d x))-a)^n}{d}\right )\)

\(\Big \downarrow \) 3487

\(\displaystyle (-\cos (c+d x))^n (a (-\cos (c+d x))-a)^{-n} (a \sec (c+d x)+a)^n \left (-\frac {\csc (c+d x) \sqrt {a (-\cos (c+d x))-a} \sqrt {a n-a n \cos (c+d x)} \int (-\cos (c+d x))^{-n} (-\cos (c+d x) a-a)^{n-\frac {1}{2}} \sqrt {a n-a n \cos (c+d x)}d\cos (c+d x)}{a d}+\frac {2^{n+\frac {1}{2}} \sin (c+d x) (-\cos (c+d x))^{-n} \cos ^n(c+d x) (\cos (c+d x)+1)^{-n-\frac {1}{2}} (a (-\cos (c+d x))-a)^n \operatorname {AppellF1}\left (\frac {1}{2},n-4,\frac {1}{2}-n,\frac {3}{2},1-\cos (c+d x),\frac {1}{2} (1-\cos (c+d x))\right )}{d}+\frac {\sin (c+d x) (-\cos (c+d x))^{1-n} (a (-\cos (c+d x))-a)^n}{d}\right )\)

\(\Big \downarrow \) 152

\(\displaystyle (-\cos (c+d x))^n (a (-\cos (c+d x))-a)^{-n} (a \sec (c+d x)+a)^n \left (-\frac {\csc (c+d x) (\cos (c+d x)+1)^{\frac {1}{2}-n} (a (-\cos (c+d x))-a)^n \sqrt {a n-a n \cos (c+d x)} \int (-\cos (c+d x))^{-n} (\cos (c+d x)+1)^{n-\frac {1}{2}} \sqrt {a n-a n \cos (c+d x)}d\cos (c+d x)}{a d}+\frac {2^{n+\frac {1}{2}} \sin (c+d x) (-\cos (c+d x))^{-n} \cos ^n(c+d x) (\cos (c+d x)+1)^{-n-\frac {1}{2}} (a (-\cos (c+d x))-a)^n \operatorname {AppellF1}\left (\frac {1}{2},n-4,\frac {1}{2}-n,\frac {3}{2},1-\cos (c+d x),\frac {1}{2} (1-\cos (c+d x))\right )}{d}+\frac {\sin (c+d x) (-\cos (c+d x))^{1-n} (a (-\cos (c+d x))-a)^n}{d}\right )\)

\(\Big \downarrow \) 152

\(\displaystyle (-\cos (c+d x))^n (a (-\cos (c+d x))-a)^{-n} (a \sec (c+d x)+a)^n \left (-\frac {\csc (c+d x) (\cos (c+d x)+1)^{\frac {1}{2}-n} (a (-\cos (c+d x))-a)^n (a n-a n \cos (c+d x)) \int \sqrt {1-\cos (c+d x)} (-\cos (c+d x))^{-n} (\cos (c+d x)+1)^{n-\frac {1}{2}}d\cos (c+d x)}{a d \sqrt {1-\cos (c+d x)}}+\frac {2^{n+\frac {1}{2}} \sin (c+d x) (-\cos (c+d x))^{-n} \cos ^n(c+d x) (\cos (c+d x)+1)^{-n-\frac {1}{2}} (a (-\cos (c+d x))-a)^n \operatorname {AppellF1}\left (\frac {1}{2},n-4,\frac {1}{2}-n,\frac {3}{2},1-\cos (c+d x),\frac {1}{2} (1-\cos (c+d x))\right )}{d}+\frac {\sin (c+d x) (-\cos (c+d x))^{1-n} (a (-\cos (c+d x))-a)^n}{d}\right )\)

\(\Big \downarrow \) 150

\(\displaystyle (-\cos (c+d x))^n (a (-\cos (c+d x))-a)^{-n} (a \sec (c+d x)+a)^n \left (\frac {2^{n+\frac {1}{2}} \sin (c+d x) (-\cos (c+d x))^{-n} \cos ^n(c+d x) (\cos (c+d x)+1)^{-n-\frac {1}{2}} (a (-\cos (c+d x))-a)^n \operatorname {AppellF1}\left (\frac {1}{2},n-4,\frac {1}{2}-n,\frac {3}{2},1-\cos (c+d x),\frac {1}{2} (1-\cos (c+d x))\right )}{d}+\frac {\csc (c+d x) (-\cos (c+d x))^{1-n} (\cos (c+d x)+1)^{\frac {1}{2}-n} (a (-\cos (c+d x))-a)^n (a n-a n \cos (c+d x)) \operatorname {AppellF1}\left (1-n,-\frac {1}{2},\frac {1}{2}-n,2-n,\cos (c+d x),-\cos (c+d x)\right )}{a d (1-n) \sqrt {1-\cos (c+d x)}}+\frac {\sin (c+d x) (-\cos (c+d x))^{1-n} (a (-\cos (c+d x))-a)^n}{d}\right )\)

input

Int[(a + a*Sec[c + d*x])^n*Sin[c + d*x]^4,x]
 

output

((-Cos[c + d*x])^n*(a + a*Sec[c + d*x])^n*((AppellF1[1 - n, -1/2, 1/2 - n, 
 2 - n, Cos[c + d*x], -Cos[c + d*x]]*(-Cos[c + d*x])^(1 - n)*(1 + Cos[c + 
d*x])^(1/2 - n)*(-a - a*Cos[c + d*x])^n*(a*n - a*n*Cos[c + d*x])*Csc[c + d 
*x])/(a*d*(1 - n)*Sqrt[1 - Cos[c + d*x]]) + ((-Cos[c + d*x])^(1 - n)*(-a - 
 a*Cos[c + d*x])^n*Sin[c + d*x])/d + (2^(1/2 + n)*AppellF1[1/2, -4 + n, 1/ 
2 - n, 3/2, 1 - Cos[c + d*x], (1 - Cos[c + d*x])/2]*Cos[c + d*x]^n*(1 + Co 
s[c + d*x])^(-1/2 - n)*(-a - a*Cos[c + d*x])^n*Sin[c + d*x])/(d*(-Cos[c + 
d*x])^n)))/(-a - a*Cos[c + d*x])^n
 

3.2.52.3.1 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 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 152

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n]) 
Int[(b*x)^m*(1 + d*(x/c))^n*(e + f*x)^p, x], x] /; FreeQ[{b, c, d, e, f, m, 
 n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !GtQ[c, 0]
 

rule 3042

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

rule 3264

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_), x_Symbol] :> Simp[(-b)*(d/b)^n*(Cos[e + f*x]/(f*Sqrt[a + b*Sin[e 
 + f*x]]*Sqrt[a - b*Sin[e + f*x]]))   Subst[Int[(a - x)^n*((2*a - x)^(m - 1 
/2)/Sqrt[x]), x], x, a - b*Sin[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, n} 
, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[m] && GtQ[a, 0] && GtQ[d/b, 0]
 

rule 3265

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*( 
x_)])^(m_), x_Symbol] :> Simp[(d/b)^IntPart[n]*((d*Sin[e + f*x])^FracPart[n 
]/(b*Sin[e + f*x])^FracPart[n])   Int[(a + b*Sin[e + f*x])^m*(b*Sin[e + f*x 
])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] &&  !I 
ntegerQ[m] && GtQ[a, 0] &&  !GtQ[d/b, 0]
 

rule 3266

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*( 
x_)])^(m_), x_Symbol] :> Simp[a^IntPart[m]*((a + b*Sin[e + f*x])^FracPart[m 
]/(1 + (b/a)*Sin[e + f*x])^FracPart[m])   Int[(1 + (b/a)*Sin[e + f*x])^m*(d 
*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^ 
2, 0] &&  !IntegerQ[m] &&  !GtQ[a, 0]
 

rule 3360

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + 
(b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[1/d^4   Int[(d*Sin[e 
 + f*x])^(n + 4)*(a + b*Sin[e + f*x])^m, x], x] + Int[(d*Sin[e + f*x])^n*(a 
 + b*Sin[e + f*x])^m*(1 - 2*Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, m, 
 n}, x] && EqQ[a^2 - b^2, 0] &&  !IGtQ[m, 0]
 

rule 3487

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + 
 (f_.)*(x_)])^(p_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] 
:> Simp[Sqrt[a + b*Sin[e + f*x]]*(Sqrt[c + d*Sin[e + f*x]]/(f*Cos[e + f*x]) 
)   Subst[Int[(a + b*x)^(m - 1/2)*(c + d*x)^(n - 1/2)*(A + B*x)^p, x], x, S 
in[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n, p}, x] && EqQ[b*c 
+ a*d, 0] && EqQ[a^2 - b^2, 0]
 

rule 3525

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_.)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> 
 Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1 
)/(d*f*(m + n + 2))), x] + Simp[1/(b*d*(m + n + 2))   Int[(a + b*Sin[e + f* 
x])^m*(c + d*Sin[e + f*x])^n*Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1 
)) + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, 
 e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 
 - d^2, 0] &&  !LtQ[m, -2^(-1)] && NeQ[m + n + 2, 0]
 

rule 4364

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + 
(a_))^(m_), x_Symbol] :> Simp[Sin[e + f*x]^FracPart[m]*((a + b*Csc[e + f*x] 
)^FracPart[m]/(b + a*Sin[e + f*x])^FracPart[m])   Int[(g*Cos[e + f*x])^p*(( 
b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x], x] /; FreeQ[{a, b, e, f, g, m, p 
}, x] && (EqQ[a^2 - b^2, 0] || IntegersQ[2*m, p])
 

3.2.52.4 Maple [F]

\[\int \left (a +a \sec \left (d x +c \right )\right )^{n} \sin \left (d x +c \right )^{4}d x\]

input

int((a+a*sec(d*x+c))^n*sin(d*x+c)^4,x)
 

output

int((a+a*sec(d*x+c))^n*sin(d*x+c)^4,x)
 

3.2.52.5 Fricas [F]

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

input

integrate((a+a*sec(d*x+c))^n*sin(d*x+c)^4,x, algorithm="fricas")
 

output

integral((cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*(a*sec(d*x + c) + a)^n, x 
)
 

3.2.52.6 Sympy [F(-1)]

Timed out. \[ \int (a+a \sec (c+d x))^n \sin ^4(c+d x) \, dx=\text {Timed out} \]

input

integrate((a+a*sec(d*x+c))**n*sin(d*x+c)**4,x)
 

output

Timed out
 

3.2.52.7 Maxima [F]

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

input

integrate((a+a*sec(d*x+c))^n*sin(d*x+c)^4,x, algorithm="maxima")
 

output

integrate((a*sec(d*x + c) + a)^n*sin(d*x + c)^4, x)
 

3.2.52.8 Giac [F]

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

input

integrate((a+a*sec(d*x+c))^n*sin(d*x+c)^4,x, algorithm="giac")
 

output

integrate((a*sec(d*x + c) + a)^n*sin(d*x + c)^4, x)
 

3.2.52.9 Mupad [F(-1)]

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

input

int(sin(c + d*x)^4*(a + a/cos(c + d*x))^n,x)
 

output

int(sin(c + d*x)^4*(a + a/cos(c + d*x))^n, x)