\(\int \cot ^4(c+d x) (a+b \sin ^2(c+d x))^p \, dx\) [482]

Optimal result

Integrand size = 23, antiderivative size = 101 \[ \int \cot ^4(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=-\frac {\operatorname {AppellF1}\left (-\frac {3}{2},-\frac {3}{2},-p,-\frac {1}{2},\sin ^2(c+d x),-\frac {b \sin ^2(c+d x)}{a}\right ) \sqrt {\cos ^2(c+d x)} \csc ^3(c+d x) \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (1+\frac {b \sin ^2(c+d x)}{a}\right )^{-p}}{3 d} \] Output:

-1/3*AppellF1(-3/2,-3/2,-p,-1/2,sin(d*x+c)^2,-b*sin(d*x+c)^2/a)*(cos(d*x+c 
)^2)^(1/2)*csc(d*x+c)^3*sec(d*x+c)*(a+b*sin(d*x+c)^2)^p/d/((1+b*sin(d*x+c) 
^2/a)^p)
 

Mathematica [A] (verified)

Time = 3.74 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.01 \[ \int \cot ^4(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=-\frac {\operatorname {AppellF1}\left (-\frac {3}{2},-\frac {3}{2},-p,-\frac {1}{2},\sin ^2(c+d x),-\frac {b \sin ^2(c+d x)}{a}\right ) \sqrt {\cos ^2(c+d x)} \csc ^3(c+d x) \sec (c+d x) \left (a+b \sin ^2(c+d x)\right )^p \left (\frac {a+b \sin ^2(c+d x)}{a}\right )^{-p}}{3 d} \] Input:

Integrate[Cot[c + d*x]^4*(a + b*Sin[c + d*x]^2)^p,x]
 

Output:

-1/3*(AppellF1[-3/2, -3/2, -p, -1/2, Sin[c + d*x]^2, -((b*Sin[c + d*x]^2)/ 
a)]*Sqrt[Cos[c + d*x]^2]*Csc[c + d*x]^3*Sec[c + d*x]*(a + b*Sin[c + d*x]^2 
)^p)/(d*((a + b*Sin[c + d*x]^2)/a)^p)
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 3675, 395, 394}

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.

\(\displaystyle \int \cot ^4(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a+b \sin (c+d x)^2\right )^p}{\tan (c+d x)^4}dx\)

\(\Big \downarrow \) 3675

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

\(\Big \downarrow \) 395

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

\(\Big \downarrow \) 394

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

Input:

Int[Cot[c + d*x]^4*(a + b*Sin[c + d*x]^2)^p,x]
 

Output:

-1/3*(AppellF1[-3/2, -3/2, -p, -1/2, Sin[c + d*x]^2, -((b*Sin[c + d*x]^2)/ 
a)]*Sqrt[Cos[c + d*x]^2]*Csc[c + d*x]^3*Sec[c + d*x]*(a + b*Sin[c + d*x]^2 
)^p)/(d*(1 + (b*Sin[c + d*x]^2)/a)^p)
 

Defintions of rubi rules used

rule 394

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/2 
, -p, -q, 1 + (m + 1)/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; FreeQ[{a, b, c, 
 d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 1] && (Int 
egerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 395

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^ 
FracPart[p])   Int[(e*x)^m*(1 + b*(x^2/a))^p*(c + d*x^2)^q, x], x] /; FreeQ 
[{a, b, c, d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 
1] &&  !(IntegerQ[p] || GtQ[a, 0])
 

rule 3042

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

rule 3675

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^ 
(m_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff^(m + 1 
)*(Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x]))   Subst[Int[x^m*((a + b*ff^2*x^2) 
^p/(1 - ff^2*x^2)^((m + 1)/2)), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b 
, e, f, p}, x] && IntegerQ[m/2] &&  !IntegerQ[p]
 

Maple [F]

\[\int \cot \left (d x +c \right )^{4} \left (a +b \sin \left (d x +c \right )^{2}\right )^{p}d x\]

Input:

int(cot(d*x+c)^4*(a+b*sin(d*x+c)^2)^p,x)
 

Output:

int(cot(d*x+c)^4*(a+b*sin(d*x+c)^2)^p,x)
 

Fricas [F]

\[ \int \cot ^4(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \cot \left (d x + c\right )^{4} \,d x } \] Input:

integrate(cot(d*x+c)^4*(a+b*sin(d*x+c)^2)^p,x, algorithm="fricas")
 

Output:

integral((-b*cos(d*x + c)^2 + a + b)^p*cot(d*x + c)^4, x)
 

Sympy [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=\text {Timed out} \] Input:

integrate(cot(d*x+c)**4*(a+b*sin(d*x+c)**2)**p,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \cot ^4(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \cot \left (d x + c\right )^{4} \,d x } \] Input:

integrate(cot(d*x+c)^4*(a+b*sin(d*x+c)^2)^p,x, algorithm="maxima")
 

Output:

integrate((b*sin(d*x + c)^2 + a)^p*cot(d*x + c)^4, x)
 

Giac [F]

\[ \int \cot ^4(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{2} + a\right )}^{p} \cot \left (d x + c\right )^{4} \,d x } \] Input:

integrate(cot(d*x+c)^4*(a+b*sin(d*x+c)^2)^p,x, algorithm="giac")
 

Output:

integrate((b*sin(d*x + c)^2 + a)^p*cot(d*x + c)^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^4\,{\left (b\,{\sin \left (c+d\,x\right )}^2+a\right )}^p \,d x \] Input:

int(cot(c + d*x)^4*(a + b*sin(c + d*x)^2)^p,x)
 

Output:

int(cot(c + d*x)^4*(a + b*sin(c + d*x)^2)^p, x)
 

Reduce [F]

\[ \int \cot ^4(c+d x) \left (a+b \sin ^2(c+d x)\right )^p \, dx=\int \left (\sin \left (d x +c \right )^{2} b +a \right )^{p} \cot \left (d x +c \right )^{4}d x \] Input:

int(cot(d*x+c)^4*(a+b*sin(d*x+c)^2)^p,x)
 

Output:

int((sin(c + d*x)**2*b + a)**p*cot(c + d*x)**4,x)