\(\int \frac {\cot ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx\) [385]

Optimal result

Integrand size = 23, antiderivative size = 71 \[ \int \frac {\cot ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {(a+b)^{3/2} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {(a+b) \cot (c+d x)}{a^2 d}-\frac {\cot ^3(c+d x)}{3 a d} \] Output:

(a+b)^(3/2)*arctan((a+b)^(1/2)*tan(d*x+c)/a^(1/2))/a^(5/2)/d+(a+b)*cot(d*x 
+c)/a^2/d-1/3*cot(d*x+c)^3/a/d
 

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.01 \[ \int \frac {\cot ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {3 (a+b)^{3/2} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )+\sqrt {a} \cot (c+d x) \left (4 a+3 b-a \csc ^2(c+d x)\right )}{3 a^{5/2} d} \] Input:

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

Output:

(3*(a + b)^(3/2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]] + Sqrt[a]*Cot[ 
c + d*x]*(4*a + 3*b - a*Csc[c + d*x]^2))/(3*a^(5/2)*d)
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 3674, 264, 264, 218}

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.

\(\displaystyle \int \frac {\cot ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\tan (c+d x)^4 \left (a+b \sin (c+d x)^2\right )}dx\)

\(\Big \downarrow \) 3674

\(\displaystyle \frac {\int \frac {\cot ^4(c+d x)}{(a+b) \tan ^2(c+d x)+a}d\tan (c+d x)}{d}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {-\frac {(a+b) \int \frac {\cot ^2(c+d x)}{(a+b) \tan ^2(c+d x)+a}d\tan (c+d x)}{a}-\frac {\cot ^3(c+d x)}{3 a}}{d}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {-\frac {(a+b) \left (-\frac {(a+b) \int \frac {1}{(a+b) \tan ^2(c+d x)+a}d\tan (c+d x)}{a}-\frac {\cot (c+d x)}{a}\right )}{a}-\frac {\cot ^3(c+d x)}{3 a}}{d}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {-\frac {(a+b) \left (-\frac {\sqrt {a+b} \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\cot (c+d x)}{a}\right )}{a}-\frac {\cot ^3(c+d x)}{3 a}}{d}\)

Input:

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

Output:

(-1/3*Cot[c + d*x]^3/a - ((a + b)*(-((Sqrt[a + b]*ArcTan[(Sqrt[a + b]*Tan[ 
c + d*x])/Sqrt[a]])/a^(3/2)) - Cot[c + d*x]/a))/a)/d
 

Defintions of rubi rules used

rule 218

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

rule 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 3042

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

rule 3674

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

Maple [A] (verified)

Time = 2.59 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.11

method	result	size
derivativedivides	\(\frac {-\frac {1}{3 a \tan \left (d x +c \right )^{3}}-\frac {-a -b}{a^{2} \tan \left (d x +c \right )}+\frac {\left (a^{2}+2 a b +b^{2}\right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{a^{2} \sqrt {a \left (a +b \right )}}}{d}\)	\(79\)
default	\(\frac {-\frac {1}{3 a \tan \left (d x +c \right )^{3}}-\frac {-a -b}{a^{2} \tan \left (d x +c \right )}+\frac {\left (a^{2}+2 a b +b^{2}\right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{a^{2} \sqrt {a \left (a +b \right )}}}{d}\)	\(79\)
risch	\(\frac {2 i \left (6 \,{\mathrm e}^{4 i \left (d x +c \right )} a +3 b \,{\mathrm e}^{4 i \left (d x +c \right )}-6 a \,{\mathrm e}^{2 i \left (d x +c \right )}-6 b \,{\mathrm e}^{2 i \left (d x +c \right )}+4 a +3 b \right )}{3 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}+\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {-a \left (a +b \right )}-2 a -b}{b}\right )}{2 a^{2} d}+\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i \sqrt {-a \left (a +b \right )}-2 a -b}{b}\right ) b}{2 a^{3} d}-\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {-a \left (a +b \right )}+2 a +b}{b}\right )}{2 a^{2} d}-\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i \sqrt {-a \left (a +b \right )}+2 a +b}{b}\right ) b}{2 a^{3} d}\)	\(275\)

Input:

int(cot(d*x+c)^4/(a+b*sin(d*x+c)^2),x,method=_RETURNVERBOSE)
 

Output:

1/d*(-1/3/a/tan(d*x+c)^3-1/a^2*(-a-b)/tan(d*x+c)+(a^2+2*a*b+b^2)/a^2/(a*(a 
+b))^(1/2)*arctan((a+b)*tan(d*x+c)/(a*(a+b))^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (61) = 122\).

Time = 0.10 (sec) , antiderivative size = 402, normalized size of antiderivative = 5.66 \[ \int \frac {\cot ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\left [\frac {4 \, {\left (4 \, a + 3 \, b\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left ({\left (a + b\right )} \cos \left (d x + c\right )^{2} - a - b\right )} \sqrt {-\frac {a + b}{a}} \log \left (\frac {{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (4 \, a^{2} + 5 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} - 4 \, {\left ({\left (2 \, a^{2} + a b\right )} \cos \left (d x + c\right )^{3} - {\left (a^{2} + a b\right )} \cos \left (d x + c\right )\right )} \sqrt {-\frac {a + b}{a}} \sin \left (d x + c\right ) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos \left (d x + c\right )^{4} - 2 \, {\left (a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}}\right ) \sin \left (d x + c\right ) - 12 \, {\left (a + b\right )} \cos \left (d x + c\right )}{12 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )} \sin \left (d x + c\right )}, \frac {2 \, {\left (4 \, a + 3 \, b\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left ({\left (a + b\right )} \cos \left (d x + c\right )^{2} - a - b\right )} \sqrt {\frac {a + b}{a}} \arctan \left (\frac {{\left ({\left (2 \, a + b\right )} \cos \left (d x + c\right )^{2} - a - b\right )} \sqrt {\frac {a + b}{a}}}{2 \, {\left (a + b\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) - 6 \, {\left (a + b\right )} \cos \left (d x + c\right )}{6 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )} \sin \left (d x + c\right )}\right ] \] Input:

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

Output:

[1/12*(4*(4*a + 3*b)*cos(d*x + c)^3 + 3*((a + b)*cos(d*x + c)^2 - a - b)*s 
qrt(-(a + b)/a)*log(((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^4 - 2*(4*a^2 + 5*a 
*b + b^2)*cos(d*x + c)^2 - 4*((2*a^2 + a*b)*cos(d*x + c)^3 - (a^2 + a*b)*c 
os(d*x + c))*sqrt(-(a + b)/a)*sin(d*x + c) + a^2 + 2*a*b + b^2)/(b^2*cos(d 
*x + c)^4 - 2*(a*b + b^2)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2))*sin(d*x + c 
) - 12*(a + b)*cos(d*x + c))/((a^2*d*cos(d*x + c)^2 - a^2*d)*sin(d*x + c)) 
, 1/6*(2*(4*a + 3*b)*cos(d*x + c)^3 - 3*((a + b)*cos(d*x + c)^2 - a - b)*s 
qrt((a + b)/a)*arctan(1/2*((2*a + b)*cos(d*x + c)^2 - a - b)*sqrt((a + b)/ 
a)/((a + b)*cos(d*x + c)*sin(d*x + c)))*sin(d*x + c) - 6*(a + b)*cos(d*x + 
 c))/((a^2*d*cos(d*x + c)^2 - a^2*d)*sin(d*x + c))]
 

Sympy [F]

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

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

Output:

Integral(cot(c + d*x)**4/(a + b*sin(c + d*x)**2), x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.07 \[ \int \frac {\cot ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {\frac {3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \arctan \left (\frac {{\left (a + b\right )} \tan \left (d x + c\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{\sqrt {{\left (a + b\right )} a} a^{2}} + \frac {3 \, {\left (a + b\right )} \tan \left (d x + c\right )^{2} - a}{a^{2} \tan \left (d x + c\right )^{3}}}{3 \, d} \] Input:

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

Output:

1/3*(3*(a^2 + 2*a*b + b^2)*arctan((a + b)*tan(d*x + c)/sqrt((a + b)*a))/(s 
qrt((a + b)*a)*a^2) + (3*(a + b)*tan(d*x + c)^2 - a)/(a^2*tan(d*x + c)^3)) 
/d
 

Giac [A] (verification not implemented)

Time = 0.61 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.69 \[ \int \frac {\cot ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {\frac {3 \, {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt {a^{2} + a b}}\right )\right )} {\left (a^{2} + 2 \, a b + b^{2}\right )}}{\sqrt {a^{2} + a b} a^{2}} + \frac {3 \, a \tan \left (d x + c\right )^{2} + 3 \, b \tan \left (d x + c\right )^{2} - a}{a^{2} \tan \left (d x + c\right )^{3}}}{3 \, d} \] Input:

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

Output:

1/3*(3*(pi*floor((d*x + c)/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(d*x + 
c) + b*tan(d*x + c))/sqrt(a^2 + a*b)))*(a^2 + 2*a*b + b^2)/(sqrt(a^2 + a*b 
)*a^2) + (3*a*tan(d*x + c)^2 + 3*b*tan(d*x + c)^2 - a)/(a^2*tan(d*x + c)^3 
))/d
 

Mupad [B] (verification not implemented)

Time = 34.73 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.90 \[ \int \frac {\cot ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {\mathrm {atan}\left (\frac {\mathrm {tan}\left (c+d\,x\right )\,\sqrt {a+b}}{\sqrt {a}}\right )\,{\left (a+b\right )}^{3/2}}{a^{5/2}\,d}-\frac {\frac {1}{3\,a}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (a+b\right )}{a^2}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^3} \] Input:

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

Output:

(atan((tan(c + d*x)*(a + b)^(1/2))/a^(1/2))*(a + b)^(3/2))/(a^(5/2)*d) - ( 
1/(3*a) - (tan(c + d*x)^2*(a + b))/a^2)/(d*tan(c + d*x)^3)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 1013, normalized size of antiderivative = 14.27 \[ \int \frac {\cot ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 6*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*at 
an((tan((c + d*x)/2)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*s 
in(c + d*x)**3*a - 6*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b 
) + a + 2*b)*atan((tan((c + d*x)/2)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) 
 + a + 2*b)))*sin(c + d*x)**3*b + 6*sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a 
 + 2*b)*atan((tan((c + d*x)/2)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a 
+ 2*b)))*sin(c + d*x)**3*a**2 + 12*sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a 
+ 2*b)*atan((tan((c + d*x)/2)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 
 2*b)))*sin(c + d*x)**3*a*b + 6*sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2 
*b)*atan((tan((c + d*x)/2)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2* 
b)))*sin(c + d*x)**3*b**2 + 8*cos(c + d*x)*sin(c + d*x)**2*a**3 + 6*cos(c 
+ d*x)*sin(c + d*x)**2*a**2*b - 2*cos(c + d*x)*a**3 - 3*sqrt(b)*sqrt(a)*sq 
rt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b)*log( - sqrt(2*sqrt(b)*sqrt 
(a + b) - a - 2*b) + sqrt(a)*tan((c + d*x)/2))*sin(c + d*x)**3*a - 3*sqrt( 
b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b)*log( - sqrt(2 
*sqrt(b)*sqrt(a + b) - a - 2*b) + sqrt(a)*tan((c + d*x)/2))*sin(c + d*x)** 
3*b + 3*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b)* 
log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + sqrt(a)*tan((c + d*x)/2))*sin( 
c + d*x)**3*a + 3*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) - 
 a - 2*b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + sqrt(a)*tan((c + ...