\(\int \frac {1}{(a+b \sin ^2(x))^4} \, dx\) [139]

Optimal result

Integrand size = 10, antiderivative size = 154 \[ \int \frac {1}{\left (a+b \sin ^2(x)\right )^4} \, dx=\frac {(2 a+b) \left (8 a^2+8 a b+5 b^2\right ) \arctan \left (\frac {\sqrt {a+b} \tan (x)}{\sqrt {a}}\right )}{16 a^{7/2} (a+b)^{7/2}}+\frac {b \cos (x) \sin (x)}{6 a (a+b) \left (a+b \sin ^2(x)\right )^3}+\frac {5 b (2 a+b) \cos (x) \sin (x)}{24 a^2 (a+b)^2 \left (a+b \sin ^2(x)\right )^2}+\frac {b \left (44 a^2+44 a b+15 b^2\right ) \cos (x) \sin (x)}{48 a^3 (a+b)^3 \left (a+b \sin ^2(x)\right )} \] Output:

1/16*(2*a+b)*(8*a^2+8*a*b+5*b^2)*arctan((a+b)^(1/2)*tan(x)/a^(1/2))/a^(7/2 
)/(a+b)^(7/2)+1/6*b*cos(x)*sin(x)/a/(a+b)/(a+b*sin(x)^2)^3+5/24*b*(2*a+b)* 
cos(x)*sin(x)/a^2/(a+b)^2/(a+b*sin(x)^2)^2+1/48*b*(44*a^2+44*a*b+15*b^2)*c 
os(x)*sin(x)/a^3/(a+b)^3/(a+b*sin(x)^2)
 

Mathematica [A] (verified)

Time = 6.13 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (a+b \sin ^2(x)\right )^4} \, dx=\frac {\frac {3 \left (16 a^3+24 a^2 b+18 a b^2+5 b^3\right ) \arctan \left (\frac {\sqrt {a+b} \tan (x)}{\sqrt {a}}\right )}{(a+b)^{7/2}}+\frac {32 a^{5/2} b \sin (2 x)}{(a+b) (2 a+b-b \cos (2 x))^3}+\frac {20 a^{3/2} b (2 a+b) \sin (2 x)}{(a+b)^2 (2 a+b-b \cos (2 x))^2}+\frac {\sqrt {a} b \left (44 a^2+44 a b+15 b^2\right ) \sin (2 x)}{(a+b)^3 (2 a+b-b \cos (2 x))}}{48 a^{7/2}} \] Input:

Integrate[(a + b*Sin[x]^2)^(-4),x]
 

Output:

((3*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*ArcTan[(Sqrt[a + b]*Tan[x])/Sqr 
t[a]])/(a + b)^(7/2) + (32*a^(5/2)*b*Sin[2*x])/((a + b)*(2*a + b - b*Cos[2 
*x])^3) + (20*a^(3/2)*b*(2*a + b)*Sin[2*x])/((a + b)^2*(2*a + b - b*Cos[2* 
x])^2) + (Sqrt[a]*b*(44*a^2 + 44*a*b + 15*b^2)*Sin[2*x])/((a + b)^3*(2*a + 
 b - b*Cos[2*x])))/(48*a^(7/2))
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.17, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {3042, 3663, 25, 3042, 3652, 3042, 3652, 27, 3042, 3660, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3663

\(\displaystyle \frac {b \sin (x) \cos (x)}{6 a (a+b) \left (a+b \sin ^2(x)\right )^3}-\frac {\int -\frac {-4 b \sin ^2(x)+6 a+5 b}{\left (b \sin ^2(x)+a\right )^3}dx}{6 a (a+b)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {-4 b \sin ^2(x)+6 a+5 b}{\left (b \sin ^2(x)+a\right )^3}dx}{6 a (a+b)}+\frac {b \sin (x) \cos (x)}{6 a (a+b) \left (a+b \sin ^2(x)\right )^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {-4 b \sin (x)^2+6 a+5 b}{\left (b \sin (x)^2+a\right )^3}dx}{6 a (a+b)}+\frac {b \sin (x) \cos (x)}{6 a (a+b) \left (a+b \sin ^2(x)\right )^3}\)

\(\Big \downarrow \) 3652

\(\displaystyle \frac {\frac {\int \frac {24 a^2+34 b a+15 b^2-10 b (2 a+b) \sin ^2(x)}{\left (b \sin ^2(x)+a\right )^2}dx}{4 a (a+b)}+\frac {5 b (2 a+b) \sin (x) \cos (x)}{4 a (a+b) \left (a+b \sin ^2(x)\right )^2}}{6 a (a+b)}+\frac {b \sin (x) \cos (x)}{6 a (a+b) \left (a+b \sin ^2(x)\right )^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {24 a^2+34 b a+15 b^2-10 b (2 a+b) \sin (x)^2}{\left (b \sin (x)^2+a\right )^2}dx}{4 a (a+b)}+\frac {5 b (2 a+b) \sin (x) \cos (x)}{4 a (a+b) \left (a+b \sin ^2(x)\right )^2}}{6 a (a+b)}+\frac {b \sin (x) \cos (x)}{6 a (a+b) \left (a+b \sin ^2(x)\right )^3}\)

\(\Big \downarrow \) 3652

\(\displaystyle \frac {\frac {\frac {\int \frac {3 (2 a+b) \left (8 a^2+8 b a+5 b^2\right )}{b \sin ^2(x)+a}dx}{2 a (a+b)}+\frac {b \left (44 a^2+44 a b+15 b^2\right ) \sin (x) \cos (x)}{2 a (a+b) \left (a+b \sin ^2(x)\right )}}{4 a (a+b)}+\frac {5 b (2 a+b) \sin (x) \cos (x)}{4 a (a+b) \left (a+b \sin ^2(x)\right )^2}}{6 a (a+b)}+\frac {b \sin (x) \cos (x)}{6 a (a+b) \left (a+b \sin ^2(x)\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {3 (2 a+b) \left (8 a^2+8 a b+5 b^2\right ) \int \frac {1}{b \sin ^2(x)+a}dx}{2 a (a+b)}+\frac {b \left (44 a^2+44 a b+15 b^2\right ) \sin (x) \cos (x)}{2 a (a+b) \left (a+b \sin ^2(x)\right )}}{4 a (a+b)}+\frac {5 b (2 a+b) \sin (x) \cos (x)}{4 a (a+b) \left (a+b \sin ^2(x)\right )^2}}{6 a (a+b)}+\frac {b \sin (x) \cos (x)}{6 a (a+b) \left (a+b \sin ^2(x)\right )^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {3 (2 a+b) \left (8 a^2+8 a b+5 b^2\right ) \int \frac {1}{b \sin (x)^2+a}dx}{2 a (a+b)}+\frac {b \left (44 a^2+44 a b+15 b^2\right ) \sin (x) \cos (x)}{2 a (a+b) \left (a+b \sin ^2(x)\right )}}{4 a (a+b)}+\frac {5 b (2 a+b) \sin (x) \cos (x)}{4 a (a+b) \left (a+b \sin ^2(x)\right )^2}}{6 a (a+b)}+\frac {b \sin (x) \cos (x)}{6 a (a+b) \left (a+b \sin ^2(x)\right )^3}\)

\(\Big \downarrow \) 3660

\(\displaystyle \frac {\frac {\frac {3 (2 a+b) \left (8 a^2+8 a b+5 b^2\right ) \int \frac {1}{(a+b) \tan ^2(x)+a}d\tan (x)}{2 a (a+b)}+\frac {b \left (44 a^2+44 a b+15 b^2\right ) \sin (x) \cos (x)}{2 a (a+b) \left (a+b \sin ^2(x)\right )}}{4 a (a+b)}+\frac {5 b (2 a+b) \sin (x) \cos (x)}{4 a (a+b) \left (a+b \sin ^2(x)\right )^2}}{6 a (a+b)}+\frac {b \sin (x) \cos (x)}{6 a (a+b) \left (a+b \sin ^2(x)\right )^3}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\frac {b \left (44 a^2+44 a b+15 b^2\right ) \sin (x) \cos (x)}{2 a (a+b) \left (a+b \sin ^2(x)\right )}+\frac {3 (2 a+b) \left (8 a^2+8 a b+5 b^2\right ) \arctan \left (\frac {\sqrt {a+b} \tan (x)}{\sqrt {a}}\right )}{2 a^{3/2} (a+b)^{3/2}}}{4 a (a+b)}+\frac {5 b (2 a+b) \sin (x) \cos (x)}{4 a (a+b) \left (a+b \sin ^2(x)\right )^2}}{6 a (a+b)}+\frac {b \sin (x) \cos (x)}{6 a (a+b) \left (a+b \sin ^2(x)\right )^3}\)

Input:

Int[(a + b*Sin[x]^2)^(-4),x]
 

Output:

(b*Cos[x]*Sin[x])/(6*a*(a + b)*(a + b*Sin[x]^2)^3) + ((5*b*(2*a + b)*Cos[x 
]*Sin[x])/(4*a*(a + b)*(a + b*Sin[x]^2)^2) + ((3*(2*a + b)*(8*a^2 + 8*a*b 
+ 5*b^2)*ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a]])/(2*a^(3/2)*(a + b)^(3/2)) + 
 (b*(44*a^2 + 44*a*b + 15*b^2)*Cos[x]*Sin[x])/(2*a*(a + b)*(a + b*Sin[x]^2 
)))/(4*a*(a + b)))/(6*a*(a + b))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 3042

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

rule 3652

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b - a*B))*Cos[e + f*x]*Sin[e + f*x 
]*((a + b*Sin[e + f*x]^2)^(p + 1)/(2*a*f*(a + b)*(p + 1))), x] - Simp[1/(2* 
a*(a + b)*(p + 1))   Int[(a + b*Sin[e + f*x]^2)^(p + 1)*Simp[a*B - A*(2*a*( 
p + 1) + b*(2*p + 3)) + 2*(A*b - a*B)*(p + 2)*Sin[e + f*x]^2, x], x], x] /; 
 FreeQ[{a, b, e, f, A, B}, x] && LtQ[p, -1] && NeQ[a + b, 0]
 

rule 3660

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = 
FreeFactors[Tan[e + f*x], x]}, Simp[ff/f   Subst[Int[1/(a + (a + b)*ff^2*x^ 
2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]
 

rule 3663

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(-b)*C 
os[e + f*x]*Sin[e + f*x]*((a + b*Sin[e + f*x]^2)^(p + 1)/(2*a*f*(p + 1)*(a 
+ b))), x] + Simp[1/(2*a*(p + 1)*(a + b))   Int[(a + b*Sin[e + f*x]^2)^(p + 
 1)*Simp[2*a*(p + 1) + b*(2*p + 3) - 2*b*(p + 2)*Sin[e + f*x]^2, x], x], x] 
 /; FreeQ[{a, b, e, f}, x] && NeQ[a + b, 0] && LtQ[p, -1]
 

Maple [A] (verified)

Time = 1.51 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.31

method	result	size
default	\(\frac {\frac {\left (24 a^{2}+18 b a +5 b^{2}\right ) b \tan \left (x \right )^{5}}{16 a^{3} \left (a +b \right )}+\frac {\left (18 a^{2}+18 b a +5 b^{2}\right ) b \tan \left (x \right )^{3}}{6 a^{2} \left (a^{2}+2 b a +b^{2}\right )}+\frac {b \left (24 a^{2}+30 b a +11 b^{2}\right ) \tan \left (x \right )}{16 a \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}}{\left (\tan \left (x \right )^{2} a +b \tan \left (x \right )^{2}+a \right )^{3}}+\frac {\left (16 a^{3}+24 a^{2} b +18 b^{2} a +5 b^{3}\right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (x \right )}{\sqrt {a \left (a +b \right )}}\right )}{16 a^{3} \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right ) \sqrt {a \left (a +b \right )}}\)	\(202\)
risch	\(-\frac {i \left (-44 a^{2} b^{3}-44 a \,b^{4}-15 b^{5}+4176 a^{3} b^{2} {\mathrm e}^{6 i x}+2744 a^{2} b^{3} {\mathrm e}^{6 i x}-3264 a^{3} b^{2} {\mathrm e}^{4 i x}-2592 a^{2} b^{3} {\mathrm e}^{4 i x}+980 a \,b^{4} {\mathrm e}^{6 i x}-1632 a^{4} b \,{\mathrm e}^{4 i x}-960 a \,b^{4} {\mathrm e}^{4 i x}+480 a^{3} b^{2} {\mathrm e}^{2 i x}+720 a^{2} b^{3} {\mathrm e}^{2 i x}+390 a \,b^{4} {\mathrm e}^{2 i x}+48 a^{3} b^{2} {\mathrm e}^{10 i x}+72 a^{2} b^{3} {\mathrm e}^{10 i x}+54 a \,b^{4} {\mathrm e}^{10 i x}-480 a^{4} b \,{\mathrm e}^{8 i x}-960 a^{3} b^{2} {\mathrm e}^{8 i x}-900 a^{2} b^{3} {\mathrm e}^{8 i x}-420 a \,b^{4} {\mathrm e}^{8 i x}+3520 a^{4} b \,{\mathrm e}^{6 i x}+75 b^{5} {\mathrm e}^{2 i x}-75 b^{5} {\mathrm e}^{8 i x}+15 b^{5} {\mathrm e}^{10 i x}+1408 a^{5} {\mathrm e}^{6 i x}-150 b^{5} {\mathrm e}^{4 i x}+150 b^{5} {\mathrm e}^{6 i x}\right )}{24 a^{3} \left (a +b \right )^{3} \left (-{\mathrm e}^{4 i x} b +4 \,{\mathrm e}^{2 i x} a +2 \,{\mathrm e}^{2 i x} b -b \right )^{3}}-\frac {\ln \left ({\mathrm e}^{2 i x}-\frac {2 i a^{2}+2 i a b +2 a \sqrt {-a^{2}-b a}+b \sqrt {-a^{2}-b a}}{b \sqrt {-a^{2}-b a}}\right )}{2 \sqrt {-a^{2}-b a}\, \left (a +b \right )^{3}}-\frac {3 \ln \left ({\mathrm e}^{2 i x}-\frac {2 i a^{2}+2 i a b +2 a \sqrt {-a^{2}-b a}+b \sqrt {-a^{2}-b a}}{b \sqrt {-a^{2}-b a}}\right ) b}{4 \sqrt {-a^{2}-b a}\, \left (a +b \right )^{3} a}-\frac {9 \ln \left ({\mathrm e}^{2 i x}-\frac {2 i a^{2}+2 i a b +2 a \sqrt {-a^{2}-b a}+b \sqrt {-a^{2}-b a}}{b \sqrt {-a^{2}-b a}}\right ) b^{2}}{16 \sqrt {-a^{2}-b a}\, \left (a +b \right )^{3} a^{2}}-\frac {5 \ln \left ({\mathrm e}^{2 i x}-\frac {2 i a^{2}+2 i a b +2 a \sqrt {-a^{2}-b a}+b \sqrt {-a^{2}-b a}}{b \sqrt {-a^{2}-b a}}\right ) b^{3}}{32 \sqrt {-a^{2}-b a}\, \left (a +b \right )^{3} a^{3}}+\frac {\ln \left ({\mathrm e}^{2 i x}+\frac {2 i a^{2}+2 i a b -2 a \sqrt {-a^{2}-b a}-b \sqrt {-a^{2}-b a}}{b \sqrt {-a^{2}-b a}}\right )}{2 \sqrt {-a^{2}-b a}\, \left (a +b \right )^{3}}+\frac {3 \ln \left ({\mathrm e}^{2 i x}+\frac {2 i a^{2}+2 i a b -2 a \sqrt {-a^{2}-b a}-b \sqrt {-a^{2}-b a}}{b \sqrt {-a^{2}-b a}}\right ) b}{4 \sqrt {-a^{2}-b a}\, \left (a +b \right )^{3} a}+\frac {9 \ln \left ({\mathrm e}^{2 i x}+\frac {2 i a^{2}+2 i a b -2 a \sqrt {-a^{2}-b a}-b \sqrt {-a^{2}-b a}}{b \sqrt {-a^{2}-b a}}\right ) b^{2}}{16 \sqrt {-a^{2}-b a}\, \left (a +b \right )^{3} a^{2}}+\frac {5 \ln \left ({\mathrm e}^{2 i x}+\frac {2 i a^{2}+2 i a b -2 a \sqrt {-a^{2}-b a}-b \sqrt {-a^{2}-b a}}{b \sqrt {-a^{2}-b a}}\right ) b^{3}}{32 \sqrt {-a^{2}-b a}\, \left (a +b \right )^{3} a^{3}}\)	\(1045\)

Input:

int(1/(a+b*sin(x)^2)^4,x,method=_RETURNVERBOSE)
 

Output:

(1/16*(24*a^2+18*a*b+5*b^2)/a^3*b/(a+b)*tan(x)^5+1/6*(18*a^2+18*a*b+5*b^2) 
/a^2*b/(a^2+2*a*b+b^2)*tan(x)^3+1/16*b*(24*a^2+30*a*b+11*b^2)/a/(a^3+3*a^2 
*b+3*a*b^2+b^3)*tan(x))/(tan(x)^2*a+b*tan(x)^2+a)^3+1/16*(16*a^3+24*a^2*b+ 
18*a*b^2+5*b^3)/a^3/(a^3+3*a^2*b+3*a*b^2+b^3)/(a*(a+b))^(1/2)*arctan((a+b) 
*tan(x)/(a*(a+b))^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 572 vs. \(2 (138) = 276\).

Time = 0.14 (sec) , antiderivative size = 1229, normalized size of antiderivative = 7.98 \[ \int \frac {1}{\left (a+b \sin ^2(x)\right )^4} \, dx=\text {Too large to display} \] Input:

integrate(1/(a+b*sin(x)^2)^4,x, algorithm="fricas")
 

Output:

[1/192*(3*((16*a^3*b^3 + 24*a^2*b^4 + 18*a*b^5 + 5*b^6)*cos(x)^6 - 16*a^6 
- 72*a^5*b - 138*a^4*b^2 - 147*a^3*b^3 - 93*a^2*b^4 - 33*a*b^5 - 5*b^6 - 3 
*(16*a^4*b^2 + 40*a^3*b^3 + 42*a^2*b^4 + 23*a*b^5 + 5*b^6)*cos(x)^4 + 3*(1 
6*a^5*b + 56*a^4*b^2 + 82*a^3*b^3 + 65*a^2*b^4 + 28*a*b^5 + 5*b^6)*cos(x)^ 
2)*sqrt(-a^2 - a*b)*log(((8*a^2 + 8*a*b + b^2)*cos(x)^4 - 2*(4*a^2 + 5*a*b 
 + b^2)*cos(x)^2 + 4*((2*a + b)*cos(x)^3 - (a + b)*cos(x))*sqrt(-a^2 - a*b 
)*sin(x) + a^2 + 2*a*b + b^2)/(b^2*cos(x)^4 - 2*(a*b + b^2)*cos(x)^2 + a^2 
 + 2*a*b + b^2)) + 4*((44*a^4*b^3 + 88*a^3*b^4 + 59*a^2*b^5 + 15*a*b^6)*co 
s(x)^5 - 2*(54*a^5*b^2 + 157*a^4*b^3 + 167*a^3*b^4 + 79*a^2*b^5 + 15*a*b^6 
)*cos(x)^3 + 3*(24*a^6*b + 90*a^5*b^2 + 131*a^4*b^3 + 93*a^3*b^4 + 33*a^2* 
b^5 + 5*a*b^6)*cos(x))*sin(x))/(a^11 + 7*a^10*b + 21*a^9*b^2 + 35*a^8*b^3 
+ 35*a^7*b^4 + 21*a^6*b^5 + 7*a^5*b^6 + a^4*b^7 - (a^8*b^3 + 4*a^7*b^4 + 6 
*a^6*b^5 + 4*a^5*b^6 + a^4*b^7)*cos(x)^6 + 3*(a^9*b^2 + 5*a^8*b^3 + 10*a^7 
*b^4 + 10*a^6*b^5 + 5*a^5*b^6 + a^4*b^7)*cos(x)^4 - 3*(a^10*b + 6*a^9*b^2 
+ 15*a^8*b^3 + 20*a^7*b^4 + 15*a^6*b^5 + 6*a^5*b^6 + a^4*b^7)*cos(x)^2), 1 
/96*(3*((16*a^3*b^3 + 24*a^2*b^4 + 18*a*b^5 + 5*b^6)*cos(x)^6 - 16*a^6 - 7 
2*a^5*b - 138*a^4*b^2 - 147*a^3*b^3 - 93*a^2*b^4 - 33*a*b^5 - 5*b^6 - 3*(1 
6*a^4*b^2 + 40*a^3*b^3 + 42*a^2*b^4 + 23*a*b^5 + 5*b^6)*cos(x)^4 + 3*(16*a 
^5*b + 56*a^4*b^2 + 82*a^3*b^3 + 65*a^2*b^4 + 28*a*b^5 + 5*b^6)*cos(x)^2)* 
sqrt(a^2 + a*b)*arctan(1/2*((2*a + b)*cos(x)^2 - a - b)/(sqrt(a^2 + a*b...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b \sin ^2(x)\right )^4} \, dx=\text {Timed out} \] Input:

integrate(1/(a+b*sin(x)**2)**4,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 346 vs. \(2 (138) = 276\).

Time = 0.12 (sec) , antiderivative size = 346, normalized size of antiderivative = 2.25 \[ \int \frac {1}{\left (a+b \sin ^2(x)\right )^4} \, dx=\frac {{\left (16 \, a^{3} + 24 \, a^{2} b + 18 \, a b^{2} + 5 \, b^{3}\right )} \arctan \left (\frac {{\left (a + b\right )} \tan \left (x\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{16 \, {\left (a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right )} \sqrt {{\left (a + b\right )} a}} + \frac {3 \, {\left (24 \, a^{4} b + 66 \, a^{3} b^{2} + 65 \, a^{2} b^{3} + 28 \, a b^{4} + 5 \, b^{5}\right )} \tan \left (x\right )^{5} + 8 \, {\left (18 \, a^{4} b + 36 \, a^{3} b^{2} + 23 \, a^{2} b^{3} + 5 \, a b^{4}\right )} \tan \left (x\right )^{3} + 3 \, {\left (24 \, a^{4} b + 30 \, a^{3} b^{2} + 11 \, a^{2} b^{3}\right )} \tan \left (x\right )}{48 \, {\left (a^{9} + 3 \, a^{8} b + 3 \, a^{7} b^{2} + a^{6} b^{3} + {\left (a^{9} + 6 \, a^{8} b + 15 \, a^{7} b^{2} + 20 \, a^{6} b^{3} + 15 \, a^{5} b^{4} + 6 \, a^{4} b^{5} + a^{3} b^{6}\right )} \tan \left (x\right )^{6} + 3 \, {\left (a^{9} + 5 \, a^{8} b + 10 \, a^{7} b^{2} + 10 \, a^{6} b^{3} + 5 \, a^{5} b^{4} + a^{4} b^{5}\right )} \tan \left (x\right )^{4} + 3 \, {\left (a^{9} + 4 \, a^{8} b + 6 \, a^{7} b^{2} + 4 \, a^{6} b^{3} + a^{5} b^{4}\right )} \tan \left (x\right )^{2}\right )}} \] Input:

integrate(1/(a+b*sin(x)^2)^4,x, algorithm="maxima")
 

Output:

1/16*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*arctan((a + b)*tan(x)/sqrt((a 
+ b)*a))/((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*sqrt((a + b)*a)) + 1/48*(3 
*(24*a^4*b + 66*a^3*b^2 + 65*a^2*b^3 + 28*a*b^4 + 5*b^5)*tan(x)^5 + 8*(18* 
a^4*b + 36*a^3*b^2 + 23*a^2*b^3 + 5*a*b^4)*tan(x)^3 + 3*(24*a^4*b + 30*a^3 
*b^2 + 11*a^2*b^3)*tan(x))/(a^9 + 3*a^8*b + 3*a^7*b^2 + a^6*b^3 + (a^9 + 6 
*a^8*b + 15*a^7*b^2 + 20*a^6*b^3 + 15*a^5*b^4 + 6*a^4*b^5 + a^3*b^6)*tan(x 
)^6 + 3*(a^9 + 5*a^8*b + 10*a^7*b^2 + 10*a^6*b^3 + 5*a^5*b^4 + a^4*b^5)*ta 
n(x)^4 + 3*(a^9 + 4*a^8*b + 6*a^7*b^2 + 4*a^6*b^3 + a^5*b^4)*tan(x)^2)
 

Giac [A] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.77 \[ \int \frac {1}{\left (a+b \sin ^2(x)\right )^4} \, dx=\frac {{\left (16 \, a^{3} + 24 \, a^{2} b + 18 \, a b^{2} + 5 \, b^{3}\right )} {\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (x\right ) + b \tan \left (x\right )}{\sqrt {a^{2} + a b}}\right )\right )}}{16 \, {\left (a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right )} \sqrt {a^{2} + a b}} + \frac {72 \, a^{4} b \tan \left (x\right )^{5} + 198 \, a^{3} b^{2} \tan \left (x\right )^{5} + 195 \, a^{2} b^{3} \tan \left (x\right )^{5} + 84 \, a b^{4} \tan \left (x\right )^{5} + 15 \, b^{5} \tan \left (x\right )^{5} + 144 \, a^{4} b \tan \left (x\right )^{3} + 288 \, a^{3} b^{2} \tan \left (x\right )^{3} + 184 \, a^{2} b^{3} \tan \left (x\right )^{3} + 40 \, a b^{4} \tan \left (x\right )^{3} + 72 \, a^{4} b \tan \left (x\right ) + 90 \, a^{3} b^{2} \tan \left (x\right ) + 33 \, a^{2} b^{3} \tan \left (x\right )}{48 \, {\left (a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right )} {\left (a \tan \left (x\right )^{2} + b \tan \left (x\right )^{2} + a\right )}^{3}} \] Input:

integrate(1/(a+b*sin(x)^2)^4,x, algorithm="giac")
 

Output:

1/16*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*(pi*floor(x/pi + 1/2)*sgn(2*a 
+ 2*b) + arctan((a*tan(x) + b*tan(x))/sqrt(a^2 + a*b)))/((a^6 + 3*a^5*b + 
3*a^4*b^2 + a^3*b^3)*sqrt(a^2 + a*b)) + 1/48*(72*a^4*b*tan(x)^5 + 198*a^3* 
b^2*tan(x)^5 + 195*a^2*b^3*tan(x)^5 + 84*a*b^4*tan(x)^5 + 15*b^5*tan(x)^5 
+ 144*a^4*b*tan(x)^3 + 288*a^3*b^2*tan(x)^3 + 184*a^2*b^3*tan(x)^3 + 40*a* 
b^4*tan(x)^3 + 72*a^4*b*tan(x) + 90*a^3*b^2*tan(x) + 33*a^2*b^3*tan(x))/(( 
a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*(a*tan(x)^2 + b*tan(x)^2 + a)^3)
 

Mupad [B] (verification not implemented)

Time = 36.43 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.98 \[ \int \frac {1}{\left (a+b \sin ^2(x)\right )^4} \, dx=\frac {\frac {{\mathrm {tan}\left (x\right )}^3\,\left (18\,a^2\,b+18\,a\,b^2+5\,b^3\right )}{6\,a^2\,\left (a^2+2\,a\,b+b^2\right )}+\frac {{\mathrm {tan}\left (x\right )}^5\,\left (24\,a^2\,b+18\,a\,b^2+5\,b^3\right )}{16\,a^3\,\left (a+b\right )}+\frac {\mathrm {tan}\left (x\right )\,\left (24\,a^2\,b+30\,a\,b^2+11\,b^3\right )}{16\,a\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}}{{\mathrm {tan}\left (x\right )}^4\,\left (3\,a^3+6\,a^2\,b+3\,a\,b^2\right )+{\mathrm {tan}\left (x\right )}^6\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )+{\mathrm {tan}\left (x\right )}^2\,\left (3\,a^3+3\,b\,a^2\right )+a^3}+\frac {\mathrm {atan}\left (\frac {\mathrm {tan}\left (x\right )\,\left (2\,a+b\right )\,\left (2\,a+2\,b\right )\,\left (8\,a^2+8\,a\,b+5\,b^2\right )\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}{2\,\sqrt {a}\,{\left (a+b\right )}^{7/2}\,\left (16\,a^3+24\,a^2\,b+18\,a\,b^2+5\,b^3\right )}\right )\,\left (2\,a+b\right )\,\left (8\,a^2+8\,a\,b+5\,b^2\right )}{16\,a^{7/2}\,{\left (a+b\right )}^{7/2}} \] Input:

int(1/(a + b*sin(x)^2)^4,x)
 

Output:

((tan(x)^3*(18*a*b^2 + 18*a^2*b + 5*b^3))/(6*a^2*(2*a*b + a^2 + b^2)) + (t 
an(x)^5*(18*a*b^2 + 24*a^2*b + 5*b^3))/(16*a^3*(a + b)) + (tan(x)*(30*a*b^ 
2 + 24*a^2*b + 11*b^3))/(16*a*(3*a*b^2 + 3*a^2*b + a^3 + b^3)))/(tan(x)^4* 
(3*a*b^2 + 6*a^2*b + 3*a^3) + tan(x)^6*(3*a*b^2 + 3*a^2*b + a^3 + b^3) + t 
an(x)^2*(3*a^2*b + 3*a^3) + a^3) + (atan((tan(x)*(2*a + b)*(2*a + 2*b)*(8* 
a*b + 8*a^2 + 5*b^2)*(3*a*b^2 + 3*a^2*b + a^3 + b^3))/(2*a^(1/2)*(a + b)^( 
7/2)*(18*a*b^2 + 24*a^2*b + 16*a^3 + 5*b^3)))*(2*a + b)*(8*a*b + 8*a^2 + 5 
*b^2))/(16*a^(7/2)*(a + b)^(7/2))
 

Reduce [B] (verification not implemented)

Time = 0.88 (sec) , antiderivative size = 6476, normalized size of antiderivative = 42.05 \[ \int \frac {1}{\left (a+b \sin ^2(x)\right )^4} \, dx =\text {Too large to display} \] Input:

int(1/(a+b*sin(x)^2)^4,x)
 

Output:

( - 96*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*a 
tan((tan(x/2)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*sin(x)** 
6*a**3*b**3 - 144*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + 
 a + 2*b)*atan((tan(x/2)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b) 
))*sin(x)**6*a**2*b**4 - 108*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sq 
rt(a + b) + a + 2*b)*atan((tan(x/2)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) 
 + a + 2*b)))*sin(x)**6*a*b**5 - 30*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqr 
t(b)*sqrt(a + b) + a + 2*b)*atan((tan(x/2)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt 
(a + b) + a + 2*b)))*sin(x)**6*b**6 - 288*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt 
(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((tan(x/2)*a)/(sqrt(a)*sqrt(2*sqrt(b 
)*sqrt(a + b) + a + 2*b)))*sin(x)**4*a**4*b**2 - 432*sqrt(b)*sqrt(a)*sqrt( 
a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((tan(x/2)*a)/(sqrt(a)*sq 
rt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*sin(x)**4*a**3*b**3 - 324*sqrt(b)*sq 
rt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((tan(x/2)*a)/ 
(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*sin(x)**4*a**2*b**4 - 90* 
sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((ta 
n(x/2)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*sin(x)**4*a*b** 
5 - 288*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)* 
atan((tan(x/2)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*sin(x)* 
*2*a**5*b - 432*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) ...