Integrand size = 14, antiderivative size = 206 \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^4} \, dx=\frac {(2 a+b) \left (8 a^2+8 a b+5 b^2\right ) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{16 a^{7/2} (a+b)^{7/2} d}+\frac {b \cos (c+d x) \sin (c+d x)}{6 a (a+b) d \left (a+b \sin ^2(c+d x)\right )^3}+\frac {5 b (2 a+b) \cos (c+d x) \sin (c+d x)}{24 a^2 (a+b)^2 d \left (a+b \sin ^2(c+d x)\right )^2}+\frac {b \left (44 a^2+44 a b+15 b^2\right ) \cos (c+d x) \sin (c+d x)}{48 a^3 (a+b)^3 d \left (a+b \sin ^2(c+d x)\right )} \]
1/16*(2*a+b)*(8*a^2+8*a*b+5*b^2)*arctan((a+b)^(1/2)*tan(d*x+c)/a^(1/2))/a^ (7/2)/(a+b)^(7/2)/d+1/6*b*cos(d*x+c)*sin(d*x+c)/a/(a+b)/d/(a+b*sin(d*x+c)^ 2)^3+5/24*b*(2*a+b)*cos(d*x+c)*sin(d*x+c)/a^2/(a+b)^2/d/(a+b*sin(d*x+c)^2) ^2+1/48*b*(44*a^2+44*a*b+15*b^2)*cos(d*x+c)*sin(d*x+c)/a^3/(a+b)^3/d/(a+b* sin(d*x+c)^2)
Time = 11.75 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^4} \, dx=\frac {\frac {3 \left (16 a^3+24 a^2 b+18 a b^2+5 b^3\right ) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{(a+b)^{7/2}}+\frac {32 a^{5/2} b \sin (2 (c+d x))}{(a+b) (2 a+b-b \cos (2 (c+d x)))^3}+\frac {20 a^{3/2} b (2 a+b) \sin (2 (c+d x))}{(a+b)^2 (2 a+b-b \cos (2 (c+d x)))^2}+\frac {\sqrt {a} b \left (44 a^2+44 a b+15 b^2\right ) \sin (2 (c+d x))}{(a+b)^3 (2 a+b-b \cos (2 (c+d x)))}}{48 a^{7/2} d} \]
((3*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*ArcTan[(Sqrt[a + b]*Tan[c + d*x ])/Sqrt[a]])/(a + b)^(7/2) + (32*a^(5/2)*b*Sin[2*(c + d*x)])/((a + b)*(2*a + b - b*Cos[2*(c + d*x)])^3) + (20*a^(3/2)*b*(2*a + b)*Sin[2*(c + d*x)])/ ((a + b)^2*(2*a + b - b*Cos[2*(c + d*x)])^2) + (Sqrt[a]*b*(44*a^2 + 44*a*b + 15*b^2)*Sin[2*(c + d*x)])/((a + b)^3*(2*a + b - b*Cos[2*(c + d*x)])))/( 48*a^(7/2)*d)
Time = 0.79 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.13, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, 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.
| \(\displaystyle \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a+b \sin (c+d x)^2\right )^4}dx\) |
| \(\Big \downarrow \) 3663 |
| \(\displaystyle \frac {b \sin (c+d x) \cos (c+d x)}{6 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^3}-\frac {\int -\frac {-4 b \sin ^2(c+d x)+6 a+5 b}{\left (b \sin ^2(c+d x)+a\right )^3}dx}{6 a (a+b)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-4 b \sin ^2(c+d x)+6 a+5 b}{\left (b \sin ^2(c+d x)+a\right )^3}dx}{6 a (a+b)}+\frac {b \sin (c+d x) \cos (c+d x)}{6 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {-4 b \sin (c+d x)^2+6 a+5 b}{\left (b \sin (c+d x)^2+a\right )^3}dx}{6 a (a+b)}+\frac {b \sin (c+d x) \cos (c+d x)}{6 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^3}\) |
| \(\Big \downarrow \) 3652 |
| \(\displaystyle \frac {\frac {\int \frac {24 a^2+34 b a+15 b^2-10 b (2 a+b) \sin ^2(c+d x)}{\left (b \sin ^2(c+d x)+a\right )^2}dx}{4 a (a+b)}+\frac {5 b (2 a+b) \sin (c+d x) \cos (c+d x)}{4 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^2}}{6 a (a+b)}+\frac {b \sin (c+d x) \cos (c+d x)}{6 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {24 a^2+34 b a+15 b^2-10 b (2 a+b) \sin (c+d x)^2}{\left (b \sin (c+d x)^2+a\right )^2}dx}{4 a (a+b)}+\frac {5 b (2 a+b) \sin (c+d x) \cos (c+d x)}{4 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^2}}{6 a (a+b)}+\frac {b \sin (c+d x) \cos (c+d x)}{6 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^3}\) |
| \(\Big \downarrow \) 3652 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {3 (2 a+b) \left (8 a^2+8 b a+5 b^2\right )}{b \sin ^2(c+d x)+a}dx}{2 a (a+b)}+\frac {b \left (44 a^2+44 a b+15 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d (a+b) \left (a+b \sin ^2(c+d x)\right )}}{4 a (a+b)}+\frac {5 b (2 a+b) \sin (c+d x) \cos (c+d x)}{4 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^2}}{6 a (a+b)}+\frac {b \sin (c+d x) \cos (c+d x)}{6 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 (2 a+b) \left (8 a^2+8 a b+5 b^2\right ) \int \frac {1}{b \sin ^2(c+d x)+a}dx}{2 a (a+b)}+\frac {b \left (44 a^2+44 a b+15 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d (a+b) \left (a+b \sin ^2(c+d x)\right )}}{4 a (a+b)}+\frac {5 b (2 a+b) \sin (c+d x) \cos (c+d x)}{4 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^2}}{6 a (a+b)}+\frac {b \sin (c+d x) \cos (c+d x)}{6 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 (2 a+b) \left (8 a^2+8 a b+5 b^2\right ) \int \frac {1}{b \sin (c+d x)^2+a}dx}{2 a (a+b)}+\frac {b \left (44 a^2+44 a b+15 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d (a+b) \left (a+b \sin ^2(c+d x)\right )}}{4 a (a+b)}+\frac {5 b (2 a+b) \sin (c+d x) \cos (c+d x)}{4 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^2}}{6 a (a+b)}+\frac {b \sin (c+d x) \cos (c+d x)}{6 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^3}\) |
| \(\Big \downarrow \) 3660 |
| \(\displaystyle \frac {\frac {\frac {3 (2 a+b) \left (8 a^2+8 a b+5 b^2\right ) \int \frac {1}{(a+b) \tan ^2(c+d x)+a}d\tan (c+d x)}{2 a d (a+b)}+\frac {b \left (44 a^2+44 a b+15 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d (a+b) \left (a+b \sin ^2(c+d x)\right )}}{4 a (a+b)}+\frac {5 b (2 a+b) \sin (c+d x) \cos (c+d x)}{4 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^2}}{6 a (a+b)}+\frac {b \sin (c+d x) \cos (c+d x)}{6 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^3}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {b \left (44 a^2+44 a b+15 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d (a+b) \left (a+b \sin ^2(c+d x)\right )}+\frac {3 (2 a+b) \left (8 a^2+8 a b+5 b^2\right ) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} d (a+b)^{3/2}}}{4 a (a+b)}+\frac {5 b (2 a+b) \sin (c+d x) \cos (c+d x)}{4 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^2}}{6 a (a+b)}+\frac {b \sin (c+d x) \cos (c+d x)}{6 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^3}\) |
(b*Cos[c + d*x]*Sin[c + d*x])/(6*a*(a + b)*d*(a + b*Sin[c + d*x]^2)^3) + ( (5*b*(2*a + b)*Cos[c + d*x]*Sin[c + d*x])/(4*a*(a + b)*d*(a + b*Sin[c + d* x]^2)^2) + ((3*(2*a + b)*(8*a^2 + 8*a*b + 5*b^2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a + b)^(3/2)*d) + (b*(44*a^2 + 44*a*b + 15* b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*a*(a + b)*d*(a + b*Sin[c + d*x]^2)))/(4 *a*(a + b)))/(6*a*(a + b))
3.2.11.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
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]
Time = 1.87 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.12
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\left (24 a^{2}+18 a b +5 b^{2}\right ) b \left (\tan ^{5}\left (d x +c \right )\right )}{16 a^{3} \left (a +b \right )}+\frac {\left (18 a^{2}+18 a b +5 b^{2}\right ) b \left (\tan ^{3}\left (d x +c \right )\right )}{6 a^{2} \left (a^{2}+2 a b +b^{2}\right )}+\frac {b \left (24 a^{2}+30 a b +11 b^{2}\right ) \tan \left (d x +c \right )}{16 a \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}}{{\left (a \left (\tan ^{2}\left (d x +c \right )\right )+\left (\tan ^{2}\left (d x +c \right )\right ) b +a \right )}^{3}}+\frac {\left (16 a^{3}+24 a^{2} b +18 a \,b^{2}+5 b^{3}\right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{16 a^{3} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \sqrt {a \left (a +b \right )}}}{d}\) | \(230\) |
| default | \(\frac {\frac {\frac {\left (24 a^{2}+18 a b +5 b^{2}\right ) b \left (\tan ^{5}\left (d x +c \right )\right )}{16 a^{3} \left (a +b \right )}+\frac {\left (18 a^{2}+18 a b +5 b^{2}\right ) b \left (\tan ^{3}\left (d x +c \right )\right )}{6 a^{2} \left (a^{2}+2 a b +b^{2}\right )}+\frac {b \left (24 a^{2}+30 a b +11 b^{2}\right ) \tan \left (d x +c \right )}{16 a \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}}{{\left (a \left (\tan ^{2}\left (d x +c \right )\right )+\left (\tan ^{2}\left (d x +c \right )\right ) b +a \right )}^{3}}+\frac {\left (16 a^{3}+24 a^{2} b +18 a \,b^{2}+5 b^{3}\right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{16 a^{3} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \sqrt {a \left (a +b \right )}}}{d}\) | \(230\) |
| risch | \(\text {Expression too large to display}\) | \(1212\) |
1/d*((1/16*(24*a^2+18*a*b+5*b^2)/a^3*b/(a+b)*tan(d*x+c)^5+1/6*(18*a^2+18*a *b+5*b^2)/a^2*b/(a^2+2*a*b+b^2)*tan(d*x+c)^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(d*x+c))/(a*tan(d*x+c)^2+tan(d*x+c)^2*b+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(d*x+c)/(a*(a+b))^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 630 vs. \(2 (190) = 380\).
Time = 0.34 (sec) , antiderivative size = 1361, normalized size of antiderivative = 6.61 \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^4} \, dx=\text {Too large to display} \]
[-1/192*(3*((16*a^3*b^3 + 24*a^2*b^4 + 18*a*b^5 + 5*b^6)*cos(d*x + c)^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(d*x + c)^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(d*x + c)^2)*sqrt(-a^2 - a*b)*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 + b)*cos(d*x + c )^3 - (a + b)*cos(d*x + c))*sqrt(-a^2 - a*b)*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)) + 4*((44*a^4*b^3 + 88*a^3*b^4 + 59*a^2*b^5 + 15*a*b^6)*cos(d*x + c)^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(d *x + c)^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(d*x + c))*sin(d*x + c))/((a^8*b^3 + 4*a^7*b^4 + 6*a^6*b^ 5 + 4*a^5*b^6 + a^4*b^7)*d*cos(d*x + c)^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)*d*cos(d*x + c)^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)*d*c os(d*x + c)^2 - (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)*d), -1/96*(3*((16*a^3*b^3 + 24*a^2*b^4 + 18*a*b^5 + 5*b^6)*cos(d*x + c)^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(d*x + c)^4 + 3*(16*a^5*b + 56*a^4*b^2 +...
Timed out. \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^4} \, dx=\text {Timed out} \]
Time = 0.40 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.83 \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^4} \, dx=\frac {\frac {3 \, {\left (16 \, a^{3} + 24 \, a^{2} b + 18 \, a b^{2} + 5 \, b^{3}\right )} \arctan \left (\frac {{\left (a + b\right )} \tan \left (d x + c\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{{\left (a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right )} \sqrt {{\left (a + b\right )} a}} + \frac {3 \, {\left (24 \, a^{4} b + 66 \, a^{3} b^{2} + 65 \, a^{2} b^{3} + 28 \, a b^{4} + 5 \, b^{5}\right )} \tan \left (d x + c\right )^{5} + 8 \, {\left (18 \, a^{4} b + 36 \, a^{3} b^{2} + 23 \, a^{2} b^{3} + 5 \, a b^{4}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (24 \, a^{4} b + 30 \, a^{3} b^{2} + 11 \, a^{2} b^{3}\right )} \tan \left (d x + c\right )}{a^{9} + 3 \, a^{8} b + 3 \, a^{7} b^{2} + a^{6} b^{3} + {\left (a^{9} + 6 \, a^{8} b + 15 \, a^{7} b^{2} + 20 \, a^{6} b^{3} + 15 \, a^{5} b^{4} + 6 \, a^{4} b^{5} + a^{3} b^{6}\right )} \tan \left (d x + c\right )^{6} + 3 \, {\left (a^{9} + 5 \, a^{8} b + 10 \, a^{7} b^{2} + 10 \, a^{6} b^{3} + 5 \, a^{5} b^{4} + a^{4} b^{5}\right )} \tan \left (d x + c\right )^{4} + 3 \, {\left (a^{9} + 4 \, a^{8} b + 6 \, a^{7} b^{2} + 4 \, a^{6} b^{3} + a^{5} b^{4}\right )} \tan \left (d x + c\right )^{2}}}{48 \, d} \]
1/48*(3*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*arctan((a + b)*tan(d*x + c) /sqrt((a + b)*a))/((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*sqrt((a + b)*a)) + (3*(24*a^4*b + 66*a^3*b^2 + 65*a^2*b^3 + 28*a*b^4 + 5*b^5)*tan(d*x + c)^ 5 + 8*(18*a^4*b + 36*a^3*b^2 + 23*a^2*b^3 + 5*a*b^4)*tan(d*x + c)^3 + 3*(2 4*a^4*b + 30*a^3*b^2 + 11*a^2*b^3)*tan(d*x + c))/(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(d*x + c)^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)*tan(d*x + c)^4 + 3*(a^9 + 4*a^8*b + 6*a^7*b^2 + 4*a^6*b^3 + a^5*b^4)*tan(d*x + c)^2))/d
Time = 0.34 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.67 \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^4} \, dx=\frac {\frac {3 \, {\left (16 \, a^{3} + 24 \, a^{2} b + 18 \, a b^{2} + 5 \, b^{3}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt {a^{2} + a b}}\right )\right )}}{{\left (a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right )} \sqrt {a^{2} + a b}} + \frac {72 \, a^{4} b \tan \left (d x + c\right )^{5} + 198 \, a^{3} b^{2} \tan \left (d x + c\right )^{5} + 195 \, a^{2} b^{3} \tan \left (d x + c\right )^{5} + 84 \, a b^{4} \tan \left (d x + c\right )^{5} + 15 \, b^{5} \tan \left (d x + c\right )^{5} + 144 \, a^{4} b \tan \left (d x + c\right )^{3} + 288 \, a^{3} b^{2} \tan \left (d x + c\right )^{3} + 184 \, a^{2} b^{3} \tan \left (d x + c\right )^{3} + 40 \, a b^{4} \tan \left (d x + c\right )^{3} + 72 \, a^{4} b \tan \left (d x + c\right ) + 90 \, a^{3} b^{2} \tan \left (d x + c\right ) + 33 \, a^{2} b^{3} \tan \left (d x + c\right )}{{\left (a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right )} {\left (a \tan \left (d x + c\right )^{2} + b \tan \left (d x + c\right )^{2} + a\right )}^{3}}}{48 \, d} \]
1/48*(3*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^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^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*sqrt(a^2 + a*b)) + (72*a^4*b*t an(d*x + c)^5 + 198*a^3*b^2*tan(d*x + c)^5 + 195*a^2*b^3*tan(d*x + c)^5 + 84*a*b^4*tan(d*x + c)^5 + 15*b^5*tan(d*x + c)^5 + 144*a^4*b*tan(d*x + c)^3 + 288*a^3*b^2*tan(d*x + c)^3 + 184*a^2*b^3*tan(d*x + c)^3 + 40*a*b^4*tan( d*x + c)^3 + 72*a^4*b*tan(d*x + c) + 90*a^3*b^2*tan(d*x + c) + 33*a^2*b^3* tan(d*x + c))/((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*(a*tan(d*x + c)^2 + b *tan(d*x + c)^2 + a)^3))/d
Time = 15.29 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.65 \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^4} \, dx=\frac {\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (24\,a^2\,b+30\,a\,b^2+11\,b^3\right )}{16\,a\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (18\,a^2\,b+18\,a\,b^2+5\,b^3\right )}{6\,a^2\,\left (a^2+2\,a\,b+b^2\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (24\,a^2\,b+18\,a\,b^2+5\,b^3\right )}{16\,a^3\,\left (a+b\right )}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^6\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )+{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (3\,a^3+3\,b\,a^2\right )+{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (3\,a^3+6\,a^2\,b+3\,a\,b^2\right )+a^3\right )}+\frac {\mathrm {atan}\left (\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (2\,a+b\right )\,\left (2\,a+2\,b\right )\,\left (8\,a^2+8\,a\,b+5\,b^2\right )\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}{2\,\sqrt {a}\,{\left (a+b\right )}^{7/2}\,\left (16\,a^3+24\,a^2\,b+18\,a\,b^2+5\,b^3\right )}\right )\,\left (2\,a+b\right )\,\left (8\,a^2+8\,a\,b+5\,b^2\right )}{16\,a^{7/2}\,d\,{\left (a+b\right )}^{7/2}} \]
((tan(c + d*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(c + d*x)^3*(18*a*b^2 + 18*a^2*b + 5*b^3))/(6*a^2*(2*a*b + a^2 + b^2)) + (tan(c + d*x)^5*(18*a*b^2 + 24*a^2*b + 5*b^3))/(16*a^3*(a + b)))/(d*(tan(c + d*x)^6*(3*a*b^2 + 3*a^2*b + a^3 + b^3) + tan(c + d*x)^ 2*(3*a^2*b + 3*a^3) + tan(c + d*x)^4*(3*a*b^2 + 6*a^2*b + 3*a^3) + a^3)) + (atan((tan(c + d*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)*d*(a + b) ^(7/2))