Integrand size = 14, antiderivative size = 279 \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^5} \, dx=\frac {\left (128 a^4+256 a^3 b+288 a^2 b^2+160 a b^3+35 b^4\right ) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{128 a^{9/2} (a+b)^{9/2} d}+\frac {b \cos (c+d x) \sin (c+d x)}{8 a (a+b) d \left (a+b \sin ^2(c+d x)\right )^4}+\frac {7 b (2 a+b) \cos (c+d x) \sin (c+d x)}{48 a^2 (a+b)^2 d \left (a+b \sin ^2(c+d x)\right )^3}+\frac {b \left (104 a^2+104 a b+35 b^2\right ) \cos (c+d x) \sin (c+d x)}{192 a^3 (a+b)^3 d \left (a+b \sin ^2(c+d x)\right )^2}+\frac {5 b (2 a+b) \left (40 a^2+40 a b+21 b^2\right ) \cos (c+d x) \sin (c+d x)}{384 a^4 (a+b)^4 d \left (a+b \sin ^2(c+d x)\right )} \]
1/128*(128*a^4+256*a^3*b+288*a^2*b^2+160*a*b^3+35*b^4)*arctan((a+b)^(1/2)* tan(d*x+c)/a^(1/2))/a^(9/2)/(a+b)^(9/2)/d+1/8*b*cos(d*x+c)*sin(d*x+c)/a/(a +b)/d/(a+b*sin(d*x+c)^2)^4+7/48*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)^3+1/192*b*(104*a^2+104*a*b+35*b^2)*cos(d*x+c)*sin(d *x+c)/a^3/(a+b)^3/d/(a+b*sin(d*x+c)^2)^2+5/384*b*(2*a+b)*(40*a^2+40*a*b+21 *b^2)*cos(d*x+c)*sin(d*x+c)/a^4/(a+b)^4/d/(a+b*sin(d*x+c)^2)
Time = 11.90 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^5} \, dx=\frac {\frac {24 \left (128 a^4+256 a^3 b+288 a^2 b^2+160 a b^3+35 b^4\right ) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{(a+b)^{9/2}}+\frac {2 \sqrt {a} b \left (24576 a^6+73728 a^5 b+97280 a^4 b^2+71680 a^3 b^3+32272 a^2 b^4+8720 a b^5+1050 b^6-b \left (27648 a^5+69120 a^4 b+73616 a^3 b^2+41304 a^2 b^3+12310 a b^4+1575 b^5\right ) \cos (2 (c+d x))+2 b^2 \left (2816 a^4+5632 a^3 b+4816 a^2 b^2+2000 a b^3+315 b^4\right ) \cos (4 (c+d x))-400 a^3 b^3 \cos (6 (c+d x))-600 a^2 b^4 \cos (6 (c+d x))-410 a b^5 \cos (6 (c+d x))-105 b^6 \cos (6 (c+d x))\right ) \sin (2 (c+d x))}{(a+b)^4 (2 a+b-b \cos (2 (c+d x)))^4}}{3072 a^{9/2} d} \]
((24*(128*a^4 + 256*a^3*b + 288*a^2*b^2 + 160*a*b^3 + 35*b^4)*ArcTan[(Sqrt [a + b]*Tan[c + d*x])/Sqrt[a]])/(a + b)^(9/2) + (2*Sqrt[a]*b*(24576*a^6 + 73728*a^5*b + 97280*a^4*b^2 + 71680*a^3*b^3 + 32272*a^2*b^4 + 8720*a*b^5 + 1050*b^6 - b*(27648*a^5 + 69120*a^4*b + 73616*a^3*b^2 + 41304*a^2*b^3 + 1 2310*a*b^4 + 1575*b^5)*Cos[2*(c + d*x)] + 2*b^2*(2816*a^4 + 5632*a^3*b + 4 816*a^2*b^2 + 2000*a*b^3 + 315*b^4)*Cos[4*(c + d*x)] - 400*a^3*b^3*Cos[6*( c + d*x)] - 600*a^2*b^4*Cos[6*(c + d*x)] - 410*a*b^5*Cos[6*(c + d*x)] - 10 5*b^6*Cos[6*(c + d*x)])*Sin[2*(c + d*x)])/((a + b)^4*(2*a + b - b*Cos[2*(c + d*x)])^4))/(3072*a^(9/2)*d)
Time = 1.13 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.14, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {3042, 3663, 25, 3042, 3652, 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.
| \(\displaystyle \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a+b \sin (c+d x)^2\right )^5}dx\) |
| \(\Big \downarrow \) 3663 |
| \(\displaystyle \frac {b \sin (c+d x) \cos (c+d x)}{8 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^4}-\frac {\int -\frac {-6 b \sin ^2(c+d x)+8 a+7 b}{\left (b \sin ^2(c+d x)+a\right )^4}dx}{8 a (a+b)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-6 b \sin ^2(c+d x)+8 a+7 b}{\left (b \sin ^2(c+d x)+a\right )^4}dx}{8 a (a+b)}+\frac {b \sin (c+d x) \cos (c+d x)}{8 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {-6 b \sin (c+d x)^2+8 a+7 b}{\left (b \sin (c+d x)^2+a\right )^4}dx}{8 a (a+b)}+\frac {b \sin (c+d x) \cos (c+d x)}{8 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^4}\) |
| \(\Big \downarrow \) 3652 |
| \(\displaystyle \frac {\frac {\int \frac {48 a^2+76 b a+35 b^2-28 b (2 a+b) \sin ^2(c+d x)}{\left (b \sin ^2(c+d x)+a\right )^3}dx}{6 a (a+b)}+\frac {7 b (2 a+b) \sin (c+d x) \cos (c+d x)}{6 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^3}}{8 a (a+b)}+\frac {b \sin (c+d x) \cos (c+d x)}{8 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {48 a^2+76 b a+35 b^2-28 b (2 a+b) \sin (c+d x)^2}{\left (b \sin (c+d x)^2+a\right )^3}dx}{6 a (a+b)}+\frac {7 b (2 a+b) \sin (c+d x) \cos (c+d x)}{6 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^3}}{8 a (a+b)}+\frac {b \sin (c+d x) \cos (c+d x)}{8 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^4}\) |
| \(\Big \downarrow \) 3652 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {192 a^3+392 b a^2+340 b^2 a+105 b^3-2 b \left (104 a^2+104 b a+35 b^2\right ) \sin ^2(c+d x)}{\left (b \sin ^2(c+d x)+a\right )^2}dx}{4 a (a+b)}+\frac {b \left (104 a^2+104 a b+35 b^2\right ) \sin (c+d x) \cos (c+d x)}{4 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^2}}{6 a (a+b)}+\frac {7 b (2 a+b) \sin (c+d x) \cos (c+d x)}{6 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^3}}{8 a (a+b)}+\frac {b \sin (c+d x) \cos (c+d x)}{8 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {192 a^3+392 b a^2+340 b^2 a+105 b^3-2 b \left (104 a^2+104 b a+35 b^2\right ) \sin (c+d x)^2}{\left (b \sin (c+d x)^2+a\right )^2}dx}{4 a (a+b)}+\frac {b \left (104 a^2+104 a b+35 b^2\right ) \sin (c+d x) \cos (c+d x)}{4 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^2}}{6 a (a+b)}+\frac {7 b (2 a+b) \sin (c+d x) \cos (c+d x)}{6 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^3}}{8 a (a+b)}+\frac {b \sin (c+d x) \cos (c+d x)}{8 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^4}\) |
| \(\Big \downarrow \) 3652 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {3 \left (128 a^4+256 b a^3+288 b^2 a^2+160 b^3 a+35 b^4\right )}{b \sin ^2(c+d x)+a}dx}{2 a (a+b)}+\frac {5 b (2 a+b) \left (40 a^2+40 a b+21 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d (a+b) \left (a+b \sin ^2(c+d x)\right )}}{4 a (a+b)}+\frac {b \left (104 a^2+104 a b+35 b^2\right ) \sin (c+d x) \cos (c+d x)}{4 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^2}}{6 a (a+b)}+\frac {7 b (2 a+b) \sin (c+d x) \cos (c+d x)}{6 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^3}}{8 a (a+b)}+\frac {b \sin (c+d x) \cos (c+d x)}{8 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (128 a^4+256 a^3 b+288 a^2 b^2+160 a b^3+35 b^4\right ) \int \frac {1}{b \sin ^2(c+d x)+a}dx}{2 a (a+b)}+\frac {5 b (2 a+b) \left (40 a^2+40 a b+21 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d (a+b) \left (a+b \sin ^2(c+d x)\right )}}{4 a (a+b)}+\frac {b \left (104 a^2+104 a b+35 b^2\right ) \sin (c+d x) \cos (c+d x)}{4 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^2}}{6 a (a+b)}+\frac {7 b (2 a+b) \sin (c+d x) \cos (c+d x)}{6 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^3}}{8 a (a+b)}+\frac {b \sin (c+d x) \cos (c+d x)}{8 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (128 a^4+256 a^3 b+288 a^2 b^2+160 a b^3+35 b^4\right ) \int \frac {1}{b \sin (c+d x)^2+a}dx}{2 a (a+b)}+\frac {5 b (2 a+b) \left (40 a^2+40 a b+21 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d (a+b) \left (a+b \sin ^2(c+d x)\right )}}{4 a (a+b)}+\frac {b \left (104 a^2+104 a b+35 b^2\right ) \sin (c+d x) \cos (c+d x)}{4 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^2}}{6 a (a+b)}+\frac {7 b (2 a+b) \sin (c+d x) \cos (c+d x)}{6 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^3}}{8 a (a+b)}+\frac {b \sin (c+d x) \cos (c+d x)}{8 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^4}\) |
| \(\Big \downarrow \) 3660 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (128 a^4+256 a^3 b+288 a^2 b^2+160 a b^3+35 b^4\right ) \int \frac {1}{(a+b) \tan ^2(c+d x)+a}d\tan (c+d x)}{2 a d (a+b)}+\frac {5 b (2 a+b) \left (40 a^2+40 a b+21 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d (a+b) \left (a+b \sin ^2(c+d x)\right )}}{4 a (a+b)}+\frac {b \left (104 a^2+104 a b+35 b^2\right ) \sin (c+d x) \cos (c+d x)}{4 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^2}}{6 a (a+b)}+\frac {7 b (2 a+b) \sin (c+d x) \cos (c+d x)}{6 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^3}}{8 a (a+b)}+\frac {b \sin (c+d x) \cos (c+d x)}{8 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^4}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {b \left (104 a^2+104 a b+35 b^2\right ) \sin (c+d x) \cos (c+d x)}{4 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^2}+\frac {\frac {5 b (2 a+b) \left (40 a^2+40 a b+21 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d (a+b) \left (a+b \sin ^2(c+d x)\right )}+\frac {3 \left (128 a^4+256 a^3 b+288 a^2 b^2+160 a b^3+35 b^4\right ) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} d (a+b)^{3/2}}}{4 a (a+b)}}{6 a (a+b)}+\frac {7 b (2 a+b) \sin (c+d x) \cos (c+d x)}{6 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^3}}{8 a (a+b)}+\frac {b \sin (c+d x) \cos (c+d x)}{8 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^4}\) |
(b*Cos[c + d*x]*Sin[c + d*x])/(8*a*(a + b)*d*(a + b*Sin[c + d*x]^2)^4) + ( (7*b*(2*a + b)*Cos[c + d*x]*Sin[c + d*x])/(6*a*(a + b)*d*(a + b*Sin[c + d* x]^2)^3) + ((b*(104*a^2 + 104*a*b + 35*b^2)*Cos[c + d*x]*Sin[c + d*x])/(4* a*(a + b)*d*(a + b*Sin[c + d*x]^2)^2) + ((3*(128*a^4 + 256*a^3*b + 288*a^2 *b^2 + 160*a*b^3 + 35*b^4)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a]])/(2* a^(3/2)*(a + b)^(3/2)*d) + (5*b*(2*a + b)*(40*a^2 + 40*a*b + 21*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)))/(8*a*(a + b))
3.2.12.3.1 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 = 3.55 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.20
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {b \left (256 a^{3}+288 a^{2} b +160 a \,b^{2}+35 b^{3}\right ) \left (\tan ^{7}\left (d x +c \right )\right )}{128 a^{4} \left (a +b \right )}+\frac {\left (2304 a^{3}+3168 a^{2} b +1760 a \,b^{2}+385 b^{3}\right ) b \left (\tan ^{5}\left (d x +c \right )\right )}{384 a^{3} \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (2304 a^{3}+3744 a^{2} b +2336 a \,b^{2}+511 b^{3}\right ) b \left (\tan ^{3}\left (d x +c \right )\right )}{384 a^{2} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {b \left (256 a^{3}+480 a^{2} b +352 a \,b^{2}+93 b^{3}\right ) \tan \left (d x +c \right )}{128 a \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right )}}{{\left (a \left (\tan ^{2}\left (d x +c \right )\right )+\left (\tan ^{2}\left (d x +c \right )\right ) b +a \right )}^{4}}+\frac {\left (128 a^{4}+256 a^{3} b +288 a^{2} b^{2}+160 a \,b^{3}+35 b^{4}\right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{128 a^{4} \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) \sqrt {a \left (a +b \right )}}}{d}\) | \(336\) |
| default | \(\frac {\frac {\frac {b \left (256 a^{3}+288 a^{2} b +160 a \,b^{2}+35 b^{3}\right ) \left (\tan ^{7}\left (d x +c \right )\right )}{128 a^{4} \left (a +b \right )}+\frac {\left (2304 a^{3}+3168 a^{2} b +1760 a \,b^{2}+385 b^{3}\right ) b \left (\tan ^{5}\left (d x +c \right )\right )}{384 a^{3} \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (2304 a^{3}+3744 a^{2} b +2336 a \,b^{2}+511 b^{3}\right ) b \left (\tan ^{3}\left (d x +c \right )\right )}{384 a^{2} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {b \left (256 a^{3}+480 a^{2} b +352 a \,b^{2}+93 b^{3}\right ) \tan \left (d x +c \right )}{128 a \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right )}}{{\left (a \left (\tan ^{2}\left (d x +c \right )\right )+\left (\tan ^{2}\left (d x +c \right )\right ) b +a \right )}^{4}}+\frac {\left (128 a^{4}+256 a^{3} b +288 a^{2} b^{2}+160 a \,b^{3}+35 b^{4}\right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{128 a^{4} \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) \sqrt {a \left (a +b \right )}}}{d}\) | \(336\) |
| risch | \(\text {Expression too large to display}\) | \(1743\) |
1/d*((1/128*b*(256*a^3+288*a^2*b+160*a*b^2+35*b^3)/a^4/(a+b)*tan(d*x+c)^7+ 1/384*(2304*a^3+3168*a^2*b+1760*a*b^2+385*b^3)/a^3*b/(a^2+2*a*b+b^2)*tan(d *x+c)^5+1/384*(2304*a^3+3744*a^2*b+2336*a*b^2+511*b^3)/a^2*b/(a^3+3*a^2*b+ 3*a*b^2+b^3)*tan(d*x+c)^3+1/128*b*(256*a^3+480*a^2*b+352*a*b^2+93*b^3)/a/( a^4+4*a^3*b+6*a^2*b^2+4*a*b^3+b^4)*tan(d*x+c))/(a*tan(d*x+c)^2+tan(d*x+c)^ 2*b+a)^4+1/128*(128*a^4+256*a^3*b+288*a^2*b^2+160*a*b^3+35*b^4)/a^4/(a^4+4 *a^3*b+6*a^2*b^2+4*a*b^3+b^4)/(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. 958 vs. \(2 (261) = 522\).
Time = 0.38 (sec) , antiderivative size = 2017, normalized size of antiderivative = 7.23 \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^5} \, dx=\text {Too large to display} \]
[-1/1536*(3*((128*a^4*b^4 + 256*a^3*b^5 + 288*a^2*b^6 + 160*a*b^7 + 35*b^8 )*cos(d*x + c)^8 + 128*a^8 + 768*a^7*b + 2080*a^6*b^2 + 3360*a^5*b^3 + 355 5*a^4*b^4 + 2508*a^3*b^5 + 1138*a^2*b^6 + 300*a*b^7 + 35*b^8 - 4*(128*a^5* b^3 + 384*a^4*b^4 + 544*a^3*b^5 + 448*a^2*b^6 + 195*a*b^7 + 35*b^8)*cos(d* x + c)^6 + 6*(128*a^6*b^2 + 512*a^5*b^3 + 928*a^4*b^4 + 992*a^3*b^5 + 643* a^2*b^6 + 230*a*b^7 + 35*b^8)*cos(d*x + c)^4 - 4*(128*a^7*b + 640*a^6*b^2 + 1440*a^5*b^3 + 1920*a^4*b^4 + 1635*a^3*b^5 + 873*a^2*b^6 + 265*a*b^7 + 3 5*b^8)*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*(5*(80*a^5*b^4 + 200*a^4*b^5 + 202*a^3*b^6 + 103*a^2*b^7 + 21*a*b ^8)*cos(d*x + c)^7 - (1408*a^6*b^3 + 4824*a^5*b^4 + 6724*a^4*b^5 + 4923*a^ 3*b^6 + 1930*a^2*b^7 + 315*a*b^8)*cos(d*x + c)^5 + (1728*a^7*b^2 + 7456*a^ 6*b^3 + 13370*a^5*b^4 + 12969*a^4*b^5 + 7327*a^3*b^6 + 2315*a^2*b^7 + 315* a*b^8)*cos(d*x + c)^3 - 3*(256*a^8*b + 1312*a^7*b^2 + 2848*a^6*b^3 + 3427* a^5*b^4 + 2508*a^4*b^5 + 1138*a^3*b^6 + 300*a^2*b^7 + 35*a*b^8)*cos(d*x + c))*sin(d*x + c))/((a^10*b^4 + 5*a^9*b^5 + 10*a^8*b^6 + 10*a^7*b^7 + 5*a^6 *b^8 + a^5*b^9)*d*cos(d*x + c)^8 - 4*(a^11*b^3 + 6*a^10*b^4 + 15*a^9*b^5 + 20*a^8*b^6 + 15*a^7*b^7 + 6*a^6*b^8 + a^5*b^9)*d*cos(d*x + c)^6 + 6*(a...
Timed out. \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^5} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 588 vs. \(2 (261) = 522\).
Time = 0.40 (sec) , antiderivative size = 588, normalized size of antiderivative = 2.11 \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^5} \, dx=\frac {\frac {3 \, {\left (128 \, a^{4} + 256 \, a^{3} b + 288 \, a^{2} b^{2} + 160 \, a b^{3} + 35 \, b^{4}\right )} \arctan \left (\frac {{\left (a + b\right )} \tan \left (d x + c\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{{\left (a^{8} + 4 \, a^{7} b + 6 \, a^{6} b^{2} + 4 \, a^{5} b^{3} + a^{4} b^{4}\right )} \sqrt {{\left (a + b\right )} a}} + \frac {3 \, {\left (256 \, a^{6} b + 1056 \, a^{5} b^{2} + 1792 \, a^{4} b^{3} + 1635 \, a^{3} b^{4} + 873 \, a^{2} b^{5} + 265 \, a b^{6} + 35 \, b^{7}\right )} \tan \left (d x + c\right )^{7} + {\left (2304 \, a^{6} b + 7776 \, a^{5} b^{2} + 10400 \, a^{4} b^{3} + 7073 \, a^{3} b^{4} + 2530 \, a^{2} b^{5} + 385 \, a b^{6}\right )} \tan \left (d x + c\right )^{5} + {\left (2304 \, a^{6} b + 6048 \, a^{5} b^{2} + 6080 \, a^{4} b^{3} + 2847 \, a^{3} b^{4} + 511 \, a^{2} b^{5}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (256 \, a^{6} b + 480 \, a^{5} b^{2} + 352 \, a^{4} b^{3} + 93 \, a^{3} b^{4}\right )} \tan \left (d x + c\right )}{a^{12} + 4 \, a^{11} b + 6 \, a^{10} b^{2} + 4 \, a^{9} b^{3} + a^{8} b^{4} + {\left (a^{12} + 8 \, a^{11} b + 28 \, a^{10} b^{2} + 56 \, a^{9} b^{3} + 70 \, a^{8} b^{4} + 56 \, a^{7} b^{5} + 28 \, a^{6} b^{6} + 8 \, a^{5} b^{7} + a^{4} b^{8}\right )} \tan \left (d x + c\right )^{8} + 4 \, {\left (a^{12} + 7 \, a^{11} b + 21 \, a^{10} b^{2} + 35 \, a^{9} b^{3} + 35 \, a^{8} b^{4} + 21 \, a^{7} b^{5} + 7 \, a^{6} b^{6} + a^{5} b^{7}\right )} \tan \left (d x + c\right )^{6} + 6 \, {\left (a^{12} + 6 \, a^{11} b + 15 \, a^{10} b^{2} + 20 \, a^{9} b^{3} + 15 \, a^{8} b^{4} + 6 \, a^{7} b^{5} + a^{6} b^{6}\right )} \tan \left (d x + c\right )^{4} + 4 \, {\left (a^{12} + 5 \, a^{11} b + 10 \, a^{10} b^{2} + 10 \, a^{9} b^{3} + 5 \, a^{8} b^{4} + a^{7} b^{5}\right )} \tan \left (d x + c\right )^{2}}}{384 \, d} \]
1/384*(3*(128*a^4 + 256*a^3*b + 288*a^2*b^2 + 160*a*b^3 + 35*b^4)*arctan(( a + b)*tan(d*x + c)/sqrt((a + b)*a))/((a^8 + 4*a^7*b + 6*a^6*b^2 + 4*a^5*b ^3 + a^4*b^4)*sqrt((a + b)*a)) + (3*(256*a^6*b + 1056*a^5*b^2 + 1792*a^4*b ^3 + 1635*a^3*b^4 + 873*a^2*b^5 + 265*a*b^6 + 35*b^7)*tan(d*x + c)^7 + (23 04*a^6*b + 7776*a^5*b^2 + 10400*a^4*b^3 + 7073*a^3*b^4 + 2530*a^2*b^5 + 38 5*a*b^6)*tan(d*x + c)^5 + (2304*a^6*b + 6048*a^5*b^2 + 6080*a^4*b^3 + 2847 *a^3*b^4 + 511*a^2*b^5)*tan(d*x + c)^3 + 3*(256*a^6*b + 480*a^5*b^2 + 352* a^4*b^3 + 93*a^3*b^4)*tan(d*x + c))/(a^12 + 4*a^11*b + 6*a^10*b^2 + 4*a^9* b^3 + a^8*b^4 + (a^12 + 8*a^11*b + 28*a^10*b^2 + 56*a^9*b^3 + 70*a^8*b^4 + 56*a^7*b^5 + 28*a^6*b^6 + 8*a^5*b^7 + a^4*b^8)*tan(d*x + c)^8 + 4*(a^12 + 7*a^11*b + 21*a^10*b^2 + 35*a^9*b^3 + 35*a^8*b^4 + 21*a^7*b^5 + 7*a^6*b^6 + a^5*b^7)*tan(d*x + c)^6 + 6*(a^12 + 6*a^11*b + 15*a^10*b^2 + 20*a^9*b^3 + 15*a^8*b^4 + 6*a^7*b^5 + a^6*b^6)*tan(d*x + c)^4 + 4*(a^12 + 5*a^11*b + 10*a^10*b^2 + 10*a^9*b^3 + 5*a^8*b^4 + a^7*b^5)*tan(d*x + c)^2))/d
Leaf count of result is larger than twice the leaf count of optimal. 524 vs. \(2 (261) = 522\).
Time = 0.38 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.88 \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^5} \, dx=\frac {\frac {3 \, {\left (128 \, a^{4} + 256 \, a^{3} b + 288 \, a^{2} b^{2} + 160 \, a b^{3} + 35 \, b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt {a^{2} + a b}}\right )\right )}}{{\left (a^{8} + 4 \, a^{7} b + 6 \, a^{6} b^{2} + 4 \, a^{5} b^{3} + a^{4} b^{4}\right )} \sqrt {a^{2} + a b}} + \frac {768 \, a^{6} b \tan \left (d x + c\right )^{7} + 3168 \, a^{5} b^{2} \tan \left (d x + c\right )^{7} + 5376 \, a^{4} b^{3} \tan \left (d x + c\right )^{7} + 4905 \, a^{3} b^{4} \tan \left (d x + c\right )^{7} + 2619 \, a^{2} b^{5} \tan \left (d x + c\right )^{7} + 795 \, a b^{6} \tan \left (d x + c\right )^{7} + 105 \, b^{7} \tan \left (d x + c\right )^{7} + 2304 \, a^{6} b \tan \left (d x + c\right )^{5} + 7776 \, a^{5} b^{2} \tan \left (d x + c\right )^{5} + 10400 \, a^{4} b^{3} \tan \left (d x + c\right )^{5} + 7073 \, a^{3} b^{4} \tan \left (d x + c\right )^{5} + 2530 \, a^{2} b^{5} \tan \left (d x + c\right )^{5} + 385 \, a b^{6} \tan \left (d x + c\right )^{5} + 2304 \, a^{6} b \tan \left (d x + c\right )^{3} + 6048 \, a^{5} b^{2} \tan \left (d x + c\right )^{3} + 6080 \, a^{4} b^{3} \tan \left (d x + c\right )^{3} + 2847 \, a^{3} b^{4} \tan \left (d x + c\right )^{3} + 511 \, a^{2} b^{5} \tan \left (d x + c\right )^{3} + 768 \, a^{6} b \tan \left (d x + c\right ) + 1440 \, a^{5} b^{2} \tan \left (d x + c\right ) + 1056 \, a^{4} b^{3} \tan \left (d x + c\right ) + 279 \, a^{3} b^{4} \tan \left (d x + c\right )}{{\left (a^{8} + 4 \, a^{7} b + 6 \, a^{6} b^{2} + 4 \, a^{5} b^{3} + a^{4} b^{4}\right )} {\left (a \tan \left (d x + c\right )^{2} + b \tan \left (d x + c\right )^{2} + a\right )}^{4}}}{384 \, d} \]
1/384*(3*(128*a^4 + 256*a^3*b + 288*a^2*b^2 + 160*a*b^3 + 35*b^4)*(pi*floo r((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^8 + 4*a^7*b + 6*a^6*b^2 + 4*a^5*b^3 + a^4*b^4) *sqrt(a^2 + a*b)) + (768*a^6*b*tan(d*x + c)^7 + 3168*a^5*b^2*tan(d*x + c)^ 7 + 5376*a^4*b^3*tan(d*x + c)^7 + 4905*a^3*b^4*tan(d*x + c)^7 + 2619*a^2*b ^5*tan(d*x + c)^7 + 795*a*b^6*tan(d*x + c)^7 + 105*b^7*tan(d*x + c)^7 + 23 04*a^6*b*tan(d*x + c)^5 + 7776*a^5*b^2*tan(d*x + c)^5 + 10400*a^4*b^3*tan( d*x + c)^5 + 7073*a^3*b^4*tan(d*x + c)^5 + 2530*a^2*b^5*tan(d*x + c)^5 + 3 85*a*b^6*tan(d*x + c)^5 + 2304*a^6*b*tan(d*x + c)^3 + 6048*a^5*b^2*tan(d*x + c)^3 + 6080*a^4*b^3*tan(d*x + c)^3 + 2847*a^3*b^4*tan(d*x + c)^3 + 511* a^2*b^5*tan(d*x + c)^3 + 768*a^6*b*tan(d*x + c) + 1440*a^5*b^2*tan(d*x + c ) + 1056*a^4*b^3*tan(d*x + c) + 279*a^3*b^4*tan(d*x + c))/((a^8 + 4*a^7*b + 6*a^6*b^2 + 4*a^5*b^3 + a^4*b^4)*(a*tan(d*x + c)^2 + b*tan(d*x + c)^2 + a)^4))/d
Time = 16.12 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.61 \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^5} \, dx=\frac {\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (256\,a^3\,b+480\,a^2\,b^2+352\,a\,b^3+93\,b^4\right )}{128\,a\,\left (a^4+4\,a^3\,b+6\,a^2\,b^2+4\,a\,b^3+b^4\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (2304\,a^3\,b+3744\,a^2\,b^2+2336\,a\,b^3+511\,b^4\right )}{384\,a^2\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (2304\,a^3\,b+3168\,a^2\,b^2+1760\,a\,b^3+385\,b^4\right )}{384\,a^3\,\left (a^2+2\,a\,b+b^2\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^7\,\left (256\,a^3\,b+288\,a^2\,b^2+160\,a\,b^3+35\,b^4\right )}{128\,a^4\,\left (a+b\right )}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^4\,\left (6\,a^4+12\,a^3\,b+6\,a^2\,b^2\right )+{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (4\,a^4+4\,b\,a^3\right )+{\mathrm {tan}\left (c+d\,x\right )}^6\,\left (4\,a^4+12\,a^3\,b+12\,a^2\,b^2+4\,a\,b^3\right )+a^4+{\mathrm {tan}\left (c+d\,x\right )}^8\,\left (a^4+4\,a^3\,b+6\,a^2\,b^2+4\,a\,b^3+b^4\right )\right )}+\frac {\mathrm {atan}\left (\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (2\,a+2\,b\right )\,\left (a^4+4\,a^3\,b+6\,a^2\,b^2+4\,a\,b^3+b^4\right )}{2\,\sqrt {a}\,{\left (a+b\right )}^{9/2}}\right )\,\left (128\,a^4+256\,a^3\,b+288\,a^2\,b^2+160\,a\,b^3+35\,b^4\right )}{128\,a^{9/2}\,d\,{\left (a+b\right )}^{9/2}} \]
((tan(c + d*x)*(352*a*b^3 + 256*a^3*b + 93*b^4 + 480*a^2*b^2))/(128*a*(4*a *b^3 + 4*a^3*b + a^4 + b^4 + 6*a^2*b^2)) + (tan(c + d*x)^3*(2336*a*b^3 + 2 304*a^3*b + 511*b^4 + 3744*a^2*b^2))/(384*a^2*(3*a*b^2 + 3*a^2*b + a^3 + b ^3)) + (tan(c + d*x)^5*(1760*a*b^3 + 2304*a^3*b + 385*b^4 + 3168*a^2*b^2)) /(384*a^3*(2*a*b + a^2 + b^2)) + (tan(c + d*x)^7*(160*a*b^3 + 256*a^3*b + 35*b^4 + 288*a^2*b^2))/(128*a^4*(a + b)))/(d*(tan(c + d*x)^4*(12*a^3*b + 6 *a^4 + 6*a^2*b^2) + tan(c + d*x)^2*(4*a^3*b + 4*a^4) + tan(c + d*x)^6*(4*a *b^3 + 12*a^3*b + 4*a^4 + 12*a^2*b^2) + a^4 + tan(c + d*x)^8*(4*a*b^3 + 4* a^3*b + a^4 + b^4 + 6*a^2*b^2))) + (atan((tan(c + d*x)*(2*a + 2*b)*(4*a*b^ 3 + 4*a^3*b + a^4 + b^4 + 6*a^2*b^2))/(2*a^(1/2)*(a + b)^(9/2)))*(160*a*b^ 3 + 256*a^3*b + 128*a^4 + 35*b^4 + 288*a^2*b^2))/(128*a^(9/2)*d*(a + b)^(9 /2))