Integrand size = 24, antiderivative size = 219 \[ \int \frac {\sin ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\frac {\left (2 \sqrt {a}-\sqrt {b}\right ) \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{5/4} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} \sqrt {b} d}-\frac {\left (2 \sqrt {a}+\sqrt {b}\right ) \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{5/4} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} \sqrt {b} d}-\frac {\tan (c+d x) \left (a+(a+b) \tan ^2(c+d x)\right )}{4 a (a-b) d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )} \]
1/8*arctan((a^(1/2)-b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))*(2*a^(1/2)-b^(1/2)) /a^(5/4)/d/(a^(1/2)-b^(1/2))^(3/2)/b^(1/2)-1/8*arctan((a^(1/2)+b^(1/2))^(1 /2)*tan(d*x+c)/a^(1/4))*(2*a^(1/2)+b^(1/2))/a^(5/4)/d/b^(1/2)/(a^(1/2)+b^( 1/2))^(3/2)-1/4*tan(d*x+c)*(a+(a+b)*tan(d*x+c)^2)/a/(a-b)/d/(a+2*a*tan(d*x +c)^2+(a-b)*tan(d*x+c)^4)
Time = 1.67 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.16 \[ \int \frac {\sin ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\frac {-\frac {\sqrt {a} \left (2 a-\sqrt {a} \sqrt {b}-b\right ) \arctan \left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a+\sqrt {a} \sqrt {b}} \sqrt {b}}-\frac {\sqrt {a} \left (2 a+\sqrt {a} \sqrt {b}-b\right ) \text {arctanh}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {-a+\sqrt {a} \sqrt {b}} \sqrt {b}}-\frac {4 \sqrt {a} (2 a+b-b \cos (2 (c+d x))) \sin (2 (c+d x))}{8 a-3 b+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))}}{8 a^{3/2} (a-b) d} \]
(-((Sqrt[a]*(2*a - Sqrt[a]*Sqrt[b] - b)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a + Sqrt[a]*Sqrt[b]]*Sqrt[b])) - (Sqrt[a]*(2*a + Sqrt[a]*Sqrt[b] - b)*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(Sqrt[-a + Sqrt[a]*Sqrt[b]]*Sqrt[b]) - (4*Sqrt[a]*(2*a + b - b*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)]))/(8*a^(3/2)*(a - b)*d)
Time = 0.45 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 3696, 1672, 27, 1480, 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.
| \(\displaystyle \int \frac {\sin ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^2}{\left (a-b \sin (c+d x)^4\right )^2}dx\) |
| \(\Big \downarrow \) 3696 |
| \(\displaystyle \frac {\int \frac {\tan ^2(c+d x) \left (\tan ^2(c+d x)+1\right )^2}{\left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2}d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 1672 |
| \(\displaystyle \frac {-\frac {\int -\frac {2 a b \left ((3 a-b) \tan ^2(c+d x)+a\right )}{(a-b) \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}d\tan (c+d x)}{8 a^2 b}-\frac {\tan (c+d x) \left ((a+b) \tan ^2(c+d x)+a\right )}{4 a (a-b) \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {(3 a-b) \tan ^2(c+d x)+a}{(a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}d\tan (c+d x)}{4 a (a-b)}-\frac {\tan (c+d x) \left ((a+b) \tan ^2(c+d x)+a\right )}{4 a (a-b) \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}}{d}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {\frac {1}{2} \left (-\frac {2 a^{3/2}}{\sqrt {b}}+3 a-b\right ) \int \frac {1}{(a-b) \tan ^2(c+d x)+\sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )}d\tan (c+d x)+\frac {1}{2} \left (\frac {2 a^{3/2}}{\sqrt {b}}+3 a-b\right ) \int \frac {1}{(a-b) \tan ^2(c+d x)+\sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )}d\tan (c+d x)}{4 a (a-b)}-\frac {\tan (c+d x) \left ((a+b) \tan ^2(c+d x)+a\right )}{4 a (a-b) \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {\left (\frac {2 a^{3/2}}{\sqrt {b}}+3 a-b\right ) \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} \left (\sqrt {a}+\sqrt {b}\right )}+\frac {\left (-\frac {2 a^{3/2}}{\sqrt {b}}+3 a-b\right ) \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {b}\right ) \sqrt {\sqrt {a}+\sqrt {b}}}}{4 a (a-b)}-\frac {\tan (c+d x) \left ((a+b) \tan ^2(c+d x)+a\right )}{4 a (a-b) \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}}{d}\) |
((((3*a + (2*a^(3/2))/Sqrt[b] - b)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(1/4)*Sqrt[Sqrt[a] - Sqrt[b]]*(Sqrt[a] + Sqrt[b])) + ((3*a - (2*a^(3/2))/Sqrt[b] - b)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(1/4)*(Sqrt[a] - Sqrt[b])*Sqrt[Sqrt[a] + Sqrt[b]]))/( 4*a*(a - b)) - (Tan[c + d*x]*(a + (a + b)*Tan[c + d*x]^2))/(4*a*(a - b)*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4)))/d
3.3.21.3.1 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[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^ 4)^(p_), x_Symbol] :> With[{f = Coeff[PolynomialRemainder[x^m*(d + e*x^2)^q , a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[x^m*(d + e*x^ 2)^q, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a *b*g - f*(b^2 - 2*a*c) - c*(b*f - 2*a*g)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c*x^4)^(p + 1)* Simp[ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[x^m*(d + e*x^ 2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)*x^2, x], x], x], x]] /; FreeQ[{a, b, c, d, e}, x ] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[q, 1] && IGtQ[m/2, 0]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^( p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff^(m + 1 )/f Subst[Int[x^m*((a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*x^2) ^(m/2 + 2*p + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] & & IntegerQ[m/2] && IntegerQ[p]
Time = 1.94 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.14
| method | result | size |
| derivativedivides | \(\frac {\frac {-\frac {\left (a +b \right ) \left (\tan ^{3}\left (d x +c \right )\right )}{4 \left (a -b \right ) a}-\frac {\tan \left (d x +c \right )}{4 \left (a -b \right )}}{\left (\tan ^{4}\left (d x +c \right )\right ) a -b \left (\tan ^{4}\left (d x +c \right )\right )+2 a \left (\tan ^{2}\left (d x +c \right )\right )+a}+\frac {\frac {\left (3 a \sqrt {a b}-\sqrt {a b}\, b +2 a^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (3 a \sqrt {a b}-\sqrt {a b}\, b -2 a^{2}\right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}}{4 a}}{d}\) | \(250\) |
| default | \(\frac {\frac {-\frac {\left (a +b \right ) \left (\tan ^{3}\left (d x +c \right )\right )}{4 \left (a -b \right ) a}-\frac {\tan \left (d x +c \right )}{4 \left (a -b \right )}}{\left (\tan ^{4}\left (d x +c \right )\right ) a -b \left (\tan ^{4}\left (d x +c \right )\right )+2 a \left (\tan ^{2}\left (d x +c \right )\right )+a}+\frac {\frac {\left (3 a \sqrt {a b}-\sqrt {a b}\, b +2 a^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (3 a \sqrt {a b}-\sqrt {a b}\, b -2 a^{2}\right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}}{4 a}}{d}\) | \(250\) |
| risch | \(-\frac {i \left (2 a \,{\mathrm e}^{6 i \left (d x +c \right )}-b \,{\mathrm e}^{6 i \left (d x +c \right )}-8 a \,{\mathrm e}^{4 i \left (d x +c \right )}+3 b \,{\mathrm e}^{4 i \left (d x +c \right )}-2 a \,{\mathrm e}^{2 i \left (d x +c \right )}-3 b \,{\mathrm e}^{2 i \left (d x +c \right )}+b \right )}{2 a \left (a -b \right ) d \left ({\mathrm e}^{8 i \left (d x +c \right )} b -4 b \,{\mathrm e}^{6 i \left (d x +c \right )}-16 a \,{\mathrm e}^{4 i \left (d x +c \right )}+6 b \,{\mathrm e}^{4 i \left (d x +c \right )}-4 b \,{\mathrm e}^{2 i \left (d x +c \right )}+b \right )}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (256 a^{8} b^{2} d^{4}-768 a^{7} b^{3} d^{4}+768 a^{6} b^{4} d^{4}-256 a^{5} b^{5} d^{4}\right ) \textit {\_Z}^{4}+\left (128 a^{5} b \,d^{2}+32 a^{4} b^{2} d^{2}-32 a^{3} b^{3} d^{2}\right ) \textit {\_Z}^{2}+16 a^{2}-8 a b +b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\frac {384 i a^{8} b^{2} d^{3}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}-\frac {1280 i a^{7} b^{3} d^{3}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}+\frac {1536 i a^{6} b^{4} d^{3}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}-\frac {768 i a^{5} b^{5} d^{3}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}+\frac {128 i a^{4} b^{6} d^{3}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}\right ) \textit {\_R}^{3}+\left (-\frac {128 d^{2} a^{7} b}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}+\frac {416 d^{2} b^{2} a^{6}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}-\frac {480 d^{2} b^{3} a^{5}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}+\frac {224 d^{2} b^{4} a^{4}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}-\frac {32 d^{2} b^{5} a^{3}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}\right ) \textit {\_R}^{2}+\left (\frac {224 i a^{5} d b}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}-\frac {88 i a^{4} b^{2} d}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}-\frac {16 i a^{3} b^{3} d}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}+\frac {8 i a^{2} b^{4} d}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}\right ) \textit {\_R} -\frac {32 a^{4}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}-\frac {32 a^{3} b}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}+\frac {38 a^{2} b^{2}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}-\frac {11 a \,b^{3}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}+\frac {b^{4}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}\right )\right )}{4}\) | \(997\) |
1/d*((-1/4*(a+b)/(a-b)/a*tan(d*x+c)^3-1/4/(a-b)*tan(d*x+c))/(tan(d*x+c)^4* a-b*tan(d*x+c)^4+2*a*tan(d*x+c)^2+a)+1/4/a*(1/2*(3*a*(a*b)^(1/2)-(a*b)^(1/ 2)*b+2*a^2)/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)+a)*(a-b))^(1/2)*arctan((a-b)*t an(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2))+1/2*(3*a*(a*b)^(1/2)-(a*b)^(1/2)* b-2*a^2)/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)-a)*(a-b))^(1/2)*arctanh((-a+b)*ta n(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 3445 vs. \(2 (169) = 338\).
Time = 0.92 (sec) , antiderivative size = 3445, normalized size of antiderivative = 15.73 \[ \int \frac {\sin ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\text {Too large to display} \]
1/32*(((a^2*b - a*b^2)*d*cos(d*x + c)^4 - 2*(a^2*b - a*b^2)*d*cos(d*x + c) ^2 - (a^3 - 2*a^2*b + a*b^2)*d)*sqrt(-((a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^ 2*b^4)*d^2*sqrt((64*a^4 - 80*a^3*b + 41*a^2*b^2 - 10*a*b^3 + b^4)/((a^11*b - 6*a^10*b^2 + 15*a^9*b^3 - 20*a^8*b^4 + 15*a^7*b^5 - 6*a^6*b^6 + a^5*b^7 )*d^4)) + 4*a^2 + a*b - b^2)/((a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d^ 2))*log(8*a^3 - 7*a^2*b + 9/4*a*b^2 - 1/4*b^3 - 1/4*(32*a^3 - 28*a^2*b + 9 *a*b^2 - b^3)*cos(d*x + c)^2 + 1/2*((3*a^8*b - 10*a^7*b^2 + 12*a^6*b^3 - 6 *a^5*b^4 + a^4*b^5)*d^3*sqrt((64*a^4 - 80*a^3*b + 41*a^2*b^2 - 10*a*b^3 + b^4)/((a^11*b - 6*a^10*b^2 + 15*a^9*b^3 - 20*a^8*b^4 + 15*a^7*b^5 - 6*a^6* b^6 + a^5*b^7)*d^4))*cos(d*x + c)*sin(d*x + c) - 2*(8*a^5 - 5*a^4*b + a^3* b^2)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d^2*sqrt((64*a^4 - 80*a^3*b + 41*a^2*b^2 - 10*a*b^3 + b^4)/((a^11 *b - 6*a^10*b^2 + 15*a^9*b^3 - 20*a^8*b^4 + 15*a^7*b^5 - 6*a^6*b^6 + a^5*b ^7)*d^4)) + 4*a^2 + a*b - b^2)/((a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)* d^2)) + 1/4*(2*(4*a^7 - 13*a^6*b + 15*a^5*b^2 - 7*a^4*b^3 + a^3*b^4)*d^2*c os(d*x + c)^2 - (4*a^7 - 13*a^6*b + 15*a^5*b^2 - 7*a^4*b^3 + a^3*b^4)*d^2) *sqrt((64*a^4 - 80*a^3*b + 41*a^2*b^2 - 10*a*b^3 + b^4)/((a^11*b - 6*a^10* b^2 + 15*a^9*b^3 - 20*a^8*b^4 + 15*a^7*b^5 - 6*a^6*b^6 + a^5*b^7)*d^4))) - ((a^2*b - a*b^2)*d*cos(d*x + c)^4 - 2*(a^2*b - a*b^2)*d*cos(d*x + c)^2 - (a^3 - 2*a^2*b + a*b^2)*d)*sqrt(-((a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*...
Timed out. \[ \int \frac {\sin ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {\sin ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\int { \frac {\sin \left (d x + c\right )^{2}}{{\left (b \sin \left (d x + c\right )^{4} - a\right )}^{2}} \,d x } \]
1/2*(2*(16*a^2 + 2*a*b - 3*b^2)*cos(4*d*x + 4*c)*sin(2*d*x + 2*c) + ((2*a* b - b^2)*sin(6*d*x + 6*c) - (8*a*b - 3*b^2)*sin(4*d*x + 4*c) - (2*a*b + 3* b^2)*sin(2*d*x + 2*c))*cos(8*d*x + 8*c) + 2*((16*a^2 + 2*a*b - 3*b^2)*sin( 4*d*x + 4*c) + 4*(2*a*b + b^2)*sin(2*d*x + 2*c))*cos(6*d*x + 6*c) + 2*((a^ 2*b^2 - a*b^3)*d*cos(8*d*x + 8*c)^2 + 16*(a^2*b^2 - a*b^3)*d*cos(6*d*x + 6 *c)^2 + 4*(64*a^4 - 112*a^3*b + 57*a^2*b^2 - 9*a*b^3)*d*cos(4*d*x + 4*c)^2 + 16*(a^2*b^2 - a*b^3)*d*cos(2*d*x + 2*c)^2 + (a^2*b^2 - a*b^3)*d*sin(8*d *x + 8*c)^2 + 16*(a^2*b^2 - a*b^3)*d*sin(6*d*x + 6*c)^2 + 4*(64*a^4 - 112* a^3*b + 57*a^2*b^2 - 9*a*b^3)*d*sin(4*d*x + 4*c)^2 + 16*(8*a^3*b - 11*a^2* b^2 + 3*a*b^3)*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(a^2*b^2 - a*b^3)* d*sin(2*d*x + 2*c)^2 - 8*(a^2*b^2 - a*b^3)*d*cos(2*d*x + 2*c) + (a^2*b^2 - a*b^3)*d - 2*(4*(a^2*b^2 - a*b^3)*d*cos(6*d*x + 6*c) + 2*(8*a^3*b - 11*a^ 2*b^2 + 3*a*b^3)*d*cos(4*d*x + 4*c) + 4*(a^2*b^2 - a*b^3)*d*cos(2*d*x + 2* c) - (a^2*b^2 - a*b^3)*d)*cos(8*d*x + 8*c) + 8*(2*(8*a^3*b - 11*a^2*b^2 + 3*a*b^3)*d*cos(4*d*x + 4*c) + 4*(a^2*b^2 - a*b^3)*d*cos(2*d*x + 2*c) - (a^ 2*b^2 - a*b^3)*d)*cos(6*d*x + 6*c) + 4*(4*(8*a^3*b - 11*a^2*b^2 + 3*a*b^3) *d*cos(2*d*x + 2*c) - (8*a^3*b - 11*a^2*b^2 + 3*a*b^3)*d)*cos(4*d*x + 4*c) - 4*(2*(a^2*b^2 - a*b^3)*d*sin(6*d*x + 6*c) + (8*a^3*b - 11*a^2*b^2 + 3*a *b^3)*d*sin(4*d*x + 4*c) + 2*(a^2*b^2 - a*b^3)*d*sin(2*d*x + 2*c))*sin(8*d *x + 8*c) + 16*((8*a^3*b - 11*a^2*b^2 + 3*a*b^3)*d*sin(4*d*x + 4*c) + 2...
Leaf count of result is larger than twice the leaf count of optimal. 1407 vs. \(2 (169) = 338\).
Time = 0.81 (sec) , antiderivative size = 1407, normalized size of antiderivative = 6.42 \[ \int \frac {\sin ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\text {Too large to display} \]
-1/8*(((9*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b - 21*sqrt(a^ 2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 + 3*sqrt(a^2 - a*b - sqrt(a *b)*(a - b))*sqrt(a*b)*a*b^3 + sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a* b)*b^4)*(a^2 - a*b)^2*abs(-a + b) - (3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b)) *a^6*b - 12*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^5*b^2 + 14*sqrt(a^2 - a* b - sqrt(a*b)*(a - b))*a^4*b^3 - 4*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3 *b^4 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^5)*abs(-a^2 + a*b)*abs(-a + b) - 2*(3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^8 - 12*sqrt(a ^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^7*b + 14*sqrt(a^2 - a*b - sqrt(a *b)*(a - b))*sqrt(a*b)*a^6*b^2 - 4*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqr t(a*b)*a^5*b^3 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^4)*ab s(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^3 - a^2*b + sqrt((a^3 - a^2*b)^2 - (a^3 - a^2*b)*(a^3 - 2*a^2*b + a*b^2)))/(a ^3 - 2*a^2*b + a*b^2))))/((3*a^10*b - 21*a^9*b^2 + 59*a^8*b^3 - 85*a^7*b^4 + 65*a^6*b^5 - 23*a^5*b^6 + a^4*b^7 + a^3*b^8)*abs(-a^2 + a*b)) - ((9*sqr t(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b - 21*sqrt(a^2 - a*b + sqr t(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 + 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))* sqrt(a*b)*a*b^3 + sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*(a^2 - a*b)^2*abs(-a + b) + (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^6*b - 12*s qrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^5*b^2 + 14*sqrt(a^2 - a*b + sqrt(a...
Time = 17.17 (sec) , antiderivative size = 3842, normalized size of antiderivative = 17.54 \[ \int \frac {\sin ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\text {Too large to display} \]
- (tan(c + d*x)/(4*(a - b)) + (tan(c + d*x)^3*(a + b))/(4*a*(a - b)))/(d*( a + 2*a*tan(c + d*x)^2 + tan(c + d*x)^4*(a - b))) - (atan(((((256*a^5*b + 256*a^3*b^3 - 512*a^4*b^2)/(64*(a^2*b - a^3)) - (tan(c + d*x)*(-(8*a^2*(a^ 5*b^3)^(1/2) + b^2*(a^5*b^3)^(1/2) - 4*a^5*b + a^3*b^3 - a^4*b^2 - 5*a*b*( a^5*b^3)^(1/2))/(256*(a^5*b^5 - 3*a^6*b^4 + 3*a^7*b^3 - a^8*b^2)))^(1/2)*( 256*a^6*b - 256*a^3*b^4 + 768*a^4*b^3 - 768*a^5*b^2))/(4*(a*b - a^2)))*(-( 8*a^2*(a^5*b^3)^(1/2) + b^2*(a^5*b^3)^(1/2) - 4*a^5*b + a^3*b^3 - a^4*b^2 - 5*a*b*(a^5*b^3)^(1/2))/(256*(a^5*b^5 - 3*a^6*b^4 + 3*a^7*b^3 - a^8*b^2)) )^(1/2) - (tan(c + d*x)*(9*a^2*b - 6*a*b^2 + 4*a^3 + b^3))/(4*(a*b - a^2)) )*(-(8*a^2*(a^5*b^3)^(1/2) + b^2*(a^5*b^3)^(1/2) - 4*a^5*b + a^3*b^3 - a^4 *b^2 - 5*a*b*(a^5*b^3)^(1/2))/(256*(a^5*b^5 - 3*a^6*b^4 + 3*a^7*b^3 - a^8* b^2)))^(1/2)*1i - (((256*a^5*b + 256*a^3*b^3 - 512*a^4*b^2)/(64*(a^2*b - a ^3)) + (tan(c + d*x)*(-(8*a^2*(a^5*b^3)^(1/2) + b^2*(a^5*b^3)^(1/2) - 4*a^ 5*b + a^3*b^3 - a^4*b^2 - 5*a*b*(a^5*b^3)^(1/2))/(256*(a^5*b^5 - 3*a^6*b^4 + 3*a^7*b^3 - a^8*b^2)))^(1/2)*(256*a^6*b - 256*a^3*b^4 + 768*a^4*b^3 - 7 68*a^5*b^2))/(4*(a*b - a^2)))*(-(8*a^2*(a^5*b^3)^(1/2) + b^2*(a^5*b^3)^(1/ 2) - 4*a^5*b + a^3*b^3 - a^4*b^2 - 5*a*b*(a^5*b^3)^(1/2))/(256*(a^5*b^5 - 3*a^6*b^4 + 3*a^7*b^3 - a^8*b^2)))^(1/2) + (tan(c + d*x)*(9*a^2*b - 6*a*b^ 2 + 4*a^3 + b^3))/(4*(a*b - a^2)))*(-(8*a^2*(a^5*b^3)^(1/2) + b^2*(a^5*b^3 )^(1/2) - 4*a^5*b + a^3*b^3 - a^4*b^2 - 5*a*b*(a^5*b^3)^(1/2))/(256*(a^...