Integrand size = 15, antiderivative size = 210 \[ \int \frac {1}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\frac {\left (4 \sqrt {a}-3 \sqrt {b}\right ) \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{7/4} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} d}+\frac {\left (4 \sqrt {a}+3 \sqrt {b}\right ) \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{7/4} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} d}-\frac {b \tan (c+d x) \left (1+2 \tan ^2(c+d x)\right )}{4 a (a-b) d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )} \]
1/8*arctan((a^(1/2)-b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))*(4*a^(1/2)-3*b^(1/2 ))/a^(7/4)/d/(a^(1/2)-b^(1/2))^(3/2)+1/8*arctan((a^(1/2)+b^(1/2))^(1/2)*ta n(d*x+c)/a^(1/4))*(4*a^(1/2)+3*b^(1/2))/a^(7/4)/d/(a^(1/2)+b^(1/2))^(3/2)- 1/4*b*tan(d*x+c)*(1+2*tan(d*x+c)^2)/a/(a-b)/d/(a+2*a*tan(d*x+c)^2+(a-b)*ta n(d*x+c)^4)
Time = 4.16 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\frac {\frac {\left (4 a-\sqrt {a} \sqrt {b}-3 b\right ) \arctan \left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a+\sqrt {a} \sqrt {b}}}-\frac {\left (4 a+\sqrt {a} \sqrt {b}-3 b\right ) \text {arctanh}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {-a+\sqrt {a} \sqrt {b}}}+\frac {2 \sqrt {a} b (-6 \sin (2 (c+d x))+\sin (4 (c+d x)))}{8 a-3 b+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))}}{8 a^{3/2} (a-b) d} \]
(((4*a - Sqrt[a]*Sqrt[b] - 3*b)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/ Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]] - ((4*a + Sqrt[a]*Sq rt[b] - 3*b)*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]* Sqrt[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]] + (2*Sqrt[a]*b*(-6*Sin[2*(c + d*x)] + Sin[4*(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 = 0.48 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.24, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 3688, 1517, 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.
| \(\displaystyle \int \frac {1}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a-b \sin (c+d x)^4\right )^2}dx\) |
| \(\Big \downarrow \) 3688 |
| \(\displaystyle \frac {\int \frac {\left (\tan ^2(c+d x)+1\right )^3}{\left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2}d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 1517 |
| \(\displaystyle \frac {-\frac {\int -\frac {2 a b \left (2 (2 a-b) \tan ^2(c+d x)+4 a-3 b\right )}{(a-b) \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}d\tan (c+d x)}{8 a^2 b}-\frac {b \tan (c+d x) \left (2 \tan ^2(c+d x)+1\right )}{4 a (a-b) \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {2 (2 a-b) \tan ^2(c+d x)+4 a-3 b}{(a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}d\tan (c+d x)}{4 a (a-b)}-\frac {b \tan (c+d x) \left (2 \tan ^2(c+d x)+1\right )}{4 a (a-b) \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}}{d}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {\frac {\left (4 \sqrt {a}-3 \sqrt {b}\right ) \left (\sqrt {a}+\sqrt {b}\right )^2 \int \frac {1}{(a-b) \tan ^2(c+d x)+\sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )}d\tan (c+d x)}{2 \sqrt {a}}+\frac {\left (4 \sqrt {a}+3 \sqrt {b}\right ) \left (-2 \sqrt {a} \sqrt {b}+a+b\right ) \int \frac {1}{(a-b) \tan ^2(c+d x)+\sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )}d\tan (c+d x)}{2 \sqrt {a}}}{4 a (a-b)}-\frac {b \tan (c+d x) \left (2 \tan ^2(c+d x)+1\right )}{4 a (a-b) \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {\left (4 \sqrt {a}-3 \sqrt {b}\right ) \left (\sqrt {a}+\sqrt {b}\right ) \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\left (4 \sqrt {a}+3 \sqrt {b}\right ) \left (-2 \sqrt {a} \sqrt {b}+a+b\right ) \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \left (\sqrt {a}-\sqrt {b}\right ) \sqrt {\sqrt {a}+\sqrt {b}}}}{4 a (a-b)}-\frac {b \tan (c+d x) \left (2 \tan ^2(c+d x)+1\right )}{4 a (a-b) \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}}{d}\) |
((((4*Sqrt[a] - 3*Sqrt[b])*(Sqrt[a] + Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] - Sqrt [b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*Sqrt[Sqrt[a] - Sqrt[b]]) + ((4*Sqr t[a] + 3*Sqrt[b])*(a - 2*Sqrt[a]*Sqrt[b] + b)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[ b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] - Sqrt[b])*Sqrt[Sqrt[a] + Sqrt[b]]))/(4*a*(a - b)) - (b*Tan[c + d*x]*(1 + 2*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
3.3.22.3.1 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[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x _Symbol] :> With[{f = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[(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)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[(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]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ [c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 1] && LtQ[p, -1]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Subst[Int[(a + 2*a*ff^2*x^2 + ( a + b)*ff^4*x^4)^p/(1 + ff^2*x^2)^(2*p + 1), x], x, Tan[e + f*x]/ff], x]] / ; FreeQ[{a, b, e, f}, x] && IntegerQ[p]
Time = 2.02 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.24
| method | result | size |
| derivativedivides | \(\frac {\frac {-\frac {b \left (\tan ^{3}\left (d x +c \right )\right )}{2 a \left (a -b \right )}-\frac {b \tan \left (d x +c \right )}{4 a \left (a -b \right )}}{\left (\tan ^{4}\left (d x +c \right )\right ) a -b \left (\tan ^{4}\left (d x +c \right )\right )+2 a \left (\tan ^{2}\left (d x +c \right )\right )+a}+\frac {\frac {\left (4 a \sqrt {a b}-2 \sqrt {a b}\, b +5 a b -3 b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (4 a \sqrt {a b}-2 \sqrt {a b}\, b -5 a b +3 b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}}{4 a}}{d}\) | \(260\) |
| default | \(\frac {\frac {-\frac {b \left (\tan ^{3}\left (d x +c \right )\right )}{2 a \left (a -b \right )}-\frac {b \tan \left (d x +c \right )}{4 a \left (a -b \right )}}{\left (\tan ^{4}\left (d x +c \right )\right ) a -b \left (\tan ^{4}\left (d x +c \right )\right )+2 a \left (\tan ^{2}\left (d x +c \right )\right )+a}+\frac {\frac {\left (4 a \sqrt {a b}-2 \sqrt {a b}\, b +5 a b -3 b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (4 a \sqrt {a b}-2 \sqrt {a b}\, b -5 a b +3 b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}}{4 a}}{d}\) | \(260\) |
| risch | \(-\frac {i \left (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 )}-5 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 )}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (65536 a^{10} d^{4}-196608 a^{9} b \,d^{4}+196608 a^{8} b^{2} d^{4}-65536 a^{7} b^{3} d^{4}\right ) \textit {\_Z}^{4}+\left (8192 a^{6} d^{2}-7680 a^{5} b \,d^{2}+1536 a^{4} b^{2} d^{2}\right ) \textit {\_Z}^{2}+256 a^{2}-288 a b +81 b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\frac {32768 i d^{3} a^{10}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}-\frac {114688 i a^{9} b \,d^{3}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {147456 i a^{8} b^{2} d^{3}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}-\frac {81920 i a^{7} b^{3} d^{3}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {16384 i a^{6} b^{4} d^{3}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}\right ) \textit {\_R}^{3}+\left (-\frac {8192 d^{2} a^{8}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {29184 a^{7} b \,d^{2}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}-\frac {38400 a^{6} b^{2} d^{2}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {22016 a^{5} b^{3} d^{2}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}-\frac {4608 a^{4} b^{4} d^{2}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}\right ) \textit {\_R}^{2}+\left (\frac {2048 i d \,a^{6}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {896 i a^{5} b d}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}-\frac {5600 i a^{4} b^{2} d}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {4032 i a^{3} b^{3} d}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}-\frac {864 i d \,b^{4} a^{2}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}\right ) \textit {\_R} -\frac {512 a^{4}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {384 a^{3} b}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {314 a^{2} b^{2}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}-\frac {351 a \,b^{3}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}+\frac {81 b^{4}}{384 a^{3} b -680 a^{2} b^{2}+405 a \,b^{3}-81 b^{4}}\right )\right )\) | \(992\) |
1/d*((-1/2*b/a/(a-b)*tan(d*x+c)^3-1/4*b/a/(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*(4*a*(a*b)^(1/2)-2*(a*b)^( 1/2)*b+5*a*b-3*b^2)/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)+a)*(a-b))^(1/2)*arctan ((a-b)*tan(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2))+1/2*(4*a*(a*b)^(1/2)-2*(a *b)^(1/2)*b-5*a*b+3*b^2)/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)-a)*(a-b))^(1/2)*a rctanh((-a+b)*tan(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. 3477 vs. \(2 (164) = 328\).
Time = 0.93 (sec) , antiderivative size = 3477, normalized size of antiderivative = 16.56 \[ \int \frac {1}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\text {Too large to display} \]
-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^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b ^3)*d^2*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5 )/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) + 16*a^2 - 15*a*b + 3*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a ^3*b^3)*d^2))*log(96*a^3*b - 170*a^2*b^2 + 405/4*a*b^3 - 81/4*b^4 - 1/4*(3 84*a^3*b - 680*a^2*b^2 + 405*a*b^3 - 81*b^4)*cos(d*x + c)^2 + 1/2*(2*(2*a^ 10 - 7*a^9*b + 9*a^8*b^2 - 5*a^7*b^3 + a^6*b^4)*d^3*sqrt((576*a^4*b - 1392 *a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11* b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4))*cos(d*x + c)*s in(d*x + c) - (120*a^5*b - 217*a^4*b^2 + 132*a^3*b^3 - 27*a^2*b^4)*d*cos(d *x + c)*sin(d*x + c))*sqrt(-((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqr t((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d ^4)) + 16*a^2 - 15*a*b + 3*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2 )) + 1/4*(2*(16*a^8 - 57*a^7*b + 75*a^6*b^2 - 43*a^5*b^3 + 9*a^4*b^4)*d^2* cos(d*x + c)^2 - (16*a^8 - 57*a^7*b + 75*a^6*b^2 - 43*a^5*b^3 + 9*a^4*b^4) *d^2)*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/ ((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a ^7*b^6)*d^4))) - ((a^2*b - a*b^2)*d*cos(d*x + c)^4 - 2*(a^2*b - a*b^2)*...
Timed out. \[ \int \frac {1}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\int { \frac {1}{{\left (b \sin \left (d x + c\right )^{4} - a\right )}^{2}} \,d x } \]
-1/2*(b^2*sin(2*d*x + 2*c) - 6*(8*a*b - 3*b^2)*cos(4*d*x + 4*c)*sin(2*d*x + 2*c) - (b^2*sin(6*d*x + 6*c) - 5*b^2*sin(2*d*x + 2*c) - (8*a*b - 3*b^2)* sin(4*d*x + 4*c))*cos(8*d*x + 8*c) - 6*(4*b^2*sin(2*d*x + 2*c) + (8*a*b - 3*b^2)*sin(4*d*x + 4*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*(a^2*b^2 - a*b^3)*d*sin...
Leaf count of result is larger than twice the leaf count of optimal. 1506 vs. \(2 (164) = 328\).
Time = 0.51 (sec) , antiderivative size = 1506, normalized size of antiderivative = 7.17 \[ \int \frac {1}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\text {Too large to display} \]
-1/8*((2*(6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3 - 15*sqrt(a^ 2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b + 4*sqrt(a^2 - a*b - sqrt(a*b )*(a - b))*sqrt(a*b)*a*b^2 + sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b) *b^3)*(a^2 - a*b)^2*abs(-a + b) - (12*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))* a^6 - 57*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^5*b + 92*sqrt(a^2 - a*b - s qrt(a*b)*(a - b))*a^4*b^2 - 58*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b^3 + 8*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^4 + 3*sqrt(a^2 - a*b - sqrt (a*b)*(a - b))*a*b^5)*abs(-a^2 + a*b)*abs(-a + b) - (15*sqrt(a^2 - a*b - s qrt(a*b)*(a - b))*sqrt(a*b)*a^7 - 69*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*s qrt(a*b)*a^6*b + 106*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^2 - 62*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^3 + 7*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^4 + 3*sqrt(a^2 - a*b - sqrt(a*b) *(a - b))*sqrt(a*b)*a^2*b^5)*abs(-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 - 21*a^9* b + 59*a^8*b^2 - 85*a^7*b^3 + 65*a^6*b^4 - 23*a^5*b^5 + a^4*b^6 + a^3*b^7) *abs(-a^2 + a*b)) - (2*(6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^ 3 - 15*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b + 4*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 + sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*(a^2 - a*b)^2*abs(-a + b) + (12*sqrt(a^2 - a*b + sq...
Time = 17.09 (sec) , antiderivative size = 3675, normalized size of antiderivative = 17.50 \[ \int \frac {1}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\text {Too large to display} \]
- (atan(((((512*a^6*b - 384*a^3*b^4 + 1280*a^4*b^3 - 1408*a^5*b^2)/(32*(a^ 3*b - a^4)) - (tan(c + d*x)*((24*a^2*(a^7*b)^(1/2) + 9*b^2*(a^7*b)^(1/2) - 15*a^5*b + 16*a^6 + 3*a^4*b^2 - 29*a*b*(a^7*b)^(1/2))/(256*(3*a^9*b - a^1 0 + a^7*b^3 - 3*a^8*b^2)))^(1/2)*(256*a^7*b - 256*a^4*b^4 + 768*a^5*b^3 - 768*a^6*b^2))/(4*(a^2*b - a^3)))*((24*a^2*(a^7*b)^(1/2) + 9*b^2*(a^7*b)^(1 /2) - 15*a^5*b + 16*a^6 + 3*a^4*b^2 - 29*a*b*(a^7*b)^(1/2))/(256*(3*a^9*b - a^10 + a^7*b^3 - 3*a^8*b^2)))^(1/2) - (tan(c + d*x)*(16*a^3*b - 26*a*b^3 + 9*b^4 + 9*a^2*b^2))/(4*(a^2*b - a^3)))*((24*a^2*(a^7*b)^(1/2) + 9*b^2*( a^7*b)^(1/2) - 15*a^5*b + 16*a^6 + 3*a^4*b^2 - 29*a*b*(a^7*b)^(1/2))/(256* (3*a^9*b - a^10 + a^7*b^3 - 3*a^8*b^2)))^(1/2)*1i - (((512*a^6*b - 384*a^3 *b^4 + 1280*a^4*b^3 - 1408*a^5*b^2)/(32*(a^3*b - a^4)) + (tan(c + d*x)*((2 4*a^2*(a^7*b)^(1/2) + 9*b^2*(a^7*b)^(1/2) - 15*a^5*b + 16*a^6 + 3*a^4*b^2 - 29*a*b*(a^7*b)^(1/2))/(256*(3*a^9*b - a^10 + a^7*b^3 - 3*a^8*b^2)))^(1/2 )*(256*a^7*b - 256*a^4*b^4 + 768*a^5*b^3 - 768*a^6*b^2))/(4*(a^2*b - a^3)) )*((24*a^2*(a^7*b)^(1/2) + 9*b^2*(a^7*b)^(1/2) - 15*a^5*b + 16*a^6 + 3*a^4 *b^2 - 29*a*b*(a^7*b)^(1/2))/(256*(3*a^9*b - a^10 + a^7*b^3 - 3*a^8*b^2))) ^(1/2) + (tan(c + d*x)*(16*a^3*b - 26*a*b^3 + 9*b^4 + 9*a^2*b^2))/(4*(a^2* b - a^3)))*((24*a^2*(a^7*b)^(1/2) + 9*b^2*(a^7*b)^(1/2) - 15*a^5*b + 16*a^ 6 + 3*a^4*b^2 - 29*a*b*(a^7*b)^(1/2))/(256*(3*a^9*b - a^10 + a^7*b^3 - 3*a ^8*b^2)))^(1/2)*1i)/((32*a^2*b - 34*a*b^2 + 9*b^3)/(16*(a^3*b - a^4)) +...