Integrand size = 24, antiderivative size = 320 \[ \int \frac {\sin ^8(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\frac {x}{b^2}-\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b^2 d}+\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 \left (\sqrt {a}-\sqrt {b}\right )^{3/2} b^{3/2} d}-\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b^2 d}-\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 \left (\sqrt {a}+\sqrt {b}\right )^{3/2} b^{3/2} d}-\frac {\tan (c+d x)}{4 (a-b) b d}+\frac {\tan ^5(c+d x)}{4 b d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )} \]
x/b^2+1/8*a^(1/4)*arctan((a^(1/2)-b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))/b^(3/ 2)/d/(a^(1/2)-b^(1/2))^(3/2)-1/8*a^(1/4)*arctan((a^(1/2)+b^(1/2))^(1/2)*ta n(d*x+c)/a^(1/4))/b^(3/2)/d/(a^(1/2)+b^(1/2))^(3/2)-1/2*a^(1/4)*arctan((a^ (1/2)-b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))/b^2/d/(a^(1/2)-b^(1/2))^(1/2)-1/2 *a^(1/4)*arctan((a^(1/2)+b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))/b^2/d/(a^(1/2) +b^(1/2))^(1/2)-1/4*tan(d*x+c)/(a-b)/b/d+1/4*tan(d*x+c)^5/b/d/(a+2*a*tan(d *x+c)^2+(a-b)*tan(d*x+c)^4)
Time = 6.69 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.82 \[ \int \frac {\sin ^8(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\frac {8 (c+d x)-\frac {\sqrt {a} \left (4 \sqrt {a}+5 \sqrt {b}\right ) \arctan \left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {a+\sqrt {a} \sqrt {b}}}+\frac {\sqrt {a} \left (4 \sqrt {a}-5 \sqrt {b}\right ) \text {arctanh}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\left (\sqrt {a}-\sqrt {b}\right ) \sqrt {-a+\sqrt {a} \sqrt {b}}}+\frac {2 a b (-6 \sin (2 (c+d x))+\sin (4 (c+d x)))}{(a-b) (8 a-3 b+4 b \cos (2 (c+d x))-b \cos (4 (c+d x)))}}{8 b^2 d} \]
(8*(c + d*x) - (Sqrt[a]*(4*Sqrt[a] + 5*Sqrt[b])*ArcTan[((Sqrt[a] + Sqrt[b] )*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a] + Sqrt[b])*Sqrt[a + Sqrt[a]*Sqrt[b]]) + (Sqrt[a]*(4*Sqrt[a] - 5*Sqrt[b])*ArcTanh[((Sqrt[a] - S qrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a] - Sqrt[b])*Sq rt[-a + Sqrt[a]*Sqrt[b]]) + (2*a*b*(-6*Sin[2*(c + d*x)] + Sin[4*(c + d*x)] ))/((a - b)*(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)])))/(8*b ^2*d)
Time = 0.77 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.18, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3696, 1650, 27, 1598, 27, 1442, 27, 1480, 218, 1610, 2009}
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.
| \(\displaystyle \int \frac {\sin ^8(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^8}{\left (a-b \sin (c+d x)^4\right )^2}dx\) |
| \(\Big \downarrow \) 3696 |
| \(\displaystyle \frac {\int \frac {\tan ^8(c+d x)}{\left (\tan ^2(c+d x)+1\right ) \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2}d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 1650 |
| \(\displaystyle \frac {\frac {\int \frac {a \tan ^4(c+d x) \left (\tan ^2(c+d x)+1\right )}{\left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2}d\tan (c+d x)}{b}-\frac {\int \frac {\tan ^4(c+d x)}{\left (\tan ^2(c+d x)+1\right ) \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}d\tan (c+d x)}{b}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {a \int \frac {\tan ^4(c+d x) \left (\tan ^2(c+d x)+1\right )}{\left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2}d\tan (c+d x)}{b}-\frac {\int \frac {\tan ^4(c+d x)}{\left (\tan ^2(c+d x)+1\right ) \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}d\tan (c+d x)}{b}}{d}\) |
| \(\Big \downarrow \) 1598 |
| \(\displaystyle \frac {\frac {a \left (\frac {\int -\frac {2 b \tan ^4(c+d x)}{(a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}d\tan (c+d x)}{8 a b}+\frac {\tan ^5(c+d x)}{4 a \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}\right )}{b}-\frac {\int \frac {\tan ^4(c+d x)}{\left (\tan ^2(c+d x)+1\right ) \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}d\tan (c+d x)}{b}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {a \left (\frac {\tan ^5(c+d x)}{4 a \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}-\frac {\int \frac {\tan ^4(c+d x)}{(a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}d\tan (c+d x)}{4 a}\right )}{b}-\frac {\int \frac {\tan ^4(c+d x)}{\left (\tan ^2(c+d x)+1\right ) \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}d\tan (c+d x)}{b}}{d}\) |
| \(\Big \downarrow \) 1442 |
| \(\displaystyle \frac {\frac {a \left (\frac {\tan ^5(c+d x)}{4 a \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}-\frac {\frac {\tan (c+d x)}{a-b}-\frac {\int \frac {a \left (2 \tan ^2(c+d x)+1\right )}{(a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}d\tan (c+d x)}{a-b}}{4 a}\right )}{b}-\frac {\int \frac {\tan ^4(c+d x)}{\left (\tan ^2(c+d x)+1\right ) \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}d\tan (c+d x)}{b}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {a \left (\frac {\tan ^5(c+d x)}{4 a \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}-\frac {\frac {\tan (c+d x)}{a-b}-\frac {a \int \frac {2 \tan ^2(c+d x)+1}{(a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a}d\tan (c+d x)}{a-b}}{4 a}\right )}{b}-\frac {\int \frac {\tan ^4(c+d x)}{\left (\tan ^2(c+d x)+1\right ) \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}d\tan (c+d x)}{b}}{d}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {a \left (\frac {\tan ^5(c+d x)}{4 a \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}-\frac {\frac {\tan (c+d x)}{a-b}-\frac {a \left (\frac {\left (\sqrt {a}+\sqrt {b}\right )^2 \int \frac {1}{(a-b) \tan ^2(c+d x)+\sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )}d\tan (c+d x)}{2 \sqrt {a} \sqrt {b}}+\frac {1}{2} \left (2-\frac {a+b}{\sqrt {a} \sqrt {b}}\right ) \int \frac {1}{(a-b) \tan ^2(c+d x)+\sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )}d\tan (c+d x)\right )}{a-b}}{4 a}\right )}{b}-\frac {\int \frac {\tan ^4(c+d x)}{\left (\tan ^2(c+d x)+1\right ) \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}d\tan (c+d x)}{b}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {a \left (\frac {\tan ^5(c+d x)}{4 a \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}-\frac {\frac {\tan (c+d x)}{a-b}-\frac {a \left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt {b} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\left (2-\frac {a+b}{\sqrt {a} \sqrt {b}}\right ) \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {b}\right ) \sqrt {\sqrt {a}+\sqrt {b}}}\right )}{a-b}}{4 a}\right )}{b}-\frac {\int \frac {\tan ^4(c+d x)}{\left (\tan ^2(c+d x)+1\right ) \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}d\tan (c+d x)}{b}}{d}\) |
| \(\Big \downarrow \) 1610 |
| \(\displaystyle \frac {\frac {a \left (\frac {\tan ^5(c+d x)}{4 a \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}-\frac {\frac {\tan (c+d x)}{a-b}-\frac {a \left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt {b} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\left (2-\frac {a+b}{\sqrt {a} \sqrt {b}}\right ) \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {b}\right ) \sqrt {\sqrt {a}+\sqrt {b}}}\right )}{a-b}}{4 a}\right )}{b}-\frac {\int \left (\frac {a \left (\tan ^2(c+d x)+1\right )}{b \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}-\frac {1}{b \left (\tan ^2(c+d x)+1\right )}\right )d\tan (c+d x)}{b}}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a \left (\frac {\tan ^5(c+d x)}{4 a \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}-\frac {\frac {\tan (c+d x)}{a-b}-\frac {a \left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt {b} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\left (2-\frac {a+b}{\sqrt {a} \sqrt {b}}\right ) \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {b}\right ) \sqrt {\sqrt {a}+\sqrt {b}}}\right )}{a-b}}{4 a}\right )}{b}-\frac {\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 b \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 b \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\arctan (\tan (c+d x))}{b}}{b}}{d}\) |
(-((-(ArcTan[Tan[c + d*x]]/b) + (a^(1/4)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*T an[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b) + (a^(1/4)*ArcTan[(Sq rt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b ))/b) + (a*(-1/4*(-((a*(((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]]*Sqrt[b]) + ( (2 - (a + b)/(Sqrt[a]*Sqrt[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]])))/(a - b)) + Tan[c + d*x]/(a - b))/a + Tan[c + d*x]^5/(4*a*(a + 2*a*Tan[c + d* x]^2 + (a - b)*Tan[c + d*x]^4))))/b)/d
3.3.18.3.1 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_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[d^3*(d*x)^(m - 3)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 1))), x] - Simp[d^4/(c*(m + 4*p + 1)) Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b*(m + 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x ] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2* p] && (IntegerQ[p] || IntegerQ[m])
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[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( x_)^4)^(p_.), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(a + b*x^2 + c*x^4)^(p + 1) *((b*d - 2*a*e - (b*e - 2*c*d)*x^2)/(2*(p + 1)*(b^2 - 4*a*c))), x] - Simp[f ^2/(2*(p + 1)*(b^2 - 4*a*c)) Int[(f*x)^(m - 2)*(a + b*x^2 + c*x^4)^(p + 1 )*Simp[(m - 1)*(b*d - 2*a*e) - (4*p + 4 + m + 1)*(b*e - 2*c*d)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*((d + e*x^2)^q/(a + b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4 *a*c, 0] && IntegerQ[q] && IntegerQ[m]
Int[(((f_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[-f^4/(c*d^2 - b*d*e + a*e^2) Int[(f*x)^ (m - 4)*(a*d + (b*d - a*e)*x^2)*(a + b*x^2 + c*x^4)^p, x], x] + Simp[d^2*(f ^4/(c*d^2 - b*d*e + a*e^2)) Int[(f*x)^(m - 4)*((a + b*x^2 + c*x^4)^(p + 1 )/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 2]
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 = 1.97 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.83
| method | result | size |
| derivativedivides | \(\frac {\frac {\arctan \left (\tan \left (d x +c \right )\right )}{b^{2}}-\frac {a \left (\frac {\frac {\left (\tan ^{3}\left (d x +c \right )\right ) b}{2 a -2 b}+\frac {\tan \left (d x +c \right ) b}{4 a -4 b}}{\left (\tan ^{4}\left (d x +c \right )\right ) a -b \left (\tan ^{4}\left (d x +c \right )\right )+2 a \left (\tan ^{2}\left (d x +c \right )\right )+a}+\frac {\left (4 a \sqrt {a b}-6 \sqrt {a b}\, b -3 a b +5 b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{8 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {\left (4 a \sqrt {a b}-6 \sqrt {a b}\, b +3 a b -5 b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{8 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{b^{2}}}{d}\) | \(266\) |
| default | \(\frac {\frac {\arctan \left (\tan \left (d x +c \right )\right )}{b^{2}}-\frac {a \left (\frac {\frac {\left (\tan ^{3}\left (d x +c \right )\right ) b}{2 a -2 b}+\frac {\tan \left (d x +c \right ) b}{4 a -4 b}}{\left (\tan ^{4}\left (d x +c \right )\right ) a -b \left (\tan ^{4}\left (d x +c \right )\right )+2 a \left (\tan ^{2}\left (d x +c \right )\right )+a}+\frac {\left (4 a \sqrt {a b}-6 \sqrt {a b}\, b -3 a b +5 b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{8 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {\left (4 a \sqrt {a b}-6 \sqrt {a b}\, b +3 a b -5 b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{8 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{b^{2}}}{d}\) | \(266\) |
| risch | \(\frac {x}{b^{2}}-\frac {i a \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 b^{2} \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 (a^{3} b^{8} d^{4}-3 a^{2} b^{9} d^{4}+3 a \,b^{10} d^{4}-b^{11} d^{4}\right ) \textit {\_Z}^{4}+\left (8192 a^{3} b^{4} d^{2}-24064 a^{2} b^{5} d^{2}+17920 a \,b^{6} d^{2}\right ) \textit {\_Z}^{2}+16777216 a^{3}-52428800 a^{2} b +40960000 a \,b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (-\frac {2 i a^{4} b^{6} d^{3}}{131072 a^{3} b -679936 a^{2} b^{2}+1152000 a \,b^{3}-640000 b^{4}}+\frac {9 i a^{3} b^{7} d^{3}}{131072 a^{3} b -679936 a^{2} b^{2}+1152000 a \,b^{3}-640000 b^{4}}-\frac {15 i a^{2} b^{8} d^{3}}{131072 a^{3} b -679936 a^{2} b^{2}+1152000 a \,b^{3}-640000 b^{4}}+\frac {11 i a \,b^{9} d^{3}}{131072 a^{3} b -679936 a^{2} b^{2}+1152000 a \,b^{3}-640000 b^{4}}-\frac {3 i b^{10} d^{3}}{131072 a^{3} b -679936 a^{2} b^{2}+1152000 a \,b^{3}-640000 b^{4}}\right ) \textit {\_R}^{3}+\left (-\frac {128 a^{4} b^{4} d^{2}}{131072 a^{3} b -679936 a^{2} b^{2}+1152000 a \,b^{3}-640000 b^{4}}+\frac {584 a^{3} b^{5} d^{2}}{131072 a^{3} b -679936 a^{2} b^{2}+1152000 a \,b^{3}-640000 b^{4}}-\frac {984 a^{2} b^{6} d^{2}}{131072 a^{3} b -679936 a^{2} b^{2}+1152000 a \,b^{3}-640000 b^{4}}+\frac {728 a \,b^{7} d^{2}}{131072 a^{3} b -679936 a^{2} b^{2}+1152000 a \,b^{3}-640000 b^{4}}-\frac {200 d^{2} b^{8}}{131072 a^{3} b -679936 a^{2} b^{2}+1152000 a \,b^{3}-640000 b^{4}}\right ) \textit {\_R}^{2}+\left (-\frac {8192 i a^{4} b^{2} d}{131072 a^{3} b -679936 a^{2} b^{2}+1152000 a \,b^{3}-640000 b^{4}}+\frac {33280 i a^{3} b^{3} d}{131072 a^{3} b -679936 a^{2} b^{2}+1152000 a \,b^{3}-640000 b^{4}}-\frac {37760 i a^{2} b^{4} d}{131072 a^{3} b -679936 a^{2} b^{2}+1152000 a \,b^{3}-640000 b^{4}}-\frac {1280 i a \,b^{5} d}{131072 a^{3} b -679936 a^{2} b^{2}+1152000 a \,b^{3}-640000 b^{4}}+\frac {16000 i d \,b^{6}}{131072 a^{3} b -679936 a^{2} b^{2}+1152000 a \,b^{3}-640000 b^{4}}\right ) \textit {\_R} -\frac {524288 a^{4}}{131072 a^{3} b -679936 a^{2} b^{2}+1152000 a \,b^{3}-640000 b^{4}}+\frac {2228224 a^{3} b}{131072 a^{3} b -679936 a^{2} b^{2}+1152000 a \,b^{3}-640000 b^{4}}-\frac {2873344 a^{2} b^{2}}{131072 a^{3} b -679936 a^{2} b^{2}+1152000 a \,b^{3}-640000 b^{4}}+\frac {640000 a \,b^{3}}{131072 a^{3} b -679936 a^{2} b^{2}+1152000 a \,b^{3}-640000 b^{4}}+\frac {640000 b^{4}}{131072 a^{3} b -679936 a^{2} b^{2}+1152000 a \,b^{3}-640000 b^{4}}\right )\right )}{256}\) | \(1005\) |
1/d*(1/b^2*arctan(tan(d*x+c))-a/b^2*((1/2*b/(a-b)*tan(d*x+c)^3+1/4*b/(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)+1/8*(4*a*( a*b)^(1/2)-6*(a*b)^(1/2)*b-3*a*b+5*b^2)/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)-a) *(a-b))^(1/2)*arctanh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))+1/8 *(4*a*(a*b)^(1/2)-6*(a*b)^(1/2)*b+3*a*b-5*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) )))
Leaf count of result is larger than twice the leaf count of optimal. 3544 vs. \(2 (240) = 480\).
Time = 0.94 (sec) , antiderivative size = 3544, normalized size of antiderivative = 11.08 \[ \int \frac {\sin ^8(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\text {Too large to display} \]
1/32*(32*(a*b - b^2)*d*x*cos(d*x + c)^4 - 64*(a*b - b^2)*d*x*cos(d*x + c)^ 2 - 32*(a^2 - 2*a*b + b^2)*d*x + ((a*b^3 - b^4)*d*cos(d*x + c)^4 - 2*(a*b^ 3 - b^4)*d*cos(d*x + c)^2 - (a^2*b^2 - 2*a*b^3 + b^4)*d)*sqrt(-((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) + 16*a^3 - 47*a^2*b + 35*a*b^2)/(( a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2))*log(32*a^3 - 166*a^2*b + 1125/4 *a*b^2 - 625/4*b^3 - 1/4*(128*a^3 - 664*a^2*b + 1125*a*b^2 - 625*b^3)*cos( d*x + c)^2 + 1/2*(2*(2*a^4*b^5 - 9*a^3*b^6 + 15*a^2*b^7 - 11*a*b^8 + 3*b^9 )*d^3*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/ ((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4))*cos(d*x + c)*sin(d*x + c) - (24*a^3*b^2 - 127*a^2*b^3 + 220* a*b^4 - 125*b^5)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^ 3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b ^11 - 6*a*b^12 + b^13)*d^4)) + 16*a^3 - 47*a^2*b + 35*a*b^2)/((a^3*b^4 - 3 *a^2*b^5 + 3*a*b^6 - b^7)*d^2)) + 1/4*(2*(16*a^4*b^3 - 73*a^3*b^4 + 123*a^ 2*b^5 - 91*a*b^6 + 25*b^7)*d^2*cos(d*x + c)^2 - (16*a^4*b^3 - 73*a^3*b^4 + 123*a^2*b^5 - 91*a*b^6 + 25*b^7)*d^2)*sqrt((64*a^5 - 464*a^4*b + 1241*a^3 *b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 2...
Timed out. \[ \int \frac {\sin ^8(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {\sin ^8(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\int { \frac {\sin \left (d x + c\right )^{8}}{{\left (b \sin \left (d x + c\right )^{4} - a\right )}^{2}} \,d x } \]
1/2*(2*(a*b^2 - b^3)*d*x*cos(8*d*x + 8*c)^2 + 32*(a*b^2 - b^3)*d*x*cos(6*d *x + 6*c)^2 + 8*(64*a^3 - 112*a^2*b + 57*a*b^2 - 9*b^3)*d*x*cos(4*d*x + 4* c)^2 + 32*(a*b^2 - b^3)*d*x*cos(2*d*x + 2*c)^2 + 2*(a*b^2 - b^3)*d*x*sin(8 *d*x + 8*c)^2 + 32*(a*b^2 - b^3)*d*x*sin(6*d*x + 6*c)^2 + 8*(64*a^3 - 112* a^2*b + 57*a*b^2 - 9*b^3)*d*x*sin(4*d*x + 4*c)^2 + 32*(a*b^2 - b^3)*d*x*si n(2*d*x + 2*c)^2 - 16*(a*b^2 - b^3)*d*x*cos(2*d*x + 2*c) - a*b^2*sin(2*d*x + 2*c) + 2*(a*b^2 - b^3)*d*x - (16*(a*b^2 - b^3)*d*x*cos(6*d*x + 6*c) + 8 *(8*a^2*b - 11*a*b^2 + 3*b^3)*d*x*cos(4*d*x + 4*c) + 16*(a*b^2 - b^3)*d*x* cos(2*d*x + 2*c) - a*b^2*sin(6*d*x + 6*c) + 5*a*b^2*sin(2*d*x + 2*c) - 4*( a*b^2 - b^3)*d*x + (8*a^2*b - 3*a*b^2)*sin(4*d*x + 4*c))*cos(8*d*x + 8*c) + 2*(16*(8*a^2*b - 11*a*b^2 + 3*b^3)*d*x*cos(4*d*x + 4*c) + 32*(a*b^2 - b^ 3)*d*x*cos(2*d*x + 2*c) + 12*a*b^2*sin(2*d*x + 2*c) - 8*(a*b^2 - b^3)*d*x + 3*(8*a^2*b - 3*a*b^2)*sin(4*d*x + 4*c))*cos(6*d*x + 6*c) + 2*(16*(8*a^2* b - 11*a*b^2 + 3*b^3)*d*x*cos(2*d*x + 2*c) - 4*(8*a^2*b - 11*a*b^2 + 3*b^3 )*d*x + 3*(8*a^2*b - 3*a*b^2)*sin(2*d*x + 2*c))*cos(4*d*x + 4*c) - 2*((a*b ^4 - b^5)*d*cos(8*d*x + 8*c)^2 + 16*(a*b^4 - b^5)*d*cos(6*d*x + 6*c)^2 + 4 *(64*a^3*b^2 - 112*a^2*b^3 + 57*a*b^4 - 9*b^5)*d*cos(4*d*x + 4*c)^2 + 16*( a*b^4 - b^5)*d*cos(2*d*x + 2*c)^2 + (a*b^4 - b^5)*d*sin(8*d*x + 8*c)^2 + 1 6*(a*b^4 - b^5)*d*sin(6*d*x + 6*c)^2 + 4*(64*a^3*b^2 - 112*a^2*b^3 + 57*a* b^4 - 9*b^5)*d*sin(4*d*x + 4*c)^2 + 16*(8*a^2*b^3 - 11*a*b^4 + 3*b^5)*d...
Leaf count of result is larger than twice the leaf count of optimal. 1563 vs. \(2 (240) = 480\).
Time = 0.93 (sec) , antiderivative size = 1563, normalized size of antiderivative = 4.88 \[ \int \frac {\sin ^8(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\text {Too large to display} \]
1/8*((2*(6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3 - 21*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b + 16*sqrt(a^2 - a*b - sqrt(a*b )*(a - b))*sqrt(a*b)*a*b^2 + 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a* b)*b^3)*(a*b^2 - b^3)^2*abs(-a + b) - (12*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^5*b^2 - 63*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^4*b^3 + 116*sqrt(a^ 2 - a*b - sqrt(a*b)*(a - b))*a^3*b^4 - 86*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^5 + 16*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^6 + 5*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*b^7)*abs(-a*b^2 + b^3)*abs(-a + b) - (9*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^4 - 51*sqrt(a^2 - a*b - sqrt(a* b)*(a - b))*sqrt(a*b)*a^4*b^5 + 102*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sq rt(a*b)*a^3*b^6 - 82*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^7 + 17*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^8 + 5*sqrt(a^2 - a *b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^9)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^2*b^2 - a*b^3 + sqrt((a^2*b^2 - a*b^ 3)^2 - (a^2*b^2 - a*b^3)*(a^2*b^2 - 2*a*b^3 + b^4)))/(a^2*b^2 - 2*a*b^3 + b^4))))/((3*a^7*b^4 - 21*a^6*b^5 + 59*a^5*b^6 - 85*a^4*b^7 + 65*a^3*b^8 - 23*a^2*b^9 + a*b^10 + b^11)*abs(-a*b^2 + b^3)) + (2*(6*sqrt(a^2 - a*b + sq rt(a*b)*(a - b))*sqrt(a*b)*a^3 - 21*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sq rt(a*b)*a^2*b + 16*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 + 3 *sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*(a*b^2 - b^3)^2*abs...
Time = 17.82 (sec) , antiderivative size = 7640, normalized size of antiderivative = 23.88 \[ \int \frac {\sin ^8(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\text {Too large to display} \]
(atan(((((5120*a^2*b^7 - 17664*a^3*b^6 + 26688*a^4*b^5 - 16320*a^5*b^4 + 3 072*a^6*b^3)/(256*(a*b^5 - b^6)) + (((20480*a^2*b^11 - 110592*a^3*b^10 + 2 08896*a^4*b^9 - 167936*a^5*b^8 + 49152*a^6*b^7)/(256*(a*b^5 - b^6)) - (tan (c + d*x)*((8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) - 35*a*b^6 + 47*a^2 *b^5 - 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^ 9 + a^3*b^8)))^(1/2)*(98304*a^2*b^12 - 196608*a^3*b^11 + 196608*a^5*b^9 - 98304*a^6*b^8))/(128*(a*b^4 - b^5)))*((8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9 )^(1/2) - 35*a*b^6 + 47*a^2*b^5 - 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256* (3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2) - (tan(c + d*x)*(21376*a^2 *b^8 - 84864*a^3*b^7 + 54912*a^4*b^6 + 20864*a^5*b^5 - 18432*a^6*b^4))/(12 8*(a*b^4 - b^5)))*((8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) - 35*a*b^6 + 47*a^2*b^5 - 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2))*((8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) - 35*a*b^6 + 47*a^2*b^5 - 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^ 10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2) + (tan(c + d*x)*(768*a^6 + 800*a^ 2*b^4 + 4832*a^3*b^3 - 5295*a^4*b^2))/(128*(a*b^4 - b^5)))*((8*a^2*(a*b^9) ^(1/2) + 25*b^2*(a*b^9)^(1/2) - 35*a*b^6 + 47*a^2*b^5 - 16*a^3*b^4 - 29*a* b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2)*1i - (((5120*a^2*b^7 - 17664*a^3*b^6 + 26688*a^4*b^5 - 16320*a^5*b^4 + 3072*a^ 6*b^3)/(256*(a*b^5 - b^6)) + (((20480*a^2*b^11 - 110592*a^3*b^10 + 2088...