Integrand size = 39, antiderivative size = 157 \[ \int \frac {a+b \sin (d+e x)}{\left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right )^2} \, dx=\frac {2 a b \text {arctanh}\left (\frac {a+b \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2} e}-\frac {\cos (d+e x)}{3 e (b+a \sin (d+e x))^3}+\frac {b \cos (d+e x)}{3 \left (a^2-b^2\right ) e (b+a \sin (d+e x))^2}-\frac {\left (2 a^2+b^2\right ) \cos (d+e x)}{3 \left (a^2-b^2\right )^2 e (b+a \sin (d+e x))} \]
2*a*b*arctanh((a+b*tan(1/2*e*x+1/2*d))/(a^2-b^2)^(1/2))/(a^2-b^2)^(5/2)/e- 1/3*cos(e*x+d)/e/(b+a*sin(e*x+d))^3+1/3*b*cos(e*x+d)/(a^2-b^2)/e/(b+a*sin( e*x+d))^2-1/3*(2*a^2+b^2)*cos(e*x+d)/(a^2-b^2)^2/e/(b+a*sin(e*x+d))
Time = 1.10 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.89 \[ \int \frac {a+b \sin (d+e x)}{\left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right )^2} \, dx=-\frac {\frac {6 a b \arctan \left (\frac {a+b \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{5/2}}+\frac {\cos (d+e x) \left (a^4-a^2 b^2+3 b^4+3 a b \left (a^2+b^2\right ) \sin (d+e x)+a^2 \left (2 a^2+b^2\right ) \sin ^2(d+e x)\right )}{(a-b)^2 (a+b)^2 (b+a \sin (d+e x))^3}}{3 e} \]
-1/3*((6*a*b*ArcTan[(a + b*Tan[(d + e*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^ 2)^(5/2) + (Cos[d + e*x]*(a^4 - a^2*b^2 + 3*b^4 + 3*a*b*(a^2 + b^2)*Sin[d + e*x] + a^2*(2*a^2 + b^2)*Sin[d + e*x]^2))/((a - b)^2*(a + b)^2*(b + a*Si n[d + e*x])^3))/e
Time = 0.82 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.37, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 3769, 27, 3042, 3233, 27, 3042, 3233, 3042, 3233, 27, 3042, 3139, 1083, 219}
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.
| \(\displaystyle \int \frac {a+b \sin (d+e x)}{\left (a^2 \sin ^2(d+e x)+2 a b \sin (d+e x)+b^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a+b \sin (d+e x)}{\left (a^2 \sin (d+e x)^2+2 a b \sin (d+e x)+b^2\right )^2}dx\) |
| \(\Big \downarrow \) 3769 |
| \(\displaystyle 16 a^4 \int \frac {a+b \sin (d+e x)}{16 \left (\sin (d+e x) a^2+b a\right )^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle a^4 \int \frac {a+b \sin (d+e x)}{\left (\sin (d+e x) a^2+b a\right )^4}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 \int \frac {a+b \sin (d+e x)}{\left (\sin (d+e x) a^2+b a\right )^4}dx\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle a^4 \left (\frac {\int \frac {2 a \left (a^2-b^2\right ) \sin (d+e x)}{\left (\sin (d+e x) a^2+b a\right )^3}dx}{3 a^2 \left (a^2-b^2\right )}-\frac {\cos (d+e x)}{3 a e \left (a^2 \sin (d+e x)+a b\right )^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle a^4 \left (\frac {2 \int \frac {\sin (d+e x)}{\left (\sin (d+e x) a^2+b a\right )^3}dx}{3 a}-\frac {\cos (d+e x)}{3 a e \left (a^2 \sin (d+e x)+a b\right )^3}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 \left (\frac {2 \int \frac {\sin (d+e x)}{\left (\sin (d+e x) a^2+b a\right )^3}dx}{3 a}-\frac {\cos (d+e x)}{3 a e \left (a^2 \sin (d+e x)+a b\right )^3}\right )\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle a^4 \left (\frac {2 \left (\frac {\int \frac {2 a^2-a b \sin (d+e x)}{\left (\sin (d+e x) a^2+b a\right )^2}dx}{2 a^2 \left (a^2-b^2\right )}+\frac {b \cos (d+e x)}{2 a e \left (a^2-b^2\right ) \left (a^2 \sin (d+e x)+a b\right )^2}\right )}{3 a}-\frac {\cos (d+e x)}{3 a e \left (a^2 \sin (d+e x)+a b\right )^3}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 \left (\frac {2 \left (\frac {\int \frac {2 a^2-a b \sin (d+e x)}{\left (\sin (d+e x) a^2+b a\right )^2}dx}{2 a^2 \left (a^2-b^2\right )}+\frac {b \cos (d+e x)}{2 a e \left (a^2-b^2\right ) \left (a^2 \sin (d+e x)+a b\right )^2}\right )}{3 a}-\frac {\cos (d+e x)}{3 a e \left (a^2 \sin (d+e x)+a b\right )^3}\right )\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle a^4 \left (\frac {2 \left (\frac {\frac {\int -\frac {3 a^3 b}{\sin (d+e x) a^2+b a}dx}{a^2 \left (a^2-b^2\right )}-\frac {\left (2 a^2+b^2\right ) \cos (d+e x)}{e \left (a^2-b^2\right ) \left (a^2 \sin (d+e x)+a b\right )}}{2 a^2 \left (a^2-b^2\right )}+\frac {b \cos (d+e x)}{2 a e \left (a^2-b^2\right ) \left (a^2 \sin (d+e x)+a b\right )^2}\right )}{3 a}-\frac {\cos (d+e x)}{3 a e \left (a^2 \sin (d+e x)+a b\right )^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle a^4 \left (\frac {2 \left (\frac {-\frac {3 a b \int \frac {1}{\sin (d+e x) a^2+b a}dx}{a^2-b^2}-\frac {\left (2 a^2+b^2\right ) \cos (d+e x)}{e \left (a^2-b^2\right ) \left (a^2 \sin (d+e x)+a b\right )}}{2 a^2 \left (a^2-b^2\right )}+\frac {b \cos (d+e x)}{2 a e \left (a^2-b^2\right ) \left (a^2 \sin (d+e x)+a b\right )^2}\right )}{3 a}-\frac {\cos (d+e x)}{3 a e \left (a^2 \sin (d+e x)+a b\right )^3}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^4 \left (\frac {2 \left (\frac {-\frac {3 a b \int \frac {1}{\sin (d+e x) a^2+b a}dx}{a^2-b^2}-\frac {\left (2 a^2+b^2\right ) \cos (d+e x)}{e \left (a^2-b^2\right ) \left (a^2 \sin (d+e x)+a b\right )}}{2 a^2 \left (a^2-b^2\right )}+\frac {b \cos (d+e x)}{2 a e \left (a^2-b^2\right ) \left (a^2 \sin (d+e x)+a b\right )^2}\right )}{3 a}-\frac {\cos (d+e x)}{3 a e \left (a^2 \sin (d+e x)+a b\right )^3}\right )\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle a^4 \left (\frac {2 \left (\frac {-\frac {6 a b \int \frac {1}{2 \tan \left (\frac {1}{2} (d+e x)\right ) a^2+b \tan ^2\left (\frac {1}{2} (d+e x)\right ) a+b a}d\tan \left (\frac {1}{2} (d+e x)\right )}{e \left (a^2-b^2\right )}-\frac {\left (2 a^2+b^2\right ) \cos (d+e x)}{e \left (a^2-b^2\right ) \left (a^2 \sin (d+e x)+a b\right )}}{2 a^2 \left (a^2-b^2\right )}+\frac {b \cos (d+e x)}{2 a e \left (a^2-b^2\right ) \left (a^2 \sin (d+e x)+a b\right )^2}\right )}{3 a}-\frac {\cos (d+e x)}{3 a e \left (a^2 \sin (d+e x)+a b\right )^3}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle a^4 \left (\frac {2 \left (\frac {\frac {12 a b \int \frac {1}{4 a^2 \left (a^2-b^2\right )-\left (2 a^2+2 b \tan \left (\frac {1}{2} (d+e x)\right ) a\right )^2}d\left (2 a^2+2 b \tan \left (\frac {1}{2} (d+e x)\right ) a\right )}{e \left (a^2-b^2\right )}-\frac {\left (2 a^2+b^2\right ) \cos (d+e x)}{e \left (a^2-b^2\right ) \left (a^2 \sin (d+e x)+a b\right )}}{2 a^2 \left (a^2-b^2\right )}+\frac {b \cos (d+e x)}{2 a e \left (a^2-b^2\right ) \left (a^2 \sin (d+e x)+a b\right )^2}\right )}{3 a}-\frac {\cos (d+e x)}{3 a e \left (a^2 \sin (d+e x)+a b\right )^3}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle a^4 \left (\frac {2 \left (\frac {\frac {6 b \text {arctanh}\left (\frac {2 a^2+2 a b \tan \left (\frac {1}{2} (d+e x)\right )}{2 a \sqrt {a^2-b^2}}\right )}{e \left (a^2-b^2\right )^{3/2}}-\frac {\left (2 a^2+b^2\right ) \cos (d+e x)}{e \left (a^2-b^2\right ) \left (a^2 \sin (d+e x)+a b\right )}}{2 a^2 \left (a^2-b^2\right )}+\frac {b \cos (d+e x)}{2 a e \left (a^2-b^2\right ) \left (a^2 \sin (d+e x)+a b\right )^2}\right )}{3 a}-\frac {\cos (d+e x)}{3 a e \left (a^2 \sin (d+e x)+a b\right )^3}\right )\) |
a^4*(-1/3*Cos[d + e*x]/(a*e*(a*b + a^2*Sin[d + e*x])^3) + (2*((b*Cos[d + e *x])/(2*a*(a^2 - b^2)*e*(a*b + a^2*Sin[d + e*x])^2) + ((6*b*ArcTanh[(2*a^2 + 2*a*b*Tan[(d + e*x)/2])/(2*a*Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*e) - ((2*a^2 + b^2)*Cos[d + e*x])/((a^2 - b^2)*e*(a*b + a^2*Sin[d + e*x])))/(2 *a^2*(a^2 - b^2))))/(3*a))
3.6.2.3.1 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[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ [a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*( m + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
Int[((A_) + (B_.)*sin[(d_.) + (e_.)*(x_)])*((a_) + (b_.)*sin[(d_.) + (e_.)* (x_)] + (c_.)*sin[(d_.) + (e_.)*(x_)]^2)^(n_), x_Symbol] :> Simp[1/(4^n*c^n ) Int[(A + B*Sin[d + e*x])*(b + 2*c*Sin[d + e*x])^(2*n), x], x] /; FreeQ[ {a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[n]
Result contains complex when optimal does not.
Time = 1.94 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.46
| method | result | size |
| risch | \(-\frac {2 \left (2 i a^{5}+i a^{3} b^{2}-6 i a^{5} {\mathrm e}^{2 i \left (e x +d \right )}-12 i a^{3} b^{2} {\mathrm e}^{2 i \left (e x +d \right )}+15 i a^{3} b^{2} {\mathrm e}^{4 i \left (e x +d \right )}-12 a^{4} b \,{\mathrm e}^{3 i \left (e x +d \right )}-14 a^{2} b^{3} {\mathrm e}^{3 i \left (e x +d \right )}-4 \,{\mathrm e}^{3 i \left (e x +d \right )} b^{5}+3 a^{4} b \,{\mathrm e}^{5 i \left (e x +d \right )}+9 a^{4} b \,{\mathrm e}^{i \left (e x +d \right )}-12 i a \,b^{4} {\mathrm e}^{2 i \left (e x +d \right )}+6 a^{2} b^{3} {\mathrm e}^{i \left (e x +d \right )}\right )}{3 \left (a \,{\mathrm e}^{2 i \left (e x +d \right )}+2 i b \,{\mathrm e}^{i \left (e x +d \right )}-a \right )^{3} \left (-a^{2}+b^{2}\right )^{2} e a}+\frac {b a \ln \left ({\mathrm e}^{i \left (e x +d \right )}+\frac {i b \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} e}-\frac {b a \ln \left ({\mathrm e}^{i \left (e x +d \right )}+\frac {i b \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} e}\) | \(386\) |
| derivativedivides | \(\frac {\frac {-\frac {2 a \left (a^{4}-2 a^{2} b^{2}+2 b^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{5}}{b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {2 \left (2 a^{6}-3 a^{4} b^{2}+5 a^{2} b^{4}+b^{6}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {4 a \left (2 a^{6}+a^{4} b^{2}+3 a^{2} b^{4}+9 b^{6}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{3 b^{3} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {4 \left (a^{6}+3 a^{2} b^{4}+b^{6}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {2 a \left (a^{4}+4 b^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {2 \left (a^{4}-a^{2} b^{2}+3 b^{4}\right )}{3 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}}{\left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2} b +2 a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+b \right )^{3}}-\frac {16 a b \arctan \left (\frac {2 b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{\left (8 a^{4}-16 a^{2} b^{2}+8 b^{4}\right ) \sqrt {-a^{2}+b^{2}}}}{e}\) | \(398\) |
| default | \(\frac {\frac {-\frac {2 a \left (a^{4}-2 a^{2} b^{2}+2 b^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{5}}{b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {2 \left (2 a^{6}-3 a^{4} b^{2}+5 a^{2} b^{4}+b^{6}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {4 a \left (2 a^{6}+a^{4} b^{2}+3 a^{2} b^{4}+9 b^{6}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{3 b^{3} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {4 \left (a^{6}+3 a^{2} b^{4}+b^{6}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {2 a \left (a^{4}+4 b^{4}\right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {2 \left (a^{4}-a^{2} b^{2}+3 b^{4}\right )}{3 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}}{\left (\tan \left (\frac {e x}{2}+\frac {d}{2}\right )^{2} b +2 a \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+b \right )^{3}}-\frac {16 a b \arctan \left (\frac {2 b \tan \left (\frac {e x}{2}+\frac {d}{2}\right )+2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{\left (8 a^{4}-16 a^{2} b^{2}+8 b^{4}\right ) \sqrt {-a^{2}+b^{2}}}}{e}\) | \(398\) |
-2/3*(2*I*a^5+I*a^3*b^2-6*I*a^5*exp(2*I*(e*x+d))-12*I*a^3*b^2*exp(2*I*(e*x +d))+15*I*a^3*b^2*exp(4*I*(e*x+d))-12*a^4*b*exp(3*I*(e*x+d))-14*a^2*b^3*ex p(3*I*(e*x+d))-4*exp(3*I*(e*x+d))*b^5+3*a^4*b*exp(5*I*(e*x+d))+9*a^4*b*exp (I*(e*x+d))-12*I*a*b^4*exp(2*I*(e*x+d))+6*a^2*b^3*exp(I*(e*x+d)))/(a*exp(2 *I*(e*x+d))+2*I*b*exp(I*(e*x+d))-a)^3/(-a^2+b^2)^2/e/a+1/(a^2-b^2)^(1/2)*b *a/(a+b)^2/(a-b)^2/e*ln(exp(I*(e*x+d))+(I*b*(a^2-b^2)^(1/2)+a^2-b^2)/(a^2- b^2)^(1/2)/a)-1/(a^2-b^2)^(1/2)*b*a/(a+b)^2/(a-b)^2/e*ln(exp(I*(e*x+d))+(I *b*(a^2-b^2)^(1/2)-a^2+b^2)/(a^2-b^2)^(1/2)/a)
Leaf count of result is larger than twice the leaf count of optimal. 364 vs. \(2 (146) = 292\).
Time = 0.29 (sec) , antiderivative size = 795, normalized size of antiderivative = 5.06 \[ \int \frac {a+b \sin (d+e x)}{\left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right )^2} \, dx=\left [-\frac {2 \, {\left (2 \, a^{6} - a^{4} b^{2} - a^{2} b^{4}\right )} \cos \left (e x + d\right )^{3} - 6 \, {\left (a^{5} b - a b^{5}\right )} \cos \left (e x + d\right ) \sin \left (e x + d\right ) - 3 \, {\left (3 \, a^{3} b^{2} \cos \left (e x + d\right )^{2} - 3 \, a^{3} b^{2} - a b^{4} + {\left (a^{4} b \cos \left (e x + d\right )^{2} - a^{4} b - 3 \, a^{2} b^{3}\right )} \sin \left (e x + d\right )\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {{\left (a^{2} - 2 \, b^{2}\right )} \cos \left (e x + d\right )^{2} + 2 \, a b \sin \left (e x + d\right ) + a^{2} + b^{2} + 2 \, {\left (b \cos \left (e x + d\right ) \sin \left (e x + d\right ) + a \cos \left (e x + d\right )\right )} \sqrt {a^{2} - b^{2}}}{a^{2} \cos \left (e x + d\right )^{2} - 2 \, a b \sin \left (e x + d\right ) - a^{2} - b^{2}}\right ) - 6 \, {\left (a^{6} - a^{4} b^{2} + a^{2} b^{4} - b^{6}\right )} \cos \left (e x + d\right )}{6 \, {\left (3 \, {\left (a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right )} e \cos \left (e x + d\right )^{2} - {\left (3 \, a^{8} b - 8 \, a^{6} b^{3} + 6 \, a^{4} b^{5} - b^{9}\right )} e + {\left ({\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} e \cos \left (e x + d\right )^{2} - {\left (a^{9} - 6 \, a^{5} b^{4} + 8 \, a^{3} b^{6} - 3 \, a b^{8}\right )} e\right )} \sin \left (e x + d\right )\right )}}, -\frac {{\left (2 \, a^{6} - a^{4} b^{2} - a^{2} b^{4}\right )} \cos \left (e x + d\right )^{3} - 3 \, {\left (a^{5} b - a b^{5}\right )} \cos \left (e x + d\right ) \sin \left (e x + d\right ) - 3 \, {\left (3 \, a^{3} b^{2} \cos \left (e x + d\right )^{2} - 3 \, a^{3} b^{2} - a b^{4} + {\left (a^{4} b \cos \left (e x + d\right )^{2} - a^{4} b - 3 \, a^{2} b^{3}\right )} \sin \left (e x + d\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \sin \left (e x + d\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \cos \left (e x + d\right )}\right ) - 3 \, {\left (a^{6} - a^{4} b^{2} + a^{2} b^{4} - b^{6}\right )} \cos \left (e x + d\right )}{3 \, {\left (3 \, {\left (a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right )} e \cos \left (e x + d\right )^{2} - {\left (3 \, a^{8} b - 8 \, a^{6} b^{3} + 6 \, a^{4} b^{5} - b^{9}\right )} e + {\left ({\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} e \cos \left (e x + d\right )^{2} - {\left (a^{9} - 6 \, a^{5} b^{4} + 8 \, a^{3} b^{6} - 3 \, a b^{8}\right )} e\right )} \sin \left (e x + d\right )\right )}}\right ] \]
[-1/6*(2*(2*a^6 - a^4*b^2 - a^2*b^4)*cos(e*x + d)^3 - 6*(a^5*b - a*b^5)*co s(e*x + d)*sin(e*x + d) - 3*(3*a^3*b^2*cos(e*x + d)^2 - 3*a^3*b^2 - a*b^4 + (a^4*b*cos(e*x + d)^2 - a^4*b - 3*a^2*b^3)*sin(e*x + d))*sqrt(a^2 - b^2) *log(((a^2 - 2*b^2)*cos(e*x + d)^2 + 2*a*b*sin(e*x + d) + a^2 + b^2 + 2*(b *cos(e*x + d)*sin(e*x + d) + a*cos(e*x + d))*sqrt(a^2 - b^2))/(a^2*cos(e*x + d)^2 - 2*a*b*sin(e*x + d) - a^2 - b^2)) - 6*(a^6 - a^4*b^2 + a^2*b^4 - b^6)*cos(e*x + d))/(3*(a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*e*cos(e*x + d)^2 - (3*a^8*b - 8*a^6*b^3 + 6*a^4*b^5 - b^9)*e + ((a^9 - 3*a^7*b^2 + 3 *a^5*b^4 - a^3*b^6)*e*cos(e*x + d)^2 - (a^9 - 6*a^5*b^4 + 8*a^3*b^6 - 3*a* b^8)*e)*sin(e*x + d)), -1/3*((2*a^6 - a^4*b^2 - a^2*b^4)*cos(e*x + d)^3 - 3*(a^5*b - a*b^5)*cos(e*x + d)*sin(e*x + d) - 3*(3*a^3*b^2*cos(e*x + d)^2 - 3*a^3*b^2 - a*b^4 + (a^4*b*cos(e*x + d)^2 - a^4*b - 3*a^2*b^3)*sin(e*x + d))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*sin(e*x + d) + a)/((a^2 - b^2)*cos(e*x + d))) - 3*(a^6 - a^4*b^2 + a^2*b^4 - b^6)*cos(e*x + d))/(3 *(a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*e*cos(e*x + d)^2 - (3*a^8*b - 8 *a^6*b^3 + 6*a^4*b^5 - b^9)*e + ((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*e *cos(e*x + d)^2 - (a^9 - 6*a^5*b^4 + 8*a^3*b^6 - 3*a*b^8)*e)*sin(e*x + d)) ]
Timed out. \[ \int \frac {a+b \sin (d+e x)}{\left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {a+b \sin (d+e x)}{\left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right )^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (146) = 292\).
Time = 0.33 (sec) , antiderivative size = 435, normalized size of antiderivative = 2.77 \[ \int \frac {a+b \sin (d+e x)}{\left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right )^2} \, dx=-\frac {2 \, {\left (\frac {3 \, {\left (\pi \left \lfloor \frac {e x + d}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + a}{\sqrt {-a^{2} + b^{2}}}\right )\right )} a b}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {3 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{5} - 6 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{5} + 6 \, a b^{6} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{5} + 6 \, a^{6} b \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{4} - 9 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{4} + 15 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{4} + 3 \, b^{7} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{4} + 4 \, a^{7} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} + 2 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} + 6 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} + 18 \, a b^{6} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{3} + 6 \, a^{6} b \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} + 18 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} + 6 \, b^{7} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} + 3 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + 12 \, a b^{6} \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + a^{4} b^{3} - a^{2} b^{5} + 3 \, b^{7}}{{\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} {\left (b \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right )^{2} + 2 \, a \tan \left (\frac {1}{2} \, e x + \frac {1}{2} \, d\right ) + b\right )}^{3}}\right )}}{3 \, e} \]
-2/3*(3*(pi*floor(1/2*(e*x + d)/pi + 1/2)*sgn(b) + arctan((b*tan(1/2*e*x + 1/2*d) + a)/sqrt(-a^2 + b^2)))*a*b/((a^4 - 2*a^2*b^2 + b^4)*sqrt(-a^2 + b ^2)) + (3*a^5*b^2*tan(1/2*e*x + 1/2*d)^5 - 6*a^3*b^4*tan(1/2*e*x + 1/2*d)^ 5 + 6*a*b^6*tan(1/2*e*x + 1/2*d)^5 + 6*a^6*b*tan(1/2*e*x + 1/2*d)^4 - 9*a^ 4*b^3*tan(1/2*e*x + 1/2*d)^4 + 15*a^2*b^5*tan(1/2*e*x + 1/2*d)^4 + 3*b^7*t an(1/2*e*x + 1/2*d)^4 + 4*a^7*tan(1/2*e*x + 1/2*d)^3 + 2*a^5*b^2*tan(1/2*e *x + 1/2*d)^3 + 6*a^3*b^4*tan(1/2*e*x + 1/2*d)^3 + 18*a*b^6*tan(1/2*e*x + 1/2*d)^3 + 6*a^6*b*tan(1/2*e*x + 1/2*d)^2 + 18*a^2*b^5*tan(1/2*e*x + 1/2*d )^2 + 6*b^7*tan(1/2*e*x + 1/2*d)^2 + 3*a^5*b^2*tan(1/2*e*x + 1/2*d) + 12*a *b^6*tan(1/2*e*x + 1/2*d) + a^4*b^3 - a^2*b^5 + 3*b^7)/((a^4*b^3 - 2*a^2*b ^5 + b^7)*(b*tan(1/2*e*x + 1/2*d)^2 + 2*a*tan(1/2*e*x + 1/2*d) + b)^3))/e
Time = 30.19 (sec) , antiderivative size = 497, normalized size of antiderivative = 3.17 \[ \int \frac {a+b \sin (d+e x)}{\left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right )^2} \, dx=\frac {2\,a\,b\,\mathrm {atanh}\left (\frac {\left (2\,a+2\,b\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}{2\,{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}}\right )}{e\,{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}}-\frac {\frac {2\,\left (a^4-a^2\,b^2+3\,b^4\right )}{3\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {4\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (a^6+3\,a^2\,b^4+b^6\right )}{b^2\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {2\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4\,\left (2\,a^6-3\,a^4\,b^2+5\,a^2\,b^4+b^6\right )}{b^2\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {2\,a\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (a^4+4\,b^4\right )}{b\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {2\,a\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^5\,\left (a^4-2\,a^2\,b^2+2\,b^4\right )}{b\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {4\,a\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\left (2\,a^2+3\,b^2\right )\,\left (a^4-a^2\,b^2+3\,b^4\right )}{3\,b^3\,\left (a^4-2\,a^2\,b^2+b^4\right )}}{e\,\left (b^3\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^6+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\left (8\,a^3+12\,a\,b^2\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (12\,a^2\,b+3\,b^3\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4\,\left (12\,a^2\,b+3\,b^3\right )+b^3+6\,a\,b^2\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )+6\,a\,b^2\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^5\right )} \]
(2*a*b*atanh(((2*a + 2*b*tan(d/2 + (e*x)/2))*(a^4 + b^4 - 2*a^2*b^2))/(2*( a + b)^(5/2)*(a - b)^(5/2))))/(e*(a + b)^(5/2)*(a - b)^(5/2)) - ((2*(a^4 + 3*b^4 - a^2*b^2))/(3*(a^4 + b^4 - 2*a^2*b^2)) + (4*tan(d/2 + (e*x)/2)^2*( a^6 + b^6 + 3*a^2*b^4))/(b^2*(a^4 + b^4 - 2*a^2*b^2)) + (2*tan(d/2 + (e*x) /2)^4*(2*a^6 + b^6 + 5*a^2*b^4 - 3*a^4*b^2))/(b^2*(a^4 + b^4 - 2*a^2*b^2)) + (2*a*tan(d/2 + (e*x)/2)*(a^4 + 4*b^4))/(b*(a^4 + b^4 - 2*a^2*b^2)) + (2 *a*tan(d/2 + (e*x)/2)^5*(a^4 + 2*b^4 - 2*a^2*b^2))/(b*(a^4 + b^4 - 2*a^2*b ^2)) + (4*a*tan(d/2 + (e*x)/2)^3*(2*a^2 + 3*b^2)*(a^4 + 3*b^4 - a^2*b^2))/ (3*b^3*(a^4 + b^4 - 2*a^2*b^2)))/(e*(b^3*tan(d/2 + (e*x)/2)^6 + tan(d/2 + (e*x)/2)^3*(12*a*b^2 + 8*a^3) + tan(d/2 + (e*x)/2)^2*(12*a^2*b + 3*b^3) + tan(d/2 + (e*x)/2)^4*(12*a^2*b + 3*b^3) + b^3 + 6*a*b^2*tan(d/2 + (e*x)/2) + 6*a*b^2*tan(d/2 + (e*x)/2)^5))