Integrand size = 25, antiderivative size = 271 \[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx=\frac {2 b^3 \arctan \left (\frac {b+3 \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {9-b^2}}\right )}{\sqrt {9-b^2} (b c-3 d)^3 f}+\frac {d \left (18 b c^3 d-9 d^2 \left (2 c^2+d^2\right )-b^2 \left (6 c^4-5 c^2 d^2+2 d^4\right )\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(b c-3 d)^3 \left (c^2-d^2\right )^{5/2} f}-\frac {d^2 \cos (e+f x)}{2 (b c-3 d) \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}-\frac {d^2 \left (5 b c^2-9 c d-2 b d^2\right ) \cos (e+f x)}{2 (b c-3 d)^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))} \]
d*(6*a*b*c^3*d-a^2*d^2*(2*c^2+d^2)-b^2*(6*c^4-5*c^2*d^2+2*d^4))*arctan((d+ c*tan(1/2*f*x+1/2*e))/(c^2-d^2)^(1/2))/(-a*d+b*c)^3/(c^2-d^2)^(5/2)/f-1/2* d^2*cos(f*x+e)/(-a*d+b*c)/(c^2-d^2)/f/(c+d*sin(f*x+e))^2-1/2*d^2*(-3*a*c*d +5*b*c^2-2*b*d^2)*cos(f*x+e)/(-a*d+b*c)^2/(c^2-d^2)^2/f/(c+d*sin(f*x+e))+2 *b^3*arctan((b+a*tan(1/2*f*x+1/2*e))/(a^2-b^2)^(1/2))/(-a*d+b*c)^3/f/(a^2- b^2)^(1/2)
Time = 1.73 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx=-\frac {-\frac {4 b^3 \arctan \left (\frac {b+3 \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {9-b^2}}\right )}{\sqrt {9-b^2}}+\frac {2 d \left (-18 b c^3 d+9 \left (2 c^2 d^2+d^4\right )+b^2 \left (6 c^4-5 c^2 d^2+2 d^4\right )\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{\left (c^2-d^2\right )^{5/2}}+\frac {(b c-3 d)^2 d^2 \cos (e+f x)}{(c-d) (c+d) (c+d \sin (e+f x))^2}+\frac {(b c-3 d) d^2 \left (5 b c^2-9 c d-2 b d^2\right ) \cos (e+f x)}{(c-d)^2 (c+d)^2 (c+d \sin (e+f x))}}{2 (b c-3 d)^3 f} \]
-1/2*((-4*b^3*ArcTan[(b + 3*Tan[(e + f*x)/2])/Sqrt[9 - b^2]])/Sqrt[9 - b^2 ] + (2*d*(-18*b*c^3*d + 9*(2*c^2*d^2 + d^4) + b^2*(6*c^4 - 5*c^2*d^2 + 2*d ^4))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(c^2 - d^2)^(5/2) + ((b*c - 3*d)^2*d^2*Cos[e + f*x])/((c - d)*(c + d)*(c + d*Sin[e + f*x])^2) + ((b*c - 3*d)*d^2*(5*b*c^2 - 9*c*d - 2*b*d^2)*Cos[e + f*x])/((c - d)^2*( c + d)^2*(c + d*Sin[e + f*x])))/((b*c - 3*d)^3*f)
Time = 1.54 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.31, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 3281, 25, 3042, 3534, 25, 3042, 3480, 3042, 3139, 1083, 217}
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.
| \(\displaystyle \int \frac {1}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^3}dx\) |
| \(\Big \downarrow \) 3281 |
| \(\displaystyle \frac {\int -\frac {-b d^2 \sin ^2(e+f x)+d (2 b c-a d) \sin (e+f x)+2 \left (a c d-b \left (c^2-d^2\right )\right )}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^2}dx}{2 \left (c^2-d^2\right ) (b c-a d)}-\frac {d^2 \cos (e+f x)}{2 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {-b d^2 \sin ^2(e+f x)+d (2 b c-a d) \sin (e+f x)+2 \left (a c d-b \left (c^2-d^2\right )\right )}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^2}dx}{2 \left (c^2-d^2\right ) (b c-a d)}-\frac {d^2 \cos (e+f x)}{2 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {-b d^2 \sin (e+f x)^2+d (2 b c-a d) \sin (e+f x)+2 \left (a c d-b \left (c^2-d^2\right )\right )}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^2}dx}{2 \left (c^2-d^2\right ) (b c-a d)}-\frac {d^2 \cos (e+f x)}{2 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle -\frac {\frac {\int -\frac {a^2 \left (2 c^2+d^2\right ) d^2-a b c \left (4 c^2-d^2\right ) d-b \left (4 b c^3-2 a d c^2-b d^2 c-a d^3\right ) \sin (e+f x) d+2 b^2 \left (c^2-d^2\right )^2}{(a+b \sin (e+f x)) (c+d \sin (e+f x))}dx}{\left (c^2-d^2\right ) (b c-a d)}+\frac {d^2 \left (-3 a c d+5 b c^2-2 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))}}{2 \left (c^2-d^2\right ) (b c-a d)}-\frac {d^2 \cos (e+f x)}{2 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {d^2 \left (-3 a c d+5 b c^2-2 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))}-\frac {\int \frac {a^2 \left (2 c^2+d^2\right ) d^2-a b c \left (4 c^2-d^2\right ) d-b \left (4 b c^3-2 a d c^2-b d^2 c-a d^3\right ) \sin (e+f x) d+2 b^2 \left (c^2-d^2\right )^2}{(a+b \sin (e+f x)) (c+d \sin (e+f x))}dx}{\left (c^2-d^2\right ) (b c-a d)}}{2 \left (c^2-d^2\right ) (b c-a d)}-\frac {d^2 \cos (e+f x)}{2 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {d^2 \left (-3 a c d+5 b c^2-2 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))}-\frac {\int \frac {a^2 \left (2 c^2+d^2\right ) d^2-a b c \left (4 c^2-d^2\right ) d-b \left (4 b c^3-2 a d c^2-b d^2 c-a d^3\right ) \sin (e+f x) d+2 b^2 \left (c^2-d^2\right )^2}{(a+b \sin (e+f x)) (c+d \sin (e+f x))}dx}{\left (c^2-d^2\right ) (b c-a d)}}{2 \left (c^2-d^2\right ) (b c-a d)}-\frac {d^2 \cos (e+f x)}{2 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle -\frac {\frac {d^2 \left (-3 a c d+5 b c^2-2 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))}-\frac {\frac {d \left (-a^2 d^2 \left (2 c^2+d^2\right )+6 a b c^3 d-b^2 \left (6 c^4-5 c^2 d^2+2 d^4\right )\right ) \int \frac {1}{c+d \sin (e+f x)}dx}{b c-a d}+\frac {2 b^3 \left (c^2-d^2\right )^2 \int \frac {1}{a+b \sin (e+f x)}dx}{b c-a d}}{\left (c^2-d^2\right ) (b c-a d)}}{2 \left (c^2-d^2\right ) (b c-a d)}-\frac {d^2 \cos (e+f x)}{2 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {d^2 \left (-3 a c d+5 b c^2-2 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))}-\frac {\frac {d \left (-a^2 d^2 \left (2 c^2+d^2\right )+6 a b c^3 d-b^2 \left (6 c^4-5 c^2 d^2+2 d^4\right )\right ) \int \frac {1}{c+d \sin (e+f x)}dx}{b c-a d}+\frac {2 b^3 \left (c^2-d^2\right )^2 \int \frac {1}{a+b \sin (e+f x)}dx}{b c-a d}}{\left (c^2-d^2\right ) (b c-a d)}}{2 \left (c^2-d^2\right ) (b c-a d)}-\frac {d^2 \cos (e+f x)}{2 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle -\frac {\frac {d^2 \left (-3 a c d+5 b c^2-2 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))}-\frac {\frac {2 d \left (-a^2 d^2 \left (2 c^2+d^2\right )+6 a b c^3 d-b^2 \left (6 c^4-5 c^2 d^2+2 d^4\right )\right ) \int \frac {1}{c \tan ^2\left (\frac {1}{2} (e+f x)\right )+2 d \tan \left (\frac {1}{2} (e+f x)\right )+c}d\tan \left (\frac {1}{2} (e+f x)\right )}{f (b c-a d)}+\frac {4 b^3 \left (c^2-d^2\right )^2 \int \frac {1}{a \tan ^2\left (\frac {1}{2} (e+f x)\right )+2 b \tan \left (\frac {1}{2} (e+f x)\right )+a}d\tan \left (\frac {1}{2} (e+f x)\right )}{f (b c-a d)}}{\left (c^2-d^2\right ) (b c-a d)}}{2 \left (c^2-d^2\right ) (b c-a d)}-\frac {d^2 \cos (e+f x)}{2 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {\frac {d^2 \left (-3 a c d+5 b c^2-2 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))}-\frac {-\frac {4 d \left (-a^2 d^2 \left (2 c^2+d^2\right )+6 a b c^3 d-b^2 \left (6 c^4-5 c^2 d^2+2 d^4\right )\right ) \int \frac {1}{-\left (2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )^2-4 \left (c^2-d^2\right )}d\left (2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )}{f (b c-a d)}-\frac {8 b^3 \left (c^2-d^2\right )^2 \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (e+f x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (e+f x)\right )\right )}{f (b c-a d)}}{\left (c^2-d^2\right ) (b c-a d)}}{2 \left (c^2-d^2\right ) (b c-a d)}-\frac {d^2 \cos (e+f x)}{2 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\frac {d^2 \left (-3 a c d+5 b c^2-2 b d^2\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))}-\frac {\frac {2 d \left (-a^2 d^2 \left (2 c^2+d^2\right )+6 a b c^3 d-b^2 \left (6 c^4-5 c^2 d^2+2 d^4\right )\right ) \arctan \left (\frac {2 c \tan \left (\frac {1}{2} (e+f x)\right )+2 d}{2 \sqrt {c^2-d^2}}\right )}{f \sqrt {c^2-d^2} (b c-a d)}+\frac {4 b^3 \left (c^2-d^2\right )^2 \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (e+f x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{f \sqrt {a^2-b^2} (b c-a d)}}{\left (c^2-d^2\right ) (b c-a d)}}{2 \left (c^2-d^2\right ) (b c-a d)}-\frac {d^2 \cos (e+f x)}{2 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^2}\) |
-1/2*(d^2*Cos[e + f*x])/((b*c - a*d)*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^2) - (-(((4*b^3*(c^2 - d^2)^2*ArcTan[(2*b + 2*a*Tan[(e + f*x)/2])/(2*Sqrt[a^ 2 - b^2])])/(Sqrt[a^2 - b^2]*(b*c - a*d)*f) + (2*d*(6*a*b*c^3*d - a^2*d^2* (2*c^2 + d^2) - b^2*(6*c^4 - 5*c^2*d^2 + 2*d^4))*ArcTan[(2*d + 2*c*Tan[(e + f*x)/2])/(2*Sqrt[c^2 - d^2])])/((b*c - a*d)*Sqrt[c^2 - d^2]*f))/((b*c - a*d)*(c^2 - d^2))) + (d^2*(5*b*c^2 - 3*a*c*d - 2*b*d^2)*Cos[e + f*x])/((b* c - a*d)*(c^2 - d^2)*f*(c + d*Sin[e + f*x])))/(2*(b*c - a*d)*(c^2 - d^2))
3.8.5.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[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_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f* x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2 ))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[a*(b*c - a*d)*(m + 1) + b^2*d*(m + n + 2) - (b^2*c + b*(b*c - a*d)*(m + 1))*Sin[e + f*x] - b^2*d*(m + n + 3)*Si n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a* d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !(IntegerQ[2* n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) || EqQ[a, 0])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(A*b - a*B)/(b*c - a*d) Int[1/(a + b*Sin[e + f*x]), x], x] + Simp[(B*c - A*d)/ (b*c - a*d) Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Leaf count of result is larger than twice the leaf count of optimal. \(624\) vs. \(2(270)=540\).
Time = 6.01 (sec) , antiderivative size = 625, normalized size of antiderivative = 2.31
| method | result | size |
| derivativedivides | \(\frac {\frac {2 d \left (\frac {\frac {d^{2} \left (5 a^{2} c^{2} d^{2}-2 a^{2} d^{4}-12 a b \,c^{3} d +6 d^{3} a b c +7 b^{2} c^{4}-4 b^{2} c^{2} d^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) c}+\frac {d \left (4 a^{2} c^{4} d^{2}+7 a^{2} c^{2} d^{4}-2 a^{2} d^{6}-10 a b \,c^{5} d -16 a b \,c^{3} d^{3}+8 a b c \,d^{5}+6 b^{2} c^{6}+9 b^{2} c^{4} d^{2}-6 b^{2} c^{2} d^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) c^{2}}+\frac {d^{2} \left (11 a^{2} c^{2} d^{2}-2 a^{2} d^{4}-28 a b \,c^{3} d +10 d^{3} a b c +17 b^{2} c^{4}-8 b^{2} c^{2} d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 c \left (c^{4}-2 c^{2} d^{2}+d^{4}\right )}+\frac {d \left (4 a^{2} c^{2} d^{2}-a^{2} d^{4}-10 a b \,c^{3} d +4 d^{3} a b c +6 b^{2} c^{4}-3 b^{2} c^{2} d^{2}\right )}{2 c^{4}-4 c^{2} d^{2}+2 d^{4}}}{{\left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )}^{2}}+\frac {\left (2 a^{2} c^{2} d^{2}+a^{2} d^{4}-6 a b \,c^{3} d +6 b^{2} c^{4}-5 b^{2} c^{2} d^{2}+2 b^{2} d^{4}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) \sqrt {c^{2}-d^{2}}}\right )}{\left (d a -c b \right )^{3}}-\frac {2 b^{3} \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (d^{3} a^{3}-3 c \,d^{2} a^{2} b +3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {a^{2}-b^{2}}}}{f}\) | \(625\) |
| default | \(\frac {\frac {2 d \left (\frac {\frac {d^{2} \left (5 a^{2} c^{2} d^{2}-2 a^{2} d^{4}-12 a b \,c^{3} d +6 d^{3} a b c +7 b^{2} c^{4}-4 b^{2} c^{2} d^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) c}+\frac {d \left (4 a^{2} c^{4} d^{2}+7 a^{2} c^{2} d^{4}-2 a^{2} d^{6}-10 a b \,c^{5} d -16 a b \,c^{3} d^{3}+8 a b c \,d^{5}+6 b^{2} c^{6}+9 b^{2} c^{4} d^{2}-6 b^{2} c^{2} d^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) c^{2}}+\frac {d^{2} \left (11 a^{2} c^{2} d^{2}-2 a^{2} d^{4}-28 a b \,c^{3} d +10 d^{3} a b c +17 b^{2} c^{4}-8 b^{2} c^{2} d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 c \left (c^{4}-2 c^{2} d^{2}+d^{4}\right )}+\frac {d \left (4 a^{2} c^{2} d^{2}-a^{2} d^{4}-10 a b \,c^{3} d +4 d^{3} a b c +6 b^{2} c^{4}-3 b^{2} c^{2} d^{2}\right )}{2 c^{4}-4 c^{2} d^{2}+2 d^{4}}}{{\left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )}^{2}}+\frac {\left (2 a^{2} c^{2} d^{2}+a^{2} d^{4}-6 a b \,c^{3} d +6 b^{2} c^{4}-5 b^{2} c^{2} d^{2}+2 b^{2} d^{4}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) \sqrt {c^{2}-d^{2}}}\right )}{\left (d a -c b \right )^{3}}-\frac {2 b^{3} \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (d^{3} a^{3}-3 c \,d^{2} a^{2} b +3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \sqrt {a^{2}-b^{2}}}}{f}\) | \(625\) |
| risch | \(\text {Expression too large to display}\) | \(1603\) |
1/f*(2*d/(a*d-b*c)^3*((1/2*d^2*(5*a^2*c^2*d^2-2*a^2*d^4-12*a*b*c^3*d+6*a*b *c*d^3+7*b^2*c^4-4*b^2*c^2*d^2)/(c^4-2*c^2*d^2+d^4)/c*tan(1/2*f*x+1/2*e)^3 +1/2*d*(4*a^2*c^4*d^2+7*a^2*c^2*d^4-2*a^2*d^6-10*a*b*c^5*d-16*a*b*c^3*d^3+ 8*a*b*c*d^5+6*b^2*c^6+9*b^2*c^4*d^2-6*b^2*c^2*d^4)/(c^4-2*c^2*d^2+d^4)/c^2 *tan(1/2*f*x+1/2*e)^2+1/2*d^2*(11*a^2*c^2*d^2-2*a^2*d^4-28*a*b*c^3*d+10*a* b*c*d^3+17*b^2*c^4-8*b^2*c^2*d^2)/c/(c^4-2*c^2*d^2+d^4)*tan(1/2*f*x+1/2*e) +1/2*d*(4*a^2*c^2*d^2-a^2*d^4-10*a*b*c^3*d+4*a*b*c*d^3+6*b^2*c^4-3*b^2*c^2 *d^2)/(c^4-2*c^2*d^2+d^4))/(tan(1/2*f*x+1/2*e)^2*c+2*d*tan(1/2*f*x+1/2*e)+ c)^2+1/2*(2*a^2*c^2*d^2+a^2*d^4-6*a*b*c^3*d+6*b^2*c^4-5*b^2*c^2*d^2+2*b^2* d^4)/(c^4-2*c^2*d^2+d^4)/(c^2-d^2)^(1/2)*arctan(1/2*(2*c*tan(1/2*f*x+1/2*e )+2*d)/(c^2-d^2)^(1/2)))-2*b^3/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^ 3)/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*f*x+1/2*e)+2*b)/(a^2-b^2)^(1/2) ))
Timed out. \[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx=\text {Exception raised: ValueError} \]
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*b^2-4*a^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 766 vs. \(2 (270) = 540\).
Time = 0.35 (sec) , antiderivative size = 766, normalized size of antiderivative = 2.83 \[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx=\frac {\frac {2 \, {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )} b^{3}}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a^{2} - b^{2}}} - \frac {{\left (6 \, b^{2} c^{4} d - 6 \, a b c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} - 5 \, b^{2} c^{2} d^{3} + a^{2} d^{5} + 2 \, b^{2} d^{5}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (c\right ) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + d}{\sqrt {c^{2} - d^{2}}}\right )\right )}}{{\left (b^{3} c^{7} - 3 \, a b^{2} c^{6} d + 3 \, a^{2} b c^{5} d^{2} - 2 \, b^{3} c^{5} d^{2} - a^{3} c^{4} d^{3} + 6 \, a b^{2} c^{4} d^{3} - 6 \, a^{2} b c^{3} d^{4} + b^{3} c^{3} d^{4} + 2 \, a^{3} c^{2} d^{5} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7}\right )} \sqrt {c^{2} - d^{2}}} - \frac {7 \, b c^{4} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 5 \, a c^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, b c^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, a c d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 \, b c^{5} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 4 \, a c^{4} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 9 \, b c^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 7 \, a c^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 6 \, b c d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, a d^{7} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 17 \, b c^{4} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 11 \, a c^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 8 \, b c^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, a c d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, b c^{5} d^{2} - 4 \, a c^{4} d^{3} - 3 \, b c^{3} d^{4} + a c^{2} d^{5}}{{\left (b^{2} c^{8} - 2 \, a b c^{7} d + a^{2} c^{6} d^{2} - 2 \, b^{2} c^{6} d^{2} + 4 \, a b c^{5} d^{3} - 2 \, a^{2} c^{4} d^{4} + b^{2} c^{4} d^{4} - 2 \, a b c^{3} d^{5} + a^{2} c^{2} d^{6}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + c\right )}^{2}}}{f} \]
(2*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*f*x + 1/2* e) + b)/sqrt(a^2 - b^2)))*b^3/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(a^2 - b^2)) - (6*b^2*c^4*d - 6*a*b*c^3*d^2 + 2*a^2*c^2*d^3 - 5*b^2*c^2*d^3 + a^2*d^5 + 2*b^2*d^5)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sg n(c) + arctan((c*tan(1/2*f*x + 1/2*e) + d)/sqrt(c^2 - d^2)))/((b^3*c^7 - 3 *a*b^2*c^6*d + 3*a^2*b*c^5*d^2 - 2*b^3*c^5*d^2 - a^3*c^4*d^3 + 6*a*b^2*c^4 *d^3 - 6*a^2*b*c^3*d^4 + b^3*c^3*d^4 + 2*a^3*c^2*d^5 - 3*a*b^2*c^2*d^5 + 3 *a^2*b*c*d^6 - a^3*d^7)*sqrt(c^2 - d^2)) - (7*b*c^4*d^3*tan(1/2*f*x + 1/2* e)^3 - 5*a*c^3*d^4*tan(1/2*f*x + 1/2*e)^3 - 4*b*c^2*d^5*tan(1/2*f*x + 1/2* e)^3 + 2*a*c*d^6*tan(1/2*f*x + 1/2*e)^3 + 6*b*c^5*d^2*tan(1/2*f*x + 1/2*e) ^2 - 4*a*c^4*d^3*tan(1/2*f*x + 1/2*e)^2 + 9*b*c^3*d^4*tan(1/2*f*x + 1/2*e) ^2 - 7*a*c^2*d^5*tan(1/2*f*x + 1/2*e)^2 - 6*b*c*d^6*tan(1/2*f*x + 1/2*e)^2 + 2*a*d^7*tan(1/2*f*x + 1/2*e)^2 + 17*b*c^4*d^3*tan(1/2*f*x + 1/2*e) - 11 *a*c^3*d^4*tan(1/2*f*x + 1/2*e) - 8*b*c^2*d^5*tan(1/2*f*x + 1/2*e) + 2*a*c *d^6*tan(1/2*f*x + 1/2*e) + 6*b*c^5*d^2 - 4*a*c^4*d^3 - 3*b*c^3*d^4 + a*c^ 2*d^5)/((b^2*c^8 - 2*a*b*c^7*d + a^2*c^6*d^2 - 2*b^2*c^6*d^2 + 4*a*b*c^5*d ^3 - 2*a^2*c^4*d^4 + b^2*c^4*d^4 - 2*a*b*c^3*d^5 + a^2*c^2*d^6)*(c*tan(1/2 *f*x + 1/2*e)^2 + 2*d*tan(1/2*f*x + 1/2*e) + c)^2))/f
Time = 33.29 (sec) , antiderivative size = 62873, normalized size of antiderivative = 232.00 \[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \]
(b^3*atan(((b^3*(b^2 - a^2)^(1/2)*((8*(4*a*b^8*c^4*d^9 - 16*a*b^8*c^6*d^7 + 24*a*b^8*c^8*d^5 - 16*a*b^8*c^10*d^3 + 4*a^4*b^5*c*d^12 + 4*a^6*b^3*c*d^ 12 + 4*a^8*b*c^3*d^10 + 4*a^8*b*c^5*d^8 - 4*a^2*b^7*c^3*d^10 + 12*a^2*b^7* c^5*d^8 + a^2*b^7*c^7*d^6 - 28*a^2*b^7*c^9*d^4 + 28*a^2*b^7*c^11*d^2 - 4*a ^3*b^6*c^2*d^11 + 24*a^3*b^6*c^4*d^9 - 98*a^3*b^6*c^6*d^7 + 164*a^3*b^6*c^ 8*d^5 - 140*a^3*b^6*c^10*d^3 - 16*a^4*b^5*c^3*d^10 + 95*a^4*b^5*c^5*d^8 - 188*a^4*b^5*c^7*d^6 + 240*a^4*b^5*c^9*d^4 - 8*a^5*b^4*c^2*d^11 - 20*a^5*b^ 4*c^4*d^9 + 64*a^5*b^4*c^6*d^7 - 216*a^5*b^4*c^8*d^5 - a^6*b^3*c^3*d^10 + 20*a^6*b^3*c^5*d^8 + 112*a^6*b^3*c^7*d^6 - 2*a^7*b^2*c^2*d^11 - 20*a^7*b^2 *c^4*d^9 - 32*a^7*b^2*c^6*d^7 + 4*a*b^8*c^12*d + a^8*b*c*d^12))/(a^6*d^14 + b^6*c^14 - 4*a^6*c^2*d^12 + 6*a^6*c^4*d^10 - 4*a^6*c^6*d^8 + a^6*c^8*d^6 + b^6*c^6*d^8 - 4*b^6*c^8*d^6 + 6*b^6*c^10*d^4 - 4*b^6*c^12*d^2 - 6*a*b^5 *c^5*d^9 + 24*a*b^5*c^7*d^7 - 36*a*b^5*c^9*d^5 + 24*a*b^5*c^11*d^3 + 24*a^ 5*b*c^3*d^11 - 36*a^5*b*c^5*d^9 + 24*a^5*b*c^7*d^7 - 6*a^5*b*c^9*d^5 + 15* a^2*b^4*c^4*d^10 - 60*a^2*b^4*c^6*d^8 + 90*a^2*b^4*c^8*d^6 - 60*a^2*b^4*c^ 10*d^4 + 15*a^2*b^4*c^12*d^2 - 20*a^3*b^3*c^3*d^11 + 80*a^3*b^3*c^5*d^9 - 120*a^3*b^3*c^7*d^7 + 80*a^3*b^3*c^9*d^5 - 20*a^3*b^3*c^11*d^3 + 15*a^4*b^ 2*c^2*d^12 - 60*a^4*b^2*c^4*d^10 + 90*a^4*b^2*c^6*d^8 - 60*a^4*b^2*c^8*d^6 + 15*a^4*b^2*c^10*d^4 - 6*a*b^5*c^13*d - 6*a^5*b*c*d^13) - (8*tan(e/2 + ( f*x)/2)*(4*a*b^8*c^13 + a^9*c*d^12 + 4*a^9*c^3*d^10 + 4*a^9*c^5*d^8 - 1...