Integrand size = 25, antiderivative size = 306 \[ \int \frac {(c+d \sin (e+f x))^4}{(a+b \sin (e+f x))^2} \, dx=-\frac {d^2 \left (16 a b c d-6 a^2 d^2-b^2 \left (12 c^2+d^2\right )\right ) x}{2 b^4}+\frac {2 (b c-a d)^3 \left (a b c+3 a^2 d-4 b^2 d\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{b^4 \left (a^2-b^2\right )^{3/2} f}+\frac {d (2 b c-a d) \left (2 a b c d-3 a^2 d^2-b^2 \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{b^3 \left (a^2-b^2\right ) f}+\frac {d^2 \left (4 a b c d-3 a^2 d^2-b^2 \left (2 c^2-d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^2 \left (a^2-b^2\right ) f}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))} \] Output:
-1/2*d^2*(16*a*b*c*d-6*a^2*d^2-b^2*(12*c^2+d^2))*x/b^4+2*(-a*d+b*c)^3*(3*a ^2*d+a*b*c-4*b^2*d)*arctan((b+a*tan(1/2*f*x+1/2*e))/(a^2-b^2)^(1/2))/b^4/( a^2-b^2)^(3/2)/f+d*(-a*d+2*b*c)*(2*a*b*c*d-3*a^2*d^2-b^2*(c^2-2*d^2))*cos( f*x+e)/b^3/(a^2-b^2)/f+1/2*d^2*(4*a*b*c*d-3*a^2*d^2-b^2*(2*c^2-d^2))*cos(f *x+e)*sin(f*x+e)/b^2/(a^2-b^2)/f+(-a*d+b*c)^2*cos(f*x+e)*(c+d*sin(f*x+e))^ 2/b/(a^2-b^2)/f/(a+b*sin(f*x+e))
Time = 7.91 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.65 \[ \int \frac {(c+d \sin (e+f x))^4}{(a+b \sin (e+f x))^2} \, dx=-\frac {-2 d^2 \left (-16 a b c d+6 a^2 d^2+b^2 \left (12 c^2+d^2\right )\right ) (e+f x)+\frac {8 (-b c+a d)^3 \left (a b c+3 a^2 d-4 b^2 d\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+8 b d^3 (2 b c-a d) \cos (e+f x)-\frac {4 b (b c-a d)^4 \cos (e+f x)}{(a-b) (a+b) (a+b \sin (e+f x))}+b^2 d^4 \sin (2 (e+f x))}{4 b^4 f} \] Input:
Integrate[(c + d*Sin[e + f*x])^4/(a + b*Sin[e + f*x])^2,x]
Output:
-1/4*(-2*d^2*(-16*a*b*c*d + 6*a^2*d^2 + b^2*(12*c^2 + d^2))*(e + f*x) + (8 *(-(b*c) + a*d)^3*(a*b*c + 3*a^2*d - 4*b^2*d)*ArcTan[(b + a*Tan[(e + f*x)/ 2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + 8*b*d^3*(2*b*c - a*d)*Cos[e + f* x] - (4*b*(b*c - a*d)^4*Cos[e + f*x])/((a - b)*(a + b)*(a + b*Sin[e + f*x] )) + b^2*d^4*Sin[2*(e + f*x)])/(b^4*f)
Time = 1.76 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.07, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 3271, 3042, 3512, 3042, 3502, 3042, 3214, 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.
| \(\displaystyle \int \frac {(c+d \sin (e+f x))^4}{(a+b \sin (e+f x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d \sin (e+f x))^4}{(a+b \sin (e+f x))^2}dx\) |
| \(\Big \downarrow \) 3271 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\int \frac {(c+d \sin (e+f x)) \left (2 a^2 d^3+4 b^2 c^2 d+\left (-\left (\left (2 c^2-d^2\right ) b^2\right )+4 a c d b-3 a^2 d^2\right ) \sin ^2(e+f x) d-\left (c d a^2+b \left (c^2+d^2\right ) a-3 b^2 c d\right ) \sin (e+f x) d-a b c \left (c^2+5 d^2\right )\right )}{a+b \sin (e+f x)}dx}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\int \frac {(c+d \sin (e+f x)) \left (2 a^2 d^3+4 b^2 c^2 d+\left (-\left (\left (2 c^2-d^2\right ) b^2\right )+4 a c d b-3 a^2 d^2\right ) \sin (e+f x)^2 d-\left (c d a^2+b \left (c^2+d^2\right ) a-3 b^2 c d\right ) \sin (e+f x) d-a b c \left (c^2+5 d^2\right )\right )}{a+b \sin (e+f x)}dx}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {\int \frac {-3 a^3 d^4+8 a^2 b c d^3+8 b^3 c^3 d+2 (2 b c-a d) \left (-\left (\left (c^2-2 d^2\right ) b^2\right )+2 a c d b-3 a^2 d^2\right ) \sin ^2(e+f x) d+b \left (-\left (\left (2 c^2 d-d^3\right ) a^2\right )-4 b c \left (c^2+2 d^2\right ) a+b^2 d \left (12 c^2+d^2\right )\right ) \sin (e+f x) d-a b^2 \left (2 c^4+12 d^2 c^2-d^4\right )}{a+b \sin (e+f x)}dx}{2 b}-\frac {d^2 \left (-3 a^2 d^2+4 a b c d-\left (b^2 \left (2 c^2-d^2\right )\right )\right ) \sin (e+f x) \cos (e+f x)}{2 b f}}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {\int \frac {-3 a^3 d^4+8 a^2 b c d^3+8 b^3 c^3 d+2 (2 b c-a d) \left (-\left (\left (c^2-2 d^2\right ) b^2\right )+2 a c d b-3 a^2 d^2\right ) \sin (e+f x)^2 d+b \left (-\left (\left (2 c^2 d-d^3\right ) a^2\right )-4 b c \left (c^2+2 d^2\right ) a+b^2 d \left (12 c^2+d^2\right )\right ) \sin (e+f x) d-a b^2 \left (2 c^4+12 d^2 c^2-d^4\right )}{a+b \sin (e+f x)}dx}{2 b}-\frac {d^2 \left (-3 a^2 d^2+4 a b c d-\left (b^2 \left (2 c^2-d^2\right )\right )\right ) \sin (e+f x) \cos (e+f x)}{2 b f}}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {\frac {\int \frac {\left (a^2-b^2\right ) \left (-\left (\left (12 c^2+d^2\right ) b^2\right )+16 a c d b-6 a^2 d^2\right ) \sin (e+f x) d^2+b \left (-3 a^3 d^4+8 a^2 b c d^3+8 b^3 c^3 d-a b^2 \left (2 c^4+12 d^2 c^2-d^4\right )\right )}{a+b \sin (e+f x)}dx}{b}-\frac {2 d (2 b c-a d) \left (-3 a^2 d^2+2 a b c d-\left (b^2 \left (c^2-2 d^2\right )\right )\right ) \cos (e+f x)}{b f}}{2 b}-\frac {d^2 \left (-3 a^2 d^2+4 a b c d-\left (b^2 \left (2 c^2-d^2\right )\right )\right ) \sin (e+f x) \cos (e+f x)}{2 b f}}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {\frac {\int \frac {\left (a^2-b^2\right ) \left (-\left (\left (12 c^2+d^2\right ) b^2\right )+16 a c d b-6 a^2 d^2\right ) \sin (e+f x) d^2+b \left (-3 a^3 d^4+8 a^2 b c d^3+8 b^3 c^3 d-a b^2 \left (2 c^4+12 d^2 c^2-d^4\right )\right )}{a+b \sin (e+f x)}dx}{b}-\frac {2 d (2 b c-a d) \left (-3 a^2 d^2+2 a b c d-\left (b^2 \left (c^2-2 d^2\right )\right )\right ) \cos (e+f x)}{b f}}{2 b}-\frac {d^2 \left (-3 a^2 d^2+4 a b c d-\left (b^2 \left (2 c^2-d^2\right )\right )\right ) \sin (e+f x) \cos (e+f x)}{2 b f}}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {\frac {\frac {d^2 x \left (a^2-b^2\right ) \left (-6 a^2 d^2+16 a b c d-\left (b^2 \left (12 c^2+d^2\right )\right )\right )}{b}-\frac {2 (b c-a d)^3 \left (3 a^2 d+a b c-4 b^2 d\right ) \int \frac {1}{a+b \sin (e+f x)}dx}{b}}{b}-\frac {2 d (2 b c-a d) \left (-3 a^2 d^2+2 a b c d-\left (b^2 \left (c^2-2 d^2\right )\right )\right ) \cos (e+f x)}{b f}}{2 b}-\frac {d^2 \left (-3 a^2 d^2+4 a b c d-\left (b^2 \left (2 c^2-d^2\right )\right )\right ) \sin (e+f x) \cos (e+f x)}{2 b f}}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {\frac {\frac {d^2 x \left (a^2-b^2\right ) \left (-6 a^2 d^2+16 a b c d-\left (b^2 \left (12 c^2+d^2\right )\right )\right )}{b}-\frac {2 (b c-a d)^3 \left (3 a^2 d+a b c-4 b^2 d\right ) \int \frac {1}{a+b \sin (e+f x)}dx}{b}}{b}-\frac {2 d (2 b c-a d) \left (-3 a^2 d^2+2 a b c d-\left (b^2 \left (c^2-2 d^2\right )\right )\right ) \cos (e+f x)}{b f}}{2 b}-\frac {d^2 \left (-3 a^2 d^2+4 a b c d-\left (b^2 \left (2 c^2-d^2\right )\right )\right ) \sin (e+f x) \cos (e+f x)}{2 b f}}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {\frac {\frac {d^2 x \left (a^2-b^2\right ) \left (-6 a^2 d^2+16 a b c d-\left (b^2 \left (12 c^2+d^2\right )\right )\right )}{b}-\frac {4 (b c-a d)^3 \left (3 a^2 d+a b c-4 b^2 d\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (e+f x)\right )+2 b \tan \left (\frac {1}{2} (e+f x)\right )+a}d\tan \left (\frac {1}{2} (e+f x)\right )}{b f}}{b}-\frac {2 d (2 b c-a d) \left (-3 a^2 d^2+2 a b c d-\left (b^2 \left (c^2-2 d^2\right )\right )\right ) \cos (e+f x)}{b f}}{2 b}-\frac {d^2 \left (-3 a^2 d^2+4 a b c d-\left (b^2 \left (2 c^2-d^2\right )\right )\right ) \sin (e+f x) \cos (e+f x)}{2 b f}}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {\frac {\frac {8 \left (3 a^2 d+a b c-4 b^2 d\right ) (b c-a d)^3 \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (e+f x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (e+f x)\right )\right )}{b f}+\frac {d^2 x \left (a^2-b^2\right ) \left (-6 a^2 d^2+16 a b c d-\left (b^2 \left (12 c^2+d^2\right )\right )\right )}{b}}{b}-\frac {2 d (2 b c-a d) \left (-3 a^2 d^2+2 a b c d-\left (b^2 \left (c^2-2 d^2\right )\right )\right ) \cos (e+f x)}{b f}}{2 b}-\frac {d^2 \left (-3 a^2 d^2+4 a b c d-\left (b^2 \left (2 c^2-d^2\right )\right )\right ) \sin (e+f x) \cos (e+f x)}{2 b f}}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {\frac {\frac {\frac {d^2 x \left (a^2-b^2\right ) \left (-6 a^2 d^2+16 a b c d-\left (b^2 \left (12 c^2+d^2\right )\right )\right )}{b}-\frac {4 (b c-a d)^3 \left (3 a^2 d+a b c-4 b^2 d\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (e+f x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{b f \sqrt {a^2-b^2}}}{b}-\frac {2 d (2 b c-a d) \left (-3 a^2 d^2+2 a b c d-\left (b^2 \left (c^2-2 d^2\right )\right )\right ) \cos (e+f x)}{b f}}{2 b}-\frac {d^2 \left (-3 a^2 d^2+4 a b c d-\left (b^2 \left (2 c^2-d^2\right )\right )\right ) \sin (e+f x) \cos (e+f x)}{2 b f}}{b \left (a^2-b^2\right )}\) |
Input:
Int[(c + d*Sin[e + f*x])^4/(a + b*Sin[e + f*x])^2,x]
Output:
((b*c - a*d)^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])) - (((((a^2 - b^2)*d^2*(16*a*b*c*d - 6*a^2*d^2 - b^2*(12*c ^2 + d^2))*x)/b - (4*(b*c - a*d)^3*(a*b*c + 3*a^2*d - 4*b^2*d)*ArcTan[(2*b + 2*a*Tan[(e + f*x)/2])/(2*Sqrt[a^2 - b^2])])/(b*Sqrt[a^2 - b^2]*f))/b - (2*d*(2*b*c - a*d)*(2*a*b*c*d - 3*a^2*d^2 - b^2*(c^2 - 2*d^2))*Cos[e + f*x ])/(b*f))/(2*b) - (d^2*(4*a*b*c*d - 3*a^2*d^2 - b^2*(2*c^2 - d^2))*Cos[e + f*x]*Sin[e + f*x])/(2*b*f))/(b*(a^2 - b^2))
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_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*Co s[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f* (n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin [e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^ 2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 + c*d*(a^2 + b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - d^2) - m*(b*c - a*d)^2 + d*n*(2*a*b*c - d*(a^2 + b^2)))*Sin[e + f*x]^2, x] , x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si n[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3)) Int[(a + b*Si n[e + f*x])^m*Simp[a*C*d + A*b*c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e + f*x]^2 , x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && !LtQ[m, -1]
Time = 21.13 (sec) , antiderivative size = 456, normalized size of antiderivative = 1.49
| method | result | size |
| derivativedivides | \(\frac {\frac {2 d^{2} \left (\frac {\frac {d^{2} b^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{2}+\left (2 a b \,d^{2}-4 b^{2} c d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-\frac {d^{2} b^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}+2 a b \,d^{2}-4 b^{2} c d}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{2}}+\frac {\left (6 a^{2} d^{2}-16 a b c d +12 b^{2} c^{2}+d^{2} b^{2}\right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )}{b^{4}}-\frac {2 \left (\frac {-\frac {b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 d^{2} a^{2} b^{2} c^{2}-4 d a \,b^{3} c^{3}+b^{4} c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (a^{2}-b^{2}\right ) a}-\frac {b \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 d^{2} a^{2} b^{2} c^{2}-4 d a \,b^{3} c^{3}+b^{4} c^{4}\right )}{a^{2}-b^{2}}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} a +2 b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+a}+\frac {\left (3 a^{5} d^{4}-8 a^{4} b c \,d^{3}+6 a^{3} b^{2} c^{2} d^{2}-4 a^{3} b^{2} d^{4}+12 a^{2} b^{3} c \,d^{3}-a \,b^{4} c^{4}-12 a \,b^{4} c^{2} d^{2}+4 b^{5} c^{3} d \right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{b^{4}}}{f}\) | \(456\) |
| default | \(\frac {\frac {2 d^{2} \left (\frac {\frac {d^{2} b^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{2}+\left (2 a b \,d^{2}-4 b^{2} c d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-\frac {d^{2} b^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}+2 a b \,d^{2}-4 b^{2} c d}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{2}}+\frac {\left (6 a^{2} d^{2}-16 a b c d +12 b^{2} c^{2}+d^{2} b^{2}\right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )}{b^{4}}-\frac {2 \left (\frac {-\frac {b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 d^{2} a^{2} b^{2} c^{2}-4 d a \,b^{3} c^{3}+b^{4} c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (a^{2}-b^{2}\right ) a}-\frac {b \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 d^{2} a^{2} b^{2} c^{2}-4 d a \,b^{3} c^{3}+b^{4} c^{4}\right )}{a^{2}-b^{2}}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} a +2 b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+a}+\frac {\left (3 a^{5} d^{4}-8 a^{4} b c \,d^{3}+6 a^{3} b^{2} c^{2} d^{2}-4 a^{3} b^{2} d^{4}+12 a^{2} b^{3} c \,d^{3}-a \,b^{4} c^{4}-12 a \,b^{4} c^{2} d^{2}+4 b^{5} c^{3} d \right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{b^{4}}}{f}\) | \(456\) |
| risch | \(\text {Expression too large to display}\) | \(1666\) |
Input:
int((c+d*sin(f*x+e))^4/(a+b*sin(f*x+e))^2,x,method=_RETURNVERBOSE)
Output:
1/f*(2*d^2/b^4*((1/2*d^2*b^2*tan(1/2*f*x+1/2*e)^3+(2*a*b*d^2-4*b^2*c*d)*ta n(1/2*f*x+1/2*e)^2-1/2*d^2*b^2*tan(1/2*f*x+1/2*e)+2*a*b*d^2-4*b^2*c*d)/(1+ tan(1/2*f*x+1/2*e)^2)^2+1/2*(6*a^2*d^2-16*a*b*c*d+12*b^2*c^2+b^2*d^2)*arct an(tan(1/2*f*x+1/2*e)))-2/b^4*((-b^2*(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2* d^2-4*a*b^3*c^3*d+b^4*c^4)/(a^2-b^2)/a*tan(1/2*f*x+1/2*e)-b*(a^4*d^4-4*a^3 *b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)/(a^2-b^2))/(tan(1/2*f*x+ 1/2*e)^2*a+2*b*tan(1/2*f*x+1/2*e)+a)+(3*a^5*d^4-8*a^4*b*c*d^3+6*a^3*b^2*c^ 2*d^2-4*a^3*b^2*d^4+12*a^2*b^3*c*d^3-a*b^4*c^4-12*a*b^4*c^2*d^2+4*b^5*c^3* d)/(a^2-b^2)^(3/2)*arctan(1/2*(2*a*tan(1/2*f*x+1/2*e)+2*b)/(a^2-b^2)^(1/2) )))
Leaf count of result is larger than twice the leaf count of optimal. 684 vs. \(2 (297) = 594\).
Time = 0.16 (sec) , antiderivative size = 1451, normalized size of antiderivative = 4.74 \[ \int \frac {(c+d \sin (e+f x))^4}{(a+b \sin (e+f x))^2} \, dx=\text {Too large to display} \] Input:
integrate((c+d*sin(f*x+e))^4/(a+b*sin(f*x+e))^2,x, algorithm="fricas")
Output:
[1/2*((a^4*b^3 - 2*a^2*b^5 + b^7)*d^4*cos(f*x + e)^3 + (12*(a^5*b^2 - 2*a^ 3*b^4 + a*b^6)*c^2*d^2 - 16*(a^6*b - 2*a^4*b^3 + a^2*b^5)*c*d^3 + (6*a^7 - 11*a^5*b^2 + 4*a^3*b^4 + a*b^6)*d^4)*f*x + (a^2*b^4*c^4 - 4*a*b^5*c^3*d - 6*(a^4*b^2 - 2*a^2*b^4)*c^2*d^2 + 4*(2*a^5*b - 3*a^3*b^3)*c*d^3 - (3*a^6 - 4*a^4*b^2)*d^4 + (a*b^5*c^4 - 4*b^6*c^3*d - 6*(a^3*b^3 - 2*a*b^5)*c^2*d^ 2 + 4*(2*a^4*b^2 - 3*a^2*b^4)*c*d^3 - (3*a^5*b - 4*a^3*b^3)*d^4)*sin(f*x + e))*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2 - 2*(a*cos(f*x + e)*sin(f*x + e) + b*cos(f*x + e))*sqrt(-a ^2 + b^2))/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2)) + (2*(a^ 2*b^5 - b^7)*c^4 - 8*(a^3*b^4 - a*b^6)*c^3*d + 12*(a^4*b^3 - a^2*b^5)*c^2* d^2 - 8*(2*a^5*b^2 - 3*a^3*b^4 + a*b^6)*c*d^3 + (6*a^6*b - 11*a^4*b^3 + 6* a^2*b^5 - b^7)*d^4)*cos(f*x + e) + ((12*(a^4*b^3 - 2*a^2*b^5 + b^7)*c^2*d^ 2 - 16*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c*d^3 + (6*a^6*b - 11*a^4*b^3 + 4*a^2 *b^5 + b^7)*d^4)*f*x - (8*(a^4*b^3 - 2*a^2*b^5 + b^7)*c*d^3 - 3*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*d^4)*cos(f*x + e))*sin(f*x + e))/((a^4*b^5 - 2*a^2*b^7 + b^9)*f*sin(f*x + e) + (a^5*b^4 - 2*a^3*b^6 + a*b^8)*f), 1/2*((a^4*b^3 - 2*a^2*b^5 + b^7)*d^4*cos(f*x + e)^3 + (12*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c ^2*d^2 - 16*(a^6*b - 2*a^4*b^3 + a^2*b^5)*c*d^3 + (6*a^7 - 11*a^5*b^2 + 4* a^3*b^4 + a*b^6)*d^4)*f*x - 2*(a^2*b^4*c^4 - 4*a*b^5*c^3*d - 6*(a^4*b^2 - 2*a^2*b^4)*c^2*d^2 + 4*(2*a^5*b - 3*a^3*b^3)*c*d^3 - (3*a^6 - 4*a^4*b^2...
Timed out. \[ \int \frac {(c+d \sin (e+f x))^4}{(a+b \sin (e+f x))^2} \, dx=\text {Timed out} \] Input:
integrate((c+d*sin(f*x+e))**4/(a+b*sin(f*x+e))**2,x)
Output:
Timed out
Exception generated. \[ \int \frac {(c+d \sin (e+f x))^4}{(a+b \sin (e+f x))^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((c+d*sin(f*x+e))^4/(a+b*sin(f*x+e))^2,x, algorithm="maxima")
Output:
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
Time = 0.40 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.64 \[ \int \frac {(c+d \sin (e+f x))^4}{(a+b \sin (e+f x))^2} \, dx=\frac {\frac {4 \, {\left (a b^{4} c^{4} - 4 \, b^{5} c^{3} d - 6 \, a^{3} b^{2} c^{2} d^{2} + 12 \, a b^{4} c^{2} d^{2} + 8 \, a^{4} b c d^{3} - 12 \, a^{2} b^{3} c d^{3} - 3 \, a^{5} d^{4} + 4 \, a^{3} b^{2} d^{4}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{2} b^{4} - b^{6}\right )} \sqrt {a^{2} - b^{2}}} + \frac {4 \, {\left (b^{5} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 4 \, a b^{4} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, a^{2} b^{3} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 4 \, a^{3} b^{2} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a^{4} b d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a b^{4} c^{4} - 4 \, a^{2} b^{3} c^{3} d + 6 \, a^{3} b^{2} c^{2} d^{2} - 4 \, a^{4} b c d^{3} + a^{5} d^{4}\right )}}{{\left (a^{3} b^{3} - a b^{5}\right )} {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a\right )}} + \frac {{\left (12 \, b^{2} c^{2} d^{2} - 16 \, a b c d^{3} + 6 \, a^{2} d^{4} + b^{2} d^{4}\right )} {\left (f x + e\right )}}{b^{4}} + \frac {2 \, {\left (b d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 8 \, b c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 4 \, a d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - b d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 8 \, b c d^{3} + 4 \, a d^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{2} b^{3}}}{2 \, f} \] Input:
integrate((c+d*sin(f*x+e))^4/(a+b*sin(f*x+e))^2,x, algorithm="giac")
Output:
1/2*(4*(a*b^4*c^4 - 4*b^5*c^3*d - 6*a^3*b^2*c^2*d^2 + 12*a*b^4*c^2*d^2 + 8 *a^4*b*c*d^3 - 12*a^2*b^3*c*d^3 - 3*a^5*d^4 + 4*a^3*b^2*d^4)*(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)))/((a^2*b^4 - b^6)*sqrt(a^2 - b^2)) + 4*(b^5*c^4*tan(1/2*f*x + 1/2 *e) - 4*a*b^4*c^3*d*tan(1/2*f*x + 1/2*e) + 6*a^2*b^3*c^2*d^2*tan(1/2*f*x + 1/2*e) - 4*a^3*b^2*c*d^3*tan(1/2*f*x + 1/2*e) + a^4*b*d^4*tan(1/2*f*x + 1 /2*e) + a*b^4*c^4 - 4*a^2*b^3*c^3*d + 6*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3 + a^5*d^4)/((a^3*b^3 - a*b^5)*(a*tan(1/2*f*x + 1/2*e)^2 + 2*b*tan(1/2*f*x + 1/2*e) + a)) + (12*b^2*c^2*d^2 - 16*a*b*c*d^3 + 6*a^2*d^4 + b^2*d^4)*(f*x + e)/b^4 + 2*(b*d^4*tan(1/2*f*x + 1/2*e)^3 - 8*b*c*d^3*tan(1/2*f*x + 1/2*e )^2 + 4*a*d^4*tan(1/2*f*x + 1/2*e)^2 - b*d^4*tan(1/2*f*x + 1/2*e) - 8*b*c* d^3 + 4*a*d^4)/((tan(1/2*f*x + 1/2*e)^2 + 1)^2*b^3))/f
Time = 27.58 (sec) , antiderivative size = 13700, normalized size of antiderivative = 44.77 \[ \int \frac {(c+d \sin (e+f x))^4}{(a+b \sin (e+f x))^2} \, dx=\text {Too large to display} \] Input:
int((c + d*sin(e + f*x))^4/(a + b*sin(e + f*x))^2,x)
Output:
((2*(3*a^4*d^4 + b^4*c^4 - 2*a^2*b^2*d^4 + 6*a^2*b^2*c^2*d^2 + 4*a*b^3*c*d ^3 - 4*a*b^3*c^3*d - 8*a^3*b*c*d^3))/(b^3*(a^2 - b^2)) + (2*tan(e/2 + (f*x )/2)^4*(3*a^4*d^4 + b^4*c^4 - b^4*d^4 - a^2*b^2*d^4 + 6*a^2*b^2*c^2*d^2 + 4*a*b^3*c*d^3 - 4*a*b^3*c^3*d - 8*a^3*b*c*d^3))/(b^3*(a^2 - b^2)) + (2*tan (e/2 + (f*x)/2)^2*(6*a^4*d^4 + 2*b^4*c^4 + b^4*d^4 - 5*a^2*b^2*d^4 + 12*a^ 2*b^2*c^2*d^2 + 8*a*b^3*c*d^3 - 8*a*b^3*c^3*d - 16*a^3*b*c*d^3))/(b^3*(a^2 - b^2)) + (tan(e/2 + (f*x)/2)^5*(3*a^4*d^4 + 2*b^4*c^4 - a^2*b^2*d^4 + 12 *a^2*b^2*c^2*d^2 - 8*a*b^3*c^3*d - 8*a^3*b*c*d^3))/(a*b^2*(a^2 - b^2)) + ( tan(e/2 + (f*x)/2)*(9*a^4*d^4 + 2*b^4*c^4 - 7*a^2*b^2*d^4 + 12*a^2*b^2*c^2 *d^2 + 16*a*b^3*c*d^3 - 8*a*b^3*c^3*d - 24*a^3*b*c*d^3))/(a*b^2*(a^2 - b^2 )) + (4*tan(e/2 + (f*x)/2)^3*(3*a^4*d^4 + b^4*c^4 - 2*a^2*b^2*d^4 + 6*a^2* b^2*c^2*d^2 + 4*a*b^3*c*d^3 - 4*a*b^3*c^3*d - 8*a^3*b*c*d^3))/(a*b^2*(a^2 - b^2)))/(f*(a + 2*b*tan(e/2 + (f*x)/2) + 3*a*tan(e/2 + (f*x)/2)^2 + 3*a*t an(e/2 + (f*x)/2)^4 + a*tan(e/2 + (f*x)/2)^6 + 4*b*tan(e/2 + (f*x)/2)^3 + 2*b*tan(e/2 + (f*x)/2)^5)) + (atan(((((8*(a^2*b^11*d^8 + 10*a^4*b^9*d^8 + 13*a^6*b^7*d^8 - 60*a^8*b^5*d^8 + 36*a^10*b^3*d^8 - 32*a^3*b^10*c*d^7 - 12 8*a^5*b^8*c*d^7 + 352*a^7*b^6*c*d^7 - 192*a^9*b^4*c*d^7 + 24*a^2*b^11*c^2* d^6 + 144*a^2*b^11*c^4*d^4 - 384*a^3*b^10*c^3*d^5 + 352*a^4*b^9*c^2*d^6 - 288*a^4*b^9*c^4*d^4 + 768*a^5*b^8*c^3*d^5 - 776*a^6*b^7*c^2*d^6 + 144*a^6* b^7*c^4*d^4 - 384*a^7*b^6*c^3*d^5 + 400*a^8*b^5*c^2*d^6))/(b^12 - 2*a^2...
Time = 0.24 (sec) , antiderivative size = 1783, normalized size of antiderivative = 5.83 \[ \int \frac {(c+d \sin (e+f x))^4}{(a+b \sin (e+f x))^2} \, dx =\text {Too large to display} \] Input:
int((c+d*sin(f*x+e))^4/(a+b*sin(f*x+e))^2,x)
Output:
( - 12*sqrt(a**2 - b**2)*atan((tan((e + f*x)/2)*a + b)/sqrt(a**2 - b**2))* sin(e + f*x)*a**5*b*d**4 + 32*sqrt(a**2 - b**2)*atan((tan((e + f*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(e + f*x)*a**4*b**2*c*d**3 - 24*sqrt(a**2 - b**2 )*atan((tan((e + f*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(e + f*x)*a**3*b**3* c**2*d**2 + 16*sqrt(a**2 - b**2)*atan((tan((e + f*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(e + f*x)*a**3*b**3*d**4 - 48*sqrt(a**2 - b**2)*atan((tan((e + f*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(e + f*x)*a**2*b**4*c*d**3 + 4*sqrt(a **2 - b**2)*atan((tan((e + f*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(e + f*x)* a*b**5*c**4 + 48*sqrt(a**2 - b**2)*atan((tan((e + f*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(e + f*x)*a*b**5*c**2*d**2 - 16*sqrt(a**2 - b**2)*atan((tan(( e + f*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(e + f*x)*b**6*c**3*d - 12*sqrt(a **2 - b**2)*atan((tan((e + f*x)/2)*a + b)/sqrt(a**2 - b**2))*a**6*d**4 + 3 2*sqrt(a**2 - b**2)*atan((tan((e + f*x)/2)*a + b)/sqrt(a**2 - b**2))*a**5* b*c*d**3 - 24*sqrt(a**2 - b**2)*atan((tan((e + f*x)/2)*a + b)/sqrt(a**2 - b**2))*a**4*b**2*c**2*d**2 + 16*sqrt(a**2 - b**2)*atan((tan((e + f*x)/2)*a + b)/sqrt(a**2 - b**2))*a**4*b**2*d**4 - 48*sqrt(a**2 - b**2)*atan((tan(( e + f*x)/2)*a + b)/sqrt(a**2 - b**2))*a**3*b**3*c*d**3 + 4*sqrt(a**2 - b** 2)*atan((tan((e + f*x)/2)*a + b)/sqrt(a**2 - b**2))*a**2*b**4*c**4 + 48*sq rt(a**2 - b**2)*atan((tan((e + f*x)/2)*a + b)/sqrt(a**2 - b**2))*a**2*b**4 *c**2*d**2 - 16*sqrt(a**2 - b**2)*atan((tan((e + f*x)/2)*a + b)/sqrt(a*...