3.715 \(\int \frac{(c+d \sin (e+f x))^4}{(a+b \sin (e+f x))^3} \, dx\)

Optimal. Leaf size=318 \[ \frac{d^2 \left (-3 a^2 d^2+2 a b c d+b^2 \left (-\left (c^2-2 d^2\right )\right )\right ) \cos (e+f x)}{2 b^3 f \left (a^2-b^2\right )}+\frac{(b c-a d)^2 \left (a^2 b^2 \left (2 c^2-15 d^2\right )+4 a^3 b c d+6 a^4 d^2-10 a b^3 c d+b^4 \left (c^2+12 d^2\right )\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (e+f x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^4 f \left (a^2-b^2\right )^{5/2}}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}+\frac{3 (b c-a d)^3 \left (a^2 d+a b c-2 b^2 d\right ) \cos (e+f x)}{2 b^3 f \left (a^2-b^2\right )^2 (a+b \sin (e+f x))}+\frac{d^3 x (4 b c-3 a d)}{b^4} \]

[Out]

(d^3*(4*b*c - 3*a*d)*x)/b^4 + ((b*c - a*d)^2*(4*a^3*b*c*d - 10*a*b^3*c*d + 6*a^4*d^2 + a^2*b^2*(2*c^2 - 15*d^2
) + b^4*(c^2 + 12*d^2))*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(b^4*(a^2 - b^2)^(5/2)*f) + (d^2*(2*
a*b*c*d - 3*a^2*d^2 - b^2*(c^2 - 2*d^2))*Cos[e + f*x])/(2*b^3*(a^2 - b^2)*f) + (3*(b*c - a*d)^3*(a*b*c + a^2*d
- 2*b^2*d)*Cos[e + f*x])/(2*b^3*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x])) + ((b*c - a*d)^2*Cos[e + f*x]*(c + d*Si
n[e + f*x])^2)/(2*b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2)

________________________________________________________________________________________

Rubi [A]  time = 0.971869, antiderivative size = 318, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {2792, 3031, 3023, 2735, 2660, 618, 204} \[ \frac{d^2 \left (-3 a^2 d^2+2 a b c d+b^2 \left (-\left (c^2-2 d^2\right )\right )\right ) \cos (e+f x)}{2 b^3 f \left (a^2-b^2\right )}+\frac{(b c-a d)^2 \left (a^2 b^2 \left (2 c^2-15 d^2\right )+4 a^3 b c d+6 a^4 d^2-10 a b^3 c d+b^4 \left (c^2+12 d^2\right )\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (e+f x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^4 f \left (a^2-b^2\right )^{5/2}}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}+\frac{3 (b c-a d)^3 \left (a^2 d+a b c-2 b^2 d\right ) \cos (e+f x)}{2 b^3 f \left (a^2-b^2\right )^2 (a+b \sin (e+f x))}+\frac{d^3 x (4 b c-3 a d)}{b^4} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*Sin[e + f*x])^4/(a + b*Sin[e + f*x])^3,x]

[Out]

(d^3*(4*b*c - 3*a*d)*x)/b^4 + ((b*c - a*d)^2*(4*a^3*b*c*d - 10*a*b^3*c*d + 6*a^4*d^2 + a^2*b^2*(2*c^2 - 15*d^2
) + b^4*(c^2 + 12*d^2))*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(b^4*(a^2 - b^2)^(5/2)*f) + (d^2*(2*
a*b*c*d - 3*a^2*d^2 - b^2*(c^2 - 2*d^2))*Cos[e + f*x])/(2*b^3*(a^2 - b^2)*f) + (3*(b*c - a*d)^3*(a*b*c + a^2*d
- 2*b^2*d)*Cos[e + f*x])/(2*b^3*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x])) + ((b*c - a*d)^2*Cos[e + f*x]*(c + d*Si
n[e + f*x])^2)/(2*b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2)

Rule 2792

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -S
imp[((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*Cos[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] + Dist[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])

Rule 3031

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[((b*c - a*d)*(A*b^2 - a*b*B + a^2*C)*
Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b^2*f*(m + 1)*(a^2 - b^2)), x] - Dist[1/(b^2*(m + 1)*(a^2 - b^2)),
Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b
^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))*Sin[e + f*x] - b*C*d*(m +
1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && Ne
Q[a^2 - b^2, 0] && LtQ[m, -1]

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[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]

Rule 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(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]

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(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]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(c+d \sin (e+f x))^4}{(a+b \sin (e+f x))^3} \, dx &=\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac{\int \frac{(c+d \sin (e+f x)) \left (2 \left (3 b^2 c^2 d+a^2 d^3-a b c \left (c^2+3 d^2\right )\right )-\left (a^2 c d^2+2 a b d \left (2 c^2+d^2\right )-b^2 \left (c^3+6 c d^2\right )\right ) \sin (e+f x)+d \left (2 a b c d-3 a^2 d^2-b^2 \left (c^2-2 d^2\right )\right ) \sin ^2(e+f x)\right )}{(a+b \sin (e+f x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=\frac{3 (b c-a d)^3 \left (a b c+a^2 d-2 b^2 d\right ) \cos (e+f x)}{2 b^3 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac{\int \frac{-b \left (4 a^3 b c d^3-3 a^4 d^4-4 a b^3 c d \left (3 c^2+4 d^2\right )+b^4 c^2 \left (c^2+12 d^2\right )+2 a^2 b^2 \left (c^4+3 c^2 d^2+3 d^4\right )\right )-\left (a^2-b^2\right ) d^2 \left (6 a^2 b c d-8 b^3 c d-3 a^3 d^2+a b^2 \left (c^2+4 d^2\right )\right ) \sin (e+f x)+b \left (a^2-b^2\right ) d^2 \left (2 a b c d-3 a^2 d^2-b^2 \left (c^2-2 d^2\right )\right ) \sin ^2(e+f x)}{a+b \sin (e+f x)} \, dx}{2 b^3 \left (a^2-b^2\right )^2}\\ &=\frac{d^2 \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)}{2 b^3 \left (a^2-b^2\right ) f}+\frac{3 (b c-a d)^3 \left (a b c+a^2 d-2 b^2 d\right ) \cos (e+f x)}{2 b^3 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac{\int \frac{-b^2 \left (4 a^3 b c d^3-3 a^4 d^4-4 a b^3 c d \left (3 c^2+4 d^2\right )+b^4 c^2 \left (c^2+12 d^2\right )+2 a^2 b^2 \left (c^4+3 c^2 d^2+3 d^4\right )\right )-2 b \left (a^2-b^2\right )^2 d^3 (4 b c-3 a d) \sin (e+f x)}{a+b \sin (e+f x)} \, dx}{2 b^4 \left (a^2-b^2\right )^2}\\ &=\frac{d^3 (4 b c-3 a d) x}{b^4}+\frac{d^2 \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)}{2 b^3 \left (a^2-b^2\right ) f}+\frac{3 (b c-a d)^3 \left (a b c+a^2 d-2 b^2 d\right ) \cos (e+f x)}{2 b^3 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac{\left ((b c-a d)^2 \left (4 a^3 b c d-10 a b^3 c d+6 a^4 d^2+a^2 b^2 \left (2 c^2-15 d^2\right )+b^4 \left (c^2+12 d^2\right )\right )\right ) \int \frac{1}{a+b \sin (e+f x)} \, dx}{2 b^4 \left (a^2-b^2\right )^2}\\ &=\frac{d^3 (4 b c-3 a d) x}{b^4}+\frac{d^2 \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)}{2 b^3 \left (a^2-b^2\right ) f}+\frac{3 (b c-a d)^3 \left (a b c+a^2 d-2 b^2 d\right ) \cos (e+f x)}{2 b^3 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac{\left ((b c-a d)^2 \left (4 a^3 b c d-10 a b^3 c d+6 a^4 d^2+a^2 b^2 \left (2 c^2-15 d^2\right )+b^4 \left (c^2+12 d^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac{1}{2} (e+f x)\right )\right )}{b^4 \left (a^2-b^2\right )^2 f}\\ &=\frac{d^3 (4 b c-3 a d) x}{b^4}+\frac{d^2 \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)}{2 b^3 \left (a^2-b^2\right ) f}+\frac{3 (b c-a d)^3 \left (a b c+a^2 d-2 b^2 d\right ) \cos (e+f x)}{2 b^3 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac{\left (2 (b c-a d)^2 \left (4 a^3 b c d-10 a b^3 c d+6 a^4 d^2+a^2 b^2 \left (2 c^2-15 d^2\right )+b^4 \left (c^2+12 d^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac{1}{2} (e+f x)\right )\right )}{b^4 \left (a^2-b^2\right )^2 f}\\ &=\frac{d^3 (4 b c-3 a d) x}{b^4}+\frac{(b c-a d)^2 \left (4 a^3 b c d-10 a b^3 c d+6 a^4 d^2+a^2 b^2 \left (2 c^2-15 d^2\right )+b^4 \left (c^2+12 d^2\right )\right ) \tan ^{-1}\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 )^{5/2} f}+\frac{d^2 \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)}{2 b^3 \left (a^2-b^2\right ) f}+\frac{3 (b c-a d)^3 \left (a b c+a^2 d-2 b^2 d\right ) \cos (e+f x)}{2 b^3 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}\\ \end{align*}

Mathematica [B]  time = 4.24157, size = 894, normalized size = 2.81 \[ \frac{\frac{4 \left (6 d^2 a^4+4 b c d a^3+b^2 \left (2 c^2-15 d^2\right ) a^2-10 b^3 c d a+b^4 \left (c^2+12 d^2\right )\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{a^2-b^2}}\right ) (b c-a d)^2}{\left (a^2-b^2\right )^{5/2}}+\frac{-12 d^4 e a^7-12 d^4 f x a^7+16 b c d^3 e a^6+16 b c d^3 f x a^6-24 b d^4 e \sin (e+f x) a^6-24 b d^4 f x \sin (e+f x) a^6+18 b^2 d^4 e a^5+18 b^2 d^4 f x a^5+32 b^2 c d^3 e \sin (e+f x) a^5+32 b^2 c d^3 f x \sin (e+f x) a^5-9 b^2 d^4 \sin (2 (e+f x)) a^5-24 b^3 c d^3 e a^4-24 b^3 c d^3 f x a^4+b^3 d^4 \cos (3 (e+f x)) a^4+48 b^3 d^4 e \sin (e+f x) a^4+48 b^3 d^4 f x \sin (e+f x) a^4+12 b^3 c d^3 \sin (2 (e+f x)) a^4-64 b^4 c d^3 e \sin (e+f x) a^3-64 b^4 c d^3 f x \sin (e+f x) a^3+16 b^4 d^4 \sin (2 (e+f x)) a^3-6 b^4 c^2 d^2 \sin (2 (e+f x)) a^3-2 b^5 d^4 \cos (3 (e+f x)) a^2-24 b^5 d^4 e \sin (e+f x) a^2-24 b^5 d^4 f x \sin (e+f x) a^2-24 b^5 c d^3 \sin (2 (e+f x)) a^2-4 b^5 c^3 d \sin (2 (e+f x)) a^2-6 b^6 d^4 e a-6 b^6 d^4 f x a+32 b^6 c d^3 e \sin (e+f x) a+32 b^6 c d^3 f x \sin (e+f x) a+3 b^6 c^4 \sin (2 (e+f x)) a-4 b^6 d^4 \sin (2 (e+f x)) a+24 b^6 c^2 d^2 \sin (2 (e+f x)) a+8 b^7 c d^3 e+8 b^7 c d^3 f x-b \left (12 d^4 a^6-16 b c d^3 a^5-21 b^2 d^4 a^4+8 b^3 c d \left (2 c^2+5 d^2\right ) a^3+2 b^4 \left (-4 c^4-18 d^2 c^2+d^4\right ) a^2+8 b^5 c^3 d a+b^6 \left (2 c^4+d^4\right )\right ) \cos (e+f x)+2 b^2 \left (a^2-b^2\right )^2 d^3 (3 a d-4 b c) (e+f x) \cos (2 (e+f x))+b^7 d^4 \cos (3 (e+f x))-8 b^7 c^3 d \sin (2 (e+f x))}{\left (a^2-b^2\right )^2 (a+b \sin (e+f x))^2}}{4 b^4 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*Sin[e + f*x])^4/(a + b*Sin[e + f*x])^3,x]

[Out]

((4*(b*c - a*d)^2*(4*a^3*b*c*d - 10*a*b^3*c*d + 6*a^4*d^2 + a^2*b^2*(2*c^2 - 15*d^2) + b^4*(c^2 + 12*d^2))*Arc
Tan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (16*a^6*b*c*d^3*e - 24*a^4*b^3*c*d^3*e + 8*
b^7*c*d^3*e - 12*a^7*d^4*e + 18*a^5*b^2*d^4*e - 6*a*b^6*d^4*e + 16*a^6*b*c*d^3*f*x - 24*a^4*b^3*c*d^3*f*x + 8*
b^7*c*d^3*f*x - 12*a^7*d^4*f*x + 18*a^5*b^2*d^4*f*x - 6*a*b^6*d^4*f*x - b*(8*a*b^5*c^3*d - 16*a^5*b*c*d^3 + 12
*a^6*d^4 - 21*a^4*b^2*d^4 + 8*a^3*b^3*c*d*(2*c^2 + 5*d^2) + b^6*(2*c^4 + d^4) + 2*a^2*b^4*(-4*c^4 - 18*c^2*d^2
+ d^4))*Cos[e + f*x] + 2*b^2*(a^2 - b^2)^2*d^3*(-4*b*c + 3*a*d)*(e + f*x)*Cos[2*(e + f*x)] + a^4*b^3*d^4*Cos[
3*(e + f*x)] - 2*a^2*b^5*d^4*Cos[3*(e + f*x)] + b^7*d^4*Cos[3*(e + f*x)] + 32*a^5*b^2*c*d^3*e*Sin[e + f*x] - 6
4*a^3*b^4*c*d^3*e*Sin[e + f*x] + 32*a*b^6*c*d^3*e*Sin[e + f*x] - 24*a^6*b*d^4*e*Sin[e + f*x] + 48*a^4*b^3*d^4*
e*Sin[e + f*x] - 24*a^2*b^5*d^4*e*Sin[e + f*x] + 32*a^5*b^2*c*d^3*f*x*Sin[e + f*x] - 64*a^3*b^4*c*d^3*f*x*Sin[
e + f*x] + 32*a*b^6*c*d^3*f*x*Sin[e + f*x] - 24*a^6*b*d^4*f*x*Sin[e + f*x] + 48*a^4*b^3*d^4*f*x*Sin[e + f*x] -
24*a^2*b^5*d^4*f*x*Sin[e + f*x] + 3*a*b^6*c^4*Sin[2*(e + f*x)] - 4*a^2*b^5*c^3*d*Sin[2*(e + f*x)] - 8*b^7*c^3
*d*Sin[2*(e + f*x)] - 6*a^3*b^4*c^2*d^2*Sin[2*(e + f*x)] + 24*a*b^6*c^2*d^2*Sin[2*(e + f*x)] + 12*a^4*b^3*c*d^
3*Sin[2*(e + f*x)] - 24*a^2*b^5*c*d^3*Sin[2*(e + f*x)] - 9*a^5*b^2*d^4*Sin[2*(e + f*x)] + 16*a^3*b^4*d^4*Sin[2
*(e + f*x)] - 4*a*b^6*d^4*Sin[2*(e + f*x)])/((a^2 - b^2)^2*(a + b*Sin[e + f*x])^2))/(4*b^4*f)

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Maple [B]  time = 0.123, size = 3683, normalized size = 11.6 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*sin(f*x+e))^4/(a+b*sin(f*x+e))^3,x)

[Out]

-2/f*d^4/b^3/(1+tan(1/2*f*x+1/2*e)^2)-6/f*d^4/b^4*arctan(tan(1/2*f*x+1/2*e))*a+8/f*d^3/b^3*arctan(tan(1/2*f*x+
1/2*e))*c-1/f/b/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a^4*tan(1/2*f*x+1/2*e)
^2*d^4+4/f*b/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a^2*tan(1/2*f*x+1/2*e)^2*
c^4+14/f*b/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a^2*tan(1/2*f*x+1/2*e)^2*d^
4+12/f*b^2/(a^4-2*a^2*b^2+b^4)/(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))*c^2*d^
2+5/f*b^2/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a*tan(1/2*f*x+1/2*e)^3*c^4-2
/f*b^4/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)/a*tan(1/2*f*x+1/2*e)^3*c^4+36/f
*b^3/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*tan(1/2*f*x+1/2*e)^2*c^2*d^2-16/f
*b^3/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*tan(1/2*f*x+1/2*e)*c^3*d+6/f/(a^4
-2*a^2*b^2+b^4)/(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))*a^2*c^2*d^2+6/f/(tan(
1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a^3*tan(1/2*f*x+1/2*e)^3*c^2*d^2-8/f/(tan(1
/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a^3*tan(1/2*f*x+1/2*e)^2*c^3*d-4/f/(tan(1/2*
f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a^3*tan(1/2*f*x+1/2*e)^2*c*d^3-6/f/(tan(1/2*f*x
+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2*a^3/(a^4-2*a^2*b^2+b^4)*tan(1/2*f*x+1/2*e)*c^2*d^2-13/f/b^2/(tan(1/2*f
*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2*a^5/(a^4-2*a^2*b^2+b^4)*tan(1/2*f*x+1/2*e)*d^4+11/f*b^2/(tan(1/2*f*x
+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2*a/(a^4-2*a^2*b^2+b^4)*tan(1/2*f*x+1/2*e)*c^4-2/f*b^4/(tan(1/2*f*x+1/2*
e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/a/(a^4-2*a^2*b^2+b^4)*tan(1/2*f*x+1/2*e)*c^4+8/f/b^2/(tan(1/2*f*x+1/2*e)^2*
a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a^5*c*d^3-8/f/b^3/(a^4-2*a^2*b^2+b^4)/(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))*a^5*c*d^3+20/f/b/(a^4-2*a^2*b^2+b^4)/(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))*a^3*c*d^3-12/f*b/(a^4-2*a^2*b^2+b^4)/(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))*a*c^3*d-24/f*b/(a^4-2*a^2*b^2+b^4)/(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))*a*c*d^3+4/f/b/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+
a)^2/(a^4-2*a^2*b^2+b^4)*a^4*tan(1/2*f*x+1/2*e)^3*c*d^3-12/f*b/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+
a)^2/(a^4-2*a^2*b^2+b^4)*a^2*tan(1/2*f*x+1/2*e)^3*c^3*d-16/f*b/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+
a)^2/(a^4-2*a^2*b^2+b^4)*a^2*tan(1/2*f*x+1/2*e)^3*c*d^3+12/f*b^2/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*
b+a)^2/(a^4-2*a^2*b^2+b^4)*a*tan(1/2*f*x+1/2*e)^3*c^2*d^2+8/f/b^2/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)
*b+a)^2/(a^4-2*a^2*b^2+b^4)*a^5*tan(1/2*f*x+1/2*e)^2*c*d^3+18/f*b/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)
*b+a)^2/(a^4-2*a^2*b^2+b^4)*a^2*tan(1/2*f*x+1/2*e)^2*c^2*d^2-20/f*b^2/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/
2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a*tan(1/2*f*x+1/2*e)^2*c^3*d+7/f*b^3/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2
*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*tan(1/2*f*x+1/2*e)^2*c^4+22/f/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)
^2*a^3/(a^4-2*a^2*b^2+b^4)*tan(1/2*f*x+1/2*e)*d^4-8/f/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4
-2*a^2*b^2+b^4)*a^3*c^3*d-20/f/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a^3*c*d
^3-40/f*b^2/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a*tan(1/2*f*x+1/2*e)^2*c*d
^3-8/f*b^4/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)/a*tan(1/2*f*x+1/2*e)^2*c^3*
d+28/f/b/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2*a^4/(a^4-2*a^2*b^2+b^4)*tan(1/2*f*x+1/2*e)*c*d^3-
20/f*b/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2*a^2/(a^4-2*a^2*b^2+b^4)*tan(1/2*f*x+1/2*e)*c^3*d-64
/f*b/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2*a^2/(a^4-2*a^2*b^2+b^4)*tan(1/2*f*x+1/2*e)*c*d^3+60/f
*b^2/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2*a/(a^4-2*a^2*b^2+b^4)*tan(1/2*f*x+1/2*e)*c^2*d^2-1/f*
b^3/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*c^4+2/f/(a^4-2*a^2*b^2+b^4)/(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))*a^2*c^4+6/f/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1
/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a^3*tan(1/2*f*x+1/2*e)^3*d^4+12/f/(a^4-2*a^2*b^2+b^4)/(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))*a^2*d^4+1/f*b^2/(a^4-2*a^2*b^2+b^4)/(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))*c^4+7/f/b/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e
)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a^4*d^4+4/f*b/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+
b^4)*a^2*c^4-4/f/b^3/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a^6*d^4+18/f*b/(t
an(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a^2*c^2*d^2-4/f*b^2/(tan(1/2*f*x+1/2*e)^
2*a+2*tan(1/2*f*x+1/2*e)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a*c^3*d-15/f/b^2/(a^4-2*a^2*b^2+b^4)/(a^2-b^2)^(1/2)*arcta
n(1/2*(2*a*tan(1/2*f*x+1/2*e)+2*b)/(a^2-b^2)^(1/2))*a^4*d^4+6/f/b^4/(a^4-2*a^2*b^2+b^4)/(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))*a^6*d^4-3/f/b^2/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e
)*b+a)^2/(a^4-2*a^2*b^2+b^4)*a^5*tan(1/2*f*x+1/2*e)^3*d^4-4/f/b^3/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)
*b+a)^2/(a^4-2*a^2*b^2+b^4)*a^6*tan(1/2*f*x+1/2*e)^2*d^4-2/f*b^5/(tan(1/2*f*x+1/2*e)^2*a+2*tan(1/2*f*x+1/2*e)*
b+a)^2/(a^4-2*a^2*b^2+b^4)/a^2*tan(1/2*f*x+1/2*e)^2*c^4

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*sin(f*x+e))^4/(a+b*sin(f*x+e))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.4938, size = 4743, normalized size = 14.92 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*sin(f*x+e))^4/(a+b*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

[-1/4*(4*(a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*d^4*cos(f*x + e)^3 - 4*(4*(a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 -
b^9)*c*d^3 - 3*(a^7*b^2 - 3*a^5*b^4 + 3*a^3*b^6 - a*b^8)*d^4)*f*x*cos(f*x + e)^2 + 4*(4*(a^8*b - 2*a^6*b^3 + 2
*a^2*b^7 - b^9)*c*d^3 - 3*(a^9 - 2*a^7*b^2 + 2*a^3*b^6 - a*b^8)*d^4)*f*x - ((2*a^4*b^4 + 3*a^2*b^6 + b^8)*c^4
- 12*(a^3*b^5 + a*b^7)*c^3*d + 6*(a^4*b^4 + 3*a^2*b^6 + 2*b^8)*c^2*d^2 - 4*(2*a^7*b - 3*a^5*b^3 + a^3*b^5 + 6*
a*b^7)*c*d^3 + 3*(2*a^8 - 3*a^6*b^2 - a^4*b^4 + 4*a^2*b^6)*d^4 + (12*a*b^7*c^3*d - (2*a^2*b^6 + b^8)*c^4 - 6*(
a^2*b^6 + 2*b^8)*c^2*d^2 + 4*(2*a^5*b^3 - 5*a^3*b^5 + 6*a*b^7)*c*d^3 - 3*(2*a^6*b^2 - 5*a^4*b^4 + 4*a^2*b^6)*d
^4)*cos(f*x + e)^2 - 2*(12*a^2*b^6*c^3*d - (2*a^3*b^5 + a*b^7)*c^4 - 6*(a^3*b^5 + 2*a*b^7)*c^2*d^2 + 4*(2*a^6*
b^2 - 5*a^4*b^4 + 6*a^2*b^6)*c*d^3 - 3*(2*a^7*b - 5*a^5*b^3 + 4*a^3*b^5)*d^4)*sin(f*x + e))*sqrt(-a^2 + b^2)*l
og(((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*((4*a^4*b^5 - 5*a^2*b^7 +
b^9)*c^4 - 4*(2*a^5*b^4 - a^3*b^6 - a*b^8)*c^3*d + 18*(a^4*b^5 - a^2*b^7)*c^2*d^2 + 4*(2*a^7*b^2 - 7*a^5*b^4 +
5*a^3*b^6)*c*d^3 - (6*a^8*b - 15*a^6*b^3 + 7*a^4*b^5 + 4*a^2*b^7 - 2*b^9)*d^4)*cos(f*x + e) + 2*(4*(4*(a^7*b^
2 - 3*a^5*b^4 + 3*a^3*b^6 - a*b^8)*c*d^3 - 3*(a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d^4)*f*x + (3*(a^3*b^6
- a*b^8)*c^4 - 4*(a^4*b^5 + a^2*b^7 - 2*b^9)*c^3*d - 6*(a^5*b^4 - 5*a^3*b^6 + 4*a*b^8)*c^2*d^2 + 12*(a^6*b^3 -
3*a^4*b^5 + 2*a^2*b^7)*c*d^3 - (9*a^7*b^2 - 25*a^5*b^4 + 20*a^3*b^6 - 4*a*b^8)*d^4)*cos(f*x + e))*sin(f*x + e
))/((a^6*b^6 - 3*a^4*b^8 + 3*a^2*b^10 - b^12)*f*cos(f*x + e)^2 - 2*(a^7*b^5 - 3*a^5*b^7 + 3*a^3*b^9 - a*b^11)*
f*sin(f*x + e) - (a^8*b^4 - 2*a^6*b^6 + 2*a^2*b^10 - b^12)*f), -1/2*(2*(a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)
*d^4*cos(f*x + e)^3 - 2*(4*(a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*c*d^3 - 3*(a^7*b^2 - 3*a^5*b^4 + 3*a^3*b^6
- a*b^8)*d^4)*f*x*cos(f*x + e)^2 + 2*(4*(a^8*b - 2*a^6*b^3 + 2*a^2*b^7 - b^9)*c*d^3 - 3*(a^9 - 2*a^7*b^2 + 2*a
^3*b^6 - a*b^8)*d^4)*f*x - ((2*a^4*b^4 + 3*a^2*b^6 + b^8)*c^4 - 12*(a^3*b^5 + a*b^7)*c^3*d + 6*(a^4*b^4 + 3*a^
2*b^6 + 2*b^8)*c^2*d^2 - 4*(2*a^7*b - 3*a^5*b^3 + a^3*b^5 + 6*a*b^7)*c*d^3 + 3*(2*a^8 - 3*a^6*b^2 - a^4*b^4 +
4*a^2*b^6)*d^4 + (12*a*b^7*c^3*d - (2*a^2*b^6 + b^8)*c^4 - 6*(a^2*b^6 + 2*b^8)*c^2*d^2 + 4*(2*a^5*b^3 - 5*a^3*
b^5 + 6*a*b^7)*c*d^3 - 3*(2*a^6*b^2 - 5*a^4*b^4 + 4*a^2*b^6)*d^4)*cos(f*x + e)^2 - 2*(12*a^2*b^6*c^3*d - (2*a^
3*b^5 + a*b^7)*c^4 - 6*(a^3*b^5 + 2*a*b^7)*c^2*d^2 + 4*(2*a^6*b^2 - 5*a^4*b^4 + 6*a^2*b^6)*c*d^3 - 3*(2*a^7*b
- 5*a^5*b^3 + 4*a^3*b^5)*d^4)*sin(f*x + e))*sqrt(a^2 - b^2)*arctan(-(a*sin(f*x + e) + b)/(sqrt(a^2 - b^2)*cos(
f*x + e))) + ((4*a^4*b^5 - 5*a^2*b^7 + b^9)*c^4 - 4*(2*a^5*b^4 - a^3*b^6 - a*b^8)*c^3*d + 18*(a^4*b^5 - a^2*b^
7)*c^2*d^2 + 4*(2*a^7*b^2 - 7*a^5*b^4 + 5*a^3*b^6)*c*d^3 - (6*a^8*b - 15*a^6*b^3 + 7*a^4*b^5 + 4*a^2*b^7 - 2*b
^9)*d^4)*cos(f*x + e) + (4*(4*(a^7*b^2 - 3*a^5*b^4 + 3*a^3*b^6 - a*b^8)*c*d^3 - 3*(a^8*b - 3*a^6*b^3 + 3*a^4*b
^5 - a^2*b^7)*d^4)*f*x + (3*(a^3*b^6 - a*b^8)*c^4 - 4*(a^4*b^5 + a^2*b^7 - 2*b^9)*c^3*d - 6*(a^5*b^4 - 5*a^3*b
^6 + 4*a*b^8)*c^2*d^2 + 12*(a^6*b^3 - 3*a^4*b^5 + 2*a^2*b^7)*c*d^3 - (9*a^7*b^2 - 25*a^5*b^4 + 20*a^3*b^6 - 4*
a*b^8)*d^4)*cos(f*x + e))*sin(f*x + e))/((a^6*b^6 - 3*a^4*b^8 + 3*a^2*b^10 - b^12)*f*cos(f*x + e)^2 - 2*(a^7*b
^5 - 3*a^5*b^7 + 3*a^3*b^9 - a*b^11)*f*sin(f*x + e) - (a^8*b^4 - 2*a^6*b^6 + 2*a^2*b^10 - b^12)*f)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*sin(f*x+e))**4/(a+b*sin(f*x+e))**3,x)

[Out]

Timed out

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Giac [B]  time = 1.38063, size = 1565, normalized size = 4.92 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*sin(f*x+e))^4/(a+b*sin(f*x+e))^3,x, algorithm="giac")

[Out]

((2*a^2*b^4*c^4 + b^6*c^4 - 12*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 + 12*b^6*c^2*d^2 - 8*a^5*b*c*d^3 + 20*a^3*b^3*c
*d^3 - 24*a*b^5*c*d^3 + 6*a^6*d^4 - 15*a^4*b^2*d^4 + 12*a^2*b^4*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^4*b^4 - 2*a^2*b^6 + b^8)*sqrt(a^2 - b^2)) - 2*d^4/
((tan(1/2*f*x + 1/2*e)^2 + 1)*b^3) + (5*a^3*b^5*c^4*tan(1/2*f*x + 1/2*e)^3 - 2*a*b^7*c^4*tan(1/2*f*x + 1/2*e)^
3 - 12*a^4*b^4*c^3*d*tan(1/2*f*x + 1/2*e)^3 + 6*a^5*b^3*c^2*d^2*tan(1/2*f*x + 1/2*e)^3 + 12*a^3*b^5*c^2*d^2*ta
n(1/2*f*x + 1/2*e)^3 + 4*a^6*b^2*c*d^3*tan(1/2*f*x + 1/2*e)^3 - 16*a^4*b^4*c*d^3*tan(1/2*f*x + 1/2*e)^3 - 3*a^
7*b*d^4*tan(1/2*f*x + 1/2*e)^3 + 6*a^5*b^3*d^4*tan(1/2*f*x + 1/2*e)^3 + 4*a^4*b^4*c^4*tan(1/2*f*x + 1/2*e)^2 +
7*a^2*b^6*c^4*tan(1/2*f*x + 1/2*e)^2 - 2*b^8*c^4*tan(1/2*f*x + 1/2*e)^2 - 8*a^5*b^3*c^3*d*tan(1/2*f*x + 1/2*e
)^2 - 20*a^3*b^5*c^3*d*tan(1/2*f*x + 1/2*e)^2 - 8*a*b^7*c^3*d*tan(1/2*f*x + 1/2*e)^2 + 18*a^4*b^4*c^2*d^2*tan(
1/2*f*x + 1/2*e)^2 + 36*a^2*b^6*c^2*d^2*tan(1/2*f*x + 1/2*e)^2 + 8*a^7*b*c*d^3*tan(1/2*f*x + 1/2*e)^2 - 4*a^5*
b^3*c*d^3*tan(1/2*f*x + 1/2*e)^2 - 40*a^3*b^5*c*d^3*tan(1/2*f*x + 1/2*e)^2 - 4*a^8*d^4*tan(1/2*f*x + 1/2*e)^2
- a^6*b^2*d^4*tan(1/2*f*x + 1/2*e)^2 + 14*a^4*b^4*d^4*tan(1/2*f*x + 1/2*e)^2 + 11*a^3*b^5*c^4*tan(1/2*f*x + 1/
2*e) - 2*a*b^7*c^4*tan(1/2*f*x + 1/2*e) - 20*a^4*b^4*c^3*d*tan(1/2*f*x + 1/2*e) - 16*a^2*b^6*c^3*d*tan(1/2*f*x
+ 1/2*e) - 6*a^5*b^3*c^2*d^2*tan(1/2*f*x + 1/2*e) + 60*a^3*b^5*c^2*d^2*tan(1/2*f*x + 1/2*e) + 28*a^6*b^2*c*d^
3*tan(1/2*f*x + 1/2*e) - 64*a^4*b^4*c*d^3*tan(1/2*f*x + 1/2*e) - 13*a^7*b*d^4*tan(1/2*f*x + 1/2*e) + 22*a^5*b^
3*d^4*tan(1/2*f*x + 1/2*e) + 4*a^4*b^4*c^4 - a^2*b^6*c^4 - 8*a^5*b^3*c^3*d - 4*a^3*b^5*c^3*d + 18*a^4*b^4*c^2*
d^2 + 8*a^7*b*c*d^3 - 20*a^5*b^3*c*d^3 - 4*a^8*d^4 + 7*a^6*b^2*d^4)/((a^6*b^3 - 2*a^4*b^5 + a^2*b^7)*(a*tan(1/
2*f*x + 1/2*e)^2 + 2*b*tan(1/2*f*x + 1/2*e) + a)^2) + (4*b*c*d^3 - 3*a*d^4)*(f*x + e)/b^4)/f