3.470 \(\int \frac{\cos ^2(c+d x)}{(a+b \sin (c+d x))^8} \, dx\)

Optimal. Leaf size=422 \[ \frac{a \left (20 a^2 b^2+8 a^4+5 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{8 d \left (a^2-b^2\right )^{13/2}}+\frac{\left (1518 a^4 b^2+1779 a^2 b^4+40 a^6+128 b^6\right ) \cos (c+d x)}{1680 b d \left (a^2-b^2\right )^6 (a+b \sin (c+d x))}+\frac{a \left (718 a^2 b^2+40 a^4+397 b^4\right ) \cos (c+d x)}{1680 b d \left (a^2-b^2\right )^5 (a+b \sin (c+d x))^2}+\frac{\left (179 a^2 b^2+20 a^4+32 b^4\right ) \cos (c+d x)}{840 b d \left (a^2-b^2\right )^4 (a+b \sin (c+d x))^3}+\frac{a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{840 b d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^4}+\frac{\left (5 a^2+6 b^2\right ) \cos (c+d x)}{210 b d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^5}+\frac{a \cos (c+d x)}{42 b d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}-\frac{\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7} \]

[Out]

(a*(8*a^4 + 20*a^2*b^2 + 5*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(8*(a^2 - b^2)^(13/2)*d) - C
os[c + d*x]/(7*b*d*(a + b*Sin[c + d*x])^7) + (a*Cos[c + d*x])/(42*b*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^6) + ((
5*a^2 + 6*b^2)*Cos[c + d*x])/(210*b*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^5) + (a*(20*a^2 + 79*b^2)*Cos[c + d*x
])/(840*b*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^4) + ((20*a^4 + 179*a^2*b^2 + 32*b^4)*Cos[c + d*x])/(840*b*(a^2
- b^2)^4*d*(a + b*Sin[c + d*x])^3) + (a*(40*a^4 + 718*a^2*b^2 + 397*b^4)*Cos[c + d*x])/(1680*b*(a^2 - b^2)^5*
d*(a + b*Sin[c + d*x])^2) + ((40*a^6 + 1518*a^4*b^2 + 1779*a^2*b^4 + 128*b^6)*Cos[c + d*x])/(1680*b*(a^2 - b^2
)^6*d*(a + b*Sin[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 0.745975, antiderivative size = 422, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2693, 2754, 12, 2660, 618, 204} \[ \frac{a \left (20 a^2 b^2+8 a^4+5 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{8 d \left (a^2-b^2\right )^{13/2}}+\frac{\left (1518 a^4 b^2+1779 a^2 b^4+40 a^6+128 b^6\right ) \cos (c+d x)}{1680 b d \left (a^2-b^2\right )^6 (a+b \sin (c+d x))}+\frac{a \left (718 a^2 b^2+40 a^4+397 b^4\right ) \cos (c+d x)}{1680 b d \left (a^2-b^2\right )^5 (a+b \sin (c+d x))^2}+\frac{\left (179 a^2 b^2+20 a^4+32 b^4\right ) \cos (c+d x)}{840 b d \left (a^2-b^2\right )^4 (a+b \sin (c+d x))^3}+\frac{a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{840 b d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^4}+\frac{\left (5 a^2+6 b^2\right ) \cos (c+d x)}{210 b d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^5}+\frac{a \cos (c+d x)}{42 b d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}-\frac{\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^2/(a + b*Sin[c + d*x])^8,x]

[Out]

(a*(8*a^4 + 20*a^2*b^2 + 5*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(8*(a^2 - b^2)^(13/2)*d) - C
os[c + d*x]/(7*b*d*(a + b*Sin[c + d*x])^7) + (a*Cos[c + d*x])/(42*b*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^6) + ((
5*a^2 + 6*b^2)*Cos[c + d*x])/(210*b*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^5) + (a*(20*a^2 + 79*b^2)*Cos[c + d*x
])/(840*b*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^4) + ((20*a^4 + 179*a^2*b^2 + 32*b^4)*Cos[c + d*x])/(840*b*(a^2
- b^2)^4*d*(a + b*Sin[c + d*x])^3) + (a*(40*a^4 + 718*a^2*b^2 + 397*b^4)*Cos[c + d*x])/(1680*b*(a^2 - b^2)^5*
d*(a + b*Sin[c + d*x])^2) + ((40*a^6 + 1518*a^4*b^2 + 1779*a^2*b^4 + 128*b^6)*Cos[c + d*x])/(1680*b*(a^2 - b^2
)^6*d*(a + b*Sin[c + d*x]))

Rule 2693

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(g*(g*
Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Dist[(g^2*(p - 1))/(b*(m + 1)), Int[(g
*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)*Sin[e + f*x], x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a
^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && IntegersQ[2*m, 2*p]

Rule 2754

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] + Dist[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]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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{\cos ^2(c+d x)}{(a+b \sin (c+d x))^8} \, dx &=-\frac{\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}-\frac{\int \frac{\sin (c+d x)}{(a+b \sin (c+d x))^7} \, dx}{7 b}\\ &=-\frac{\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac{a \cos (c+d x)}{42 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}+\frac{\int \frac{6 b-5 a \sin (c+d x)}{(a+b \sin (c+d x))^6} \, dx}{42 b \left (a^2-b^2\right )}\\ &=-\frac{\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac{a \cos (c+d x)}{42 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}+\frac{\left (5 a^2+6 b^2\right ) \cos (c+d x)}{210 b \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^5}-\frac{\int \frac{-55 a b+4 \left (5 a^2+6 b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^5} \, dx}{210 b \left (a^2-b^2\right )^2}\\ &=-\frac{\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac{a \cos (c+d x)}{42 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}+\frac{\left (5 a^2+6 b^2\right ) \cos (c+d x)}{210 b \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^5}+\frac{a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^4}+\frac{\int \frac{12 b \left (25 a^2+8 b^2\right )-3 a \left (20 a^2+79 b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^4} \, dx}{840 b \left (a^2-b^2\right )^3}\\ &=-\frac{\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac{a \cos (c+d x)}{42 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}+\frac{\left (5 a^2+6 b^2\right ) \cos (c+d x)}{210 b \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^5}+\frac{a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^4}+\frac{\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^3}-\frac{\int \frac{-27 a b \left (40 a^2+37 b^2\right )+6 \left (20 a^4+179 a^2 b^2+32 b^4\right ) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx}{2520 b \left (a^2-b^2\right )^4}\\ &=-\frac{\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac{a \cos (c+d x)}{42 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}+\frac{\left (5 a^2+6 b^2\right ) \cos (c+d x)}{210 b \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^5}+\frac{a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^4}+\frac{\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^3}+\frac{a \left (40 a^4+718 a^2 b^2+397 b^4\right ) \cos (c+d x)}{1680 b \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^2}+\frac{\int \frac{6 b \left (400 a^4+691 a^2 b^2+64 b^4\right )-3 a \left (40 a^4+718 a^2 b^2+397 b^4\right ) \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx}{5040 b \left (a^2-b^2\right )^5}\\ &=-\frac{\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac{a \cos (c+d x)}{42 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}+\frac{\left (5 a^2+6 b^2\right ) \cos (c+d x)}{210 b \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^5}+\frac{a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^4}+\frac{\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^3}+\frac{a \left (40 a^4+718 a^2 b^2+397 b^4\right ) \cos (c+d x)}{1680 b \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^2}+\frac{\left (40 a^6+1518 a^4 b^2+1779 a^2 b^4+128 b^6\right ) \cos (c+d x)}{1680 b \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))}-\frac{\int -\frac{315 a b \left (8 a^4+20 a^2 b^2+5 b^4\right )}{a+b \sin (c+d x)} \, dx}{5040 b \left (a^2-b^2\right )^6}\\ &=-\frac{\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac{a \cos (c+d x)}{42 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}+\frac{\left (5 a^2+6 b^2\right ) \cos (c+d x)}{210 b \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^5}+\frac{a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^4}+\frac{\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^3}+\frac{a \left (40 a^4+718 a^2 b^2+397 b^4\right ) \cos (c+d x)}{1680 b \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^2}+\frac{\left (40 a^6+1518 a^4 b^2+1779 a^2 b^4+128 b^6\right ) \cos (c+d x)}{1680 b \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))}+\frac{\left (a \left (8 a^4+20 a^2 b^2+5 b^4\right )\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{16 \left (a^2-b^2\right )^6}\\ &=-\frac{\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac{a \cos (c+d x)}{42 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}+\frac{\left (5 a^2+6 b^2\right ) \cos (c+d x)}{210 b \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^5}+\frac{a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^4}+\frac{\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^3}+\frac{a \left (40 a^4+718 a^2 b^2+397 b^4\right ) \cos (c+d x)}{1680 b \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^2}+\frac{\left (40 a^6+1518 a^4 b^2+1779 a^2 b^4+128 b^6\right ) \cos (c+d x)}{1680 b \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))}+\frac{\left (a \left (8 a^4+20 a^2 b^2+5 b^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{8 \left (a^2-b^2\right )^6 d}\\ &=-\frac{\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac{a \cos (c+d x)}{42 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}+\frac{\left (5 a^2+6 b^2\right ) \cos (c+d x)}{210 b \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^5}+\frac{a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^4}+\frac{\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^3}+\frac{a \left (40 a^4+718 a^2 b^2+397 b^4\right ) \cos (c+d x)}{1680 b \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^2}+\frac{\left (40 a^6+1518 a^4 b^2+1779 a^2 b^4+128 b^6\right ) \cos (c+d x)}{1680 b \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))}-\frac{\left (a \left (8 a^4+20 a^2 b^2+5 b^4\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} (c+d x)\right )\right )}{4 \left (a^2-b^2\right )^6 d}\\ &=\frac{a \left (8 a^4+20 a^2 b^2+5 b^4\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{8 \left (a^2-b^2\right )^{13/2} d}-\frac{\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac{a \cos (c+d x)}{42 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}+\frac{\left (5 a^2+6 b^2\right ) \cos (c+d x)}{210 b \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^5}+\frac{a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^4}+\frac{\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^3}+\frac{a \left (40 a^4+718 a^2 b^2+397 b^4\right ) \cos (c+d x)}{1680 b \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^2}+\frac{\left (40 a^6+1518 a^4 b^2+1779 a^2 b^4+128 b^6\right ) \cos (c+d x)}{1680 b \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))}\\ \end{align*}

Mathematica [B]  time = 6.19127, size = 1896, normalized size = 4.49 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cos[c + d*x]^2/(a + b*Sin[c + d*x])^8,x]

[Out]

Cos[c + d*x]^3/(3*(a - b)*d*(a + b*Sin[c + d*x])^7) + (a*Cos[c + d*x]*(-(b*(1 - Sin[c + d*x])^(3/2)*(1 + Sin[c
+ d*x])^(5/2))/(7*(-a + b)*(a + b)*(a + b*Sin[c + d*x])^7) - (-((3*a*b + (7*a - b)*b)*(1 - Sin[c + d*x])^(3/2
)*(1 + Sin[c + d*x])^(5/2))/(6*(-a + b)*(a + b)*(a + b*Sin[c + d*x])^6) - (-((2*a*(10*a - b)*b + b*(42*a^2 - 1
6*a*b + 19*b^2))*(1 - Sin[c + d*x])^(3/2)*(1 + Sin[c + d*x])^(5/2))/(5*(-a + b)*(a + b)*(a + b*Sin[c + d*x])^5
) - (-((a*b*(62*a^2 - 18*a*b + 19*b^2) + b*(210*a^3 - 142*a^2*b + 213*a*b^2 - 29*b^3))*(1 - Sin[c + d*x])^(3/2
)*(1 + Sin[c + d*x])^(5/2))/(4*(-a + b)*(a + b)*(a + b*Sin[c + d*x])^4) - (105*(8*a^4 - 8*a^3*b + 12*a^2*b^2 -
4*a*b^3 + b^4)*(-(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(5/2))/(3*(-a + b)*(a + b*Sin[c + d*x])^3) - (-(S
qrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(3/2))/(2*(a + b)*(a + b*Sin[c + d*x])^2) + (3*((-2*ArcTan[(Sqrt[-a +
b]*Sqrt[1 - Sin[c + d*x]])/(Sqrt[-a - b]*Sqrt[1 + Sin[c + d*x]])])/((-a - b)^(3/2)*Sqrt[-a + b]) + (Sqrt[1 -
Sin[c + d*x]]*Sqrt[1 + Sin[c + d*x]])/((-a - b)*(a + b*Sin[c + d*x]))))/(2*(a + b)))/(3*(-a + b))))/(4*(-a + b
)*(a + b)))/(5*(-a + b)*(a + b)))/(6*(-a + b)*(a + b)))/(7*(-a + b)*(a + b))))/((a - b)*d*Sqrt[1 - Sin[c + d*x
]]*Sqrt[1 + Sin[c + d*x]]) + (4*b*(Cos[c + d*x]^5/(5*(a - b)*d*(a + b*Sin[c + d*x])^7) + (a*Cos[c + d*x]*(-(b*
(1 - Sin[c + d*x])^(5/2)*(1 + Sin[c + d*x])^(7/2))/(7*(-a + b)*(a + b)*(a + b*Sin[c + d*x])^7) - (-((a*b + (7*
a - b)*b)*(1 - Sin[c + d*x])^(5/2)*(1 + Sin[c + d*x])^(7/2))/(6*(-a + b)*(a + b)*(a + b*Sin[c + d*x])^6) - (7*
(6*a^2 - 2*a*b + b^2)*(-((1 - Sin[c + d*x])^(3/2)*(1 + Sin[c + d*x])^(7/2))/(5*(-a + b)*(a + b*Sin[c + d*x])^5
) - (3*(-(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(7/2))/(4*(-a + b)*(a + b*Sin[c + d*x])^4) - (-(Sqrt[1 - S
in[c + d*x]]*(1 + Sin[c + d*x])^(5/2))/(3*(a + b)*(a + b*Sin[c + d*x])^3) + (5*(-(Sqrt[1 - Sin[c + d*x]]*(1 +
Sin[c + d*x])^(3/2))/(2*(a + b)*(a + b*Sin[c + d*x])^2) + (3*((-2*ArcTan[(Sqrt[-a + b]*Sqrt[1 - Sin[c + d*x]])
/(Sqrt[-a - b]*Sqrt[1 + Sin[c + d*x]])])/((-a - b)^(3/2)*Sqrt[-a + b]) + (Sqrt[1 - Sin[c + d*x]]*Sqrt[1 + Sin[
c + d*x]])/((-a - b)*(a + b*Sin[c + d*x]))))/(2*(a + b))))/(3*(a + b)))/(4*(-a + b))))/(5*(-a + b))))/(6*(-a +
b)*(a + b)))/(7*(-a + b)*(a + b))))/((a - b)*d*Sqrt[1 - Sin[c + d*x]]*Sqrt[1 + Sin[c + d*x]]) + (2*b*(Cos[c +
d*x]^7/(7*(a - b)*d*(a + b*Sin[c + d*x])^7) + (a*Cos[c + d*x]*(-((1 - Sin[c + d*x])^(5/2)*(1 + Sin[c + d*x])^
(9/2))/(7*(-a + b)*(a + b*Sin[c + d*x])^7) - (5*(-((1 - Sin[c + d*x])^(3/2)*(1 + Sin[c + d*x])^(9/2))/(6*(-a +
b)*(a + b*Sin[c + d*x])^6) - (-(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(9/2))/(5*(-a + b)*(a + b*Sin[c + d
*x])^5) - (-(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(7/2))/(4*(a + b)*(a + b*Sin[c + d*x])^4) + (7*(-(Sqrt[
1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(5/2))/(3*(a + b)*(a + b*Sin[c + d*x])^3) + (5*(-(Sqrt[1 - Sin[c + d*x]]*
(1 + Sin[c + d*x])^(3/2))/(2*(a + b)*(a + b*Sin[c + d*x])^2) + (3*((-2*ArcTan[(Sqrt[-a + b]*Sqrt[1 - Sin[c + d
*x]])/(Sqrt[-a - b]*Sqrt[1 + Sin[c + d*x]])])/((-a - b)^(3/2)*Sqrt[-a + b]) + (Sqrt[1 - Sin[c + d*x]]*Sqrt[1 +
Sin[c + d*x]])/((-a - b)*(a + b*Sin[c + d*x]))))/(2*(a + b))))/(3*(a + b))))/(4*(a + b)))/(5*(-a + b)))/(2*(-
a + b))))/(7*(-a + b))))/((a - b)*d*Sqrt[1 - Sin[c + d*x]]*Sqrt[1 + Sin[c + d*x]])))/(5*(a - b))))/(3*(a - b))

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Maple [B]  time = 0.206, size = 11250, normalized size = 26.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^2/(a+b*sin(d*x+c))^8,x)

[Out]

result too large to display

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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(cos(d*x+c)^2/(a+b*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 7.42321, size = 7096, normalized size = 16.82 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2/(a+b*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

[1/3360*(2*(40*a^8*b^5 + 1478*a^6*b^7 + 261*a^4*b^9 - 1651*a^2*b^11 - 128*b^13)*cos(d*x + c)^7 - 28*(60*a^10*b
^3 + 1837*a^8*b^5 + 176*a^6*b^7 - 1680*a^4*b^9 - 361*a^2*b^11 - 32*b^13)*cos(d*x + c)^5 + 70*(40*a^12*b + 900*
a^10*b^3 + 1111*a^8*b^5 - 501*a^6*b^7 - 1395*a^4*b^9 - 139*a^2*b^11 - 16*b^13)*cos(d*x + c)^3 + 105*(8*a^12 +
188*a^10*b^2 + 705*a^8*b^4 + 861*a^6*b^6 + 315*a^4*b^8 + 35*a^2*b^10 - 7*(8*a^6*b^6 + 20*a^4*b^8 + 5*a^2*b^10)
*cos(d*x + c)^6 + 7*(40*a^8*b^4 + 124*a^6*b^6 + 85*a^4*b^8 + 15*a^2*b^10)*cos(d*x + c)^4 - 7*(24*a^10*b^2 + 14
0*a^8*b^4 + 239*a^6*b^6 + 110*a^4*b^8 + 15*a^2*b^10)*cos(d*x + c)^2 + (56*a^11*b + 420*a^9*b^3 + 903*a^7*b^5 +
603*a^5*b^7 + 125*a^3*b^9 + 5*a*b^11 - (8*a^5*b^7 + 20*a^3*b^9 + 5*a*b^11)*cos(d*x + c)^6 + 3*(56*a^7*b^5 + 1
48*a^5*b^7 + 55*a^3*b^9 + 5*a*b^11)*cos(d*x + c)^4 - (280*a^9*b^3 + 1036*a^7*b^5 + 1039*a^5*b^7 + 270*a^3*b^9
+ 15*a*b^11)*cos(d*x + c)^2)*sin(d*x + c))*sqrt(-a^2 + b^2)*log(((2*a^2 - b^2)*cos(d*x + c)^2 - 2*a*b*sin(d*x
+ c) - a^2 - b^2 + 2*(a*cos(d*x + c)*sin(d*x + c) + b*cos(d*x + c))*sqrt(-a^2 + b^2))/(b^2*cos(d*x + c)^2 - 2*
a*b*sin(d*x + c) - a^2 - b^2)) - 420*(24*a^12*b + 116*a^10*b^3 + 99*a^8*b^5 - 129*a^6*b^7 - 95*a^4*b^9 - 15*a^
2*b^11)*cos(d*x + c) - 14*((40*a^9*b^4 + 1358*a^7*b^6 + 81*a^5*b^8 - 1426*a^3*b^10 - 53*a*b^12)*cos(d*x + c)^5
- 10*(20*a^11*b^2 + 535*a^9*b^4 + 147*a^7*b^6 - 407*a^5*b^8 - 283*a^3*b^10 - 12*a*b^12)*cos(d*x + c)^3 + 15*(
8*a^13 + 132*a^11*b^2 + 285*a^9*b^4 - 42*a^7*b^6 - 288*a^5*b^8 - 90*a^3*b^10 - 5*a*b^12)*cos(d*x + c))*sin(d*x
+ c))/(7*(a^15*b^6 - 7*a^13*b^8 + 21*a^11*b^10 - 35*a^9*b^12 + 35*a^7*b^14 - 21*a^5*b^16 + 7*a^3*b^18 - a*b^2
0)*d*cos(d*x + c)^6 - 7*(5*a^17*b^4 - 32*a^15*b^6 + 84*a^13*b^8 - 112*a^11*b^10 + 70*a^9*b^12 - 28*a^5*b^16 +
16*a^3*b^18 - 3*a*b^20)*d*cos(d*x + c)^4 + 7*(3*a^19*b^2 - 11*a^17*b^4 - 4*a^15*b^6 + 84*a^13*b^8 - 182*a^11*b
^10 + 182*a^9*b^12 - 84*a^7*b^14 + 4*a^5*b^16 + 11*a^3*b^18 - 3*a*b^20)*d*cos(d*x + c)^2 - (a^21 + 14*a^19*b^2
- 91*a^17*b^4 + 168*a^15*b^6 - 14*a^13*b^8 - 364*a^11*b^10 + 546*a^9*b^12 - 344*a^7*b^14 + 77*a^5*b^16 + 14*a
^3*b^18 - 7*a*b^20)*d + ((a^14*b^7 - 7*a^12*b^9 + 21*a^10*b^11 - 35*a^8*b^13 + 35*a^6*b^15 - 21*a^4*b^17 + 7*a
^2*b^19 - b^21)*d*cos(d*x + c)^6 - 3*(7*a^16*b^5 - 48*a^14*b^7 + 140*a^12*b^9 - 224*a^10*b^11 + 210*a^8*b^13 -
112*a^6*b^15 + 28*a^4*b^17 - b^21)*d*cos(d*x + c)^4 + (35*a^18*b^3 - 203*a^16*b^5 + 444*a^14*b^7 - 364*a^12*b
^9 - 182*a^10*b^11 + 630*a^8*b^13 - 532*a^6*b^15 + 196*a^4*b^17 - 21*a^2*b^19 - 3*b^21)*d*cos(d*x + c)^2 - (7*
a^20*b - 14*a^18*b^3 - 77*a^16*b^5 + 344*a^14*b^7 - 546*a^12*b^9 + 364*a^10*b^11 + 14*a^8*b^13 - 168*a^6*b^15
+ 91*a^4*b^17 - 14*a^2*b^19 - b^21)*d)*sin(d*x + c)), 1/1680*((40*a^8*b^5 + 1478*a^6*b^7 + 261*a^4*b^9 - 1651*
a^2*b^11 - 128*b^13)*cos(d*x + c)^7 - 14*(60*a^10*b^3 + 1837*a^8*b^5 + 176*a^6*b^7 - 1680*a^4*b^9 - 361*a^2*b^
11 - 32*b^13)*cos(d*x + c)^5 + 35*(40*a^12*b + 900*a^10*b^3 + 1111*a^8*b^5 - 501*a^6*b^7 - 1395*a^4*b^9 - 139*
a^2*b^11 - 16*b^13)*cos(d*x + c)^3 + 105*(8*a^12 + 188*a^10*b^2 + 705*a^8*b^4 + 861*a^6*b^6 + 315*a^4*b^8 + 35
*a^2*b^10 - 7*(8*a^6*b^6 + 20*a^4*b^8 + 5*a^2*b^10)*cos(d*x + c)^6 + 7*(40*a^8*b^4 + 124*a^6*b^6 + 85*a^4*b^8
+ 15*a^2*b^10)*cos(d*x + c)^4 - 7*(24*a^10*b^2 + 140*a^8*b^4 + 239*a^6*b^6 + 110*a^4*b^8 + 15*a^2*b^10)*cos(d*
x + c)^2 + (56*a^11*b + 420*a^9*b^3 + 903*a^7*b^5 + 603*a^5*b^7 + 125*a^3*b^9 + 5*a*b^11 - (8*a^5*b^7 + 20*a^3
*b^9 + 5*a*b^11)*cos(d*x + c)^6 + 3*(56*a^7*b^5 + 148*a^5*b^7 + 55*a^3*b^9 + 5*a*b^11)*cos(d*x + c)^4 - (280*a
^9*b^3 + 1036*a^7*b^5 + 1039*a^5*b^7 + 270*a^3*b^9 + 15*a*b^11)*cos(d*x + c)^2)*sin(d*x + c))*sqrt(a^2 - b^2)*
arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) - 210*(24*a^12*b + 116*a^10*b^3 + 99*a^8*b^5 - 12
9*a^6*b^7 - 95*a^4*b^9 - 15*a^2*b^11)*cos(d*x + c) - 7*((40*a^9*b^4 + 1358*a^7*b^6 + 81*a^5*b^8 - 1426*a^3*b^1
0 - 53*a*b^12)*cos(d*x + c)^5 - 10*(20*a^11*b^2 + 535*a^9*b^4 + 147*a^7*b^6 - 407*a^5*b^8 - 283*a^3*b^10 - 12*
a*b^12)*cos(d*x + c)^3 + 15*(8*a^13 + 132*a^11*b^2 + 285*a^9*b^4 - 42*a^7*b^6 - 288*a^5*b^8 - 90*a^3*b^10 - 5*
a*b^12)*cos(d*x + c))*sin(d*x + c))/(7*(a^15*b^6 - 7*a^13*b^8 + 21*a^11*b^10 - 35*a^9*b^12 + 35*a^7*b^14 - 21*
a^5*b^16 + 7*a^3*b^18 - a*b^20)*d*cos(d*x + c)^6 - 7*(5*a^17*b^4 - 32*a^15*b^6 + 84*a^13*b^8 - 112*a^11*b^10 +
70*a^9*b^12 - 28*a^5*b^16 + 16*a^3*b^18 - 3*a*b^20)*d*cos(d*x + c)^4 + 7*(3*a^19*b^2 - 11*a^17*b^4 - 4*a^15*b
^6 + 84*a^13*b^8 - 182*a^11*b^10 + 182*a^9*b^12 - 84*a^7*b^14 + 4*a^5*b^16 + 11*a^3*b^18 - 3*a*b^20)*d*cos(d*x
+ c)^2 - (a^21 + 14*a^19*b^2 - 91*a^17*b^4 + 168*a^15*b^6 - 14*a^13*b^8 - 364*a^11*b^10 + 546*a^9*b^12 - 344*
a^7*b^14 + 77*a^5*b^16 + 14*a^3*b^18 - 7*a*b^20)*d + ((a^14*b^7 - 7*a^12*b^9 + 21*a^10*b^11 - 35*a^8*b^13 + 35
*a^6*b^15 - 21*a^4*b^17 + 7*a^2*b^19 - b^21)*d*cos(d*x + c)^6 - 3*(7*a^16*b^5 - 48*a^14*b^7 + 140*a^12*b^9 - 2
24*a^10*b^11 + 210*a^8*b^13 - 112*a^6*b^15 + 28*a^4*b^17 - b^21)*d*cos(d*x + c)^4 + (35*a^18*b^3 - 203*a^16*b^
5 + 444*a^14*b^7 - 364*a^12*b^9 - 182*a^10*b^11 + 630*a^8*b^13 - 532*a^6*b^15 + 196*a^4*b^17 - 21*a^2*b^19 - 3
*b^21)*d*cos(d*x + c)^2 - (7*a^20*b - 14*a^18*b^3 - 77*a^16*b^5 + 344*a^14*b^7 - 546*a^12*b^9 + 364*a^10*b^11
+ 14*a^8*b^13 - 168*a^6*b^15 + 91*a^4*b^17 - 14*a^2*b^19 - b^21)*d)*sin(d*x + c))]

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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(cos(d*x+c)**2/(a+b*sin(d*x+c))**8,x)

[Out]

Timed out

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Giac [B]  time = 2.09701, size = 2979, normalized size = 7.06 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2/(a+b*sin(d*x+c))^8,x, algorithm="giac")

[Out]

1/840*(105*(8*a^5 + 20*a^3*b^2 + 5*a*b^4)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*d*x + 1
/2*c) + b)/sqrt(a^2 - b^2)))/((a^12 - 6*a^10*b^2 + 15*a^8*b^4 - 20*a^6*b^6 + 15*a^4*b^8 - 6*a^2*b^10 + b^12)*s
qrt(a^2 - b^2)) - (840*a^18*tan(1/2*d*x + 1/2*c)^13 - 12180*a^16*b^2*tan(1/2*d*x + 1/2*c)^13 + 24675*a^14*b^4*
tan(1/2*d*x + 1/2*c)^13 - 33600*a^12*b^6*tan(1/2*d*x + 1/2*c)^13 + 25200*a^10*b^8*tan(1/2*d*x + 1/2*c)^13 - 10
080*a^8*b^10*tan(1/2*d*x + 1/2*c)^13 + 1680*a^6*b^12*tan(1/2*d*x + 1/2*c)^13 - 840*a^17*b*tan(1/2*d*x + 1/2*c)
^12 - 87780*a^15*b^3*tan(1/2*d*x + 1/2*c)^12 + 144375*a^13*b^5*tan(1/2*d*x + 1/2*c)^12 - 201600*a^11*b^7*tan(1
/2*d*x + 1/2*c)^12 + 151200*a^9*b^9*tan(1/2*d*x + 1/2*c)^12 - 60480*a^7*b^11*tan(1/2*d*x + 1/2*c)^12 + 10080*a
^5*b^13*tan(1/2*d*x + 1/2*c)^12 + 3360*a^18*tan(1/2*d*x + 1/2*c)^11 - 94080*a^16*b^2*tan(1/2*d*x + 1/2*c)^11 -
220500*a^14*b^4*tan(1/2*d*x + 1/2*c)^11 + 287350*a^12*b^6*tan(1/2*d*x + 1/2*c)^11 - 537600*a^10*b^8*tan(1/2*d
*x + 1/2*c)^11 + 450240*a^8*b^10*tan(1/2*d*x + 1/2*c)^11 - 192640*a^6*b^12*tan(1/2*d*x + 1/2*c)^11 + 33600*a^4
*b^14*tan(1/2*d*x + 1/2*c)^11 - 13440*a^17*b*tan(1/2*d*x + 1/2*c)^10 - 554400*a^15*b^3*tan(1/2*d*x + 1/2*c)^10
- 165900*a^13*b^5*tan(1/2*d*x + 1/2*c)^10 - 66850*a^11*b^7*tan(1/2*d*x + 1/2*c)^10 - 621600*a^9*b^9*tan(1/2*d
*x + 1/2*c)^10 + 719040*a^7*b^11*tan(1/2*d*x + 1/2*c)^10 - 355040*a^5*b^13*tan(1/2*d*x + 1/2*c)^10 + 67200*a^3
*b^15*tan(1/2*d*x + 1/2*c)^10 + 4200*a^18*tan(1/2*d*x + 1/2*c)^9 - 304500*a^16*b^2*tan(1/2*d*x + 1/2*c)^9 - 14
18025*a^14*b^4*tan(1/2*d*x + 1/2*c)^9 + 147070*a^12*b^6*tan(1/2*d*x + 1/2*c)^9 - 1316700*a^10*b^8*tan(1/2*d*x
+ 1/2*c)^9 + 242592*a^8*b^10*tan(1/2*d*x + 1/2*c)^9 + 439376*a^6*b^12*tan(1/2*d*x + 1/2*c)^9 - 352128*a^4*b^14
*tan(1/2*d*x + 1/2*c)^9 + 80640*a^2*b^16*tan(1/2*d*x + 1/2*c)^9 - 49000*a^17*b*tan(1/2*d*x + 1/2*c)^8 - 135730
0*a^15*b^3*tan(1/2*d*x + 1/2*c)^8 - 1726305*a^13*b^5*tan(1/2*d*x + 1/2*c)^8 - 346570*a^11*b^7*tan(1/2*d*x + 1/
2*c)^8 - 1972600*a^9*b^9*tan(1/2*d*x + 1/2*c)^8 + 1360128*a^7*b^11*tan(1/2*d*x + 1/2*c)^8 - 298816*a^5*b^13*ta
n(1/2*d*x + 1/2*c)^8 - 122752*a^3*b^15*tan(1/2*d*x + 1/2*c)^8 + 53760*a*b^17*tan(1/2*d*x + 1/2*c)^8 - 509600*a
^16*b^2*tan(1/2*d*x + 1/2*c)^7 - 2685200*a^14*b^4*tan(1/2*d*x + 1/2*c)^7 - 900900*a^12*b^6*tan(1/2*d*x + 1/2*c
)^7 - 2070320*a^10*b^8*tan(1/2*d*x + 1/2*c)^7 - 278096*a^8*b^10*tan(1/2*d*x + 1/2*c)^7 + 952320*a^6*b^12*tan(1
/2*d*x + 1/2*c)^7 - 538112*a^4*b^14*tan(1/2*d*x + 1/2*c)^7 + 68608*a^2*b^16*tan(1/2*d*x + 1/2*c)^7 + 15360*b^1
8*tan(1/2*d*x + 1/2*c)^7 - 78400*a^17*b*tan(1/2*d*x + 1/2*c)^6 - 1607200*a^15*b^3*tan(1/2*d*x + 1/2*c)^6 - 232
6800*a^13*b^5*tan(1/2*d*x + 1/2*c)^6 - 823060*a^11*b^7*tan(1/2*d*x + 1/2*c)^6 - 2094400*a^9*b^9*tan(1/2*d*x +
1/2*c)^6 + 1351728*a^7*b^11*tan(1/2*d*x + 1/2*c)^6 - 298816*a^5*b^13*tan(1/2*d*x + 1/2*c)^6 - 122752*a^3*b^15*
tan(1/2*d*x + 1/2*c)^6 + 53760*a*b^17*tan(1/2*d*x + 1/2*c)^6 - 4200*a^18*tan(1/2*d*x + 1/2*c)^5 - 459900*a^16*
b^2*tan(1/2*d*x + 1/2*c)^5 - 2100175*a^14*b^4*tan(1/2*d*x + 1/2*c)^5 - 647780*a^12*b^6*tan(1/2*d*x + 1/2*c)^5
- 1643880*a^10*b^8*tan(1/2*d*x + 1/2*c)^5 + 228592*a^8*b^10*tan(1/2*d*x + 1/2*c)^5 + 439376*a^6*b^12*tan(1/2*d
*x + 1/2*c)^5 - 352128*a^4*b^14*tan(1/2*d*x + 1/2*c)^5 + 80640*a^2*b^16*tan(1/2*d*x + 1/2*c)^5 - 63000*a^17*b*
tan(1/2*d*x + 1/2*c)^4 - 918540*a^15*b^3*tan(1/2*d*x + 1/2*c)^4 - 858683*a^13*b^5*tan(1/2*d*x + 1/2*c)^4 - 434
644*a^11*b^7*tan(1/2*d*x + 1/2*c)^4 - 634368*a^9*b^9*tan(1/2*d*x + 1/2*c)^4 + 719600*a^7*b^11*tan(1/2*d*x + 1/
2*c)^4 - 355040*a^5*b^13*tan(1/2*d*x + 1/2*c)^4 + 67200*a^3*b^15*tan(1/2*d*x + 1/2*c)^4 - 3360*a^18*tan(1/2*d*
x + 1/2*c)^3 - 211680*a^16*b^2*tan(1/2*d*x + 1/2*c)^3 - 575260*a^14*b^4*tan(1/2*d*x + 1/2*c)^3 + 43918*a^12*b^
6*tan(1/2*d*x + 1/2*c)^3 - 534576*a^10*b^8*tan(1/2*d*x + 1/2*c)^3 + 449008*a^8*b^10*tan(1/2*d*x + 1/2*c)^3 - 1
92640*a^6*b^12*tan(1/2*d*x + 1/2*c)^3 + 33600*a^4*b^14*tan(1/2*d*x + 1/2*c)^3 - 24640*a^17*b*tan(1/2*d*x + 1/2
*c)^2 - 199360*a^15*b^3*tan(1/2*d*x + 1/2*c)^2 + 44604*a^13*b^5*tan(1/2*d*x + 1/2*c)^2 - 186410*a^11*b^7*tan(1
/2*d*x + 1/2*c)^2 + 144928*a^9*b^9*tan(1/2*d*x + 1/2*c)^2 - 59472*a^7*b^11*tan(1/2*d*x + 1/2*c)^2 + 10080*a^5*
b^13*tan(1/2*d*x + 1/2*c)^2 - 840*a^18*tan(1/2*d*x + 1/2*c) - 38780*a^16*b^2*tan(1/2*d*x + 1/2*c) + 12565*a^14
*b^4*tan(1/2*d*x + 1/2*c) - 35322*a^12*b^6*tan(1/2*d*x + 1/2*c) + 25844*a^10*b^8*tan(1/2*d*x + 1/2*c) - 10192*
a^8*b^10*tan(1/2*d*x + 1/2*c) + 1680*a^6*b^12*tan(1/2*d*x + 1/2*c) - 3640*a^17*b + 2660*a^15*b^3 - 4923*a^13*b
^5 + 3646*a^11*b^7 - 1448*a^9*b^9 + 240*a^7*b^11)/((a^19 - 6*a^17*b^2 + 15*a^15*b^4 - 20*a^13*b^6 + 15*a^11*b^
8 - 6*a^9*b^10 + a^7*b^12)*(a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)^7))/d