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

Optimal. Leaf size=82 \[ -\frac {b}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {2 a b \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^2}+\frac {x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^2} \]

[Out]

(a^2-b^2)*x/(a^2+b^2)^2+2*a*b*ln(a*cos(d*x+c)+b*sin(d*x+c))/(a^2+b^2)^2/d-b/(a^2+b^2)/d/(a+b*tan(d*x+c))

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Rubi [A]  time = 0.14, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3086, 3483, 3531, 3530} \[ -\frac {b}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {2 a b \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^2}+\frac {x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a^2 - b^2)*x)/(a^2 + b^2)^2 + (2*a*b*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^2*d) - b/((a^2 + b^2
)*d*(a + b*Tan[c + d*x]))

Rule 3086

Int[cos[(c_.) + (d_.)*(x_)]^(m_)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_.), x_Symb
ol] :> Int[(a + b*Tan[c + d*x])^n, x] /; FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && IntegerQ[n] && NeQ[a^2 + b
^2, 0]

Rule 3483

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a + b*Tan[c + d*x])^(n + 1))/(d*(n + 1)
*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a - b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ
[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]

Rule 3530

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c*Log[Re
moveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]])/(b*f), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d,
0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]

Rule 3531

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((a*c +
b*d)*x)/(a^2 + b^2), x] + Dist[(b*c - a*d)/(a^2 + b^2), Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[a*c + b*d, 0]

Rubi steps

\begin {align*} \int \frac {\cos ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx &=\int \frac {1}{(a+b \tan (c+d x))^2} \, dx\\ &=-\frac {b}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\int \frac {a-b \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^2+b^2}\\ &=\frac {\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}-\frac {b}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {(2 a b) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{\left (a^2+b^2\right )^2}\\ &=\frac {\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}+\frac {2 a b \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {b}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}\\ \end {align*}

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Mathematica [C]  time = 0.40, size = 192, normalized size = 2.34 \[ \frac {b \sin (c+d x) \left (a^2 b \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right )+(a+i b) \left (a^2 (c+d x)+a b (i c+i d x+1)-i b^2\right )\right )+a^2 \cos (c+d x) \left (a b \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right )+(a+i b)^2 (c+d x)\right )-2 i a^2 b \tan ^{-1}(\tan (c+d x)) (a \cos (c+d x)+b \sin (c+d x))}{a d \left (a^2+b^2\right )^2 (a \cos (c+d x)+b \sin (c+d x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*Cos[c + d*x]*((a + I*b)^2*(c + d*x) + a*b*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2]) + b*((a + I*b)*((-I)*
b^2 + a*b*(1 + I*c + I*d*x) + a^2*(c + d*x)) + a^2*b*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2])*Sin[c + d*x] -
(2*I)*a^2*b*ArcTan[Tan[c + d*x]]*(a*Cos[c + d*x] + b*Sin[c + d*x]))/(a*(a^2 + b^2)^2*d*(a*Cos[c + d*x] + b*Sin
[c + d*x]))

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fricas [B]  time = 0.54, size = 173, normalized size = 2.11 \[ -\frac {{\left (b^{3} - {\left (a^{3} - a b^{2}\right )} d x\right )} \cos \left (d x + c\right ) - {\left (a^{2} b \cos \left (d x + c\right ) + a b^{2} \sin \left (d x + c\right )\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) - {\left (a b^{2} + {\left (a^{2} b - b^{3}\right )} d x\right )} \sin \left (d x + c\right )}{{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d \cos \left (d x + c\right ) + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d \sin \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((b^3 - (a^3 - a*b^2)*d*x)*cos(d*x + c) - (a^2*b*cos(d*x + c) + a*b^2*sin(d*x + c))*log(2*a*b*cos(d*x + c)*si
n(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 + b^2) - (a*b^2 + (a^2*b - b^3)*d*x)*sin(d*x + c))/((a^5 + 2*a^3*b^2 +
a*b^4)*d*cos(d*x + c) + (a^4*b + 2*a^2*b^3 + b^5)*d*sin(d*x + c))

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giac [A]  time = 1.96, size = 159, normalized size = 1.94 \[ \frac {\frac {2 \, a b^{2} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{4} b + 2 \, a^{2} b^{3} + b^{5}} - \frac {a b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (a^{2} - b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, a b^{2} \tan \left (d x + c\right ) + 3 \, a^{2} b + b^{3}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*a*b^2*log(abs(b*tan(d*x + c) + a))/(a^4*b + 2*a^2*b^3 + b^5) - a*b*log(tan(d*x + c)^2 + 1)/(a^4 + 2*a^2*b^2
+ b^4) + (a^2 - b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) - (2*a*b^2*tan(d*x + c) + 3*a^2*b + b^3)/((a^4 + 2*a^2
*b^2 + b^4)*(b*tan(d*x + c) + a)))/d

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maple [A]  time = 0.21, size = 130, normalized size = 1.59 \[ -\frac {b}{\left (a^{2}+b^{2}\right ) d \left (a +b \tan \left (d x +c \right )\right )}+\frac {2 b a \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )^{2}}-\frac {a b \ln \left (\tan ^{2}\left (d x +c \right )+1\right )}{d \left (a^{2}+b^{2}\right )^{2}}+\frac {\arctan \left (\tan \left (d x +c \right )\right ) a^{2}}{d \left (a^{2}+b^{2}\right )^{2}}-\frac {\arctan \left (\tan \left (d x +c \right )\right ) b^{2}}{d \left (a^{2}+b^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b/(a^2+b^2)/d/(a+b*tan(d*x+c))+2/d*b*a/(a^2+b^2)^2*ln(a+b*tan(d*x+c))-1/d/(a^2+b^2)^2*a*b*ln(tan(d*x+c)^2+1)+
1/d/(a^2+b^2)^2*arctan(tan(d*x+c))*a^2-1/d/(a^2+b^2)^2*arctan(tan(d*x+c))*b^2

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maxima [A]  time = 0.43, size = 131, normalized size = 1.60 \[ \frac {\frac {2 \, a b \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {a b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (a^{2} - b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {b}{a^{3} + a b^{2} + {\left (a^{2} b + b^{3}\right )} \tan \left (d x + c\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*a*b*log(b*tan(d*x + c) + a)/(a^4 + 2*a^2*b^2 + b^4) - a*b*log(tan(d*x + c)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) +
(a^2 - b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) - b/(a^3 + a*b^2 + (a^2*b + b^3)*tan(d*x + c)))/d

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mupad [B]  time = 4.87, size = 3114, normalized size = 37.98 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*a*b*log(a + 2*b*tan(c/2 + (d*x)/2) - a*tan(c/2 + (d*x)/2)^2))/(d*(a^4 + b^4 + 2*a^2*b^2)) - (2*a*b*log(1/(c
os(c + d*x) + 1)))/(d*(a^4 + b^4 + 2*a^2*b^2)) + (2*atan((tan(c/2 + (d*x)/2)*((((2*a*b*((((32*(6*a^8*b + 6*a^4
*b^5 + 12*a^6*b^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (64*a*b*(3*a*b^10 + 12*a^3*b^8 + 18*a^5*b^6 + 12*a^7
*b^4 + 3*a^9*b^2))/((a^4 + b^4 + 2*a^2*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))*(a + b)*(a - b))/(a^4 + b^4
+ 2*a^2*b^2) - (64*a*b*(a + b)*(a - b)*(3*a*b^10 + 12*a^3*b^8 + 18*a^5*b^6 + 12*a^7*b^4 + 3*a^9*b^2))/((a^4 +
b^4 + 2*a^2*b^2)^2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))))/(a^4 + b^4 + 2*a^2*b^2) - ((a + b)*((32*(2*a*b^6 + a
^7 - 7*a^3*b^4 - 8*a^5*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (2*a*b*((32*(6*a^8*b + 6*a^4*b^5 + 12*a^6*b
^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (64*a*b*(3*a*b^10 + 12*a^3*b^8 + 18*a^5*b^6 + 12*a^7*b^4 + 3*a^9*b^
2))/((a^4 + b^4 + 2*a^2*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))))/(a^4 + b^4 + 2*a^2*b^2))*(a - b))/(a^4 + b
^4 + 2*a^2*b^2) + (32*(a + b)^3*(a - b)^3*(3*a*b^10 + 12*a^3*b^8 + 18*a^5*b^6 + 12*a^7*b^4 + 3*a^9*b^2))/((a^4
+ b^4 + 2*a^2*b^2)^3*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))*(a^6 - b^6 + 35*a^2*b^4 - 35*a^4*b^2))/(a^6 + b^6
+ 15*a^2*b^4 + 15*a^4*b^2)^2 - (2*a*b*(5*a^4 + 5*b^4 - 26*a^2*b^2)*((32*(2*a^4*b + 4*a^2*b^3))/(a^6 + b^6 + 3*
a^2*b^4 + 3*a^4*b^2) + ((a + b)*(a - b)*((((32*(6*a^8*b + 6*a^4*b^5 + 12*a^6*b^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*
a^4*b^2) - (64*a*b*(3*a*b^10 + 12*a^3*b^8 + 18*a^5*b^6 + 12*a^7*b^4 + 3*a^9*b^2))/((a^4 + b^4 + 2*a^2*b^2)*(a^
6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))*(a + b)*(a - b))/(a^4 + b^4 + 2*a^2*b^2) - (64*a*b*(a + b)*(a - b)*(3*a*b^1
0 + 12*a^3*b^8 + 18*a^5*b^6 + 12*a^7*b^4 + 3*a^9*b^2))/((a^4 + b^4 + 2*a^2*b^2)^2*(a^6 + b^6 + 3*a^2*b^4 + 3*a
^4*b^2))))/(a^4 + b^4 + 2*a^2*b^2) + (2*a*b*((32*(2*a*b^6 + a^7 - 7*a^3*b^4 - 8*a^5*b^2))/(a^6 + b^6 + 3*a^2*b
^4 + 3*a^4*b^2) - (2*a*b*((32*(6*a^8*b + 6*a^4*b^5 + 12*a^6*b^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (64*a*
b*(3*a*b^10 + 12*a^3*b^8 + 18*a^5*b^6 + 12*a^7*b^4 + 3*a^9*b^2))/((a^4 + b^4 + 2*a^2*b^2)*(a^6 + b^6 + 3*a^2*b
^4 + 3*a^4*b^2))))/(a^4 + b^4 + 2*a^2*b^2)))/(a^4 + b^4 + 2*a^2*b^2) - (64*a*b*(a + b)^2*(a - b)^2*(3*a*b^10 +
12*a^3*b^8 + 18*a^5*b^6 + 12*a^7*b^4 + 3*a^9*b^2))/((a^4 + b^4 + 2*a^2*b^2)^3*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*
b^2))))/(a^6 + b^6 + 15*a^2*b^4 + 15*a^4*b^2)^2)*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^
2))/(32*a*b^2 - 32*a^3) + ((((a + b)*(a - b)*((32*(3*a^6*b + 3*a^2*b^5 + 6*a^4*b^3))/(a^6 + b^6 + 3*a^2*b^4 +
3*a^4*b^2) - (2*a*b*((32*(2*a^3*b^6 - a^9 - a*b^8 + 6*a^5*b^4 + 2*a^7*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2
) + (64*a*b*(3*a^10*b + 3*a^2*b^9 + 12*a^4*b^7 + 18*a^6*b^5 + 12*a^8*b^3))/((a^4 + b^4 + 2*a^2*b^2)*(a^6 + b^6
+ 3*a^2*b^4 + 3*a^4*b^2))))/(a^4 + b^4 + 2*a^2*b^2)))/(a^4 + b^4 + 2*a^2*b^2) - (2*a*b*((((32*(2*a^3*b^6 - a^
9 - a*b^8 + 6*a^5*b^4 + 2*a^7*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (64*a*b*(3*a^10*b + 3*a^2*b^9 + 12*a
^4*b^7 + 18*a^6*b^5 + 12*a^8*b^3))/((a^4 + b^4 + 2*a^2*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))*(a + b)*(a -
b))/(a^4 + b^4 + 2*a^2*b^2) + (64*a*b*(a + b)*(a - b)*(3*a^10*b + 3*a^2*b^9 + 12*a^4*b^7 + 18*a^6*b^5 + 12*a^
8*b^3))/((a^4 + b^4 + 2*a^2*b^2)^2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))))/(a^4 + b^4 + 2*a^2*b^2) + (32*(a + b
)^3*(a - b)^3*(3*a^10*b + 3*a^2*b^9 + 12*a^4*b^7 + 18*a^6*b^5 + 12*a^8*b^3))/((a^4 + b^4 + 2*a^2*b^2)^3*(a^6 +
b^6 + 3*a^2*b^4 + 3*a^4*b^2)))*(a^6 - b^6 + 35*a^2*b^4 - 35*a^4*b^2)*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 +
10*a^6*b^4 + 5*a^8*b^2))/((32*a*b^2 - 32*a^3)*(a^6 + b^6 + 15*a^2*b^4 + 15*a^4*b^2)^2) + (2*a*b*(5*a^4 + 5*b^4
- 26*a^2*b^2)*((64*a^3*b^2)/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (2*a*b*((32*(3*a^6*b + 3*a^2*b^5 + 6*a^4*b^
3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (2*a*b*((32*(2*a^3*b^6 - a^9 - a*b^8 + 6*a^5*b^4 + 2*a^7*b^2))/(a^6
+ b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (64*a*b*(3*a^10*b + 3*a^2*b^9 + 12*a^4*b^7 + 18*a^6*b^5 + 12*a^8*b^3))/((a^4
+ b^4 + 2*a^2*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))))/(a^4 + b^4 + 2*a^2*b^2)))/(a^4 + b^4 + 2*a^2*b^2) +
((a + b)*(a - b)*((((32*(2*a^3*b^6 - a^9 - a*b^8 + 6*a^5*b^4 + 2*a^7*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)
+ (64*a*b*(3*a^10*b + 3*a^2*b^9 + 12*a^4*b^7 + 18*a^6*b^5 + 12*a^8*b^3))/((a^4 + b^4 + 2*a^2*b^2)*(a^6 + b^6
+ 3*a^2*b^4 + 3*a^4*b^2)))*(a + b)*(a - b))/(a^4 + b^4 + 2*a^2*b^2) + (64*a*b*(a + b)*(a - b)*(3*a^10*b + 3*a^
2*b^9 + 12*a^4*b^7 + 18*a^6*b^5 + 12*a^8*b^3))/((a^4 + b^4 + 2*a^2*b^2)^2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))
))/(a^4 + b^4 + 2*a^2*b^2) + (64*a*b*(a + b)^2*(a - b)^2*(3*a^10*b + 3*a^2*b^9 + 12*a^4*b^7 + 18*a^6*b^5 + 12*
a^8*b^3))/((a^4 + b^4 + 2*a^2*b^2)^3*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b
^6 + 10*a^6*b^4 + 5*a^8*b^2))/((32*a*b^2 - 32*a^3)*(a^6 + b^6 + 15*a^2*b^4 + 15*a^4*b^2)^2))*(a + b)*(a - b))/
(d*(a^4 + b^4 + 2*a^2*b^2)) + (2*b^2*tan(c/2 + (d*x)/2))/(a*d*(a^2 + b^2)*(a + 2*b*tan(c/2 + (d*x)/2) - a*tan(
c/2 + (d*x)/2)^2))

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sympy [A]  time = 6.38, size = 1552, normalized size = 18.93 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x*cos(c)**2/sin(c)**2, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), ((-x - cos(c + d*x)/(d*sin(c + d*x)))/b
**2, Eq(a, 0)), (2*d*x*sin(c + d*x)**2/(-8*b**2*d*sin(c + d*x)**2 + 16*I*b**2*d*sin(c + d*x)*cos(c + d*x) + 8*
b**2*d*cos(c + d*x)**2) - 4*I*d*x*sin(c + d*x)*cos(c + d*x)/(-8*b**2*d*sin(c + d*x)**2 + 16*I*b**2*d*sin(c + d
*x)*cos(c + d*x) + 8*b**2*d*cos(c + d*x)**2) - 2*d*x*cos(c + d*x)**2/(-8*b**2*d*sin(c + d*x)**2 + 16*I*b**2*d*
sin(c + d*x)*cos(c + d*x) + 8*b**2*d*cos(c + d*x)**2) - I*sin(c + d*x)**2/(-8*b**2*d*sin(c + d*x)**2 + 16*I*b*
*2*d*sin(c + d*x)*cos(c + d*x) + 8*b**2*d*cos(c + d*x)**2) - 3*I*cos(c + d*x)**2/(-8*b**2*d*sin(c + d*x)**2 +
16*I*b**2*d*sin(c + d*x)*cos(c + d*x) + 8*b**2*d*cos(c + d*x)**2), Eq(a, -I*b)), (2*I*d*x*sin(c + d*x)**2/(-8*
I*b**2*d*sin(c + d*x)**2 + 16*b**2*d*sin(c + d*x)*cos(c + d*x) + 8*I*b**2*d*cos(c + d*x)**2) - 4*d*x*sin(c + d
*x)*cos(c + d*x)/(-8*I*b**2*d*sin(c + d*x)**2 + 16*b**2*d*sin(c + d*x)*cos(c + d*x) + 8*I*b**2*d*cos(c + d*x)*
*2) - 2*I*d*x*cos(c + d*x)**2/(-8*I*b**2*d*sin(c + d*x)**2 + 16*b**2*d*sin(c + d*x)*cos(c + d*x) + 8*I*b**2*d*
cos(c + d*x)**2) - sin(c + d*x)**2/(-8*I*b**2*d*sin(c + d*x)**2 + 16*b**2*d*sin(c + d*x)*cos(c + d*x) + 8*I*b*
*2*d*cos(c + d*x)**2) - 3*cos(c + d*x)**2/(-8*I*b**2*d*sin(c + d*x)**2 + 16*b**2*d*sin(c + d*x)*cos(c + d*x) +
8*I*b**2*d*cos(c + d*x)**2), Eq(a, I*b)), (x*cos(c)**2/(a*cos(c) + b*sin(c))**2, Eq(d, 0)), (a**3*d*x*cos(c +
d*x)/(a**5*d*cos(c + d*x) + a**4*b*d*sin(c + d*x) + 2*a**3*b**2*d*cos(c + d*x) + 2*a**2*b**3*d*sin(c + d*x) +
a*b**4*d*cos(c + d*x) + b**5*d*sin(c + d*x)) + a**2*b*d*x*sin(c + d*x)/(a**5*d*cos(c + d*x) + a**4*b*d*sin(c
+ d*x) + 2*a**3*b**2*d*cos(c + d*x) + 2*a**2*b**3*d*sin(c + d*x) + a*b**4*d*cos(c + d*x) + b**5*d*sin(c + d*x)
) + 2*a**2*b*log(cos(c + d*x) + b*sin(c + d*x)/a)*cos(c + d*x)/(a**5*d*cos(c + d*x) + a**4*b*d*sin(c + d*x) +
2*a**3*b**2*d*cos(c + d*x) + 2*a**2*b**3*d*sin(c + d*x) + a*b**4*d*cos(c + d*x) + b**5*d*sin(c + d*x)) - a**2*
b*cos(c + d*x)/(a**5*d*cos(c + d*x) + a**4*b*d*sin(c + d*x) + 2*a**3*b**2*d*cos(c + d*x) + 2*a**2*b**3*d*sin(c
+ d*x) + a*b**4*d*cos(c + d*x) + b**5*d*sin(c + d*x)) - a*b**2*d*x*cos(c + d*x)/(a**5*d*cos(c + d*x) + a**4*b
*d*sin(c + d*x) + 2*a**3*b**2*d*cos(c + d*x) + 2*a**2*b**3*d*sin(c + d*x) + a*b**4*d*cos(c + d*x) + b**5*d*sin
(c + d*x)) + 2*a*b**2*log(cos(c + d*x) + b*sin(c + d*x)/a)*sin(c + d*x)/(a**5*d*cos(c + d*x) + a**4*b*d*sin(c
+ d*x) + 2*a**3*b**2*d*cos(c + d*x) + 2*a**2*b**3*d*sin(c + d*x) + a*b**4*d*cos(c + d*x) + b**5*d*sin(c + d*x)
) - b**3*d*x*sin(c + d*x)/(a**5*d*cos(c + d*x) + a**4*b*d*sin(c + d*x) + 2*a**3*b**2*d*cos(c + d*x) + 2*a**2*b
**3*d*sin(c + d*x) + a*b**4*d*cos(c + d*x) + b**5*d*sin(c + d*x)) - b**3*cos(c + d*x)/(a**5*d*cos(c + d*x) + a
**4*b*d*sin(c + d*x) + 2*a**3*b**2*d*cos(c + d*x) + 2*a**2*b**3*d*sin(c + d*x) + a*b**4*d*cos(c + d*x) + b**5*
d*sin(c + d*x)), True))

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