\(\int \frac {\cos ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx\) [124]

Optimal result

Integrand size = 28, antiderivative size = 82 \[ \int \frac {\cos ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx=\frac {\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}+\frac {2 a b \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {b}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))} \]

[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))

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3165, 3564, 3612, 3611} \[ \int \frac {\cos ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx=-\frac {b}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {2 a b \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^2}+\frac {x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^2} \]

[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 3165

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 3564

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 3611

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c/(b*f))
*Log[RemoveContent[a*Cos[e + f*x] + b*Sin[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] && EqQ[a*c + b*d, 0]

Rule 3612

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*} \text {integral}& = \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*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.88 (sec) , antiderivative size = 192, normalized size of antiderivative = 2.34 \[ \int \frac {\cos ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx=\frac {a^2 \cos (c+d x) \left ((a+i b)^2 (c+d x)+a b \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right )\right )+b \left ((a+i b) \left (-i b^2+a b (1+i c+i d x)+a^2 (c+d x)\right )+a^2 b \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right )\right ) \sin (c+d x)-2 i a^2 b \arctan (\tan (c+d x)) (a \cos (c+d x)+b \sin (c+d x))}{a \left (a^2+b^2\right )^2 d (a \cos (c+d x)+b \sin (c+d x))} \]

[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]))

Maple [A] (verified)

Time = 0.59 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.18

method	result	size
derivativedivides	\(\frac {-\frac {b}{\left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}+\frac {2 a b \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {-a b \ln \left (1+\tan \left (d x +c \right )^{2}\right )+\left (a^{2}-b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(97\)
default	\(\frac {-\frac {b}{\left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}+\frac {2 a b \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {-a b \ln \left (1+\tan \left (d x +c \right )^{2}\right )+\left (a^{2}-b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(97\)
risch	\(-\frac {x}{2 i b a -a^{2}+b^{2}}-\frac {4 i a b x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {4 i a b c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 i b^{2}}{\left (-i a +b \right ) d \left (i a +b \right )^{2} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )}+\frac {2 a b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\)	\(182\)
parallelrisch	\(\frac {2 b a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )-2 b a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right ) \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+\left (a^{3} x d -a \,b^{2} d x -a^{2} b -b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 b d x \left (a -b \right ) \left (a +b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a^{3} x d +a \,b^{2} d x +a^{2} b +b^{3}}{\left (a^{2}+b^{2}\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right ) d}\)	\(236\)
norman	\(\frac {\frac {\left (a^{2}-b^{2}\right ) a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{\left (a^{2}+b^{2}\right )^{2}}+\frac {\left (a^{2}-b^{2}\right ) a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{\left (a^{2}+b^{2}\right )^{2}}-\frac {\left (a^{2}-b^{2}\right ) a x}{\left (a^{2}+b^{2}\right )^{2}}-\frac {2 b \left (a^{2}-b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {4 b \left (a^{2}-b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{\left (a^{2}+b^{2}\right )^{2}}-\frac {2 b \left (a^{2}-b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{\left (a^{2}+b^{2}\right )^{2}}-\frac {\left (a^{2}-b^{2}\right ) a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{\left (a^{2}+b^{2}\right )^{2}}-\frac {2 b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d \left (a^{2}+b^{2}\right )}-\frac {4 b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{a d \left (a^{2}+b^{2}\right )}-\frac {2 b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{a d \left (a^{2}+b^{2}\right )}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}-\frac {2 a b \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 a b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\)	\(449\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (82) = 164\).

Time = 0.27 (sec) , antiderivative size = 173, normalized size of antiderivative = 2.11 \[ \int \frac {\cos ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx=-\frac {{\left (b^{3} - {\left (a^{3} - a b^{2}\right )} d x\right )} \cos \left (d x + c\right ) - {\left (a^{2} b \cos \left (d x + c\right ) + a b^{2} \sin \left (d x + c\right )\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) - {\left (a b^{2} + {\left (a^{2} b - b^{3}\right )} d x\right )} \sin \left (d x + c\right )}{{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d \cos \left (d x + c\right ) + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d \sin \left (d x + c\right )} \]

[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))

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.20 (sec) , antiderivative size = 1545, normalized size of antiderivative = 18.84 \[ \int \frac {\cos ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]

[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*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)), (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*si
n(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))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.60 \[ \int \frac {\cos ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx=\frac {\frac {2 \, a b \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {a b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (a^{2} - b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {b}{a^{3} + a b^{2} + {\left (a^{2} b + b^{3}\right )} \tan \left (d x + c\right )}}{d} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.94 \[ \int \frac {\cos ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx=\frac {\frac {2 \, a b^{2} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{4} b + 2 \, a^{2} b^{3} + b^{5}} - \frac {a b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (a^{2} - b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, a b^{2} \tan \left (d x + c\right ) + 3 \, a^{2} b + b^{3}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}}}{d} \]

[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

Mupad [B] (verification not implemented)

Time = 27.38 (sec) , antiderivative size = 3114, normalized size of antiderivative = 37.98 \[ \int \frac {\cos ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]

[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))