\(\int \cos ^2(\frac {a+b x}{c+d x}) \, dx\) [52]

Optimal result

Integrand size = 16, antiderivative size = 107 \[ \int \cos ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {(c+d x) \cos ^2\left (\frac {a+b x}{c+d x}\right )}{d}-\frac {(b c-a d) \operatorname {CosIntegral}\left (\frac {2 (b c-a d)}{d (c+d x)}\right ) \sin \left (\frac {2 b}{d}\right )}{d^2}+\frac {(b c-a d) \cos \left (\frac {2 b}{d}\right ) \text {Si}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )}{d^2} \]

[Out]

(d*x+c)*cos((b*x+a)/(d*x+c))^2/d+(-a*d+b*c)*cos(2*b/d)*Si(2*(-a*d+b*c)/d/(d*x+c))/d^2-(-a*d+b*c)*Ci(2*(-a*d+b*
c)/d/(d*x+c))*sin(2*b/d)/d^2

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4660, 3394, 12, 3384, 3380, 3383} \[ \int \cos ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=-\frac {\sin \left (\frac {2 b}{d}\right ) (b c-a d) \operatorname {CosIntegral}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )}{d^2}+\frac {\cos \left (\frac {2 b}{d}\right ) (b c-a d) \text {Si}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \cos ^2\left (\frac {a+b x}{c+d x}\right )}{d} \]

[In]

Int[Cos[(a + b*x)/(c + d*x)]^2,x]

[Out]

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

Rule 12

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

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3394

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]^
n/(d*(m + 1))), x] - Dist[f*(n/(d*(m + 1))), Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]
^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] && LtQ[m, -1]

Rule 4660

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\cos ^2\left (\frac {b}{d}-\frac {(b c-a d) x}{d}\right )}{x^2} \, dx,x,\frac {1}{c+d x}\right )}{d} \\ & = \frac {(c+d x) \cos ^2\left (\frac {a+b x}{c+d x}\right )}{d}+\frac {(2 (b c-a d)) \text {Subst}\left (\int -\frac {\sin \left (\frac {2 b}{d}-\frac {2 (b c-a d) x}{d}\right )}{2 x} \, dx,x,\frac {1}{c+d x}\right )}{d^2} \\ & = \frac {(c+d x) \cos ^2\left (\frac {a+b x}{c+d x}\right )}{d}-\frac {(b c-a d) \text {Subst}\left (\int \frac {\sin \left (\frac {2 b}{d}-\frac {2 (b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2} \\ & = \frac {(c+d x) \cos ^2\left (\frac {a+b x}{c+d x}\right )}{d}+\frac {\left ((b c-a d) \cos \left (\frac {2 b}{d}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 (b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2}-\frac {\left ((b c-a d) \sin \left (\frac {2 b}{d}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 (b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2} \\ & = \frac {(c+d x) \cos ^2\left (\frac {a+b x}{c+d x}\right )}{d}-\frac {(b c-a d) \operatorname {CosIntegral}\left (\frac {2 (b c-a d)}{d (c+d x)}\right ) \sin \left (\frac {2 b}{d}\right )}{d^2}+\frac {(b c-a d) \cos \left (\frac {2 b}{d}\right ) \text {Si}\left (\frac {2 (b c-a d)}{d (c+d x)}\right )}{d^2} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.57 (sec) , antiderivative size = 497, normalized size of antiderivative = 4.64 \[ \int \cos ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {c d e^{-\frac {2 i (a+b x)}{c+d x}}+c d e^{\frac {2 i (a+b x)}{c+d x}}+2 d^2 x+2 d^2 x \cos \left (\frac {2 b}{d}\right ) \cos \left (\frac {2 (-b c+a d)}{d (c+d x)}\right )-2 i (b c-a d) \operatorname {CosIntegral}\left (\frac {2 (-b c+a d)}{d (c+d x)}\right ) \left (\cos \left (\frac {2 b}{d}\right )-i \sin \left (\frac {2 b}{d}\right )\right )+2 i (b c-a d) \operatorname {CosIntegral}\left (\frac {2 b c-2 a d}{c d+d^2 x}\right ) \left (\cos \left (\frac {2 b}{d}\right )+i \sin \left (\frac {2 b}{d}\right )\right )-2 d^2 x \sin \left (\frac {2 b}{d}\right ) \sin \left (\frac {2 (-b c+a d)}{d (c+d x)}\right )-2 b c \cos \left (\frac {2 b}{d}\right ) \text {Si}\left (\frac {2 (-b c+a d)}{d (c+d x)}\right )+2 a d \cos \left (\frac {2 b}{d}\right ) \text {Si}\left (\frac {2 (-b c+a d)}{d (c+d x)}\right )-2 i b c \sin \left (\frac {2 b}{d}\right ) \text {Si}\left (\frac {2 (-b c+a d)}{d (c+d x)}\right )+2 i a d \sin \left (\frac {2 b}{d}\right ) \text {Si}\left (\frac {2 (-b c+a d)}{d (c+d x)}\right )+2 b c \cos \left (\frac {2 b}{d}\right ) \text {Si}\left (\frac {2 b c-2 a d}{c d+d^2 x}\right )-2 a d \cos \left (\frac {2 b}{d}\right ) \text {Si}\left (\frac {2 b c-2 a d}{c d+d^2 x}\right )-2 i b c \sin \left (\frac {2 b}{d}\right ) \text {Si}\left (\frac {2 b c-2 a d}{c d+d^2 x}\right )+2 i a d \sin \left (\frac {2 b}{d}\right ) \text {Si}\left (\frac {2 b c-2 a d}{c d+d^2 x}\right )}{4 d^2} \]

[In]

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

[Out]

((c*d)/E^(((2*I)*(a + b*x))/(c + d*x)) + c*d*E^(((2*I)*(a + b*x))/(c + d*x)) + 2*d^2*x + 2*d^2*x*Cos[(2*b)/d]*
Cos[(2*(-(b*c) + a*d))/(d*(c + d*x))] - (2*I)*(b*c - a*d)*CosIntegral[(2*(-(b*c) + a*d))/(d*(c + d*x))]*(Cos[(
2*b)/d] - I*Sin[(2*b)/d]) + (2*I)*(b*c - a*d)*CosIntegral[(2*b*c - 2*a*d)/(c*d + d^2*x)]*(Cos[(2*b)/d] + I*Sin
[(2*b)/d]) - 2*d^2*x*Sin[(2*b)/d]*Sin[(2*(-(b*c) + a*d))/(d*(c + d*x))] - 2*b*c*Cos[(2*b)/d]*SinIntegral[(2*(-
(b*c) + a*d))/(d*(c + d*x))] + 2*a*d*Cos[(2*b)/d]*SinIntegral[(2*(-(b*c) + a*d))/(d*(c + d*x))] - (2*I)*b*c*Si
n[(2*b)/d]*SinIntegral[(2*(-(b*c) + a*d))/(d*(c + d*x))] + (2*I)*a*d*Sin[(2*b)/d]*SinIntegral[(2*(-(b*c) + a*d
))/(d*(c + d*x))] + 2*b*c*Cos[(2*b)/d]*SinIntegral[(2*b*c - 2*a*d)/(c*d + d^2*x)] - 2*a*d*Cos[(2*b)/d]*SinInte
gral[(2*b*c - 2*a*d)/(c*d + d^2*x)] - (2*I)*b*c*Sin[(2*b)/d]*SinIntegral[(2*b*c - 2*a*d)/(c*d + d^2*x)] + (2*I
)*a*d*Sin[(2*b)/d]*SinIntegral[(2*b*c - 2*a*d)/(c*d + d^2*x)])/(4*d^2)

Maple [A] (verified)

Time = 1.66 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.82

method	result	size
derivativedivides	\(-\frac {\left (a d -c b \right ) \left (\frac {d^{2} \left (-\frac {2 \cos \left (\frac {2 a d -2 c b}{d \left (d x +c \right )}+\frac {2 b}{d}\right )}{\left (\left (\frac {b}{d}+\frac {a d -c b}{d \left (d x +c \right )}\right ) d -b \right ) d}-\frac {2 \left (\frac {2 \,\operatorname {Si}\left (\frac {2 a d -2 c b}{d \left (d x +c \right )}\right ) \cos \left (\frac {2 b}{d}\right )}{d}+\frac {2 \,\operatorname {Ci}\left (\frac {2 a d -2 c b}{d \left (d x +c \right )}\right ) \sin \left (\frac {2 b}{d}\right )}{d}\right )}{d}\right )}{4}-\frac {d}{2 \left (\left (\frac {b}{d}+\frac {a d -c b}{d \left (d x +c \right )}\right ) d -b \right )}\right )}{d^{2}}\)	\(195\)
default	\(-\frac {\left (a d -c b \right ) \left (\frac {d^{2} \left (-\frac {2 \cos \left (\frac {2 a d -2 c b}{d \left (d x +c \right )}+\frac {2 b}{d}\right )}{\left (\left (\frac {b}{d}+\frac {a d -c b}{d \left (d x +c \right )}\right ) d -b \right ) d}-\frac {2 \left (\frac {2 \,\operatorname {Si}\left (\frac {2 a d -2 c b}{d \left (d x +c \right )}\right ) \cos \left (\frac {2 b}{d}\right )}{d}+\frac {2 \,\operatorname {Ci}\left (\frac {2 a d -2 c b}{d \left (d x +c \right )}\right ) \sin \left (\frac {2 b}{d}\right )}{d}\right )}{d}\right )}{4}-\frac {d}{2 \left (\left (\frac {b}{d}+\frac {a d -c b}{d \left (d x +c \right )}\right ) d -b \right )}\right )}{d^{2}}\)	\(195\)
risch	\(-\frac {{\mathrm e}^{-\frac {2 i b}{d}} \operatorname {csgn}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) \pi a}{2 d}+\frac {{\mathrm e}^{-\frac {2 i b}{d}} \operatorname {csgn}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) \pi b c}{2 d^{2}}+\frac {{\mathrm e}^{-\frac {2 i b}{d}} \operatorname {Si}\left (\frac {2 a d -2 c b}{d \left (d x +c \right )}\right ) a}{d}-\frac {{\mathrm e}^{-\frac {2 i b}{d}} \operatorname {Si}\left (\frac {2 a d -2 c b}{d \left (d x +c \right )}\right ) b c}{d^{2}}-\frac {i {\mathrm e}^{-\frac {2 i b}{d}} \operatorname {Ei}_{1}\left (-\frac {2 i \left (a d -c b \right )}{d \left (d x +c \right )}\right ) a}{2 d}+\frac {i {\mathrm e}^{-\frac {2 i b}{d}} \operatorname {Ei}_{1}\left (-\frac {2 i \left (a d -c b \right )}{d \left (d x +c \right )}\right ) b c}{2 d^{2}}+\frac {i \operatorname {Ei}_{1}\left (-\frac {2 i \left (a d -c b \right )}{d \left (d x +c \right )}\right ) {\mathrm e}^{\frac {2 i b}{d}} a}{2 d}-\frac {i \operatorname {Ei}_{1}\left (-\frac {2 i \left (a d -c b \right )}{d \left (d x +c \right )}\right ) {\mathrm e}^{\frac {2 i b}{d}} c b}{2 d^{2}}+\frac {x}{2}+\frac {\cos \left (\frac {2 x b +2 a}{d x +c}\right ) x}{2}+\frac {\cos \left (\frac {2 x b +2 a}{d x +c}\right ) c}{2 d}\)	\(339\)

[In]

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

[Out]

-1/d^2*(a*d-b*c)*(1/4*d^2*(-2*cos(2*(a*d-b*c)/d/(d*x+c)+2*b/d)/((b/d+(a*d-b*c)/d/(d*x+c))*d-b)/d-2*(2*Si(2*(a*
d-b*c)/d/(d*x+c))*cos(2*b/d)/d+2*Ci(2*(a*d-b*c)/d/(d*x+c))*sin(2*b/d)/d)/d)-1/2*d/((b/d+(a*d-b*c)/d/(d*x+c))*d
-b))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.02 \[ \int \cos ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {{\left (d^{2} x + c d\right )} \cos \left (\frac {b x + a}{d x + c}\right )^{2} - {\left (b c - a d\right )} \operatorname {Ci}\left (-\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) \sin \left (\frac {2 \, b}{d}\right ) - {\left (b c - a d\right )} \cos \left (\frac {2 \, b}{d}\right ) \operatorname {Si}\left (-\frac {2 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right )}{d^{2}} \]

[In]

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

[Out]

((d^2*x + c*d)*cos((b*x + a)/(d*x + c))^2 - (b*c - a*d)*cos_integral(-2*(b*c - a*d)/(d^2*x + c*d))*sin(2*b/d)
- (b*c - a*d)*cos(2*b/d)*sin_integral(-2*(b*c - a*d)/(d^2*x + c*d)))/d^2

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/2*x + 1/2*integrate(cos(2*(b*x + a)/(d*x + c)), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 683 vs. \(2 (107) = 214\).

Time = 26.79 (sec) , antiderivative size = 683, normalized size of antiderivative = 6.38 \[ \int \cos ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=-\frac {{\left (2 \, b^{3} c^{2} \operatorname {Ci}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) \sin \left (\frac {2 \, b}{d}\right ) - 4 \, a b^{2} c d \operatorname {Ci}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) \sin \left (\frac {2 \, b}{d}\right ) - \frac {2 \, {\left (b x + a\right )} b^{2} c^{2} d \operatorname {Ci}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) \sin \left (\frac {2 \, b}{d}\right )}{d x + c} + 2 \, a^{2} b d^{2} \operatorname {Ci}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) \sin \left (\frac {2 \, b}{d}\right ) + \frac {4 \, {\left (b x + a\right )} a b c d^{2} \operatorname {Ci}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) \sin \left (\frac {2 \, b}{d}\right )}{d x + c} - \frac {2 \, {\left (b x + a\right )} a^{2} d^{3} \operatorname {Ci}\left (-\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) \sin \left (\frac {2 \, b}{d}\right )}{d x + c} - 2 \, b^{3} c^{2} \cos \left (\frac {2 \, b}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) + 4 \, a b^{2} c d \cos \left (\frac {2 \, b}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) + \frac {2 \, {\left (b x + a\right )} b^{2} c^{2} d \cos \left (\frac {2 \, b}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right )}{d x + c} - 2 \, a^{2} b d^{2} \cos \left (\frac {2 \, b}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right ) - \frac {4 \, {\left (b x + a\right )} a b c d^{2} \cos \left (\frac {2 \, b}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right )}{d x + c} + \frac {2 \, {\left (b x + a\right )} a^{2} d^{3} \cos \left (\frac {2 \, b}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}}{d}\right )}{d x + c} - b^{2} c^{2} d \cos \left (\frac {2 \, {\left (b x + a\right )}}{d x + c}\right ) + 2 \, a b c d^{2} \cos \left (\frac {2 \, {\left (b x + a\right )}}{d x + c}\right ) - a^{2} d^{3} \cos \left (\frac {2 \, {\left (b x + a\right )}}{d x + c}\right ) - b^{2} c^{2} d + 2 \, a b c d^{2} - a^{2} d^{3}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}}{2 \, {\left (b d^{2} - \frac {{\left (b x + a\right )} d^{3}}{d x + c}\right )}} \]

[In]

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

[Out]

-1/2*(2*b^3*c^2*cos_integral(-2*(b - (b*x + a)*d/(d*x + c))/d)*sin(2*b/d) - 4*a*b^2*c*d*cos_integral(-2*(b - (
b*x + a)*d/(d*x + c))/d)*sin(2*b/d) - 2*(b*x + a)*b^2*c^2*d*cos_integral(-2*(b - (b*x + a)*d/(d*x + c))/d)*sin
(2*b/d)/(d*x + c) + 2*a^2*b*d^2*cos_integral(-2*(b - (b*x + a)*d/(d*x + c))/d)*sin(2*b/d) + 4*(b*x + a)*a*b*c*
d^2*cos_integral(-2*(b - (b*x + a)*d/(d*x + c))/d)*sin(2*b/d)/(d*x + c) - 2*(b*x + a)*a^2*d^3*cos_integral(-2*
(b - (b*x + a)*d/(d*x + c))/d)*sin(2*b/d)/(d*x + c) - 2*b^3*c^2*cos(2*b/d)*sin_integral(2*(b - (b*x + a)*d/(d*
x + c))/d) + 4*a*b^2*c*d*cos(2*b/d)*sin_integral(2*(b - (b*x + a)*d/(d*x + c))/d) + 2*(b*x + a)*b^2*c^2*d*cos(
2*b/d)*sin_integral(2*(b - (b*x + a)*d/(d*x + c))/d)/(d*x + c) - 2*a^2*b*d^2*cos(2*b/d)*sin_integral(2*(b - (b
*x + a)*d/(d*x + c))/d) - 4*(b*x + a)*a*b*c*d^2*cos(2*b/d)*sin_integral(2*(b - (b*x + a)*d/(d*x + c))/d)/(d*x
+ c) + 2*(b*x + a)*a^2*d^3*cos(2*b/d)*sin_integral(2*(b - (b*x + a)*d/(d*x + c))/d)/(d*x + c) - b^2*c^2*d*cos(
2*(b*x + a)/(d*x + c)) + 2*a*b*c*d^2*cos(2*(b*x + a)/(d*x + c)) - a^2*d^3*cos(2*(b*x + a)/(d*x + c)) - b^2*c^2
*d + 2*a*b*c*d^2 - a^2*d^3)*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)/(b*d^2 - (b*x + a)*d^3/(d*x + c))

Mupad [F(-1)]

Timed out. \[ \int \cos ^2\left (\frac {a+b x}{c+d x}\right ) \, dx=\int {\cos \left (\frac {a+b\,x}{c+d\,x}\right )}^2 \,d x \]

[In]

int(cos((a + b*x)/(c + d*x))^2,x)

[Out]

int(cos((a + b*x)/(c + d*x))^2, x)