\(\int \frac {\cos (c+d x) (A+C \cos ^2(c+d x))}{a+b \cos (c+d x)} \, dx\) [564]

Optimal result

Integrand size = 31, antiderivative size = 128 \[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\frac {\left (2 a^2 C+b^2 (2 A+C)\right ) x}{2 b^3}-\frac {2 a \left (A b^2+a^2 C\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^3 \sqrt {a+b} d}-\frac {a C \sin (c+d x)}{b^2 d}+\frac {C \cos (c+d x) \sin (c+d x)}{2 b d} \]

[Out]

1/2*(2*a^2*C+b^2*(2*A+C))*x/b^3-a*C*sin(d*x+c)/b^2/d+1/2*C*cos(d*x+c)*sin(d*x+c)/b/d-2*a*(A*b^2+C*a^2)*arctan(
(a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/b^3/d/(a-b)^(1/2)/(a+b)^(1/2)

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3129, 3102, 2814, 2738, 211} \[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=-\frac {2 a \left (a^2 C+A b^2\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^3 d \sqrt {a-b} \sqrt {a+b}}+\frac {x \left (\frac {2 a^2 C}{b^2}+2 A+C\right )}{2 b}-\frac {a C \sin (c+d x)}{b^2 d}+\frac {C \sin (c+d x) \cos (c+d x)}{2 b d} \]

[In]

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

[Out]

((2*A + C + (2*a^2*C)/b^2)*x)/(2*b) - (2*a*(A*b^2 + a^2*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])
/(Sqrt[a - b]*b^3*Sqrt[a + b]*d) - (a*C*Sin[c + d*x])/(b^2*d) + (C*Cos[c + d*x]*Sin[c + d*x])/(2*b*d)

Rule 211

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

Rule 2738

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[2*(e/d), Subst[Int[1/(a + b + (a - b)*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 2814

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 3102

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 3129

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)
*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^
(n + 1)/(d*f*(m + n + 2))), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])
^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e +
f*x] + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a,
0] && NeQ[c, 0])))

Rubi steps \begin{align*} \text {integral}& = \frac {C \cos (c+d x) \sin (c+d x)}{2 b d}+\frac {\int \frac {a C+b (2 A+C) \cos (c+d x)-2 a C \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{2 b} \\ & = -\frac {a C \sin (c+d x)}{b^2 d}+\frac {C \cos (c+d x) \sin (c+d x)}{2 b d}+\frac {\int \frac {a b C+\left (2 a^2 C+b^2 (2 A+C)\right ) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{2 b^2} \\ & = \frac {\left (2 a^2 C+b^2 (2 A+C)\right ) x}{2 b^3}-\frac {a C \sin (c+d x)}{b^2 d}+\frac {C \cos (c+d x) \sin (c+d x)}{2 b d}-\frac {\left (a \left (A b^2+a^2 C\right )\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{b^3} \\ & = \frac {\left (2 a^2 C+b^2 (2 A+C)\right ) x}{2 b^3}-\frac {a C \sin (c+d x)}{b^2 d}+\frac {C \cos (c+d x) \sin (c+d x)}{2 b d}-\frac {\left (2 a \left (A b^2+a^2 C\right )\right ) \text {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^3 d} \\ & = \frac {\left (2 a^2 C+b^2 (2 A+C)\right ) x}{2 b^3}-\frac {2 a \left (A b^2+a^2 C\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^3 \sqrt {a+b} d}-\frac {a C \sin (c+d x)}{b^2 d}+\frac {C \cos (c+d x) \sin (c+d x)}{2 b d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.13 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.91 \[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\frac {2 \left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) (c+d x)+\frac {8 a \left (A b^2+a^2 C\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}-4 a b C \sin (c+d x)+b^2 C \sin (2 (c+d x))}{4 b^3 d} \]

[In]

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

[Out]

(2*(2*A*b^2 + (2*a^2 + b^2)*C)*(c + d*x) + (8*a*(A*b^2 + a^2*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 +
b^2]])/Sqrt[-a^2 + b^2] - 4*a*b*C*Sin[c + d*x] + b^2*C*Sin[2*(c + d*x)])/(4*b^3*d)

Maple [A] (verified)

Time = 1.67 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.25

method	result	size
derivativedivides	\(\frac {\frac {\frac {2 \left (\left (-C a b -\frac {1}{2} b^{2} C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-C a b +\frac {1}{2} b^{2} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\left (2 A \,b^{2}+2 a^{2} C +b^{2} C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{3}}-\frac {2 a \left (A \,b^{2}+a^{2} C \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{3} \sqrt {\left (a -b \right ) \left (a +b \right )}}}{d}\)	\(160\)
default	\(\frac {\frac {\frac {2 \left (\left (-C a b -\frac {1}{2} b^{2} C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-C a b +\frac {1}{2} b^{2} C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\left (2 A \,b^{2}+2 a^{2} C +b^{2} C \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{3}}-\frac {2 a \left (A \,b^{2}+a^{2} C \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{3} \sqrt {\left (a -b \right ) \left (a +b \right )}}}{d}\)	\(160\)
risch	\(\frac {x A}{b}+\frac {x C \,a^{2}}{b^{3}}+\frac {C x}{2 b}+\frac {i a \,{\mathrm e}^{i \left (d x +c \right )} C}{2 b^{2} d}-\frac {i a \,{\mathrm e}^{-i \left (d x +c \right )} C}{2 b^{2} d}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+a \sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) A}{\sqrt {-a^{2}+b^{2}}\, d b}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+a \sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) C}{\sqrt {-a^{2}+b^{2}}\, d \,b^{3}}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right ) A}{\sqrt {-a^{2}+b^{2}}\, d b}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right ) C}{\sqrt {-a^{2}+b^{2}}\, d \,b^{3}}+\frac {\sin \left (2 d x +2 c \right ) C}{4 b d}\)	\(376\)

[In]

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

[Out]

1/d*(2/b^3*(((-C*a*b-1/2*b^2*C)*tan(1/2*d*x+1/2*c)^3+(-C*a*b+1/2*b^2*C)*tan(1/2*d*x+1/2*c))/(1+tan(1/2*d*x+1/2
*c)^2)^2+1/2*(2*A*b^2+2*C*a^2+C*b^2)*arctan(tan(1/2*d*x+1/2*c)))-2*a*(A*b^2+C*a^2)/b^3/((a-b)*(a+b))^(1/2)*arc
tan((a-b)*tan(1/2*d*x+1/2*c)/((a-b)*(a+b))^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 385, normalized size of antiderivative = 3.01 \[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\left [\frac {{\left (2 \, C a^{4} + {\left (2 \, A - C\right )} a^{2} b^{2} - {\left (2 \, A + C\right )} b^{4}\right )} d x - {\left (C a^{3} + A a b^{2}\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right ) - {\left (2 \, C a^{3} b - 2 \, C a b^{3} - {\left (C a^{2} b^{2} - C b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left (a^{2} b^{3} - b^{5}\right )} d}, \frac {{\left (2 \, C a^{4} + {\left (2 \, A - C\right )} a^{2} b^{2} - {\left (2 \, A + C\right )} b^{4}\right )} d x - 2 \, {\left (C a^{3} + A a b^{2}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cos \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (d x + c\right )}\right ) - {\left (2 \, C a^{3} b - 2 \, C a b^{3} - {\left (C a^{2} b^{2} - C b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left (a^{2} b^{3} - b^{5}\right )} d}\right ] \]

[In]

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

[Out]

[1/2*((2*C*a^4 + (2*A - C)*a^2*b^2 - (2*A + C)*b^4)*d*x - (C*a^3 + A*a*b^2)*sqrt(-a^2 + b^2)*log((2*a*b*cos(d*
x + c) + (2*a^2 - b^2)*cos(d*x + c)^2 - 2*sqrt(-a^2 + b^2)*(a*cos(d*x + c) + b)*sin(d*x + c) - a^2 + 2*b^2)/(b
^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + a^2)) - (2*C*a^3*b - 2*C*a*b^3 - (C*a^2*b^2 - C*b^4)*cos(d*x + c))*si
n(d*x + c))/((a^2*b^3 - b^5)*d), 1/2*((2*C*a^4 + (2*A - C)*a^2*b^2 - (2*A + C)*b^4)*d*x - 2*(C*a^3 + A*a*b^2)*
sqrt(a^2 - b^2)*arctan(-(a*cos(d*x + c) + b)/(sqrt(a^2 - b^2)*sin(d*x + c))) - (2*C*a^3*b - 2*C*a*b^3 - (C*a^2
*b^2 - C*b^4)*cos(d*x + c))*sin(d*x + c))/((a^2*b^3 - b^5)*d)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 9489 vs. \(2 (112) = 224\).

Time = 164.05 (sec) , antiderivative size = 9489, normalized size of antiderivative = 74.13 \[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((zoo*x*(A + C*cos(c)**2), Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), (2*A*d*x*tan(c/2 + d*x/2)**4/(2*b*d*tan(c
/2 + d*x/2)**4 + 4*b*d*tan(c/2 + d*x/2)**2 + 2*b*d) + 4*A*d*x*tan(c/2 + d*x/2)**2/(2*b*d*tan(c/2 + d*x/2)**4 +
4*b*d*tan(c/2 + d*x/2)**2 + 2*b*d) + 2*A*d*x/(2*b*d*tan(c/2 + d*x/2)**4 + 4*b*d*tan(c/2 + d*x/2)**2 + 2*b*d)
- 2*A*tan(c/2 + d*x/2)**5/(2*b*d*tan(c/2 + d*x/2)**4 + 4*b*d*tan(c/2 + d*x/2)**2 + 2*b*d) - 4*A*tan(c/2 + d*x/
2)**3/(2*b*d*tan(c/2 + d*x/2)**4 + 4*b*d*tan(c/2 + d*x/2)**2 + 2*b*d) - 2*A*tan(c/2 + d*x/2)/(2*b*d*tan(c/2 +
d*x/2)**4 + 4*b*d*tan(c/2 + d*x/2)**2 + 2*b*d) + 3*C*d*x*tan(c/2 + d*x/2)**4/(2*b*d*tan(c/2 + d*x/2)**4 + 4*b*
d*tan(c/2 + d*x/2)**2 + 2*b*d) + 6*C*d*x*tan(c/2 + d*x/2)**2/(2*b*d*tan(c/2 + d*x/2)**4 + 4*b*d*tan(c/2 + d*x/
2)**2 + 2*b*d) + 3*C*d*x/(2*b*d*tan(c/2 + d*x/2)**4 + 4*b*d*tan(c/2 + d*x/2)**2 + 2*b*d) - 2*C*tan(c/2 + d*x/2
)**5/(2*b*d*tan(c/2 + d*x/2)**4 + 4*b*d*tan(c/2 + d*x/2)**2 + 2*b*d) - 10*C*tan(c/2 + d*x/2)**3/(2*b*d*tan(c/2
+ d*x/2)**4 + 4*b*d*tan(c/2 + d*x/2)**2 + 2*b*d) - 4*C*tan(c/2 + d*x/2)/(2*b*d*tan(c/2 + d*x/2)**4 + 4*b*d*ta
n(c/2 + d*x/2)**2 + 2*b*d), Eq(a, b)), (2*A*d*x*tan(c/2 + d*x/2)**5/(2*b*d*tan(c/2 + d*x/2)**5 + 4*b*d*tan(c/2
+ d*x/2)**3 + 2*b*d*tan(c/2 + d*x/2)) + 4*A*d*x*tan(c/2 + d*x/2)**3/(2*b*d*tan(c/2 + d*x/2)**5 + 4*b*d*tan(c/
2 + d*x/2)**3 + 2*b*d*tan(c/2 + d*x/2)) + 2*A*d*x*tan(c/2 + d*x/2)/(2*b*d*tan(c/2 + d*x/2)**5 + 4*b*d*tan(c/2
+ d*x/2)**3 + 2*b*d*tan(c/2 + d*x/2)) + 2*A*tan(c/2 + d*x/2)**4/(2*b*d*tan(c/2 + d*x/2)**5 + 4*b*d*tan(c/2 + d
*x/2)**3 + 2*b*d*tan(c/2 + d*x/2)) + 4*A*tan(c/2 + d*x/2)**2/(2*b*d*tan(c/2 + d*x/2)**5 + 4*b*d*tan(c/2 + d*x/
2)**3 + 2*b*d*tan(c/2 + d*x/2)) + 2*A/(2*b*d*tan(c/2 + d*x/2)**5 + 4*b*d*tan(c/2 + d*x/2)**3 + 2*b*d*tan(c/2 +
d*x/2)) + 3*C*d*x*tan(c/2 + d*x/2)**5/(2*b*d*tan(c/2 + d*x/2)**5 + 4*b*d*tan(c/2 + d*x/2)**3 + 2*b*d*tan(c/2
+ d*x/2)) + 6*C*d*x*tan(c/2 + d*x/2)**3/(2*b*d*tan(c/2 + d*x/2)**5 + 4*b*d*tan(c/2 + d*x/2)**3 + 2*b*d*tan(c/2
+ d*x/2)) + 3*C*d*x*tan(c/2 + d*x/2)/(2*b*d*tan(c/2 + d*x/2)**5 + 4*b*d*tan(c/2 + d*x/2)**3 + 2*b*d*tan(c/2 +
d*x/2)) + 4*C*tan(c/2 + d*x/2)**4/(2*b*d*tan(c/2 + d*x/2)**5 + 4*b*d*tan(c/2 + d*x/2)**3 + 2*b*d*tan(c/2 + d*
x/2)) + 10*C*tan(c/2 + d*x/2)**2/(2*b*d*tan(c/2 + d*x/2)**5 + 4*b*d*tan(c/2 + d*x/2)**3 + 2*b*d*tan(c/2 + d*x/
2)) + 2*C/(2*b*d*tan(c/2 + d*x/2)**5 + 4*b*d*tan(c/2 + d*x/2)**3 + 2*b*d*tan(c/2 + d*x/2)), Eq(a, -b)), ((A*si
n(c + d*x)/d + 2*C*sin(c + d*x)**3/(3*d) + C*sin(c + d*x)*cos(c + d*x)**2/d)/a, Eq(b, 0)), (x*(A + C*cos(c)**2
)*cos(c)/(a + b*cos(c)), Eq(d, 0)), (2*A*a*b**2*d*x*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4/(2*a*b**3
*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 + 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)
**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b)) - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 - 4*
b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) + 4*A*a*b**2*
d*x*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2/(2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)
**4 + 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b)) -
2*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 - 4*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*
x/2)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) + 2*A*a*b**2*d*x*sqrt(-a/(a - b) - b/(a - b))/(2*a*b**3*d*sqr
t(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 + 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 +
2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b)) - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 - 4*b**4*d
*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) - 2*A*a*b**2*log(-s
qrt(-a/(a - b) - b/(a - b)) + tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**4/(2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*t
an(c/2 + d*x/2)**4 + 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + 2*a*b**3*d*sqrt(-a/(a - b)
- b/(a - b)) - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 - 4*b**4*d*sqrt(-a/(a - b) - b/(a - b
))*tan(c/2 + d*x/2)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) - 4*A*a*b**2*log(-sqrt(-a/(a - b) - b/(a - b))
+ tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**2/(2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 + 4*a*b*
*3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b)) - 2*b**4*d*sqr
t(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 - 4*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - 2*
b**4*d*sqrt(-a/(a - b) - b/(a - b))) - 2*A*a*b**2*log(-sqrt(-a/(a - b) - b/(a - b)) + tan(c/2 + d*x/2))/(2*a*b
**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 + 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x
/2)**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b)) - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 -
4*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) + 2*A*a*b*
*2*log(sqrt(-a/(a - b) - b/(a - b)) + tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**4/(2*a*b**3*d*sqrt(-a/(a - b) - b/(a
- b))*tan(c/2 + d*x/2)**4 + 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + 2*a*b**3*d*sqrt(-a/
(a - b) - b/(a - b)) - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 - 4*b**4*d*sqrt(-a/(a - b) -
b/(a - b))*tan(c/2 + d*x/2)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) + 4*A*a*b**2*log(sqrt(-a/(a - b) - b/(
a - b)) + tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**2/(2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 +
4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b)) - 2*b**
4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 - 4*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)*
*2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) + 2*A*a*b**2*log(sqrt(-a/(a - b) - b/(a - b)) + tan(c/2 + d*x/2))/
(2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 + 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2
+ d*x/2)**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b)) - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2
)**4 - 4*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) - 2*
A*b**3*d*x*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4/(2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 +
d*x/2)**4 + 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a -
b)) - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 - 4*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c
/2 + d*x/2)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) - 4*A*b**3*d*x*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 +
d*x/2)**2/(2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 + 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b
))*tan(c/2 + d*x/2)**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b)) - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c
/2 + d*x/2)**4 - 4*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a -
b))) - 2*A*b**3*d*x*sqrt(-a/(a - b) - b/(a - b))/(2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4
+ 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b)) - 2*b
**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 - 4*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2
)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) + 2*C*a**3*d*x*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4/
(2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 + 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2
+ d*x/2)**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b)) - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2
)**4 - 4*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) + 4*
C*a**3*d*x*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2/(2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 +
d*x/2)**4 + 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a -
b)) - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 - 4*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c
/2 + d*x/2)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) + 2*C*a**3*d*x*sqrt(-a/(a - b) - b/(a - b))/(2*a*b**3*
d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 + 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)*
*2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b)) - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 - 4*b
**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) - 2*C*a**3*log
(-sqrt(-a/(a - b) - b/(a - b)) + tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**4/(2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b)
)*tan(c/2 + d*x/2)**4 + 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + 2*a*b**3*d*sqrt(-a/(a -
b) - b/(a - b)) - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 - 4*b**4*d*sqrt(-a/(a - b) - b/(a
- b))*tan(c/2 + d*x/2)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) - 4*C*a**3*log(-sqrt(-a/(a - b) - b/(a - b)
) + tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**2/(2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 + 4*a*b
**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b)) - 2*b**4*d*sq
rt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 - 4*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - 2
*b**4*d*sqrt(-a/(a - b) - b/(a - b))) - 2*C*a**3*log(-sqrt(-a/(a - b) - b/(a - b)) + tan(c/2 + d*x/2))/(2*a*b*
*3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 + 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/
2)**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b)) - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 -
4*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) + 2*C*a**3*
log(sqrt(-a/(a - b) - b/(a - b)) + tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**4/(2*a*b**3*d*sqrt(-a/(a - b) - b/(a -
b))*tan(c/2 + d*x/2)**4 + 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + 2*a*b**3*d*sqrt(-a/(a
- b) - b/(a - b)) - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 - 4*b**4*d*sqrt(-a/(a - b) - b/(
a - b))*tan(c/2 + d*x/2)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) + 4*C*a**3*log(sqrt(-a/(a - b) - b/(a - b
)) + tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**2/(2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 + 4*a*
b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b)) - 2*b**4*d*s
qrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 - 4*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 -
2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) + 2*C*a**3*log(sqrt(-a/(a - b) - b/(a - b)) + tan(c/2 + d*x/2))/(2*a*b*
*3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 + 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/
2)**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b)) - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 -
4*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) - 2*C*a**2*
b*d*x*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4/(2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/
2)**4 + 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))
- 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 - 4*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 +
d*x/2)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) - 4*C*a**2*b*d*x*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x
/2)**2/(2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 + 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*
tan(c/2 + d*x/2)**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b)) - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2
+ d*x/2)**4 - 4*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b)
)) - 2*C*a**2*b*d*x*sqrt(-a/(a - b) - b/(a - b))/(2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4
+ 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b)) - 2*b*
*4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 - 4*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)
**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) - 4*C*a**2*b*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**3/(2*
a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 + 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 +
d*x/2)**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b)) - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**
4 - 4*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) - 4*C*a
**2*b*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)/(2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)*
*4 + 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b)) - 2
*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 - 4*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x
/2)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) + C*a*b**2*d*x*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**
4/(2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 + 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c
/2 + d*x/2)**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b)) - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x
/2)**4 - 4*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) +
2*C*a*b**2*d*x*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2/(2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c
/2 + d*x/2)**4 + 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/
(a - b)) - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 - 4*b**4*d*sqrt(-a/(a - b) - b/(a - b))*t
an(c/2 + d*x/2)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) + C*a*b**2*d*x*sqrt(-a/(a - b) - b/(a - b))/(2*a*b
**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 + 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x
/2)**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b)) - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 -
4*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) + 2*C*a*b*
*2*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**3/(2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)*
*4 + 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b)) - 2
*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 - 4*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x
/2)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) + 6*C*a*b**2*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)/(2*
a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 + 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 +
d*x/2)**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b)) - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**
4 - 4*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) - C*b**
3*d*x*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4/(2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/
2)**4 + 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))
- 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 - 4*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 +
d*x/2)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) - 2*C*b**3*d*x*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2
)**2/(2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 + 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*ta
n(c/2 + d*x/2)**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b)) - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 +
d*x/2)**4 - 4*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b)))
- C*b**3*d*x*sqrt(-a/(a - b) - b/(a - b))/(2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 + 4*a*
b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b)) - 2*b**4*d*s
qrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 - 4*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 -
2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) + 2*C*b**3*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**3/(2*a*b**3*d
*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 + 4*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**
2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b)) - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 - 4*b*
*4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - 2*b**4*d*sqrt(-a/(a - b) - b/(a - b))) - 2*C*b**3*sqrt
(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)/(2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 + 4*a*b
**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + 2*a*b**3*d*sqrt(-a/(a - b) - b/(a - b)) - 2*b**4*d*sq
rt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**4 - 4*b**4*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - 2
*b**4*d*sqrt(-a/(a - b) - b/(a - b))), True))

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more de

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.55 \[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\frac {\frac {{\left (2 \, C a^{2} + 2 \, A b^{2} + C b^{2}\right )} {\left (d x + c\right )}}{b^{3}} + \frac {4 \, {\left (C a^{3} + A a b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} b^{3}} - \frac {2 \, {\left (2 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} b^{2}}}{2 \, d} \]

[In]

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

[Out]

1/2*((2*C*a^2 + 2*A*b^2 + C*b^2)*(d*x + c)/b^3 + 4*(C*a^3 + A*a*b^2)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*
a + 2*b) + arctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*b^3) -
2*(2*C*a*tan(1/2*d*x + 1/2*c)^3 + C*b*tan(1/2*d*x + 1/2*c)^3 + 2*C*a*tan(1/2*d*x + 1/2*c) - C*b*tan(1/2*d*x +
1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^2*b^2))/d

Mupad [B] (verification not implemented)

Time = 5.17 (sec) , antiderivative size = 2398, normalized size of antiderivative = 18.73 \[ \int \frac {\cos (c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

- ((tan(c/2 + (d*x)/2)*(2*C*a - C*b))/b^2 + (tan(c/2 + (d*x)/2)^3*(2*C*a + C*b))/b^2)/(d*(2*tan(c/2 + (d*x)/2)
^2 + tan(c/2 + (d*x)/2)^4 + 1)) - (atan(((((8*tan(c/2 + (d*x)/2)*(4*A^2*b^7 - 8*C^2*a^7 + C^2*b^7 - 12*A^2*a*b
^6 - 3*C^2*a*b^6 + 16*C^2*a^6*b + 16*A^2*a^2*b^5 - 8*A^2*a^3*b^4 + 7*C^2*a^2*b^5 - 13*C^2*a^3*b^4 + 16*C^2*a^4
*b^3 - 16*C^2*a^5*b^2 + 4*A*C*b^7 - 12*A*C*a*b^6 + 20*A*C*a^2*b^5 - 28*A*C*a^3*b^4 + 32*A*C*a^4*b^3 - 16*A*C*a
^5*b^2))/b^4 + ((C*a^2*1i + b^2*(A*1i + (C*1i)/2))*((8*(4*A*b^10 + 2*C*b^10 + 4*A*a^2*b^8 + 2*C*a^2*b^8 - 6*C*
a^3*b^7 + 4*C*a^4*b^6 - 8*A*a*b^9 - 2*C*a*b^9))/b^6 - (8*tan(c/2 + (d*x)/2)*(C*a^2*1i + b^2*(A*1i + (C*1i)/2))
*(8*a*b^8 - 16*a^2*b^7 + 8*a^3*b^6))/b^7))/b^3)*(C*a^2*1i + b^2*(A*1i + (C*1i)/2))*1i)/b^3 + (((8*tan(c/2 + (d
*x)/2)*(4*A^2*b^7 - 8*C^2*a^7 + C^2*b^7 - 12*A^2*a*b^6 - 3*C^2*a*b^6 + 16*C^2*a^6*b + 16*A^2*a^2*b^5 - 8*A^2*a
^3*b^4 + 7*C^2*a^2*b^5 - 13*C^2*a^3*b^4 + 16*C^2*a^4*b^3 - 16*C^2*a^5*b^2 + 4*A*C*b^7 - 12*A*C*a*b^6 + 20*A*C*
a^2*b^5 - 28*A*C*a^3*b^4 + 32*A*C*a^4*b^3 - 16*A*C*a^5*b^2))/b^4 - ((C*a^2*1i + b^2*(A*1i + (C*1i)/2))*((8*(4*
A*b^10 + 2*C*b^10 + 4*A*a^2*b^8 + 2*C*a^2*b^8 - 6*C*a^3*b^7 + 4*C*a^4*b^6 - 8*A*a*b^9 - 2*C*a*b^9))/b^6 + (8*t
an(c/2 + (d*x)/2)*(C*a^2*1i + b^2*(A*1i + (C*1i)/2))*(8*a*b^8 - 16*a^2*b^7 + 8*a^3*b^6))/b^7))/b^3)*(C*a^2*1i
+ b^2*(A*1i + (C*1i)/2))*1i)/b^3)/((16*(4*C^3*a^8 - 4*A^3*a*b^7 - 6*C^3*a^7*b + 4*A^3*a^2*b^6 - C^3*a^3*b^5 +
2*C^3*a^4*b^4 - 5*C^3*a^5*b^3 + 6*C^3*a^6*b^2 - A*C^2*a*b^7 - 4*A^2*C*a*b^7 + 2*A*C^2*a^2*b^6 - 9*A*C^2*a^3*b^
5 + 12*A*C^2*a^4*b^4 - 16*A*C^2*a^5*b^3 + 12*A*C^2*a^6*b^2 + 6*A^2*C*a^2*b^6 - 14*A^2*C*a^3*b^5 + 12*A^2*C*a^4
*b^4))/b^6 + (((8*tan(c/2 + (d*x)/2)*(4*A^2*b^7 - 8*C^2*a^7 + C^2*b^7 - 12*A^2*a*b^6 - 3*C^2*a*b^6 + 16*C^2*a^
6*b + 16*A^2*a^2*b^5 - 8*A^2*a^3*b^4 + 7*C^2*a^2*b^5 - 13*C^2*a^3*b^4 + 16*C^2*a^4*b^3 - 16*C^2*a^5*b^2 + 4*A*
C*b^7 - 12*A*C*a*b^6 + 20*A*C*a^2*b^5 - 28*A*C*a^3*b^4 + 32*A*C*a^4*b^3 - 16*A*C*a^5*b^2))/b^4 + ((C*a^2*1i +
b^2*(A*1i + (C*1i)/2))*((8*(4*A*b^10 + 2*C*b^10 + 4*A*a^2*b^8 + 2*C*a^2*b^8 - 6*C*a^3*b^7 + 4*C*a^4*b^6 - 8*A*
a*b^9 - 2*C*a*b^9))/b^6 - (8*tan(c/2 + (d*x)/2)*(C*a^2*1i + b^2*(A*1i + (C*1i)/2))*(8*a*b^8 - 16*a^2*b^7 + 8*a
^3*b^6))/b^7))/b^3)*(C*a^2*1i + b^2*(A*1i + (C*1i)/2)))/b^3 - (((8*tan(c/2 + (d*x)/2)*(4*A^2*b^7 - 8*C^2*a^7 +
C^2*b^7 - 12*A^2*a*b^6 - 3*C^2*a*b^6 + 16*C^2*a^6*b + 16*A^2*a^2*b^5 - 8*A^2*a^3*b^4 + 7*C^2*a^2*b^5 - 13*C^2
*a^3*b^4 + 16*C^2*a^4*b^3 - 16*C^2*a^5*b^2 + 4*A*C*b^7 - 12*A*C*a*b^6 + 20*A*C*a^2*b^5 - 28*A*C*a^3*b^4 + 32*A
*C*a^4*b^3 - 16*A*C*a^5*b^2))/b^4 - ((C*a^2*1i + b^2*(A*1i + (C*1i)/2))*((8*(4*A*b^10 + 2*C*b^10 + 4*A*a^2*b^8
+ 2*C*a^2*b^8 - 6*C*a^3*b^7 + 4*C*a^4*b^6 - 8*A*a*b^9 - 2*C*a*b^9))/b^6 + (8*tan(c/2 + (d*x)/2)*(C*a^2*1i + b
^2*(A*1i + (C*1i)/2))*(8*a*b^8 - 16*a^2*b^7 + 8*a^3*b^6))/b^7))/b^3)*(C*a^2*1i + b^2*(A*1i + (C*1i)/2)))/b^3))
*(C*a^2*1i + b^2*(A*1i + (C*1i)/2))*2i)/(b^3*d) - (log(a*tan(c/2 + (d*x)/2) - b*tan(c/2 + (d*x)/2) + (b^2 - a^
2)^(1/2))*(C*a^3*(b^2 - a^2)^(1/2) + A*a*b^2*(b^2 - a^2)^(1/2)))/(b^3*d*(a^2 - b^2)) - (a*log((8*a*(a - b)*(4*
A^3*b^6 + 4*C^3*a^6 + A*C^2*b^6 + 4*A^2*C*b^6 - 2*C^3*a^5*b + C^3*a^2*b^4 - C^3*a^3*b^3 + 4*C^3*a^4*b^2 - A*C^
2*a*b^5 - 2*A^2*C*a*b^5 + 8*A*C^2*a^2*b^4 - 4*A*C^2*a^3*b^3 + 12*A*C^2*a^4*b^2 + 12*A^2*C*a^2*b^4))/b^6 + (a*(
A*b^2 + C*a^2)*(b^2 - a^2)^(1/2)*((8*tan(c/2 + (d*x)/2)*(a - b)*(4*A^2*b^6 + 8*C^2*a^6 + C^2*b^6 - 8*A^2*a*b^5
- 2*C^2*a*b^5 - 8*C^2*a^5*b + 8*A^2*a^2*b^4 + 5*C^2*a^2*b^4 - 8*C^2*a^3*b^3 + 8*C^2*a^4*b^2 + 4*A*C*b^6 - 8*A
*C*a*b^5 + 12*A*C*a^2*b^4 - 16*A*C*a^3*b^3 + 16*A*C*a^4*b^2))/b^4 + (a*(A*b^2 + C*a^2)*(b^2 - a^2)^(1/2)*(16*(
a - b)^2*(2*A*b^2 + 2*C*a^2 + C*b^2 + C*a*b) + (64*a^2*b^2*tan(c/2 + (d*x)/2)*(A*b^2 + C*a^2)*(b^2 - a^2)^(1/2
)*(a - b)^2)/(b^5 - a^2*b^3)))/(b^5 - a^2*b^3)))/(b^5 - a^2*b^3))*(-(a + b)*(a - b))^(1/2)*(A*b^2 + C*a^2))/(d
*(b^5 - a^2*b^3))