\(\int \frac {A+B \cos (c+d x)}{(a+b \cos (c+d x))^2} \, dx\) [261]

Optimal result

Integrand size = 23, antiderivative size = 100 \[ \int \frac {A+B \cos (c+d x)}{(a+b \cos (c+d x))^2} \, dx=\frac {2 (a A-b B) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} (a+b)^{3/2} d}-\frac {(A b-a B) \sin (c+d x)}{\left (a^2-b^2\right ) d (a+b \cos (c+d x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2833, 12, 2738, 211} \[ \int \frac {A+B \cos (c+d x)}{(a+b \cos (c+d x))^2} \, dx=\frac {2 (a A-b B) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d (a-b)^{3/2} (a+b)^{3/2}}-\frac {(A b-a B) \sin (c+d x)}{d \left (a^2-b^2\right ) (a+b \cos (c+d x))} \]

[In]

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

[Out]

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

Rule 12

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

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 2833

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]

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

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.97 \[ \int \frac {A+B \cos (c+d x)}{(a+b \cos (c+d x))^2} \, dx=\frac {\frac {2 (a A-b B) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{3/2}}+\frac {(-A b+a B) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}}{d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.05 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.28

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (A b -B a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}+\frac {2 \left (a A -B b \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}}{d}\)	\(128\)
default	\(\frac {-\frac {2 \left (A b -B a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}+\frac {2 \left (a A -B b \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}}{d}\)	\(128\)
risch	\(\frac {2 i \left (A b -B a \right ) \left (a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )}{b \left (-a^{2}+b^{2}\right ) d \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right ) a A}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right ) B b}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+a \sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) a A}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+a \sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) B b}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}\)	\(396\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 379, normalized size of antiderivative = 3.79 \[ \int \frac {A+B \cos (c+d x)}{(a+b \cos (c+d x))^2} \, dx=\left [-\frac {{\left (A a^{2} - B a b + {\left (A a b - B b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right ) - 2 \, {\left (B a^{3} - A a^{2} b - B a b^{2} + A b^{3}\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d \cos \left (d x + c\right ) + {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d\right )}}, \frac {{\left (A a^{2} - B a b + {\left (A a b - B b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cos \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (d x + c\right )}\right ) + {\left (B a^{3} - A a^{2} b - B a b^{2} + A b^{3}\right )} \sin \left (d x + c\right )}{{\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d \cos \left (d x + c\right ) + {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d}\right ] \]

[In]

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

[Out]

[-1/2*((A*a^2 - B*a*b + (A*a*b - B*b^2)*cos(d*x + c))*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*(B*a^3 - A*a^2*b - B*a*b^2 + A*b^3)*sin(d*x + c))/((a^4*b - 2*a^2*b^3 + b^5)*d*co
s(d*x + c) + (a^5 - 2*a^3*b^2 + a*b^4)*d), ((A*a^2 - B*a*b + (A*a*b - B*b^2)*cos(d*x + c))*sqrt(a^2 - b^2)*arc
tan(-(a*cos(d*x + c) + b)/(sqrt(a^2 - b^2)*sin(d*x + c))) + (B*a^3 - A*a^2*b - B*a*b^2 + A*b^3)*sin(d*x + c))/
((a^4*b - 2*a^2*b^3 + b^5)*d*cos(d*x + c) + (a^5 - 2*a^3*b^2 + a*b^4)*d)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4974 vs. \(2 (82) = 164\).

Time = 157.10 (sec) , antiderivative size = 4974, normalized size of antiderivative = 49.74 \[ \int \frac {A+B \cos (c+d x)}{(a+b \cos (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((A+B*cos(d*x+c))/(a+b*cos(d*x+c))^2,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 = 0.33 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.59 \[ \int \frac {A+B \cos (c+d x)}{(a+b \cos (c+d x))^2} \, dx=-\frac {2 \, {\left (\frac {{\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )} {\left (A a - B b\right )}}{{\left (a^{2} - b^{2}\right )}^{\frac {3}{2}}} - \frac {B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b\right )} {\left (a^{2} - b^{2}\right )}}\right )}}{d} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.92 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.13 \[ \int \frac {A+B \cos (c+d x)}{(a+b \cos (c+d x))^2} \, dx=\frac {2\,\mathrm {atan}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a-2\,b\right )}{2\,\sqrt {a+b}\,\sqrt {a-b}}\right )\,\left (A\,a-B\,b\right )}{d\,{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}}-\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A\,b-B\,a\right )}{d\,\left (a+b\right )\,\left (a-b\right )\,\left (\left (a-b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a+b\right )} \]

[In]

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

[Out]

(2*atan((tan(c/2 + (d*x)/2)*(2*a - 2*b))/(2*(a + b)^(1/2)*(a - b)^(1/2)))*(A*a - B*b))/(d*(a + b)^(3/2)*(a - b
)^(3/2)) - (2*tan(c/2 + (d*x)/2)*(A*b - B*a))/(d*(a + b)*(a - b)*(a + b + tan(c/2 + (d*x)/2)^2*(a - b)))