3.4 \(\int \frac {b+c+\sin (x)}{a+b \cos (x)} \, dx\)

Optimal. Leaf size=58 \[ \frac {2 (b+c) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}}-\frac {\log (a+b \cos (x))}{b} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {4401, 2659, 205, 2668, 31} \[ \frac {2 (b+c) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}}-\frac {\log (a+b \cos (x))}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b + c)*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]) - Log[a + b*Cos[x]]/b

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 205

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

Rule 2659

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 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 4401

Int[u_, x_Symbol] :> With[{v = ExpandTrig[u, x]}, Int[v, x] /; SumQ[v]] /;  !InertTrigFreeQ[u]

Rubi steps

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

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Mathematica [A]  time = 0.08, size = 57, normalized size = 0.98 \[ -\frac {2 (b+c) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {x}{2}\right )}{\sqrt {b^2-a^2}}\right )}{\sqrt {b^2-a^2}}-\frac {\log (a+b \cos (x))}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.54, size = 231, normalized size = 3.98 \[ \left [-\frac {\sqrt {-a^{2} + b^{2}} {\left (b^{2} + b c\right )} \log \left (\frac {2 \, a b \cos \relax (x) + {\left (2 \, a^{2} - b^{2}\right )} \cos \relax (x)^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \relax (x) + b\right )} \sin \relax (x) - a^{2} + 2 \, b^{2}}{b^{2} \cos \relax (x)^{2} + 2 \, a b \cos \relax (x) + a^{2}}\right ) + {\left (a^{2} - b^{2}\right )} \log \left (b^{2} \cos \relax (x)^{2} + 2 \, a b \cos \relax (x) + a^{2}\right )}{2 \, {\left (a^{2} b - b^{3}\right )}}, \frac {2 \, \sqrt {a^{2} - b^{2}} {\left (b^{2} + b c\right )} \arctan \left (-\frac {a \cos \relax (x) + b}{\sqrt {a^{2} - b^{2}} \sin \relax (x)}\right ) - {\left (a^{2} - b^{2}\right )} \log \left (b^{2} \cos \relax (x)^{2} + 2 \, a b \cos \relax (x) + a^{2}\right )}{2 \, {\left (a^{2} b - b^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(sqrt(-a^2 + b^2)*(b^2 + b*c)*log((2*a*b*cos(x) + (2*a^2 - b^2)*cos(x)^2 + 2*sqrt(-a^2 + b^2)*(a*cos(x)
+ b)*sin(x) - a^2 + 2*b^2)/(b^2*cos(x)^2 + 2*a*b*cos(x) + a^2)) + (a^2 - b^2)*log(b^2*cos(x)^2 + 2*a*b*cos(x)
+ a^2))/(a^2*b - b^3), 1/2*(2*sqrt(a^2 - b^2)*(b^2 + b*c)*arctan(-(a*cos(x) + b)/(sqrt(a^2 - b^2)*sin(x))) - (
a^2 - b^2)*log(b^2*cos(x)^2 + 2*a*b*cos(x) + a^2))/(a^2*b - b^3)]

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giac [B]  time = 0.47, size = 407, normalized size = 7.02 \[ -\frac {{\left (a + b\right )} {\left (a - b\right )}^{2} \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + \frac {2 \, a + \sqrt {-4 \, {\left (a + b\right )} {\left (a - b\right )} + 4 \, a^{2}}}{2 \, {\left (a - b\right )}}\right )}{{\left (a^{2} - 2 \, a b + b^{2}\right )} b^{2} + {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} {\left | b \right |}} + \frac {{\left (\sqrt {a^{2} - b^{2}} b^{2} {\left | a - b \right |} + \sqrt {a^{2} - b^{2}} b c {\left | a - b \right |} + \sqrt {a^{2} - b^{2}} b {\left | a - b \right |} {\left | b \right |} + \sqrt {a^{2} - b^{2}} c {\left | a - b \right |} {\left | b \right |}\right )} {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \tan \left (\frac {1}{2} \, x\right )}{\sqrt {\frac {2 \, a + \sqrt {-4 \, {\left (a + b\right )} {\left (a - b\right )} + 4 \, a^{2}}}{a - b}}}\right )\right )}}{{\left (a^{2} - 2 \, a b + b^{2}\right )} b^{2} + {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} {\left | b \right |}} + \frac {{\left (b^{2} + b c - b {\left | b \right |} - c {\left | b \right |}\right )} {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \tan \left (\frac {1}{2} \, x\right )}{\sqrt {\frac {2 \, a - \sqrt {-4 \, {\left (a + b\right )} {\left (a - b\right )} + 4 \, a^{2}}}{a - b}}}\right )\right )}}{b^{2} - a {\left | b \right |}} - \frac {{\left (a - b\right )} \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + \frac {2 \, a - \sqrt {-4 \, {\left (a + b\right )} {\left (a - b\right )} + 4 \, a^{2}}}{2 \, {\left (a - b\right )}}\right )}{b^{2} - a {\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a + b)*(a - b)^2*log(tan(1/2*x)^2 + 1/2*(2*a + sqrt(-4*(a + b)*(a - b) + 4*a^2))/(a - b))/((a^2 - 2*a*b + b^
2)*b^2 + (a^3 - 2*a^2*b + a*b^2)*abs(b)) + (sqrt(a^2 - b^2)*b^2*abs(a - b) + sqrt(a^2 - b^2)*b*c*abs(a - b) +
sqrt(a^2 - b^2)*b*abs(a - b)*abs(b) + sqrt(a^2 - b^2)*c*abs(a - b)*abs(b))*(pi*floor(1/2*x/pi + 1/2) + arctan(
2*sqrt(1/2)*tan(1/2*x)/sqrt((2*a + sqrt(-4*(a + b)*(a - b) + 4*a^2))/(a - b))))/((a^2 - 2*a*b + b^2)*b^2 + (a^
3 - 2*a^2*b + a*b^2)*abs(b)) + (b^2 + b*c - b*abs(b) - c*abs(b))*(pi*floor(1/2*x/pi + 1/2) + arctan(2*sqrt(1/2
)*tan(1/2*x)/sqrt((2*a - sqrt(-4*(a + b)*(a - b) + 4*a^2))/(a - b))))/(b^2 - a*abs(b)) - (a - b)*log(tan(1/2*x
)^2 + 1/2*(2*a - sqrt(-4*(a + b)*(a - b) + 4*a^2))/(a - b))/(b^2 - a*abs(b))

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maple [B]  time = 0.06, size = 150, normalized size = 2.59 \[ -\frac {\ln \left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-\left (\tan ^{2}\left (\frac {x}{2}\right )\right ) b +a +b \right ) a}{b \left (a -b \right )}+\frac {\ln \left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-\left (\tan ^{2}\left (\frac {x}{2}\right )\right ) b +a +b \right )}{a -b}+\frac {2 b \arctan \left (\frac {\tan \left (\frac {x}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {2 \arctan \left (\frac {\tan \left (\frac {x}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right ) c}{\sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {\ln \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b+c+sin(x))/(a+b*cos(x)),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 details)Is 4*b^2-4*a^2 positive or negative?

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mupad [B]  time = 11.81, size = 2219, normalized size = 38.26 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(tan(x/2)^2 + 1)/b - (2*atan((tan(x/2)*((4*b*(b + c)*(32*a*b^3 - 128*a*b - 64*b^3*c + 64*a^2 + 64*b^2 - 32*
b^4 - 32*b^2*c^2 + ((b + c)*(((b + c)*(64*b^4 - 128*a*b^3 + 64*a^2*b^2 + ((2*a^2*b - 2*b^3)*(64*a*b^4 - 128*a^
2*b^3 + 64*a^3*b^2))/(2*(b^4 - a^2*b^2))))/(a^2 - b^2)^(1/2) + ((2*a^2*b - 2*b^3)*(b + c)*(64*a*b^4 - 128*a^2*
b^3 + 64*a^3*b^2))/(2*(b^4 - a^2*b^2)*(a^2 - b^2)^(1/2))))/(a^2 - b^2)^(1/2) + ((2*a^2*b - 2*b^3)*(64*a*b^2 -
128*a^2*b - 32*a*b^4 + 64*b^4*c + 64*a^3 + 32*b^5 + 32*b^3*c^2 - ((2*a^2*b - 2*b^3)*(64*b^4 - 128*a*b^3 + 64*a
^2*b^2 + ((2*a^2*b - 2*b^3)*(64*a*b^4 - 128*a^2*b^3 + 64*a^3*b^2))/(2*(b^4 - a^2*b^2))))/(2*(b^4 - a^2*b^2)) -
32*a*b^2*c^2 - 64*a*b^3*c))/(2*(b^4 - a^2*b^2)) + 32*a*b*c^2 + 64*a*b^2*c + ((2*a^2*b - 2*b^3)*(b + c)^2*(64*
a*b^4 - 128*a^2*b^3 + 64*a^3*b^2))/(2*(b^4 - a^2*b^2)*(a^2 - b^2))))/((a - b)*(2*b^3*c + 4*a^2 - 4*b^2 + b^4 +
b^2*c^2)^2) - ((((b + c)^3*(64*a*b^4 - 128*a^2*b^3 + 64*a^3*b^2))/(a^2 - b^2)^(3/2) - ((2*a^2*b - 2*b^3)*(((b
+ c)*(64*b^4 - 128*a*b^3 + 64*a^2*b^2 + ((2*a^2*b - 2*b^3)*(64*a*b^4 - 128*a^2*b^3 + 64*a^3*b^2))/(2*(b^4 - a
^2*b^2))))/(a^2 - b^2)^(1/2) + ((2*a^2*b - 2*b^3)*(b + c)*(64*a*b^4 - 128*a^2*b^3 + 64*a^3*b^2))/(2*(b^4 - a^2
*b^2)*(a^2 - b^2)^(1/2))))/(2*(b^4 - a^2*b^2)) + ((b + c)*(64*a*b^2 - 128*a^2*b - 32*a*b^4 + 64*b^4*c + 64*a^3
+ 32*b^5 + 32*b^3*c^2 - ((2*a^2*b - 2*b^3)*(64*b^4 - 128*a*b^3 + 64*a^2*b^2 + ((2*a^2*b - 2*b^3)*(64*a*b^4 -
128*a^2*b^3 + 64*a^3*b^2))/(2*(b^4 - a^2*b^2))))/(2*(b^4 - a^2*b^2)) - 32*a*b^2*c^2 - 64*a*b^3*c))/(a^2 - b^2)
^(1/2))*(2*b^3*c - 4*a^2 + 4*b^2 + b^4 + b^2*c^2))/((a^2 - b^2)^(1/2)*(a - b)*(2*b^3*c + 4*a^2 - 4*b^2 + b^4 +
b^2*c^2)^2))*(a^2 - b^2)^(3/2))/(32*a*b + 32*a*c - 32*b*c - 32*b^2) + ((((b + c)*(64*b^3*c - 128*a*b^3 + 64*b
^4 + 64*a^2*b^2 - ((2*a^2*b - 2*b^3)*(32*b^4*c - 64*a*b^4 + 32*b^5 + 32*a^2*b^3 + 32*a^2*b^2*c - 64*a*b^3*c))/
(2*(b^4 - a^2*b^2)) - 128*a*b^2*c + 64*a^2*b*c))/(a^2 - b^2)^(1/2) - ((2*a^2*b - 2*b^3)*(b + c)*(32*b^4*c - 64
*a*b^4 + 32*b^5 + 32*a^2*b^3 + 32*a^2*b^2*c - 64*a*b^3*c))/(2*(b^4 - a^2*b^2)*(a^2 - b^2)^(1/2)))*(a^2 - b^2)*
(2*b^3*c - 4*a^2 + 4*b^2 + b^4 + b^2*c^2))/((a - b)*(32*a*b + 32*a*c - 32*b*c - 32*b^2)*(2*b^3*c + 4*a^2 - 4*b
^2 + b^4 + b^2*c^2)^2) - (4*b*(b + c)*(a^2 - b^2)^(3/2)*(64*a*b^2 - 32*a^2*b - 32*a^2*c - 32*b^2*c - 32*b^3 +
((b + c)^2*(32*b^4*c - 64*a*b^4 + 32*b^5 + 32*a^2*b^3 + 32*a^2*b^2*c - 64*a*b^3*c))/(a^2 - b^2) + ((2*a^2*b -
2*b^3)*(64*b^3*c - 128*a*b^3 + 64*b^4 + 64*a^2*b^2 - ((2*a^2*b - 2*b^3)*(32*b^4*c - 64*a*b^4 + 32*b^5 + 32*a^2
*b^3 + 32*a^2*b^2*c - 64*a*b^3*c))/(2*(b^4 - a^2*b^2)) - 128*a*b^2*c + 64*a^2*b*c))/(2*(b^4 - a^2*b^2)) + 64*a
*b*c))/((a - b)*(32*a*b + 32*a*c - 32*b*c - 32*b^2)*(2*b^3*c + 4*a^2 - 4*b^2 + b^4 + b^2*c^2)^2))*(b + c))/(a^
2 - b^2)^(1/2) + (log((32*a^2*b - 64*a*b^2 + 32*a^2*c + 32*b^2*c + 32*b^3 + ((b*(-(b + c)^2/(a^2 - b^2))^(1/2)
+ 1)*(64*b^3*c - 128*a*b^3 + 64*b^4 + 64*a^2*b^2 - 128*a*b^2*c + 64*a^2*b*c + 32*(b*(-(b + c)^2/(a^2 - b^2))^
(1/2) + 1)*(a - b)^2*(2*a*tan(x/2) - 2*b*tan(x/2) + b*c + b^2 + 2*a*b*tan(x/2)*(-(b + c)^2/(a^2 - b^2))^(1/2))
+ 32*tan(x/2)*(a - b)*(2*a*b + 2*b^3*c - 2*a^2 + b^4 + b^2*c^2)))/b - 64*a*b*c + 32*tan(x/2)*(a - b)*(2*a - 2
*b + b*c^2 + 2*b^2*c + b^3))*(32*a^2*b - 64*a*b^2 + 32*a^2*c + 32*b^2*c + 32*b^3 - ((b*(-(b + c)^2/(a^2 - b^2)
)^(1/2) - 1)*(64*b^3*c - 128*a*b^3 + 64*b^4 + 64*a^2*b^2 - 128*a*b^2*c + 64*a^2*b*c - 32*(b*(-(b + c)^2/(a^2 -
b^2))^(1/2) - 1)*(a - b)^2*(2*a*tan(x/2) - 2*b*tan(x/2) + b*c + b^2 - 2*a*b*tan(x/2)*(-(b + c)^2/(a^2 - b^2))
^(1/2)) + 32*tan(x/2)*(a - b)*(2*a*b + 2*b^3*c - 2*a^2 + b^4 + b^2*c^2)))/b - 64*a*b*c + 32*tan(x/2)*(a - b)*(
2*a - 2*b + b*c^2 + 2*b^2*c + b^3)))*(2*a^2*b - 2*b^3))/(2*(b^4 - a^2*b^2))

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sympy [A]  time = 23.96, size = 804, normalized size = 13.86 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*(-c*log(tan(x/2) - 1) + c*log(tan(x/2) + 1) - log(tan(x/2) - 1) - log(tan(x/2) + 1) + log(tan(x
/2)**2 + 1)), Eq(a, 0) & Eq(b, 0)), (tan(x/2) + c*tan(x/2)/b + log(tan(x/2)**2 + 1)/b, Eq(a, b)), (1/tan(x/2)
+ c/(b*tan(x/2)) + log(tan(x/2)**2 + 1)/b - 2*log(tan(x/2))/b, Eq(a, -b)), ((c*x - cos(x))/a, Eq(b, 0)), (-a*s
qrt(-a/(a - b) - b/(a - b))*log(-sqrt(-a/(a - b) - b/(a - b)) + tan(x/2))/(a*b*sqrt(-a/(a - b) - b/(a - b)) -
b**2*sqrt(-a/(a - b) - b/(a - b))) - a*sqrt(-a/(a - b) - b/(a - b))*log(sqrt(-a/(a - b) - b/(a - b)) + tan(x/2
))/(a*b*sqrt(-a/(a - b) - b/(a - b)) - b**2*sqrt(-a/(a - b) - b/(a - b))) + a*sqrt(-a/(a - b) - b/(a - b))*log
(tan(x/2)**2 + 1)/(a*b*sqrt(-a/(a - b) - b/(a - b)) - b**2*sqrt(-a/(a - b) - b/(a - b))) + b**2*log(-sqrt(-a/(
a - b) - b/(a - b)) + tan(x/2))/(a*b*sqrt(-a/(a - b) - b/(a - b)) - b**2*sqrt(-a/(a - b) - b/(a - b))) - b**2*
log(sqrt(-a/(a - b) - b/(a - b)) + tan(x/2))/(a*b*sqrt(-a/(a - b) - b/(a - b)) - b**2*sqrt(-a/(a - b) - b/(a -
b))) + b*c*log(-sqrt(-a/(a - b) - b/(a - b)) + tan(x/2))/(a*b*sqrt(-a/(a - b) - b/(a - b)) - b**2*sqrt(-a/(a
- b) - b/(a - b))) - b*c*log(sqrt(-a/(a - b) - b/(a - b)) + tan(x/2))/(a*b*sqrt(-a/(a - b) - b/(a - b)) - b**2
*sqrt(-a/(a - b) - b/(a - b))) + b*sqrt(-a/(a - b) - b/(a - b))*log(-sqrt(-a/(a - b) - b/(a - b)) + tan(x/2))/
(a*b*sqrt(-a/(a - b) - b/(a - b)) - b**2*sqrt(-a/(a - b) - b/(a - b))) + b*sqrt(-a/(a - b) - b/(a - b))*log(sq
rt(-a/(a - b) - b/(a - b)) + tan(x/2))/(a*b*sqrt(-a/(a - b) - b/(a - b)) - b**2*sqrt(-a/(a - b) - b/(a - b)))
- b*sqrt(-a/(a - b) - b/(a - b))*log(tan(x/2)**2 + 1)/(a*b*sqrt(-a/(a - b) - b/(a - b)) - b**2*sqrt(-a/(a - b)
- b/(a - b))), True))

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