3.174 \(\int (c+d x) \cos (a+b x) \cot ^2(a+b x) \, dx\)
Optimal. Leaf size=58 \[ -\frac{d \cos (a+b x)}{b^2}-\frac{d \tanh ^{-1}(\cos (a+b x))}{b^2}-\frac{(c+d x) \sin (a+b x)}{b}-\frac{(c+d x) \csc (a+b x)}{b} \]
[Out]
-((d*ArcTanh[Cos[a + b*x]])/b^2) - (d*Cos[a + b*x])/b^2 - ((c + d*x)*Csc[a + b*x])/b - ((c + d*x)*Sin[a + b*x]
)/b
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Rubi [A] time = 0.0631534, antiderivative size = 58, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used =
{4408, 3296, 2638, 4410, 3770} \[ -\frac{d \cos (a+b x)}{b^2}-\frac{d \tanh ^{-1}(\cos (a+b x))}{b^2}-\frac{(c+d x) \sin (a+b x)}{b}-\frac{(c+d x) \csc (a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*x)*Cos[a + b*x]*Cot[a + b*x]^2,x]
[Out]
-((d*ArcTanh[Cos[a + b*x]])/b^2) - (d*Cos[a + b*x])/b^2 - ((c + d*x)*Csc[a + b*x])/b - ((c + d*x)*Sin[a + b*x]
)/b
Rule 4408
Int[Cos[(a_.) + (b_.)*(x_)]^(n_.)*Cot[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Int[
(c + d*x)^m*Cos[a + b*x]^n*Cot[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Cos[a + b*x]^(n - 2)*Cot[a + b*x]^p, x]
/; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
Rule 3296
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Rule 2638
Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Rule 4410
Int[Cot[(a_.) + (b_.)*(x_)]^(p_.)*Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp
[((c + d*x)^m*Csc[a + b*x]^n)/(b*n), x] + Dist[(d*m)/(b*n), Int[(c + d*x)^(m - 1)*Csc[a + b*x]^n, x], x] /; Fr
eeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
Rule 3770
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Rubi steps
\begin{align*} \int (c+d x) \cos (a+b x) \cot ^2(a+b x) \, dx &=-\int (c+d x) \cos (a+b x) \, dx+\int (c+d x) \cot (a+b x) \csc (a+b x) \, dx\\ &=-\frac{(c+d x) \csc (a+b x)}{b}-\frac{(c+d x) \sin (a+b x)}{b}+\frac{d \int \csc (a+b x) \, dx}{b}+\frac{d \int \sin (a+b x) \, dx}{b}\\ &=-\frac{d \tanh ^{-1}(\cos (a+b x))}{b^2}-\frac{d \cos (a+b x)}{b^2}-\frac{(c+d x) \csc (a+b x)}{b}-\frac{(c+d x) \sin (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.706704, size = 104, normalized size = 1.79 \[ -\frac{2 b c \sin (a+b x)+2 b c \csc (a+b x)+2 b d x \sin (a+b x)+2 d \cos (a+b x)+b d x \tan \left (\frac{1}{2} (a+b x)\right )+b d x \cot \left (\frac{1}{2} (a+b x)\right )-2 d \log \left (\sin \left (\frac{1}{2} (a+b x)\right )\right )+2 d \log \left (\cos \left (\frac{1}{2} (a+b x)\right )\right )}{2 b^2} \]
Antiderivative was successfully verified.
[In]
Integrate[(c + d*x)*Cos[a + b*x]*Cot[a + b*x]^2,x]
[Out]
-(2*d*Cos[a + b*x] + b*d*x*Cot[(a + b*x)/2] + 2*b*c*Csc[a + b*x] + 2*d*Log[Cos[(a + b*x)/2]] - 2*d*Log[Sin[(a
+ b*x)/2]] + 2*b*c*Sin[a + b*x] + 2*b*d*x*Sin[a + b*x] + b*d*x*Tan[(a + b*x)/2])/(2*b^2)
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Maple [C] time = 0.119, size = 124, normalized size = 2.1 \begin{align*}{\frac{{\frac{i}{2}} \left ( dxb+bc+id \right ){{\rm e}^{i \left ( bx+a \right ) }}}{{b}^{2}}}-{\frac{{\frac{i}{2}} \left ( dxb+bc-id \right ){{\rm e}^{-i \left ( bx+a \right ) }}}{{b}^{2}}}-{\frac{2\,i{{\rm e}^{i \left ( bx+a \right ) }} \left ( dx+c \right ) }{b \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) }}-{\frac{d\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}+1 \right ) }{{b}^{2}}}+{\frac{d\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}-1 \right ) }{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x+c)*cos(b*x+a)*cot(b*x+a)^2,x)
[Out]
1/2*I*(d*x*b+b*c+I*d)/b^2*exp(I*(b*x+a))-1/2*I*(d*x*b+b*c-I*d)/b^2*exp(-I*(b*x+a))-2*I*exp(I*(b*x+a))*(d*x+c)/
b/(exp(2*I*(b*x+a))-1)-d/b^2*ln(exp(I*(b*x+a))+1)+d/b^2*ln(exp(I*(b*x+a))-1)
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Maxima [B] time = 1.18379, size = 2849, normalized size = 49.12 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)*cos(b*x+a)*cot(b*x+a)^2,x, algorithm="maxima")
[Out]
-1/2*(2*c*(1/sin(b*x + a) + sin(b*x + a)) - 2*a*d*(1/sin(b*x + a) + sin(b*x + a))/b - (((b*x + a)*sin(2*b*x +
2*a) - cos(2*b*x + 2*a) + 1)*cos(3*b*x + 3*a)^3 + (b*x - (b*x + a)*cos(2*b*x + 2*a) + a - sin(2*b*x + 2*a))*si
n(3*b*x + 3*a)^3 - 6*(b*x + a)*sin(b*x + a)^3 - 2*(4*(b*x + a)*cos(b*x + a)*sin(2*b*x + 2*a) - (3*(b*x + a)*si
n(b*x + a) + cos(b*x + a))*cos(2*b*x + 2*a) + 3*(b*x + a)*sin(b*x + a) + cos(b*x + a))*cos(3*b*x + 3*a)^2 - ((
b*x + a)*sin(b*x + a) + cos(b*x + a))*cos(2*b*x + 2*a)^2 + (8*(b*x + a)*cos(2*b*x + 2*a)*sin(b*x + a) + ((b*x
+ a)*sin(2*b*x + 2*a) - cos(2*b*x + 2*a) + 1)*cos(3*b*x + 3*a) - 2*(3*(b*x + a)*cos(b*x + a) - sin(b*x + a))*s
in(2*b*x + 2*a) - 8*(b*x + a)*sin(b*x + a))*sin(3*b*x + 3*a)^2 - ((b*x + a)*sin(b*x + a) + cos(b*x + a))*sin(2
*b*x + 2*a)^2 + (12*(b*x + a)*cos(b*x + a)*sin(b*x + a) - (12*(b*x + a)*cos(b*x + a)*sin(b*x + a) + cos(b*x +
a)^2 + sin(b*x + a)^2 + 2)*cos(2*b*x + 2*a) + cos(2*b*x + 2*a)^2 + cos(b*x + a)^2 + (13*(b*x + a)*cos(b*x + a)
^2 + (b*x + a)*sin(b*x + a)^2)*sin(2*b*x + 2*a) + sin(2*b*x + 2*a)^2 + sin(b*x + a)^2 + 1)*cos(3*b*x + 3*a) +
2*(3*(b*x + a)*sin(b*x + a)^3 + (3*(b*x + a)*cos(b*x + a)^2 + b*x + a)*sin(b*x + a) + cos(b*x + a))*cos(2*b*x
+ 2*a) - ((cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)*cos(3*b*x + 3*a)^2 + (cos(b*x + a
)^2 + sin(b*x + a)^2)*cos(2*b*x + 2*a)^2 + (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)*
sin(3*b*x + 3*a)^2 + (cos(b*x + a)^2 + sin(b*x + a)^2)*sin(2*b*x + 2*a)^2 - 2*(cos(2*b*x + 2*a)^2*cos(b*x + a)
+ cos(b*x + a)*sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a)*cos(b*x + a) + cos(b*x + a))*cos(3*b*x + 3*a) - 2*(cos
(b*x + a)^2 + sin(b*x + a)^2)*cos(2*b*x + 2*a) + cos(b*x + a)^2 - 2*(cos(2*b*x + 2*a)^2*sin(b*x + a) + sin(2*b
*x + 2*a)^2*sin(b*x + a) - 2*cos(2*b*x + 2*a)*sin(b*x + a) + sin(b*x + a))*sin(3*b*x + 3*a) + sin(b*x + a)^2)*
log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) + ((cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(
2*b*x + 2*a) + 1)*cos(3*b*x + 3*a)^2 + (cos(b*x + a)^2 + sin(b*x + a)^2)*cos(2*b*x + 2*a)^2 + (cos(2*b*x + 2*a
)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)*sin(3*b*x + 3*a)^2 + (cos(b*x + a)^2 + sin(b*x + a)^2)*sin(
2*b*x + 2*a)^2 - 2*(cos(2*b*x + 2*a)^2*cos(b*x + a) + cos(b*x + a)*sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a)*cos
(b*x + a) + cos(b*x + a))*cos(3*b*x + 3*a) - 2*(cos(b*x + a)^2 + sin(b*x + a)^2)*cos(2*b*x + 2*a) + cos(b*x +
a)^2 - 2*(cos(2*b*x + 2*a)^2*sin(b*x + a) + sin(2*b*x + 2*a)^2*sin(b*x + a) - 2*cos(2*b*x + 2*a)*sin(b*x + a)
+ sin(b*x + a))*sin(3*b*x + 3*a) + sin(b*x + a)^2)*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + 1) +
((b*x - (b*x + a)*cos(2*b*x + 2*a) + a - sin(2*b*x + 2*a))*cos(3*b*x + 3*a)^2 + (b*x + a)*cos(2*b*x + 2*a)^2
+ (b*x + a)*cos(b*x + a)^2 + (b*x + a)*sin(2*b*x + 2*a)^2 + 13*(b*x + a)*sin(b*x + a)^2 + b*x + 2*(((b*x + a)*
cos(b*x + a) + sin(b*x + a))*cos(2*b*x + 2*a) - (b*x + a)*cos(b*x + a) - ((b*x + a)*sin(b*x + a) - cos(b*x + a
))*sin(2*b*x + 2*a) - sin(b*x + a))*cos(3*b*x + 3*a) - ((b*x + a)*cos(b*x + a)^2 + 13*(b*x + a)*sin(b*x + a)^2
+ 2*b*x + 2*a)*cos(2*b*x + 2*a) + (12*(b*x + a)*cos(b*x + a)*sin(b*x + a) - cos(b*x + a)^2 - sin(b*x + a)^2)*
sin(2*b*x + 2*a) + a)*sin(3*b*x + 3*a) - 6*((b*x + a)*cos(b*x + a)^3 + (b*x + a)*cos(b*x + a)*sin(b*x + a)^2)*
sin(2*b*x + 2*a) - (6*(b*x + a)*cos(b*x + a)^2 + b*x + a)*sin(b*x + a) - cos(b*x + a))*d/(((cos(2*b*x + 2*a)^2
+ sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)*cos(3*b*x + 3*a)^2 + (cos(b*x + a)^2 + sin(b*x + a)^2)*cos(2*b
*x + 2*a)^2 + (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 - 2*cos(2*b*x + 2*a) + 1)*sin(3*b*x + 3*a)^2 + (cos(b*x
+ a)^2 + sin(b*x + a)^2)*sin(2*b*x + 2*a)^2 - 2*(cos(2*b*x + 2*a)^2*cos(b*x + a) + cos(b*x + a)*sin(2*b*x + 2
*a)^2 - 2*cos(2*b*x + 2*a)*cos(b*x + a) + cos(b*x + a))*cos(3*b*x + 3*a) - 2*(cos(b*x + a)^2 + sin(b*x + a)^2)
*cos(2*b*x + 2*a) + cos(b*x + a)^2 - 2*(cos(2*b*x + 2*a)^2*sin(b*x + a) + sin(2*b*x + 2*a)^2*sin(b*x + a) - 2*
cos(2*b*x + 2*a)*sin(b*x + a) + sin(b*x + a))*sin(3*b*x + 3*a) + sin(b*x + a)^2)*b))/b
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Fricas [A] time = 0.521035, size = 269, normalized size = 4.64 \begin{align*} -\frac{4 \, b d x - 2 \,{\left (b d x + b c\right )} \cos \left (b x + a\right )^{2} + 2 \, d \cos \left (b x + a\right ) \sin \left (b x + a\right ) + d \log \left (\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2}\right ) \sin \left (b x + a\right ) - d \log \left (-\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2}\right ) \sin \left (b x + a\right ) + 4 \, b c}{2 \, b^{2} \sin \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)*cos(b*x+a)*cot(b*x+a)^2,x, algorithm="fricas")
[Out]
-1/2*(4*b*d*x - 2*(b*d*x + b*c)*cos(b*x + a)^2 + 2*d*cos(b*x + a)*sin(b*x + a) + d*log(1/2*cos(b*x + a) + 1/2)
*sin(b*x + a) - d*log(-1/2*cos(b*x + a) + 1/2)*sin(b*x + a) + 4*b*c)/(b^2*sin(b*x + a))
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c + d x\right ) \cos{\left (a + b x \right )} \cot ^{2}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)*cos(b*x+a)*cot(b*x+a)**2,x)
[Out]
Integral((c + d*x)*cos(a + b*x)*cot(a + b*x)**2, x)
________________________________________________________________________________________
Giac [B] time = 2.22117, size = 2655, normalized size = 45.78 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)*cos(b*x+a)*cot(b*x+a)^2,x, algorithm="giac")
[Out]
1/2*(b*d*x*tan(1/2*b*x)^4*tan(1/2*a)^4 + b*c*tan(1/2*b*x)^4*tan(1/2*a)^4 + 6*b*d*x*tan(1/2*b*x)^4*tan(1/2*a)^2
+ 8*b*d*x*tan(1/2*b*x)^3*tan(1/2*a)^3 - d*log(4*(tan(1/2*a)^2 + 1)/(tan(1/2*b*x)^4*tan(1/2*a)^2 - 2*tan(1/2*b
*x)^3*tan(1/2*a) + tan(1/2*b*x)^2*tan(1/2*a)^2 + tan(1/2*b*x)^2 - 2*tan(1/2*b*x)*tan(1/2*a) + 1))*tan(1/2*b*x)
^4*tan(1/2*a)^3 + d*log(4*(tan(1/2*a)^2 + 1)/(tan(1/2*b*x)^4 + 2*tan(1/2*b*x)^3*tan(1/2*a) + tan(1/2*b*x)^2*ta
n(1/2*a)^2 + tan(1/2*b*x)^2 + 2*tan(1/2*b*x)*tan(1/2*a) + tan(1/2*a)^2))*tan(1/2*b*x)^4*tan(1/2*a)^3 + 6*b*d*x
*tan(1/2*b*x)^2*tan(1/2*a)^4 - d*log(4*(tan(1/2*a)^2 + 1)/(tan(1/2*b*x)^4*tan(1/2*a)^2 - 2*tan(1/2*b*x)^3*tan(
1/2*a) + tan(1/2*b*x)^2*tan(1/2*a)^2 + tan(1/2*b*x)^2 - 2*tan(1/2*b*x)*tan(1/2*a) + 1))*tan(1/2*b*x)^3*tan(1/2
*a)^4 + d*log(4*(tan(1/2*a)^2 + 1)/(tan(1/2*b*x)^4 + 2*tan(1/2*b*x)^3*tan(1/2*a) + tan(1/2*b*x)^2*tan(1/2*a)^2
+ tan(1/2*b*x)^2 + 2*tan(1/2*b*x)*tan(1/2*a) + tan(1/2*a)^2))*tan(1/2*b*x)^3*tan(1/2*a)^4 + 6*b*c*tan(1/2*b*x
)^4*tan(1/2*a)^2 + 8*b*c*tan(1/2*b*x)^3*tan(1/2*a)^3 - 2*d*tan(1/2*b*x)^4*tan(1/2*a)^3 + 6*b*c*tan(1/2*b*x)^2*
tan(1/2*a)^4 - 2*d*tan(1/2*b*x)^3*tan(1/2*a)^4 + b*d*x*tan(1/2*b*x)^4 - 8*b*d*x*tan(1/2*b*x)^3*tan(1/2*a) - d*
log(4*(tan(1/2*a)^2 + 1)/(tan(1/2*b*x)^4*tan(1/2*a)^2 - 2*tan(1/2*b*x)^3*tan(1/2*a) + tan(1/2*b*x)^2*tan(1/2*a
)^2 + tan(1/2*b*x)^2 - 2*tan(1/2*b*x)*tan(1/2*a) + 1))*tan(1/2*b*x)^4*tan(1/2*a) + d*log(4*(tan(1/2*a)^2 + 1)/
(tan(1/2*b*x)^4 + 2*tan(1/2*b*x)^3*tan(1/2*a) + tan(1/2*b*x)^2*tan(1/2*a)^2 + tan(1/2*b*x)^2 + 2*tan(1/2*b*x)*
tan(1/2*a) + tan(1/2*a)^2))*tan(1/2*b*x)^4*tan(1/2*a) - 12*b*d*x*tan(1/2*b*x)^2*tan(1/2*a)^2 - 8*b*d*x*tan(1/2
*b*x)*tan(1/2*a)^3 + b*d*x*tan(1/2*a)^4 - d*log(4*(tan(1/2*a)^2 + 1)/(tan(1/2*b*x)^4*tan(1/2*a)^2 - 2*tan(1/2*
b*x)^3*tan(1/2*a) + tan(1/2*b*x)^2*tan(1/2*a)^2 + tan(1/2*b*x)^2 - 2*tan(1/2*b*x)*tan(1/2*a) + 1))*tan(1/2*b*x
)*tan(1/2*a)^4 + d*log(4*(tan(1/2*a)^2 + 1)/(tan(1/2*b*x)^4 + 2*tan(1/2*b*x)^3*tan(1/2*a) + tan(1/2*b*x)^2*tan
(1/2*a)^2 + tan(1/2*b*x)^2 + 2*tan(1/2*b*x)*tan(1/2*a) + tan(1/2*a)^2))*tan(1/2*b*x)*tan(1/2*a)^4 + b*c*tan(1/
2*b*x)^4 - 8*b*c*tan(1/2*b*x)^3*tan(1/2*a) + 2*d*tan(1/2*b*x)^4*tan(1/2*a) - 12*b*c*tan(1/2*b*x)^2*tan(1/2*a)^
2 + 12*d*tan(1/2*b*x)^3*tan(1/2*a)^2 - 8*b*c*tan(1/2*b*x)*tan(1/2*a)^3 + 12*d*tan(1/2*b*x)^2*tan(1/2*a)^3 + b*
c*tan(1/2*a)^4 + 2*d*tan(1/2*b*x)*tan(1/2*a)^4 + 6*b*d*x*tan(1/2*b*x)^2 + d*log(4*(tan(1/2*a)^2 + 1)/(tan(1/2*
b*x)^4*tan(1/2*a)^2 - 2*tan(1/2*b*x)^3*tan(1/2*a) + tan(1/2*b*x)^2*tan(1/2*a)^2 + tan(1/2*b*x)^2 - 2*tan(1/2*b
*x)*tan(1/2*a) + 1))*tan(1/2*b*x)^3 - d*log(4*(tan(1/2*a)^2 + 1)/(tan(1/2*b*x)^4 + 2*tan(1/2*b*x)^3*tan(1/2*a)
+ tan(1/2*b*x)^2*tan(1/2*a)^2 + tan(1/2*b*x)^2 + 2*tan(1/2*b*x)*tan(1/2*a) + tan(1/2*a)^2))*tan(1/2*b*x)^3 +
8*b*d*x*tan(1/2*b*x)*tan(1/2*a) + 6*b*d*x*tan(1/2*a)^2 + d*log(4*(tan(1/2*a)^2 + 1)/(tan(1/2*b*x)^4*tan(1/2*a)
^2 - 2*tan(1/2*b*x)^3*tan(1/2*a) + tan(1/2*b*x)^2*tan(1/2*a)^2 + tan(1/2*b*x)^2 - 2*tan(1/2*b*x)*tan(1/2*a) +
1))*tan(1/2*a)^3 - d*log(4*(tan(1/2*a)^2 + 1)/(tan(1/2*b*x)^4 + 2*tan(1/2*b*x)^3*tan(1/2*a) + tan(1/2*b*x)^2*t
an(1/2*a)^2 + tan(1/2*b*x)^2 + 2*tan(1/2*b*x)*tan(1/2*a) + tan(1/2*a)^2))*tan(1/2*a)^3 + 6*b*c*tan(1/2*b*x)^2
- 2*d*tan(1/2*b*x)^3 + 8*b*c*tan(1/2*b*x)*tan(1/2*a) - 12*d*tan(1/2*b*x)^2*tan(1/2*a) + 6*b*c*tan(1/2*a)^2 - 1
2*d*tan(1/2*b*x)*tan(1/2*a)^2 - 2*d*tan(1/2*a)^3 + b*d*x + d*log(4*(tan(1/2*a)^2 + 1)/(tan(1/2*b*x)^4*tan(1/2*
a)^2 - 2*tan(1/2*b*x)^3*tan(1/2*a) + tan(1/2*b*x)^2*tan(1/2*a)^2 + tan(1/2*b*x)^2 - 2*tan(1/2*b*x)*tan(1/2*a)
+ 1))*tan(1/2*b*x) - d*log(4*(tan(1/2*a)^2 + 1)/(tan(1/2*b*x)^4 + 2*tan(1/2*b*x)^3*tan(1/2*a) + tan(1/2*b*x)^2
*tan(1/2*a)^2 + tan(1/2*b*x)^2 + 2*tan(1/2*b*x)*tan(1/2*a) + tan(1/2*a)^2))*tan(1/2*b*x) + d*log(4*(tan(1/2*a)
^2 + 1)/(tan(1/2*b*x)^4*tan(1/2*a)^2 - 2*tan(1/2*b*x)^3*tan(1/2*a) + tan(1/2*b*x)^2*tan(1/2*a)^2 + tan(1/2*b*x
)^2 - 2*tan(1/2*b*x)*tan(1/2*a) + 1))*tan(1/2*a) - d*log(4*(tan(1/2*a)^2 + 1)/(tan(1/2*b*x)^4 + 2*tan(1/2*b*x)
^3*tan(1/2*a) + tan(1/2*b*x)^2*tan(1/2*a)^2 + tan(1/2*b*x)^2 + 2*tan(1/2*b*x)*tan(1/2*a) + tan(1/2*a)^2))*tan(
1/2*a) + b*c + 2*d*tan(1/2*b*x) + 2*d*tan(1/2*a))/(b^2*tan(1/2*b*x)^4*tan(1/2*a)^3 + b^2*tan(1/2*b*x)^3*tan(1/
2*a)^4 + b^2*tan(1/2*b*x)^4*tan(1/2*a) + b^2*tan(1/2*b*x)*tan(1/2*a)^4 - b^2*tan(1/2*b*x)^3 - b^2*tan(1/2*a)^3
- b^2*tan(1/2*b*x) - b^2*tan(1/2*a))