3.540 \(\int \cos (c+d x) (a+b \tan (c+d x))^3 \, dx\)
Optimal. Leaf size=84 \[ -\frac{b \sec (c+d x) \left (2 \left (a^2-b^2\right )+a b \tan (c+d x)\right )}{d}+\frac{3 a b^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{\cos (c+d x) (b-a \tan (c+d x)) (a+b \tan (c+d x))^2}{d} \]
[Out]
(3*a*b^2*ArcTanh[Sin[c + d*x]])/d - (Cos[c + d*x]*(b - a*Tan[c + d*x])*(a + b*Tan[c + d*x])^2)/d - (b*Sec[c +
d*x]*(2*(a^2 - b^2) + a*b*Tan[c + d*x]))/d
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Rubi [A] time = 0.0801492, antiderivative size = 102, normalized size of antiderivative =
1.21, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules
used = {3512, 739, 780, 215} \[ -\frac{b \sec (c+d x) \left (2 \left (a^2-b^2\right )+a b \tan (c+d x)\right )}{d}+\frac{3 a b^2 \cos (c+d x) \sqrt{\sec ^2(c+d x)} \sinh ^{-1}(\tan (c+d x))}{d}-\frac{\cos (c+d x) (b-a \tan (c+d x)) (a+b \tan (c+d x))^2}{d} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]*(a + b*Tan[c + d*x])^3,x]
[Out]
(3*a*b^2*ArcSinh[Tan[c + d*x]]*Cos[c + d*x]*Sqrt[Sec[c + d*x]^2])/d - (Cos[c + d*x]*(b - a*Tan[c + d*x])*(a +
b*Tan[c + d*x])^2)/d - (b*Sec[c + d*x]*(2*(a^2 - b^2) + a*b*Tan[c + d*x]))/d
Rule 3512
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(d^(2
*IntPart[m/2])*(d*Sec[e + f*x])^(2*FracPart[m/2]))/(b*f*(Sec[e + f*x]^2)^FracPart[m/2]), Subst[Int[(a + x)^n*(
1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && NeQ[a^2 + b^2, 0] &&
!IntegerQ[m/2]
Rule 739
Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(a*e - c*d*x)*(a
+ c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] + Dist[1/((p + 1)*(-2*a*c)), Int[(d + e*x)^(m - 2)*Simp[a*e^2*(m - 1) -
c*d^2*(2*p + 3) - d*c*e*(m + 2*p + 2)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^
2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]
Rule 780
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && !LeQ[p, -1]
Rule 215
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]
Rubi steps
\begin{align*} \int \cos (c+d x) (a+b \tan (c+d x))^3 \, dx &=\frac{\left (\cos (c+d x) \sqrt{\sec ^2(c+d x)}\right ) \operatorname{Subst}\left (\int \frac{(a+x)^3}{\left (1+\frac{x^2}{b^2}\right )^{3/2}} \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=-\frac{\cos (c+d x) (b-a \tan (c+d x)) (a+b \tan (c+d x))^2}{d}+\frac{\left (b \cos (c+d x) \sqrt{\sec ^2(c+d x)}\right ) \operatorname{Subst}\left (\int \frac{(a+x) \left (2-\frac{2 a x}{b^2}\right )}{\sqrt{1+\frac{x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=-\frac{\cos (c+d x) (b-a \tan (c+d x)) (a+b \tan (c+d x))^2}{d}-\frac{b \sec (c+d x) \left (2 \left (a^2-b^2\right )+a b \tan (c+d x)\right )}{d}+\frac{\left (3 a b \cos (c+d x) \sqrt{\sec ^2(c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac{3 a b^2 \sinh ^{-1}(\tan (c+d x)) \cos (c+d x) \sqrt{\sec ^2(c+d x)}}{d}-\frac{\cos (c+d x) (b-a \tan (c+d x)) (a+b \tan (c+d x))^2}{d}-\frac{b \sec (c+d x) \left (2 \left (a^2-b^2\right )+a b \tan (c+d x)\right )}{d}\\ \end{align*}
Mathematica [A] time = 1.06517, size = 131, normalized size = 1.56 \[ \frac{\sec (c+d x) \left (\left (b^3-3 a^2 b\right ) \cos (2 (c+d x))-3 a^2 b+a^3 \sin (2 (c+d x))-3 a b^2 \sin (2 (c+d x))-6 a b^2 \cos (c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+3 b^3\right )}{2 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]*(a + b*Tan[c + d*x])^3,x]
[Out]
(Sec[c + d*x]*(-3*a^2*b + 3*b^3 + (-3*a^2*b + b^3)*Cos[2*(c + d*x)] - 6*a*b^2*Cos[c + d*x]*(Log[Cos[(c + d*x)/
2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + a^3*Sin[2*(c + d*x)] - 3*a*b^2*Sin[2*(c +
d*x)]))/(2*d)
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Maple [A] time = 0.049, size = 126, normalized size = 1.5 \begin{align*}{\frac{{b}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d\cos \left ( dx+c \right ) }}+{\frac{{b}^{3}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d}}+2\,{\frac{{b}^{3}\cos \left ( dx+c \right ) }{d}}-3\,{\frac{a{b}^{2}\sin \left ( dx+c \right ) }{d}}+3\,{\frac{a{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}-3\,{\frac{b{a}^{2}\cos \left ( dx+c \right ) }{d}}+{\frac{{a}^{3}\sin \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)*(a+b*tan(d*x+c))^3,x)
[Out]
1/d*b^3*sin(d*x+c)^4/cos(d*x+c)+1/d*b^3*cos(d*x+c)*sin(d*x+c)^2+2/d*b^3*cos(d*x+c)-3/d*a*b^2*sin(d*x+c)+3/d*a*
b^2*ln(sec(d*x+c)+tan(d*x+c))-3/d*b*a^2*cos(d*x+c)+a^3*sin(d*x+c)/d
________________________________________________________________________________________
Maxima [A] time = 1.1417, size = 113, normalized size = 1.35 \begin{align*} \frac{2 \, b^{3}{\left (\frac{1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )} + 3 \, a b^{2}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right ) - 2 \, \sin \left (d x + c\right )\right )} - 6 \, a^{2} b \cos \left (d x + c\right ) + 2 \, a^{3} \sin \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)*(a+b*tan(d*x+c))^3,x, algorithm="maxima")
[Out]
1/2*(2*b^3*(1/cos(d*x + c) + cos(d*x + c)) + 3*a*b^2*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1) - 2*sin(d*
x + c)) - 6*a^2*b*cos(d*x + c) + 2*a^3*sin(d*x + c))/d
________________________________________________________________________________________
Fricas [A] time = 2.04388, size = 273, normalized size = 3.25 \begin{align*} \frac{3 \, a b^{2} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, a b^{2} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, b^{3} - 2 \,{\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)*(a+b*tan(d*x+c))^3,x, algorithm="fricas")
[Out]
1/2*(3*a*b^2*cos(d*x + c)*log(sin(d*x + c) + 1) - 3*a*b^2*cos(d*x + c)*log(-sin(d*x + c) + 1) + 2*b^3 - 2*(3*a
^2*b - b^3)*cos(d*x + c)^2 + 2*(a^3 - 3*a*b^2)*cos(d*x + c)*sin(d*x + c))/(d*cos(d*x + c))
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tan{\left (c + d x \right )}\right )^{3} \cos{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)*(a+b*tan(d*x+c))**3,x)
[Out]
Integral((a + b*tan(c + d*x))**3*cos(c + d*x), x)
________________________________________________________________________________________
Giac [B] time = 6.00224, size = 6400, normalized size = 76.19 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)*(a+b*tan(d*x+c))^3,x, algorithm="giac")
[Out]
-1/4*(3*pi*a^2*b*sgn(tan(1/2*d*x)^2*tan(1/2*c)^2 - tan(1/2*d*x)^2 - 4*tan(1/2*d*x)*tan(1/2*c) - tan(1/2*c)^2 +
1)*tan(1/2*d*x)^4*tan(1/2*c)^4 - 3*pi*a^2*b*tan(1/2*d*x)^4*tan(1/2*c)^4 - 6*a^2*b*arctan((tan(1/2*d*x)*tan(1/
2*c) + tan(1/2*d*x) + tan(1/2*c) - 1)/(tan(1/2*d*x)*tan(1/2*c) - tan(1/2*d*x) - tan(1/2*c) - 1))*tan(1/2*d*x)^
4*tan(1/2*c)^4 - 6*a^2*b*arctan((tan(1/2*d*x)*tan(1/2*c) - tan(1/2*d*x) - tan(1/2*c) - 1)/(tan(1/2*d*x)*tan(1/
2*c) + tan(1/2*d*x) + tan(1/2*c) - 1))*tan(1/2*d*x)^4*tan(1/2*c)^4 + 6*a*b^2*log(2*(tan(1/2*c)^2 + 1)/(tan(1/2
*d*x)^4*tan(1/2*c)^2 + 2*tan(1/2*d*x)^4*tan(1/2*c) + 2*tan(1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/
2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)^3 + 2*tan(1/2*d*x)*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 - 2*t
an(1/2*d*x) - 2*tan(1/2*c) + 1))*tan(1/2*d*x)^4*tan(1/2*c)^4 - 6*a*b^2*log(2*(tan(1/2*c)^2 + 1)/(tan(1/2*d*x)^
4*tan(1/2*c)^2 - 2*tan(1/2*d*x)^4*tan(1/2*c) - 2*tan(1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*x)
^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^3 - 2*tan(1/2*d*x)*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/2
*d*x) + 2*tan(1/2*c) + 1))*tan(1/2*d*x)^4*tan(1/2*c)^4 - 12*pi*a^2*b*sgn(tan(1/2*d*x)^2*tan(1/2*c)^2 - tan(1/2
*d*x)^2 - 4*tan(1/2*d*x)*tan(1/2*c) - tan(1/2*c)^2 + 1)*tan(1/2*d*x)^3*tan(1/2*c)^3 + 12*a^2*b*tan(1/2*d*x)^4*
tan(1/2*c)^4 - 8*b^3*tan(1/2*d*x)^4*tan(1/2*c)^4 + 12*pi*a^2*b*tan(1/2*d*x)^3*tan(1/2*c)^3 + 24*a^2*b*arctan((
tan(1/2*d*x)*tan(1/2*c) + tan(1/2*d*x) + tan(1/2*c) - 1)/(tan(1/2*d*x)*tan(1/2*c) - tan(1/2*d*x) - tan(1/2*c)
- 1))*tan(1/2*d*x)^3*tan(1/2*c)^3 + 24*a^2*b*arctan((tan(1/2*d*x)*tan(1/2*c) - tan(1/2*d*x) - tan(1/2*c) - 1)/
(tan(1/2*d*x)*tan(1/2*c) + tan(1/2*d*x) + tan(1/2*c) - 1))*tan(1/2*d*x)^3*tan(1/2*c)^3 - 24*a*b^2*log(2*(tan(1
/2*c)^2 + 1)/(tan(1/2*d*x)^4*tan(1/2*c)^2 + 2*tan(1/2*d*x)^4*tan(1/2*c) + 2*tan(1/2*d*x)^3*tan(1/2*c)^2 + tan(
1/2*d*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)^3 + 2*tan(1/2*d*x)*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2
+ tan(1/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1))*tan(1/2*d*x)^3*tan(1/2*c)^3 + 24*a*b^2*log(2*(tan(1/2*c)
^2 + 1)/(tan(1/2*d*x)^4*tan(1/2*c)^2 - 2*tan(1/2*d*x)^4*tan(1/2*c) - 2*tan(1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d
*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^3 - 2*tan(1/2*d*x)*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2 + ta
n(1/2*c)^2 + 2*tan(1/2*d*x) + 2*tan(1/2*c) + 1))*tan(1/2*d*x)^3*tan(1/2*c)^3 + 8*a^3*tan(1/2*d*x)^4*tan(1/2*c)
^3 - 24*a*b^2*tan(1/2*d*x)^4*tan(1/2*c)^3 + 8*a^3*tan(1/2*d*x)^3*tan(1/2*c)^4 - 24*a*b^2*tan(1/2*d*x)^3*tan(1/
2*c)^4 - 3*pi*a^2*b*sgn(tan(1/2*d*x)^2*tan(1/2*c)^2 - tan(1/2*d*x)^2 - 4*tan(1/2*d*x)*tan(1/2*c) - tan(1/2*c)^
2 + 1)*tan(1/2*d*x)^4 - 12*pi*a^2*b*sgn(tan(1/2*d*x)^2*tan(1/2*c)^2 - tan(1/2*d*x)^2 - 4*tan(1/2*d*x)*tan(1/2*
c) - tan(1/2*c)^2 + 1)*tan(1/2*d*x)^3*tan(1/2*c) - 24*a^2*b*tan(1/2*d*x)^4*tan(1/2*c)^2 - 12*pi*a^2*b*sgn(tan(
1/2*d*x)^2*tan(1/2*c)^2 - tan(1/2*d*x)^2 - 4*tan(1/2*d*x)*tan(1/2*c) - tan(1/2*c)^2 + 1)*tan(1/2*d*x)*tan(1/2*
c)^3 - 96*a^2*b*tan(1/2*d*x)^3*tan(1/2*c)^3 + 32*b^3*tan(1/2*d*x)^3*tan(1/2*c)^3 - 3*pi*a^2*b*sgn(tan(1/2*d*x)
^2*tan(1/2*c)^2 - tan(1/2*d*x)^2 - 4*tan(1/2*d*x)*tan(1/2*c) - tan(1/2*c)^2 + 1)*tan(1/2*c)^4 - 24*a^2*b*tan(1
/2*d*x)^2*tan(1/2*c)^4 + 3*pi*a^2*b*tan(1/2*d*x)^4 + 6*a^2*b*arctan((tan(1/2*d*x)*tan(1/2*c) + tan(1/2*d*x) +
tan(1/2*c) - 1)/(tan(1/2*d*x)*tan(1/2*c) - tan(1/2*d*x) - tan(1/2*c) - 1))*tan(1/2*d*x)^4 + 6*a^2*b*arctan((ta
n(1/2*d*x)*tan(1/2*c) - tan(1/2*d*x) - tan(1/2*c) - 1)/(tan(1/2*d*x)*tan(1/2*c) + tan(1/2*d*x) + tan(1/2*c) -
1))*tan(1/2*d*x)^4 - 6*a*b^2*log(2*(tan(1/2*c)^2 + 1)/(tan(1/2*d*x)^4*tan(1/2*c)^2 + 2*tan(1/2*d*x)^4*tan(1/2*
c) + 2*tan(1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)^3 + 2*tan
(1/2*d*x)*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1))*tan(1/2*d*x)^4
+ 6*a*b^2*log(2*(tan(1/2*c)^2 + 1)/(tan(1/2*d*x)^4*tan(1/2*c)^2 - 2*tan(1/2*d*x)^4*tan(1/2*c) - 2*tan(1/2*d*x)
^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^3 - 2*tan(1/2*d*x)*tan(1/2*c
)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/2*d*x) + 2*tan(1/2*c) + 1))*tan(1/2*d*x)^4 + 12*pi*a^2*b*tan(1
/2*d*x)^3*tan(1/2*c) + 24*a^2*b*arctan((tan(1/2*d*x)*tan(1/2*c) + tan(1/2*d*x) + tan(1/2*c) - 1)/(tan(1/2*d*x)
*tan(1/2*c) - tan(1/2*d*x) - tan(1/2*c) - 1))*tan(1/2*d*x)^3*tan(1/2*c) + 24*a^2*b*arctan((tan(1/2*d*x)*tan(1/
2*c) - tan(1/2*d*x) - tan(1/2*c) - 1)/(tan(1/2*d*x)*tan(1/2*c) + tan(1/2*d*x) + tan(1/2*c) - 1))*tan(1/2*d*x)^
3*tan(1/2*c) - 24*a*b^2*log(2*(tan(1/2*c)^2 + 1)/(tan(1/2*d*x)^4*tan(1/2*c)^2 + 2*tan(1/2*d*x)^4*tan(1/2*c) +
2*tan(1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)^3 + 2*tan(1/2*
d*x)*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1))*tan(1/2*d*x)^3*tan(1
/2*c) + 24*a*b^2*log(2*(tan(1/2*c)^2 + 1)/(tan(1/2*d*x)^4*tan(1/2*c)^2 - 2*tan(1/2*d*x)^4*tan(1/2*c) - 2*tan(1
/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^3 - 2*tan(1/2*d*x)*ta
n(1/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/2*d*x) + 2*tan(1/2*c) + 1))*tan(1/2*d*x)^3*tan(1/2*c) -
8*a^3*tan(1/2*d*x)^4*tan(1/2*c) + 24*a*b^2*tan(1/2*d*x)^4*tan(1/2*c) - 48*a^3*tan(1/2*d*x)^3*tan(1/2*c)^2 + 1
44*a*b^2*tan(1/2*d*x)^3*tan(1/2*c)^2 + 12*pi*a^2*b*tan(1/2*d*x)*tan(1/2*c)^3 + 24*a^2*b*arctan((tan(1/2*d*x)*t
an(1/2*c) + tan(1/2*d*x) + tan(1/2*c) - 1)/(tan(1/2*d*x)*tan(1/2*c) - tan(1/2*d*x) - tan(1/2*c) - 1))*tan(1/2*
d*x)*tan(1/2*c)^3 + 24*a^2*b*arctan((tan(1/2*d*x)*tan(1/2*c) - tan(1/2*d*x) - tan(1/2*c) - 1)/(tan(1/2*d*x)*ta
n(1/2*c) + tan(1/2*d*x) + tan(1/2*c) - 1))*tan(1/2*d*x)*tan(1/2*c)^3 - 24*a*b^2*log(2*(tan(1/2*c)^2 + 1)/(tan(
1/2*d*x)^4*tan(1/2*c)^2 + 2*tan(1/2*d*x)^4*tan(1/2*c) + 2*tan(1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan
(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)^3 + 2*tan(1/2*d*x)*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 -
2*tan(1/2*d*x) - 2*tan(1/2*c) + 1))*tan(1/2*d*x)*tan(1/2*c)^3 + 24*a*b^2*log(2*(tan(1/2*c)^2 + 1)/(tan(1/2*d*x
)^4*tan(1/2*c)^2 - 2*tan(1/2*d*x)^4*tan(1/2*c) - 2*tan(1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*
x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^3 - 2*tan(1/2*d*x)*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1
/2*d*x) + 2*tan(1/2*c) + 1))*tan(1/2*d*x)*tan(1/2*c)^3 - 48*a^3*tan(1/2*d*x)^2*tan(1/2*c)^3 + 144*a*b^2*tan(1/
2*d*x)^2*tan(1/2*c)^3 + 3*pi*a^2*b*tan(1/2*c)^4 + 6*a^2*b*arctan((tan(1/2*d*x)*tan(1/2*c) + tan(1/2*d*x) + tan
(1/2*c) - 1)/(tan(1/2*d*x)*tan(1/2*c) - tan(1/2*d*x) - tan(1/2*c) - 1))*tan(1/2*c)^4 + 6*a^2*b*arctan((tan(1/2
*d*x)*tan(1/2*c) - tan(1/2*d*x) - tan(1/2*c) - 1)/(tan(1/2*d*x)*tan(1/2*c) + tan(1/2*d*x) + tan(1/2*c) - 1))*t
an(1/2*c)^4 - 6*a*b^2*log(2*(tan(1/2*c)^2 + 1)/(tan(1/2*d*x)^4*tan(1/2*c)^2 + 2*tan(1/2*d*x)^4*tan(1/2*c) + 2*
tan(1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)^3 + 2*tan(1/2*d*
x)*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1))*tan(1/2*c)^4 + 6*a*b^2
*log(2*(tan(1/2*c)^2 + 1)/(tan(1/2*d*x)^4*tan(1/2*c)^2 - 2*tan(1/2*d*x)^4*tan(1/2*c) - 2*tan(1/2*d*x)^3*tan(1/
2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^3 - 2*tan(1/2*d*x)*tan(1/2*c)^2 + 2*t
an(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/2*d*x) + 2*tan(1/2*c) + 1))*tan(1/2*c)^4 - 8*a^3*tan(1/2*d*x)*tan(1/2*c
)^4 + 24*a*b^2*tan(1/2*d*x)*tan(1/2*c)^4 + 12*a^2*b*tan(1/2*d*x)^4 - 8*b^3*tan(1/2*d*x)^4 - 12*pi*a^2*b*sgn(ta
n(1/2*d*x)^2*tan(1/2*c)^2 - tan(1/2*d*x)^2 - 4*tan(1/2*d*x)*tan(1/2*c) - tan(1/2*c)^2 + 1)*tan(1/2*d*x)*tan(1/
2*c) + 96*a^2*b*tan(1/2*d*x)^3*tan(1/2*c) - 32*b^3*tan(1/2*d*x)^3*tan(1/2*c) + 240*a^2*b*tan(1/2*d*x)^2*tan(1/
2*c)^2 - 96*b^3*tan(1/2*d*x)^2*tan(1/2*c)^2 + 96*a^2*b*tan(1/2*d*x)*tan(1/2*c)^3 - 32*b^3*tan(1/2*d*x)*tan(1/2
*c)^3 + 12*a^2*b*tan(1/2*c)^4 - 8*b^3*tan(1/2*c)^4 + 8*a^3*tan(1/2*d*x)^3 - 24*a*b^2*tan(1/2*d*x)^3 + 12*pi*a^
2*b*tan(1/2*d*x)*tan(1/2*c) + 24*a^2*b*arctan((tan(1/2*d*x)*tan(1/2*c) + tan(1/2*d*x) + tan(1/2*c) - 1)/(tan(1
/2*d*x)*tan(1/2*c) - tan(1/2*d*x) - tan(1/2*c) - 1))*tan(1/2*d*x)*tan(1/2*c) + 24*a^2*b*arctan((tan(1/2*d*x)*t
an(1/2*c) - tan(1/2*d*x) - tan(1/2*c) - 1)/(tan(1/2*d*x)*tan(1/2*c) + tan(1/2*d*x) + tan(1/2*c) - 1))*tan(1/2*
d*x)*tan(1/2*c) - 24*a*b^2*log(2*(tan(1/2*c)^2 + 1)/(tan(1/2*d*x)^4*tan(1/2*c)^2 + 2*tan(1/2*d*x)^4*tan(1/2*c)
+ 2*tan(1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)^3 + 2*tan(1
/2*d*x)*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1))*tan(1/2*d*x)*tan(
1/2*c) + 24*a*b^2*log(2*(tan(1/2*c)^2 + 1)/(tan(1/2*d*x)^4*tan(1/2*c)^2 - 2*tan(1/2*d*x)^4*tan(1/2*c) - 2*tan(
1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^3 - 2*tan(1/2*d*x)*t
an(1/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/2*d*x) + 2*tan(1/2*c) + 1))*tan(1/2*d*x)*tan(1/2*c) +
48*a^3*tan(1/2*d*x)^2*tan(1/2*c) - 144*a*b^2*tan(1/2*d*x)^2*tan(1/2*c) + 48*a^3*tan(1/2*d*x)*tan(1/2*c)^2 - 14
4*a*b^2*tan(1/2*d*x)*tan(1/2*c)^2 + 8*a^3*tan(1/2*c)^3 - 24*a*b^2*tan(1/2*c)^3 + 3*pi*a^2*b*sgn(tan(1/2*d*x)^2
*tan(1/2*c)^2 - tan(1/2*d*x)^2 - 4*tan(1/2*d*x)*tan(1/2*c) - tan(1/2*c)^2 + 1) - 24*a^2*b*tan(1/2*d*x)^2 - 96*
a^2*b*tan(1/2*d*x)*tan(1/2*c) + 32*b^3*tan(1/2*d*x)*tan(1/2*c) - 24*a^2*b*tan(1/2*c)^2 - 3*pi*a^2*b - 6*a^2*b*
arctan((tan(1/2*d*x)*tan(1/2*c) + tan(1/2*d*x) + tan(1/2*c) - 1)/(tan(1/2*d*x)*tan(1/2*c) - tan(1/2*d*x) - tan
(1/2*c) - 1)) - 6*a^2*b*arctan((tan(1/2*d*x)*tan(1/2*c) - tan(1/2*d*x) - tan(1/2*c) - 1)/(tan(1/2*d*x)*tan(1/2
*c) + tan(1/2*d*x) + tan(1/2*c) - 1)) + 6*a*b^2*log(2*(tan(1/2*c)^2 + 1)/(tan(1/2*d*x)^4*tan(1/2*c)^2 + 2*tan(
1/2*d*x)^4*tan(1/2*c) + 2*tan(1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan
(1/2*d*x)^3 + 2*tan(1/2*d*x)*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) +
1)) - 6*a*b^2*log(2*(tan(1/2*c)^2 + 1)/(tan(1/2*d*x)^4*tan(1/2*c)^2 - 2*tan(1/2*d*x)^4*tan(1/2*c) - 2*tan(1/2*
d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^3 - 2*tan(1/2*d*x)*tan(1
/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/2*d*x) + 2*tan(1/2*c) + 1)) - 8*a^3*tan(1/2*d*x) + 24*a*b^
2*tan(1/2*d*x) - 8*a^3*tan(1/2*c) + 24*a*b^2*tan(1/2*c) + 12*a^2*b - 8*b^3)/(d*tan(1/2*d*x)^4*tan(1/2*c)^4 - 4
*d*tan(1/2*d*x)^3*tan(1/2*c)^3 - d*tan(1/2*d*x)^4 - 4*d*tan(1/2*d*x)^3*tan(1/2*c) - 4*d*tan(1/2*d*x)*tan(1/2*c
)^3 - d*tan(1/2*c)^4 - 4*d*tan(1/2*d*x)*tan(1/2*c) + d)