3.22 \(\int \sin ^4(c+d x) (a+b \tan (c+d x))^2 \, dx\)
Optimal. Leaf size=113 \[ \frac{3}{8} x \left (a^2-5 b^2\right )+\frac{\cos ^2(c+d x) (7 b-5 a \tan (c+d x)) (a+b \tan (c+d x))}{8 d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{\sin (c+d x) \cos ^3(c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac{b^2 \tan (c+d x)}{d} \]
[Out]
(3*(a^2 - 5*b^2)*x)/8 - (2*a*b*Log[Cos[c + d*x]])/d + (b^2*Tan[c + d*x])/d + (Cos[c + d*x]^2*(7*b - 5*a*Tan[c
+ d*x])*(a + b*Tan[c + d*x]))/(8*d) + (Cos[c + d*x]^3*Sin[c + d*x]*(a + b*Tan[c + d*x])^2)/(4*d)
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Rubi [A] time = 0.185813, antiderivative size = 113, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used =
{3516, 1645, 1810, 635, 203, 260} \[ \frac{3}{8} x \left (a^2-5 b^2\right )+\frac{\cos ^2(c+d x) (7 b-5 a \tan (c+d x)) (a+b \tan (c+d x))}{8 d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{\sin (c+d x) \cos ^3(c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac{b^2 \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
Int[Sin[c + d*x]^4*(a + b*Tan[c + d*x])^2,x]
[Out]
(3*(a^2 - 5*b^2)*x)/8 - (2*a*b*Log[Cos[c + d*x]])/d + (b^2*Tan[c + d*x])/d + (Cos[c + d*x]^2*(7*b - 5*a*Tan[c
+ d*x])*(a + b*Tan[c + d*x]))/(8*d) + (Cos[c + d*x]^3*Sin[c + d*x]*(a + b*Tan[c + d*x])^2)/(4*d)
Rule 3516
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[b/f, Subst[Int
[(x^m*(a + x)^n)/(b^2 + x^2)^(m/2 + 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[m/
2]
Rule 1645
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq,
a + c*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + c
*x^2, x], x, 1]}, Simp[((d + e*x)^m*(a + c*x^2)^(p + 1)*(a*g - c*f*x))/(2*a*c*(p + 1)), x] + Dist[1/(2*a*c*(p
+ 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*ExpandToSum[2*a*c*(p + 1)*(d + e*x)*Q - a*e*g*m + c*d*f*(2*p
+ 3) + c*e*f*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] &
& LtQ[p, -1] && GtQ[m, 0] && !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Rule 1810
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Rule 635
Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] && !NiceSqrtQ[-(a*c)]
Rule 203
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rule 260
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]
Rubi steps
\begin{align*} \int \sin ^4(c+d x) (a+b \tan (c+d x))^2 \, dx &=\frac{b \operatorname{Subst}\left (\int \frac{x^4 (a+x)^2}{\left (b^2+x^2\right )^3} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac{\cos ^3(c+d x) \sin (c+d x) (a+b \tan (c+d x))^2}{4 d}-\frac{\operatorname{Subst}\left (\int \frac{(a+x) \left (a b^4+3 b^4 x-4 a b^2 x^2-4 b^2 x^3\right )}{\left (b^2+x^2\right )^2} \, dx,x,b \tan (c+d x)\right )}{4 b d}\\ &=\frac{\cos ^2(c+d x) (7 b-5 a \tan (c+d x)) (a+b \tan (c+d x))}{8 d}+\frac{\cos ^3(c+d x) \sin (c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac{\operatorname{Subst}\left (\int \frac{b^4 \left (3 a^2-7 b^2\right )+16 a b^4 x+8 b^4 x^2}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{8 b^3 d}\\ &=\frac{\cos ^2(c+d x) (7 b-5 a \tan (c+d x)) (a+b \tan (c+d x))}{8 d}+\frac{\cos ^3(c+d x) \sin (c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac{\operatorname{Subst}\left (\int \left (8 b^4+\frac{3 b^4 \left (a^2-5 b^2\right )+16 a b^4 x}{b^2+x^2}\right ) \, dx,x,b \tan (c+d x)\right )}{8 b^3 d}\\ &=\frac{b^2 \tan (c+d x)}{d}+\frac{\cos ^2(c+d x) (7 b-5 a \tan (c+d x)) (a+b \tan (c+d x))}{8 d}+\frac{\cos ^3(c+d x) \sin (c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac{\operatorname{Subst}\left (\int \frac{3 b^4 \left (a^2-5 b^2\right )+16 a b^4 x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{8 b^3 d}\\ &=\frac{b^2 \tan (c+d x)}{d}+\frac{\cos ^2(c+d x) (7 b-5 a \tan (c+d x)) (a+b \tan (c+d x))}{8 d}+\frac{\cos ^3(c+d x) \sin (c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac{(2 a b) \operatorname{Subst}\left (\int \frac{x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{d}+\frac{\left (3 b \left (a^2-5 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{8 d}\\ &=\frac{3}{8} \left (a^2-5 b^2\right ) x-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}+\frac{\cos ^2(c+d x) (7 b-5 a \tan (c+d x)) (a+b \tan (c+d x))}{8 d}+\frac{\cos ^3(c+d x) \sin (c+d x) (a+b \tan (c+d x))^2}{4 d}\\ \end{align*}
Mathematica [B] time = 3.19675, size = 240, normalized size = 2.12 \[ \frac{b \left (\frac{2 \left (3 b^2-2 a^2\right ) \sin (2 (c+d x))}{b}+\frac{4 \left (3 b^2-2 a^2\right ) \tan ^{-1}(\tan (c+d x))}{b}+4 \left (\frac{a^2-3 b^2}{\sqrt{-b^2}}+2 a\right ) \log \left (\sqrt{-b^2}-b \tan (c+d x)\right )+4 \left (\frac{3 b^2-a^2}{\sqrt{-b^2}}+2 a\right ) \log \left (\sqrt{-b^2}+b \tan (c+d x)\right )+\frac{3 \left (a^2-b^2\right ) \left (\sin (2 (c+d x))+2 \tan ^{-1}(\tan (c+d x))\right )}{2 b}+\frac{2 \left (a^2-b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{b}-4 a \cos ^4(c+d x)+16 a \cos ^2(c+d x)+8 b \tan (c+d x)\right )}{8 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Sin[c + d*x]^4*(a + b*Tan[c + d*x])^2,x]
[Out]
(b*((4*(-2*a^2 + 3*b^2)*ArcTan[Tan[c + d*x]])/b + 16*a*Cos[c + d*x]^2 - 4*a*Cos[c + d*x]^4 + 4*(2*a + (a^2 - 3
*b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]] + 4*(2*a + (-a^2 + 3*b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Ta
n[c + d*x]] + (2*(a^2 - b^2)*Cos[c + d*x]^3*Sin[c + d*x])/b + (2*(-2*a^2 + 3*b^2)*Sin[2*(c + d*x)])/b + (3*(a^
2 - b^2)*(2*ArcTan[Tan[c + d*x]] + Sin[2*(c + d*x)]))/(2*b) + 8*b*Tan[c + d*x]))/(8*d)
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Maple [A] time = 0.04, size = 204, normalized size = 1.8 \begin{align*}{\frac{{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{d\cos \left ( dx+c \right ) }}+{\frac{{b}^{2}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{d}}+{\frac{5\,{b}^{2}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{15\,{b}^{2}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{8\,d}}-{\frac{15\,{b}^{2}x}{8}}-{\frac{15\,{b}^{2}c}{8\,d}}-{\frac{ab \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{2\,d}}-{\frac{ab \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d}}-2\,{\frac{ab\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{2}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{4\,d}}-{\frac{3\,{a}^{2}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{8\,d}}+{\frac{3\,{a}^{2}x}{8}}+{\frac{3\,{a}^{2}c}{8\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sin(d*x+c)^4*(a+b*tan(d*x+c))^2,x)
[Out]
1/d*b^2*sin(d*x+c)^7/cos(d*x+c)+1/d*b^2*cos(d*x+c)*sin(d*x+c)^5+5/4/d*b^2*cos(d*x+c)*sin(d*x+c)^3+15/8/d*b^2*c
os(d*x+c)*sin(d*x+c)-15/8*b^2*x-15/8/d*b^2*c-1/2/d*a*b*sin(d*x+c)^4-1/d*a*b*sin(d*x+c)^2-2*a*b*ln(cos(d*x+c))/
d-1/4/d*a^2*cos(d*x+c)*sin(d*x+c)^3-3/8/d*a^2*cos(d*x+c)*sin(d*x+c)+3/8*a^2*x+3/8/d*a^2*c
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Maxima [A] time = 1.47685, size = 173, normalized size = 1.53 \begin{align*} \frac{8 \, a b \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 8 \, b^{2} \tan \left (d x + c\right ) + 3 \,{\left (a^{2} - 5 \, b^{2}\right )}{\left (d x + c\right )} + \frac{16 \, a b \tan \left (d x + c\right )^{2} -{\left (5 \, a^{2} - 9 \, b^{2}\right )} \tan \left (d x + c\right )^{3} + 12 \, a b -{\left (3 \, a^{2} - 7 \, b^{2}\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(d*x+c)^4*(a+b*tan(d*x+c))^2,x, algorithm="maxima")
[Out]
1/8*(8*a*b*log(tan(d*x + c)^2 + 1) + 8*b^2*tan(d*x + c) + 3*(a^2 - 5*b^2)*(d*x + c) + (16*a*b*tan(d*x + c)^2 -
(5*a^2 - 9*b^2)*tan(d*x + c)^3 + 12*a*b - (3*a^2 - 7*b^2)*tan(d*x + c))/(tan(d*x + c)^4 + 2*tan(d*x + c)^2 +
1))/d
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Fricas [A] time = 2.06341, size = 333, normalized size = 2.95 \begin{align*} -\frac{8 \, a b \cos \left (d x + c\right )^{5} - 32 \, a b \cos \left (d x + c\right )^{3} + 32 \, a b \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) -{\left (6 \,{\left (a^{2} - 5 \, b^{2}\right )} d x - 13 \, a b\right )} \cos \left (d x + c\right ) - 2 \,{\left (2 \,{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{4} -{\left (5 \, a^{2} - 9 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 8 \, b^{2}\right )} \sin \left (d x + c\right )}{16 \, d \cos \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(d*x+c)^4*(a+b*tan(d*x+c))^2,x, algorithm="fricas")
[Out]
-1/16*(8*a*b*cos(d*x + c)^5 - 32*a*b*cos(d*x + c)^3 + 32*a*b*cos(d*x + c)*log(-cos(d*x + c)) - (6*(a^2 - 5*b^2
)*d*x - 13*a*b)*cos(d*x + c) - 2*(2*(a^2 - b^2)*cos(d*x + c)^4 - (5*a^2 - 9*b^2)*cos(d*x + c)^2 + 8*b^2)*sin(d
*x + c))/(d*cos(d*x + c))
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(d*x+c)**4*(a+b*tan(d*x+c))**2,x)
[Out]
Timed out
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Giac [B] time = 6.66882, size = 7808, normalized size = 69.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(d*x+c)^4*(a+b*tan(d*x+c))^2,x, algorithm="giac")
[Out]
1/64*(3*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*
tan(c))*tan(d*x)^5*tan(c)^5 + 24*a^2*d*x*tan(d*x)^5*tan(c)^5 - 120*b^2*d*x*tan(d*x)^5*tan(c)^5 + 3*pi*b^2*sgn(
-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^5*tan(c)^5 + 6*pi*b^2*sgn(2*tan(d
*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^5*tan(c)^
3 - 3*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*ta
n(c))*tan(d*x)^4*tan(c)^4 + 6*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(
c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^3*tan(c)^5 + 6*b^2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*ta
n(d*x)^5*tan(c)^5 - 6*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^5*tan(c)^5 - 64*a*b*log(
4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*ta
n(c) + 1))*tan(d*x)^5*tan(c)^5 + 48*a^2*d*x*tan(d*x)^5*tan(c)^3 - 240*b^2*d*x*tan(d*x)^5*tan(c)^3 + 6*pi*b^2*s
gn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^5*tan(c)^3 - 24*a^2*d*x*tan(d*
x)^4*tan(c)^4 + 120*b^2*d*x*tan(d*x)^4*tan(c)^4 - 3*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*
tan(d*x) - 2*tan(c))*tan(d*x)^4*tan(c)^4 + 48*a^2*d*x*tan(d*x)^3*tan(c)^5 - 240*b^2*d*x*tan(d*x)^3*tan(c)^5 +
6*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^3*tan(c)^5 + 44*a*b*
tan(d*x)^5*tan(c)^5 + 3*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 +
2*tan(d*x) - 2*tan(c))*tan(d*x)^5*tan(c) - 6*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) +
2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^4*tan(c)^2 + 12*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*s
gn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^3*tan(c)^3 + 12*b^2*arctan((ta
n(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^5*tan(c)^3 - 12*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan
(c) + 1))*tan(d*x)^5*tan(c)^3 - 128*a*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(
d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^5*tan(c)^3 - 6*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^
2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^2*tan(c)^4 - 6*b^2*arc
tan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^4*tan(c)^4 + 6*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*
x)*tan(c) + 1))*tan(d*x)^4*tan(c)^4 + 64*a*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) +
tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^4*tan(c)^4 + 24*a^2*tan(d*x)^5*tan(c)^4 -
120*b^2*tan(d*x)^5*tan(c)^4 + 3*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*t
an(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)*tan(c)^5 + 12*b^2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*
tan(d*x)^3*tan(c)^5 - 12*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^3*tan(c)^5 - 128*a*b*
log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x
)*tan(c) + 1))*tan(d*x)^3*tan(c)^5 + 24*a^2*tan(d*x)^4*tan(c)^5 - 120*b^2*tan(d*x)^4*tan(c)^5 + 24*a^2*d*x*tan
(d*x)^5*tan(c) - 120*b^2*d*x*tan(d*x)^5*tan(c) + 3*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*t
an(d*x) - 2*tan(c))*tan(d*x)^5*tan(c) - 48*a^2*d*x*tan(d*x)^4*tan(c)^2 + 240*b^2*d*x*tan(d*x)^4*tan(c)^2 - 6*p
i*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^4*tan(c)^2 + 96*a^2*d*x
*tan(d*x)^3*tan(c)^3 - 480*b^2*d*x*tan(d*x)^3*tan(c)^3 + 12*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c
)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^3*tan(c)^3 + 24*a*b*tan(d*x)^5*tan(c)^3 - 48*a^2*d*x*tan(d*x)^2*tan(c)^4
+ 240*b^2*d*x*tan(d*x)^2*tan(c)^4 - 6*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*
tan(c))*tan(d*x)^2*tan(c)^4 - 172*a*b*tan(d*x)^4*tan(c)^4 + 24*a^2*d*x*tan(d*x)*tan(c)^5 - 120*b^2*d*x*tan(d*x
)*tan(c)^5 + 3*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)*tan(c)^
5 + 24*a*b*tan(d*x)^3*tan(c)^5 - 3*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)
*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^4 + 6*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(
c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^3*tan(c) + 6*b^2*arctan((tan(d*x) + tan(c))/(tan(d*
x)*tan(c) - 1))*tan(d*x)^5*tan(c) - 6*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^5*tan(c)
- 64*a*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 -
2*tan(d*x)*tan(c) + 1))*tan(d*x)^5*tan(c) - 12*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c)
+ 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^2*tan(c)^2 - 12*b^2*arctan((tan(d*x) + tan(c))/(tan(d
*x)*tan(c) - 1))*tan(d*x)^4*tan(c)^2 + 12*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^4*ta
n(c)^2 + 128*a*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d
*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^4*tan(c)^2 + 40*a^2*tan(d*x)^5*tan(c)^2 - 200*b^2*tan(d*x)^5*tan(c)^2
+ 6*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan
(c))*tan(d*x)*tan(c)^3 + 24*b^2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^3*tan(c)^3 - 24*b^2
*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^3*tan(c)^3 - 256*a*b*log(4*(tan(c)^2 + 1)/(tan(d*
x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^3*ta
n(c)^3 + 24*a^2*tan(d*x)^4*tan(c)^3 - 120*b^2*tan(d*x)^4*tan(c)^3 - 3*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sg
n(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(c)^4 - 12*b^2*arctan((tan(d*x) + tan
(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^2*tan(c)^4 + 12*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*ta
n(d*x)^2*tan(c)^4 + 128*a*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c
)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^2*tan(c)^4 + 24*a^2*tan(d*x)^3*tan(c)^4 - 120*b^2*tan(d*x)
^3*tan(c)^4 + 6*b^2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)*tan(c)^5 - 6*b^2*arctan(-(tan(d
*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)*tan(c)^5 - 64*a*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*
tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)*tan(c)^5 + 40*a^2*tan(
d*x)^2*tan(c)^5 - 200*b^2*tan(d*x)^2*tan(c)^5 - 24*a^2*d*x*tan(d*x)^4 + 120*b^2*d*x*tan(d*x)^4 - 3*pi*b^2*sgn(
-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^4 + 48*a^2*d*x*tan(d*x)^3*tan(c)
- 240*b^2*d*x*tan(d*x)^3*tan(c) + 6*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan
(c))*tan(d*x)^3*tan(c) - 52*a*b*tan(d*x)^5*tan(c) - 96*a^2*d*x*tan(d*x)^2*tan(c)^2 + 480*b^2*d*x*tan(d*x)^2*ta
n(c)^2 - 12*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^2*tan(c)^2
- 280*a*b*tan(d*x)^4*tan(c)^2 + 48*a^2*d*x*tan(d*x)*tan(c)^3 - 240*b^2*d*x*tan(d*x)*tan(c)^3 + 6*pi*b^2*sgn(-
2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)*tan(c)^3 - 16*a*b*tan(d*x)^3*tan(c
)^3 - 24*a^2*d*x*tan(c)^4 + 120*b^2*d*x*tan(c)^4 - 3*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2
*tan(d*x) - 2*tan(c))*tan(c)^4 - 280*a*b*tan(d*x)^2*tan(c)^4 - 52*a*b*tan(d*x)*tan(c)^5 - 6*pi*b^2*sgn(2*tan(d
*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^2 - 6*b^2
*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^4 + 6*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*ta
n(c) + 1))*tan(d*x)^4 + 64*a*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*ta
n(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^4 - 64*b^2*tan(d*x)^5 + 3*pi*b^2*sgn(2*tan(d*x)^2*tan(c
)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)*tan(c) + 12*b^2*arct
an((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^3*tan(c) - 12*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)
*tan(c) + 1))*tan(d*x)^3*tan(c) - 128*a*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + ta
n(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^3*tan(c) - 80*a^2*tan(d*x)^4*tan(c) + 80*b^2
*tan(d*x)^4*tan(c) - 6*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 +
2*tan(d*x) - 2*tan(c))*tan(c)^2 - 24*b^2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^2*tan(c)^2
+ 24*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^2*tan(c)^2 + 256*a*b*log(4*(tan(c)^2 + 1
)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(
d*x)^2*tan(c)^2 - 96*a^2*tan(d*x)^3*tan(c)^2 - 160*b^2*tan(d*x)^3*tan(c)^2 + 12*b^2*arctan((tan(d*x) + tan(c))
/(tan(d*x)*tan(c) - 1))*tan(d*x)*tan(c)^3 - 12*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)
*tan(c)^3 - 128*a*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + ta
n(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)*tan(c)^3 - 96*a^2*tan(d*x)^2*tan(c)^3 - 160*b^2*tan(d*x)^2*tan(c)^
3 - 6*b^2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(c)^4 + 6*b^2*arctan(-(tan(d*x) - tan(c))/(tan(
d*x)*tan(c) + 1))*tan(c)^4 + 64*a*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)
^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(c)^4 - 80*a^2*tan(d*x)*tan(c)^4 + 80*b^2*tan(d*x)*tan(c
)^4 - 64*b^2*tan(c)^5 - 48*a^2*d*x*tan(d*x)^2 + 240*b^2*d*x*tan(d*x)^2 - 6*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2
*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^2 + 52*a*b*tan(d*x)^4 + 24*a^2*d*x*tan(d*x)*tan(c) - 120*
b^2*d*x*tan(d*x)*tan(c) + 3*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan
(d*x)*tan(c) + 280*a*b*tan(d*x)^3*tan(c) - 48*a^2*d*x*tan(c)^2 + 240*b^2*d*x*tan(c)^2 - 6*pi*b^2*sgn(-2*tan(d*
x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(c)^2 + 16*a*b*tan(d*x)^2*tan(c)^2 + 280*a*b*tan
(d*x)*tan(c)^3 + 52*a*b*tan(c)^4 - 3*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*
x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c)) - 12*b^2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^2 + 1
2*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^2 + 128*a*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4
*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^2 + 40*a
^2*tan(d*x)^3 - 200*b^2*tan(d*x)^3 + 6*b^2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)*tan(c) -
6*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)*tan(c) - 64*a*b*log(4*(tan(c)^2 + 1)/(tan(d
*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)*tan
(c) + 24*a^2*tan(d*x)^2*tan(c) - 120*b^2*tan(d*x)^2*tan(c) - 12*b^2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c
) - 1))*tan(c)^2 + 12*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(c)^2 + 128*a*b*log(4*(tan(c)^
2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))
*tan(c)^2 + 24*a^2*tan(d*x)*tan(c)^2 - 120*b^2*tan(d*x)*tan(c)^2 + 40*a^2*tan(c)^3 - 200*b^2*tan(c)^3 - 24*a^2
*d*x + 120*b^2*d*x - 3*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c)) - 24*a*b
*tan(d*x)^2 + 172*a*b*tan(d*x)*tan(c) - 24*a*b*tan(c)^2 - 6*b^2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) -
1)) + 6*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1)) + 64*a*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c
)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)) + 24*a^2*tan(d*x) - 120
*b^2*tan(d*x) + 24*a^2*tan(c) - 120*b^2*tan(c) - 44*a*b)/(d*tan(d*x)^5*tan(c)^5 + 2*d*tan(d*x)^5*tan(c)^3 - d*
tan(d*x)^4*tan(c)^4 + 2*d*tan(d*x)^3*tan(c)^5 + d*tan(d*x)^5*tan(c) - 2*d*tan(d*x)^4*tan(c)^2 + 4*d*tan(d*x)^3
*tan(c)^3 - 2*d*tan(d*x)^2*tan(c)^4 + d*tan(d*x)*tan(c)^5 - d*tan(d*x)^4 + 2*d*tan(d*x)^3*tan(c) - 4*d*tan(d*x
)^2*tan(c)^2 + 2*d*tan(d*x)*tan(c)^3 - d*tan(c)^4 - 2*d*tan(d*x)^2 + d*tan(d*x)*tan(c) - 2*d*tan(c)^2 - d)