3.42 \(\int \sin ^2(c+d x) (a+b \tan (c+d x))^4 \, dx\)

Optimal. Leaf size=139 \[ \frac{b^2 \left (18 a^2-5 b^2\right ) \tan (c+d x)}{2 d}-\frac{4 a b \left (a^2-2 b^2\right ) \log (\cos (c+d x))}{d}+\frac{1}{2} x \left (-18 a^2 b^2+a^4+5 b^4\right )+\frac{4 a b^3 \tan ^2(c+d x)}{d}-\frac{\sin (c+d x) \cos (c+d x) (a+b \tan (c+d x))^4}{2 d}+\frac{5 b^4 \tan ^3(c+d x)}{6 d} \]

[Out]

((a^4 - 18*a^2*b^2 + 5*b^4)*x)/2 - (4*a*b*(a^2 - 2*b^2)*Log[Cos[c + d*x]])/d + (b^2*(18*a^2 - 5*b^2)*Tan[c + d
*x])/(2*d) + (4*a*b^3*Tan[c + d*x]^2)/d + (5*b^4*Tan[c + d*x]^3)/(6*d) - (Cos[c + d*x]*Sin[c + d*x]*(a + b*Tan
[c + d*x])^4)/(2*d)

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Rubi [A]  time = 0.174975, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3516, 1645, 801, 635, 203, 260} \[ \frac{b^2 \left (18 a^2-5 b^2\right ) \tan (c+d x)}{2 d}-\frac{4 a b \left (a^2-2 b^2\right ) \log (\cos (c+d x))}{d}+\frac{1}{2} x \left (-18 a^2 b^2+a^4+5 b^4\right )+\frac{4 a b^3 \tan ^2(c+d x)}{d}-\frac{\sin (c+d x) \cos (c+d x) (a+b \tan (c+d x))^4}{2 d}+\frac{5 b^4 \tan ^3(c+d x)}{6 d} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^2*(a + b*Tan[c + d*x])^4,x]

[Out]

((a^4 - 18*a^2*b^2 + 5*b^4)*x)/2 - (4*a*b*(a^2 - 2*b^2)*Log[Cos[c + d*x]])/d + (b^2*(18*a^2 - 5*b^2)*Tan[c + d
*x])/(2*d) + (4*a*b^3*Tan[c + d*x]^2)/d + (5*b^4*Tan[c + d*x]^3)/(6*d) - (Cos[c + d*x]*Sin[c + d*x]*(a + b*Tan
[c + d*x])^4)/(2*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 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

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 ^2(c+d x) (a+b \tan (c+d x))^4 \, dx &=\frac{b \operatorname{Subst}\left (\int \frac{x^2 (a+x)^4}{\left (b^2+x^2\right )^2} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=-\frac{\cos (c+d x) \sin (c+d x) (a+b \tan (c+d x))^4}{2 d}-\frac{\operatorname{Subst}\left (\int \frac{(a+x)^3 \left (-a b^2-5 b^2 x\right )}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{2 b d}\\ &=-\frac{\cos (c+d x) \sin (c+d x) (a+b \tan (c+d x))^4}{2 d}-\frac{\operatorname{Subst}\left (\int \left (-18 a^2 b^2+5 b^4-16 a b^2 x-5 b^2 x^2-\frac{b^2 \left (a^4-18 a^2 b^2+5 b^4\right )+8 a b^2 \left (a^2-2 b^2\right ) x}{b^2+x^2}\right ) \, dx,x,b \tan (c+d x)\right )}{2 b d}\\ &=\frac{b^2 \left (18 a^2-5 b^2\right ) \tan (c+d x)}{2 d}+\frac{4 a b^3 \tan ^2(c+d x)}{d}+\frac{5 b^4 \tan ^3(c+d x)}{6 d}-\frac{\cos (c+d x) \sin (c+d x) (a+b \tan (c+d x))^4}{2 d}+\frac{\operatorname{Subst}\left (\int \frac{b^2 \left (a^4-18 a^2 b^2+5 b^4\right )+8 a b^2 \left (a^2-2 b^2\right ) x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{2 b d}\\ &=\frac{b^2 \left (18 a^2-5 b^2\right ) \tan (c+d x)}{2 d}+\frac{4 a b^3 \tan ^2(c+d x)}{d}+\frac{5 b^4 \tan ^3(c+d x)}{6 d}-\frac{\cos (c+d x) \sin (c+d x) (a+b \tan (c+d x))^4}{2 d}+\frac{\left (4 a b \left (a^2-2 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{d}+\frac{\left (b \left (a^4-18 a^2 b^2+5 b^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{2 d}\\ &=\frac{1}{2} \left (a^4-18 a^2 b^2+5 b^4\right ) x-\frac{4 a b \left (a^2-2 b^2\right ) \log (\cos (c+d x))}{d}+\frac{b^2 \left (18 a^2-5 b^2\right ) \tan (c+d x)}{2 d}+\frac{4 a b^3 \tan ^2(c+d x)}{d}+\frac{5 b^4 \tan ^3(c+d x)}{6 d}-\frac{\cos (c+d x) \sin (c+d x) (a+b \tan (c+d x))^4}{2 d}\\ \end{align*}

Mathematica [A]  time = 6.26561, size = 263, normalized size = 1.89 \[ \frac{b \left (-\frac{\left (-6 a^2 b^2+a^4+b^4\right ) \tan ^{-1}(\tan (c+d x))}{2 b}+2 b \left (3 a^2-b^2\right ) \tan (c+d x)+\frac{1}{2} \left (\frac{-12 a^2 b^2+a^4+3 b^4}{\sqrt{-b^2}}+4 a^3-8 a b^2\right ) \log \left (\sqrt{-b^2}-b \tan (c+d x)\right )+\frac{1}{2} \left (-\frac{-12 a^2 b^2+a^4+3 b^4}{\sqrt{-b^2}}+4 a^3-8 a b^2\right ) \log \left (\sqrt{-b^2}+b \tan (c+d x)\right )-\frac{\left (-6 a^2 b^2+a^4+b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b}+2 a b^2 \tan ^2(c+d x)+2 a (a-b) (a+b) \cos ^2(c+d x)+\frac{1}{3} b^3 \tan ^3(c+d x)\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^2*(a + b*Tan[c + d*x])^4,x]

[Out]

(b*(-((a^4 - 6*a^2*b^2 + b^4)*ArcTan[Tan[c + d*x]])/(2*b) + 2*a*(a - b)*(a + b)*Cos[c + d*x]^2 + ((4*a^3 - 8*a
*b^2 + (a^4 - 12*a^2*b^2 + 3*b^4)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]])/2 + ((4*a^3 - 8*a*b^2 - (a^4 -
12*a^2*b^2 + 3*b^4)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]])/2 - ((a^4 - 6*a^2*b^2 + b^4)*Cos[c + d*x]*S
in[c + d*x])/(2*b) + 2*b*(3*a^2 - b^2)*Tan[c + d*x] + 2*a*b^2*Tan[c + d*x]^2 + (b^3*Tan[c + d*x]^3)/3))/d

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Maple [B]  time = 0.057, size = 368, normalized size = 2.7 \begin{align*}{\frac{{b}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{3\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-{\frac{4\,{b}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{3\,d\cos \left ( dx+c \right ) }}-{\frac{4\,{b}^{4}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{3\,d}}-{\frac{5\,{b}^{4}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{5\,{b}^{4}\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }{2\,d}}+{\frac{5\,{b}^{4}x}{2}}+{\frac{5\,{b}^{4}c}{2\,d}}+2\,{\frac{{b}^{3}a \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+2\,{\frac{{b}^{3}a \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d}}+4\,{\frac{{b}^{3}a \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d}}+8\,{\frac{{b}^{3}a\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+6\,{\frac{{a}^{2}{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{d\cos \left ( dx+c \right ) }}+6\,{\frac{{a}^{2}{b}^{2}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{d}}+9\,{\frac{{a}^{2}{b}^{2}\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }{d}}-9\,{a}^{2}{b}^{2}x-9\,{\frac{{a}^{2}{b}^{2}c}{d}}-2\,{\frac{b{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d}}-4\,{\frac{b{a}^{3}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{4}\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }{2\,d}}+{\frac{{a}^{4}x}{2}}+{\frac{{a}^{4}c}{2\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^2*(a+b*tan(d*x+c))^4,x)

[Out]

1/3/d*b^4*sin(d*x+c)^7/cos(d*x+c)^3-4/3/d*b^4*sin(d*x+c)^7/cos(d*x+c)-4/3/d*b^4*cos(d*x+c)*sin(d*x+c)^5-5/3/d*
b^4*cos(d*x+c)*sin(d*x+c)^3-5/2/d*b^4*sin(d*x+c)*cos(d*x+c)+5/2*b^4*x+5/2/d*b^4*c+2/d*b^3*a*sin(d*x+c)^6/cos(d
*x+c)^2+2/d*b^3*a*sin(d*x+c)^4+4/d*b^3*a*sin(d*x+c)^2+8/d*b^3*a*ln(cos(d*x+c))+6/d*a^2*b^2*sin(d*x+c)^5/cos(d*
x+c)+6/d*a^2*b^2*cos(d*x+c)*sin(d*x+c)^3+9/d*a^2*b^2*sin(d*x+c)*cos(d*x+c)-9*a^2*b^2*x-9/d*a^2*b^2*c-2/d*b*a^3
*sin(d*x+c)^2-4/d*b*a^3*ln(cos(d*x+c))-1/2/d*a^4*sin(d*x+c)*cos(d*x+c)+1/2*a^4*x+1/2/d*a^4*c

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Maxima [A]  time = 1.70871, size = 208, normalized size = 1.5 \begin{align*} \frac{2 \, b^{4} \tan \left (d x + c\right )^{3} + 12 \, a b^{3} \tan \left (d x + c\right )^{2} + 3 \,{\left (a^{4} - 18 \, a^{2} b^{2} + 5 \, b^{4}\right )}{\left (d x + c\right )} + 12 \,{\left (a^{3} b - 2 \, a b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 12 \,{\left (3 \, a^{2} b^{2} - b^{4}\right )} \tan \left (d x + c\right ) + \frac{3 \,{\left (4 \, a^{3} b - 4 \, a b^{3} -{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right )\right )}}{\tan \left (d x + c\right )^{2} + 1}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^2*(a+b*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

1/6*(2*b^4*tan(d*x + c)^3 + 12*a*b^3*tan(d*x + c)^2 + 3*(a^4 - 18*a^2*b^2 + 5*b^4)*(d*x + c) + 12*(a^3*b - 2*a
*b^3)*log(tan(d*x + c)^2 + 1) + 12*(3*a^2*b^2 - b^4)*tan(d*x + c) + 3*(4*a^3*b - 4*a*b^3 - (a^4 - 6*a^2*b^2 +
b^4)*tan(d*x + c))/(tan(d*x + c)^2 + 1))/d

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Fricas [A]  time = 2.24796, size = 431, normalized size = 3.1 \begin{align*} \frac{12 \,{\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{5} + 12 \, a b^{3} \cos \left (d x + c\right ) - 24 \,{\left (a^{3} b - 2 \, a b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (-\cos \left (d x + c\right )\right ) - 3 \,{\left (2 \, a^{3} b - 2 \, a b^{3} -{\left (a^{4} - 18 \, a^{2} b^{2} + 5 \, b^{4}\right )} d x\right )} \cos \left (d x + c\right )^{3} -{\left (3 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} - 2 \, b^{4} - 2 \,{\left (18 \, a^{2} b^{2} - 7 \, b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^2*(a+b*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

1/6*(12*(a^3*b - a*b^3)*cos(d*x + c)^5 + 12*a*b^3*cos(d*x + c) - 24*(a^3*b - 2*a*b^3)*cos(d*x + c)^3*log(-cos(
d*x + c)) - 3*(2*a^3*b - 2*a*b^3 - (a^4 - 18*a^2*b^2 + 5*b^4)*d*x)*cos(d*x + c)^3 - (3*(a^4 - 6*a^2*b^2 + b^4)
*cos(d*x + c)^4 - 2*b^4 - 2*(18*a^2*b^2 - 7*b^4)*cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^3)

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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)**2*(a+b*tan(d*x+c))**4,x)

[Out]

Timed out

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Giac [B]  time = 7.19152, size = 5307, normalized size = 38.18 \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)^2*(a+b*tan(d*x+c))^4,x, algorithm="giac")

[Out]

1/6*(3*a^4*d*x*tan(d*x)^5*tan(c)^5 - 54*a^2*b^2*d*x*tan(d*x)^5*tan(c)^5 + 15*b^4*d*x*tan(d*x)^5*tan(c)^5 - 12*
a^3*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*t
an(d*x)*tan(c) + 1))*tan(d*x)^5*tan(c)^5 + 24*a*b^3*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*t
an(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)^5 + 3*a^4*d*x*tan(d*x)^5*
tan(c)^3 - 54*a^2*b^2*d*x*tan(d*x)^5*tan(c)^3 + 15*b^4*d*x*tan(d*x)^5*tan(c)^3 - 9*a^4*d*x*tan(d*x)^4*tan(c)^4
+ 162*a^2*b^2*d*x*tan(d*x)^4*tan(c)^4 - 45*b^4*d*x*tan(d*x)^4*tan(c)^4 + 3*a^4*d*x*tan(d*x)^3*tan(c)^5 - 54*a
^2*b^2*d*x*tan(d*x)^3*tan(c)^5 + 15*b^4*d*x*tan(d*x)^3*tan(c)^5 + 6*a^3*b*tan(d*x)^5*tan(c)^5 + 6*a*b^3*tan(d*
x)^5*tan(c)^5 - 12*a^3*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 + 24*a*b^3*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 + 36
*a^3*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 - 72*a*b^3*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 + 3*a^4*tan(d*x)^5*tan
(c)^4 - 54*a^2*b^2*tan(d*x)^5*tan(c)^4 + 15*b^4*tan(d*x)^5*tan(c)^4 - 12*a^3*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*b^3*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 + 3*a^4*tan(d*x)^4*tan(c)^5 - 54*a^2*b^2*tan(d*x)^4*tan(c)^5
+ 15*b^4*tan(d*x)^4*tan(c)^5 - 9*a^4*d*x*tan(d*x)^4*tan(c)^2 + 162*a^2*b^2*d*x*tan(d*x)^4*tan(c)^2 - 45*b^4*d
*x*tan(d*x)^4*tan(c)^2 + 12*a^4*d*x*tan(d*x)^3*tan(c)^3 - 216*a^2*b^2*d*x*tan(d*x)^3*tan(c)^3 + 60*b^4*d*x*tan
(d*x)^3*tan(c)^3 - 6*a^3*b*tan(d*x)^5*tan(c)^3 + 30*a*b^3*tan(d*x)^5*tan(c)^3 - 9*a^4*d*x*tan(d*x)^2*tan(c)^4
+ 162*a^2*b^2*d*x*tan(d*x)^2*tan(c)^4 - 45*b^4*d*x*tan(d*x)^2*tan(c)^4 - 42*a^3*b*tan(d*x)^4*tan(c)^4 + 30*a*b
^3*tan(d*x)^4*tan(c)^4 - 6*a^3*b*tan(d*x)^3*tan(c)^5 + 30*a*b^3*tan(d*x)^3*tan(c)^5 + 36*a^3*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 - 72*a*b^3*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*tan(c)^2 - 36*a^2*b^2*tan(d*x)^5*tan(c)^2 + 10*b^4*ta
n(d*x)^5*tan(c)^2 - 48*a^3*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)^3 + 96*a*b^3*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*ta
n(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)^3
- 12*a^4*tan(d*x)^4*tan(c)^3 + 108*a^2*b^2*tan(d*x)^4*tan(c)^3 - 30*b^4*tan(d*x)^4*tan(c)^3 + 36*a^3*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 - 72*a*b^3*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 - 12*a^4*tan(d*x)^3*tan(c)^4 + 108*a
^2*b^2*tan(d*x)^3*tan(c)^4 - 30*b^4*tan(d*x)^3*tan(c)^4 - 36*a^2*b^2*tan(d*x)^2*tan(c)^5 + 10*b^4*tan(d*x)^2*t
an(c)^5 + 9*a^4*d*x*tan(d*x)^3*tan(c) - 162*a^2*b^2*d*x*tan(d*x)^3*tan(c) + 45*b^4*d*x*tan(d*x)^3*tan(c) + 12*
a*b^3*tan(d*x)^5*tan(c) - 12*a^4*d*x*tan(d*x)^2*tan(c)^2 + 216*a^2*b^2*d*x*tan(d*x)^2*tan(c)^2 - 60*b^4*d*x*ta
n(d*x)^2*tan(c)^2 + 18*a^3*b*tan(d*x)^4*tan(c)^2 - 42*a*b^3*tan(d*x)^4*tan(c)^2 + 9*a^4*d*x*tan(d*x)*tan(c)^3
- 162*a^2*b^2*d*x*tan(d*x)*tan(c)^3 + 45*b^4*d*x*tan(d*x)*tan(c)^3 + 96*a^3*b*tan(d*x)^3*tan(c)^3 - 48*a*b^3*t
an(d*x)^3*tan(c)^3 + 18*a^3*b*tan(d*x)^2*tan(c)^4 - 42*a*b^3*tan(d*x)^2*tan(c)^4 + 12*a*b^3*tan(d*x)*tan(c)^5
- 2*b^4*tan(d*x)^5 - 36*a^3*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) + 72*a*b^3*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) + 7
2*a^2*b^2*tan(d*x)^4*tan(c) - 30*b^4*tan(d*x)^4*tan(c) + 48*a^3*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*b^
3*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 + 18*a^4*tan(d*x)^3*tan(c)^2 - 108*a^2*b^2*tan(d*x)^3*tan(c)^2 + 10*b^4*t
an(d*x)^3*tan(c)^2 - 36*a^3*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)^3 + 72*a*b^3*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)^3 + 1
8*a^4*tan(d*x)^2*tan(c)^3 - 108*a^2*b^2*tan(d*x)^2*tan(c)^3 + 10*b^4*tan(d*x)^2*tan(c)^3 + 72*a^2*b^2*tan(d*x)
*tan(c)^4 - 30*b^4*tan(d*x)*tan(c)^4 - 2*b^4*tan(c)^5 - 3*a^4*d*x*tan(d*x)^2 + 54*a^2*b^2*d*x*tan(d*x)^2 - 15*
b^4*d*x*tan(d*x)^2 - 12*a*b^3*tan(d*x)^4 + 9*a^4*d*x*tan(d*x)*tan(c) - 162*a^2*b^2*d*x*tan(d*x)*tan(c) + 45*b^
4*d*x*tan(d*x)*tan(c) - 18*a^3*b*tan(d*x)^3*tan(c) + 42*a*b^3*tan(d*x)^3*tan(c) - 3*a^4*d*x*tan(c)^2 + 54*a^2*
b^2*d*x*tan(c)^2 - 15*b^4*d*x*tan(c)^2 - 96*a^3*b*tan(d*x)^2*tan(c)^2 + 48*a*b^3*tan(d*x)^2*tan(c)^2 - 18*a^3*
b*tan(d*x)*tan(c)^3 + 42*a*b^3*tan(d*x)*tan(c)^3 - 12*a*b^3*tan(c)^4 + 12*a^3*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 - 24
*a*b^3*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 - 36*a^2*b^2*tan(d*x)^3 + 10*b^4*tan(d*x)^3 - 36*a^3*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) + 72*a*b^3*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) - 12*a^4*tan(d*x)^2*tan(c) + 108*a^2*b^2*tan(d*x)^2*tan(c)
- 30*b^4*tan(d*x)^2*tan(c) + 12*a^3*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*b^3*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 - 12*a^4*tan(
d*x)*tan(c)^2 + 108*a^2*b^2*tan(d*x)*tan(c)^2 - 30*b^4*tan(d*x)*tan(c)^2 - 36*a^2*b^2*tan(c)^3 + 10*b^4*tan(c)
^3 - 3*a^4*d*x + 54*a^2*b^2*d*x - 15*b^4*d*x + 6*a^3*b*tan(d*x)^2 - 30*a*b^3*tan(d*x)^2 + 42*a^3*b*tan(d*x)*ta
n(c) - 30*a*b^3*tan(d*x)*tan(c) + 6*a^3*b*tan(c)^2 - 30*a*b^3*tan(c)^2 + 12*a^3*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*b^3*lo
g(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)) + 3*a^4*tan(d*x) - 54*a^2*b^2*tan(d*x) + 15*b^4*tan(d*x) + 3*a^4*tan(c) - 54*a^2*b^2*tan(c) + 15*
b^4*tan(c) - 6*a^3*b - 6*a*b^3)/(d*tan(d*x)^5*tan(c)^5 + d*tan(d*x)^5*tan(c)^3 - 3*d*tan(d*x)^4*tan(c)^4 + d*t
an(d*x)^3*tan(c)^5 - 3*d*tan(d*x)^4*tan(c)^2 + 4*d*tan(d*x)^3*tan(c)^3 - 3*d*tan(d*x)^2*tan(c)^4 + 3*d*tan(d*x
)^3*tan(c) - 4*d*tan(d*x)^2*tan(c)^2 + 3*d*tan(d*x)*tan(c)^3 - d*tan(d*x)^2 + 3*d*tan(d*x)*tan(c) - d*tan(c)^2
- d)