3.445 \(\int \tan ^3(c+d x) (a+b \tan (c+d x))^4 \, dx\)

Optimal. Leaf size=181 \[ -\frac{\left (a^2-b^2\right ) (a+b \tan (c+d x))^2}{2 d}-\frac{a b \left (a^2-3 b^2\right ) \tan (c+d x)}{d}+\frac{\left (-6 a^2 b^2+a^4+b^4\right ) \log (\cos (c+d x))}{d}+4 a b x \left (a^2-b^2\right )-\frac{a (a+b \tan (c+d x))^5}{30 b^2 d}+\frac{\tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}-\frac{(a+b \tan (c+d x))^4}{4 d}-\frac{a (a+b \tan (c+d x))^3}{3 d} \]

[Out]

4*a*b*(a^2 - b^2)*x + ((a^4 - 6*a^2*b^2 + b^4)*Log[Cos[c + d*x]])/d - (a*b*(a^2 - 3*b^2)*Tan[c + d*x])/d - ((a
^2 - b^2)*(a + b*Tan[c + d*x])^2)/(2*d) - (a*(a + b*Tan[c + d*x])^3)/(3*d) - (a + b*Tan[c + d*x])^4/(4*d) - (a
*(a + b*Tan[c + d*x])^5)/(30*b^2*d) + (Tan[c + d*x]*(a + b*Tan[c + d*x])^5)/(6*b*d)

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Rubi [A]  time = 0.236407, antiderivative size = 181, 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 = {3566, 3630, 12, 3528, 3525, 3475} \[ -\frac{\left (a^2-b^2\right ) (a+b \tan (c+d x))^2}{2 d}-\frac{a b \left (a^2-3 b^2\right ) \tan (c+d x)}{d}+\frac{\left (-6 a^2 b^2+a^4+b^4\right ) \log (\cos (c+d x))}{d}+4 a b x \left (a^2-b^2\right )-\frac{a (a+b \tan (c+d x))^5}{30 b^2 d}+\frac{\tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}-\frac{(a+b \tan (c+d x))^4}{4 d}-\frac{a (a+b \tan (c+d x))^3}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^3*(a + b*Tan[c + d*x])^4,x]

[Out]

4*a*b*(a^2 - b^2)*x + ((a^4 - 6*a^2*b^2 + b^4)*Log[Cos[c + d*x]])/d - (a*b*(a^2 - 3*b^2)*Tan[c + d*x])/d - ((a
^2 - b^2)*(a + b*Tan[c + d*x])^2)/(2*d) - (a*(a + b*Tan[c + d*x])^3)/(3*d) - (a + b*Tan[c + d*x])^4/(4*d) - (a
*(a + b*Tan[c + d*x])^5)/(30*b^2*d) + (Tan[c + d*x]*(a + b*Tan[c + d*x])^5)/(6*b*d)

Rule 3566

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(m + n - 1)), x] + Dist[1/(d*(m + n -
1)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(
1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2,
x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || IntegerQ[m]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0]
&& NeQ[a, 0])))

Rule 3630

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(C*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Int[(a + b*Tan[e + f*x])
^m*Simp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0]
&&  !LeQ[m, -1]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3528

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(d
*(a + b*Tan[e + f*x])^m)/(f*m), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]

Rule 3525

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*c - b
*d)*x, x] + (Dist[b*c + a*d, Int[Tan[e + f*x], x], x] + Simp[(b*d*Tan[e + f*x])/f, x]) /; FreeQ[{a, b, c, d, e
, f}, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*d, 0]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \tan ^3(c+d x) (a+b \tan (c+d x))^4 \, dx &=\frac{\tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}+\frac{\int (a+b \tan (c+d x))^4 \left (-a-6 b \tan (c+d x)-a \tan ^2(c+d x)\right ) \, dx}{6 b}\\ &=-\frac{a (a+b \tan (c+d x))^5}{30 b^2 d}+\frac{\tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}+\frac{\int -6 b \tan (c+d x) (a+b \tan (c+d x))^4 \, dx}{6 b}\\ &=-\frac{a (a+b \tan (c+d x))^5}{30 b^2 d}+\frac{\tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}-\int \tan (c+d x) (a+b \tan (c+d x))^4 \, dx\\ &=-\frac{(a+b \tan (c+d x))^4}{4 d}-\frac{a (a+b \tan (c+d x))^5}{30 b^2 d}+\frac{\tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}-\int (-b+a \tan (c+d x)) (a+b \tan (c+d x))^3 \, dx\\ &=-\frac{a (a+b \tan (c+d x))^3}{3 d}-\frac{(a+b \tan (c+d x))^4}{4 d}-\frac{a (a+b \tan (c+d x))^5}{30 b^2 d}+\frac{\tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}-\int (a+b \tan (c+d x))^2 \left (-2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=-\frac{\left (a^2-b^2\right ) (a+b \tan (c+d x))^2}{2 d}-\frac{a (a+b \tan (c+d x))^3}{3 d}-\frac{(a+b \tan (c+d x))^4}{4 d}-\frac{a (a+b \tan (c+d x))^5}{30 b^2 d}+\frac{\tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}-\int (a+b \tan (c+d x)) \left (-b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right ) \, dx\\ &=4 a b \left (a^2-b^2\right ) x-\frac{a b \left (a^2-3 b^2\right ) \tan (c+d x)}{d}-\frac{\left (a^2-b^2\right ) (a+b \tan (c+d x))^2}{2 d}-\frac{a (a+b \tan (c+d x))^3}{3 d}-\frac{(a+b \tan (c+d x))^4}{4 d}-\frac{a (a+b \tan (c+d x))^5}{30 b^2 d}+\frac{\tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}-\left (a^4-6 a^2 b^2+b^4\right ) \int \tan (c+d x) \, dx\\ &=4 a b \left (a^2-b^2\right ) x+\frac{\left (a^4-6 a^2 b^2+b^4\right ) \log (\cos (c+d x))}{d}-\frac{a b \left (a^2-3 b^2\right ) \tan (c+d x)}{d}-\frac{\left (a^2-b^2\right ) (a+b \tan (c+d x))^2}{2 d}-\frac{a (a+b \tan (c+d x))^3}{3 d}-\frac{(a+b \tan (c+d x))^4}{4 d}-\frac{a (a+b \tan (c+d x))^5}{30 b^2 d}+\frac{\tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}\\ \end{align*}

Mathematica [C]  time = 1.1308, size = 190, normalized size = 1.05 \[ \frac{-15 b^4 \left (b^2-6 a^2\right ) \tan ^4(c+d x)+80 a b^3 \left (a^2-b^2\right ) \tan ^3(c+d x)+30 b^2 \left (-6 a^2 b^2+a^4+b^4\right ) \tan ^2(c+d x)-240 a b^3 \left (a^2-b^2\right ) \tan (c+d x)-2 \left (a^6+15 b^2 (a+i b)^4 \log (-\tan (c+d x)+i)+15 b^2 (a-i b)^4 \log (\tan (c+d x)+i)\right )+48 a b^5 \tan ^5(c+d x)+10 b^6 \tan ^6(c+d x)}{60 b^2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^3*(a + b*Tan[c + d*x])^4,x]

[Out]

(-2*(a^6 + 15*(a + I*b)^4*b^2*Log[I - Tan[c + d*x]] + 15*(a - I*b)^4*b^2*Log[I + Tan[c + d*x]]) - 240*a*b^3*(a
^2 - b^2)*Tan[c + d*x] + 30*b^2*(a^4 - 6*a^2*b^2 + b^4)*Tan[c + d*x]^2 + 80*a*b^3*(a^2 - b^2)*Tan[c + d*x]^3 -
15*b^4*(-6*a^2 + b^2)*Tan[c + d*x]^4 + 48*a*b^5*Tan[c + d*x]^5 + 10*b^6*Tan[c + d*x]^6)/(60*b^2*d)

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Maple [A]  time = 0.006, size = 277, normalized size = 1.5 \begin{align*}{\frac{{b}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{6}}{6\,d}}+{\frac{4\,{b}^{3}a \left ( \tan \left ( dx+c \right ) \right ) ^{5}}{5\,d}}+{\frac{3\,{a}^{2}{b}^{2} \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{2\,d}}-{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{4}{b}^{4}}{4\,d}}+{\frac{4\,b{a}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{4\, \left ( \tan \left ( dx+c \right ) \right ) ^{3}a{b}^{3}}{3\,d}}+{\frac{{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-3\,{\frac{{a}^{2}{b}^{2} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{d}}+{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{2}{b}^{4}}{2\,d}}-4\,{\frac{{a}^{3}\tan \left ( dx+c \right ) b}{d}}+4\,{\frac{{b}^{3}a\tan \left ( dx+c \right ) }{d}}+3\,{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){a}^{2}{b}^{2}}{d}}-{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){b}^{4}}{2\,d}}-{\frac{{a}^{4}\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{2\,d}}+4\,{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{3}b}{d}}-4\,{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ) a{b}^{3}}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^3*(a+b*tan(d*x+c))^4,x)

[Out]

1/6/d*b^4*tan(d*x+c)^6+4/5/d*b^3*a*tan(d*x+c)^5+3/2/d*a^2*b^2*tan(d*x+c)^4-1/4/d*tan(d*x+c)^4*b^4+4/3/d*b*a^3*
tan(d*x+c)^3-4/3/d*tan(d*x+c)^3*a*b^3+1/2*a^4*tan(d*x+c)^2/d-3/d*a^2*b^2*tan(d*x+c)^2+1/2/d*tan(d*x+c)^2*b^4-4
/d*tan(d*x+c)*a^3*b+4*a*b^3*tan(d*x+c)/d+3/d*ln(1+tan(d*x+c)^2)*a^2*b^2-1/2/d*ln(1+tan(d*x+c)^2)*b^4-1/2/d*a^4
*ln(1+tan(d*x+c)^2)+4/d*arctan(tan(d*x+c))*a^3*b-4/d*arctan(tan(d*x+c))*a*b^3

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Maxima [A]  time = 1.57625, size = 231, normalized size = 1.28 \begin{align*} \frac{10 \, b^{4} \tan \left (d x + c\right )^{6} + 48 \, a b^{3} \tan \left (d x + c\right )^{5} + 15 \,{\left (6 \, a^{2} b^{2} - b^{4}\right )} \tan \left (d x + c\right )^{4} + 80 \,{\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{3} + 30 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right )^{2} + 240 \,{\left (a^{3} b - a b^{3}\right )}{\left (d x + c\right )} - 30 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 240 \,{\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )}{60 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^3*(a+b*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

1/60*(10*b^4*tan(d*x + c)^6 + 48*a*b^3*tan(d*x + c)^5 + 15*(6*a^2*b^2 - b^4)*tan(d*x + c)^4 + 80*(a^3*b - a*b^
3)*tan(d*x + c)^3 + 30*(a^4 - 6*a^2*b^2 + b^4)*tan(d*x + c)^2 + 240*(a^3*b - a*b^3)*(d*x + c) - 30*(a^4 - 6*a^
2*b^2 + b^4)*log(tan(d*x + c)^2 + 1) - 240*(a^3*b - a*b^3)*tan(d*x + c))/d

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Fricas [A]  time = 1.6941, size = 397, normalized size = 2.19 \begin{align*} \frac{10 \, b^{4} \tan \left (d x + c\right )^{6} + 48 \, a b^{3} \tan \left (d x + c\right )^{5} + 15 \,{\left (6 \, a^{2} b^{2} - b^{4}\right )} \tan \left (d x + c\right )^{4} + 80 \,{\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{3} + 240 \,{\left (a^{3} b - a b^{3}\right )} d x + 30 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right )^{2} + 30 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 240 \,{\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )}{60 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^3*(a+b*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

1/60*(10*b^4*tan(d*x + c)^6 + 48*a*b^3*tan(d*x + c)^5 + 15*(6*a^2*b^2 - b^4)*tan(d*x + c)^4 + 80*(a^3*b - a*b^
3)*tan(d*x + c)^3 + 240*(a^3*b - a*b^3)*d*x + 30*(a^4 - 6*a^2*b^2 + b^4)*tan(d*x + c)^2 + 30*(a^4 - 6*a^2*b^2
+ b^4)*log(1/(tan(d*x + c)^2 + 1)) - 240*(a^3*b - a*b^3)*tan(d*x + c))/d

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Sympy [A]  time = 1.38796, size = 277, normalized size = 1.53 \begin{align*} \begin{cases} - \frac{a^{4} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{a^{4} \tan ^{2}{\left (c + d x \right )}}{2 d} + 4 a^{3} b x + \frac{4 a^{3} b \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac{4 a^{3} b \tan{\left (c + d x \right )}}{d} + \frac{3 a^{2} b^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac{3 a^{2} b^{2} \tan ^{4}{\left (c + d x \right )}}{2 d} - \frac{3 a^{2} b^{2} \tan ^{2}{\left (c + d x \right )}}{d} - 4 a b^{3} x + \frac{4 a b^{3} \tan ^{5}{\left (c + d x \right )}}{5 d} - \frac{4 a b^{3} \tan ^{3}{\left (c + d x \right )}}{3 d} + \frac{4 a b^{3} \tan{\left (c + d x \right )}}{d} - \frac{b^{4} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{b^{4} \tan ^{6}{\left (c + d x \right )}}{6 d} - \frac{b^{4} \tan ^{4}{\left (c + d x \right )}}{4 d} + \frac{b^{4} \tan ^{2}{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (a + b \tan{\left (c \right )}\right )^{4} \tan ^{3}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**3*(a+b*tan(d*x+c))**4,x)

[Out]

Piecewise((-a**4*log(tan(c + d*x)**2 + 1)/(2*d) + a**4*tan(c + d*x)**2/(2*d) + 4*a**3*b*x + 4*a**3*b*tan(c + d
*x)**3/(3*d) - 4*a**3*b*tan(c + d*x)/d + 3*a**2*b**2*log(tan(c + d*x)**2 + 1)/d + 3*a**2*b**2*tan(c + d*x)**4/
(2*d) - 3*a**2*b**2*tan(c + d*x)**2/d - 4*a*b**3*x + 4*a*b**3*tan(c + d*x)**5/(5*d) - 4*a*b**3*tan(c + d*x)**3
/(3*d) + 4*a*b**3*tan(c + d*x)/d - b**4*log(tan(c + d*x)**2 + 1)/(2*d) + b**4*tan(c + d*x)**6/(6*d) - b**4*tan
(c + d*x)**4/(4*d) + b**4*tan(c + d*x)**2/(2*d), Ne(d, 0)), (x*(a + b*tan(c))**4*tan(c)**3, True))

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Giac [B]  time = 12.0378, size = 4570, normalized size = 25.25 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^3*(a+b*tan(d*x+c))^4,x, algorithm="giac")

[Out]

1/60*(240*a^3*b*d*x*tan(d*x)^6*tan(c)^6 - 240*a*b^3*d*x*tan(d*x)^6*tan(c)^6 + 30*a^4*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)^6
*tan(c)^6 - 180*a^2*b^2*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)^6*tan(c)^6 + 30*b^4*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)^6*tan(c)^6 - 1440*
a^3*b*d*x*tan(d*x)^5*tan(c)^5 + 1440*a*b^3*d*x*tan(d*x)^5*tan(c)^5 + 30*a^4*tan(d*x)^6*tan(c)^6 - 270*a^2*b^2*
tan(d*x)^6*tan(c)^6 + 55*b^4*tan(d*x)^6*tan(c)^6 - 180*a^4*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)^5 + 1080*a^2*b^2*l
og(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)^5 - 180*b^4*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)^5 + 240*a^3*b*tan(d*x)^6*tan(c)^5
- 240*a*b^3*tan(d*x)^6*tan(c)^5 + 240*a^3*b*tan(d*x)^5*tan(c)^6 - 240*a*b^3*tan(d*x)^5*tan(c)^6 + 3600*a^3*b*
d*x*tan(d*x)^4*tan(c)^4 - 3600*a*b^3*d*x*tan(d*x)^4*tan(c)^4 + 30*a^4*tan(d*x)^6*tan(c)^4 - 180*a^2*b^2*tan(d*
x)^6*tan(c)^4 + 30*b^4*tan(d*x)^6*tan(c)^4 - 120*a^4*tan(d*x)^5*tan(c)^5 + 1260*a^2*b^2*tan(d*x)^5*tan(c)^5 -
270*b^4*tan(d*x)^5*tan(c)^5 + 30*a^4*tan(d*x)^4*tan(c)^6 - 180*a^2*b^2*tan(d*x)^4*tan(c)^6 + 30*b^4*tan(d*x)^4
*tan(c)^6 - 80*a^3*b*tan(d*x)^6*tan(c)^3 + 80*a*b^3*tan(d*x)^6*tan(c)^3 + 450*a^4*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*ta
n(c)^4 - 2700*a^2*b^2*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 + 450*b^4*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 - 1440*a
^3*b*tan(d*x)^5*tan(c)^4 + 1440*a*b^3*tan(d*x)^5*tan(c)^4 - 1440*a^3*b*tan(d*x)^4*tan(c)^5 + 1440*a*b^3*tan(d*
x)^4*tan(c)^5 - 80*a^3*b*tan(d*x)^3*tan(c)^6 + 80*a*b^3*tan(d*x)^3*tan(c)^6 + 90*a^2*b^2*tan(d*x)^6*tan(c)^2 -
15*b^4*tan(d*x)^6*tan(c)^2 - 4800*a^3*b*d*x*tan(d*x)^3*tan(c)^3 + 4800*a*b^3*d*x*tan(d*x)^3*tan(c)^3 - 120*a^
4*tan(d*x)^5*tan(c)^3 + 1080*a^2*b^2*tan(d*x)^5*tan(c)^3 - 180*b^4*tan(d*x)^5*tan(c)^3 + 210*a^4*tan(d*x)^4*ta
n(c)^4 - 2070*a^2*b^2*tan(d*x)^4*tan(c)^4 + 495*b^4*tan(d*x)^4*tan(c)^4 - 120*a^4*tan(d*x)^3*tan(c)^5 + 1080*a
^2*b^2*tan(d*x)^3*tan(c)^5 - 180*b^4*tan(d*x)^3*tan(c)^5 + 90*a^2*b^2*tan(d*x)^2*tan(c)^6 - 15*b^4*tan(d*x)^2*
tan(c)^6 - 48*a*b^3*tan(d*x)^6*tan(c) + 240*a^3*b*tan(d*x)^5*tan(c)^2 - 480*a*b^3*tan(d*x)^5*tan(c)^2 - 600*a^
4*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 + 3600*a^2*b^2*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)^3*tan(c)^3 - 600*b^4*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 + 3120*a^3*b*tan(d*x)^4*tan(c)^3 - 3600*a*b^3*tan(d*x)^4*tan(c)^3 + 3120*a^3*b*tan(d*x)^
3*tan(c)^4 - 3600*a*b^3*tan(d*x)^3*tan(c)^4 + 240*a^3*b*tan(d*x)^2*tan(c)^5 - 480*a*b^3*tan(d*x)^2*tan(c)^5 -
48*a*b^3*tan(d*x)*tan(c)^6 + 10*b^4*tan(d*x)^6 - 180*a^2*b^2*tan(d*x)^5*tan(c) + 90*b^4*tan(d*x)^5*tan(c) + 36
00*a^3*b*d*x*tan(d*x)^2*tan(c)^2 - 3600*a*b^3*d*x*tan(d*x)^2*tan(c)^2 + 180*a^4*tan(d*x)^4*tan(c)^2 - 1800*a^2
*b^2*tan(d*x)^4*tan(c)^2 + 450*b^4*tan(d*x)^4*tan(c)^2 - 240*a^4*tan(d*x)^3*tan(c)^3 + 2160*a^2*b^2*tan(d*x)^3
*tan(c)^3 - 360*b^4*tan(d*x)^3*tan(c)^3 + 180*a^4*tan(d*x)^2*tan(c)^4 - 1800*a^2*b^2*tan(d*x)^2*tan(c)^4 + 450
*b^4*tan(d*x)^2*tan(c)^4 - 180*a^2*b^2*tan(d*x)*tan(c)^5 + 90*b^4*tan(d*x)*tan(c)^5 + 10*b^4*tan(c)^6 + 48*a*b
^3*tan(d*x)^5 - 240*a^3*b*tan(d*x)^4*tan(c) + 480*a*b^3*tan(d*x)^4*tan(c) + 450*a^4*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 - 2700*a^2*b^2*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 + 450*b^4*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 - 3120
*a^3*b*tan(d*x)^3*tan(c)^2 + 3600*a*b^3*tan(d*x)^3*tan(c)^2 - 3120*a^3*b*tan(d*x)^2*tan(c)^3 + 3600*a*b^3*tan(
d*x)^2*tan(c)^3 - 240*a^3*b*tan(d*x)*tan(c)^4 + 480*a*b^3*tan(d*x)*tan(c)^4 + 48*a*b^3*tan(c)^5 + 90*a^2*b^2*t
an(d*x)^4 - 15*b^4*tan(d*x)^4 - 1440*a^3*b*d*x*tan(d*x)*tan(c) + 1440*a*b^3*d*x*tan(d*x)*tan(c) - 120*a^4*tan(
d*x)^3*tan(c) + 1080*a^2*b^2*tan(d*x)^3*tan(c) - 180*b^4*tan(d*x)^3*tan(c) + 210*a^4*tan(d*x)^2*tan(c)^2 - 207
0*a^2*b^2*tan(d*x)^2*tan(c)^2 + 495*b^4*tan(d*x)^2*tan(c)^2 - 120*a^4*tan(d*x)*tan(c)^3 + 1080*a^2*b^2*tan(d*x
)*tan(c)^3 - 180*b^4*tan(d*x)*tan(c)^3 + 90*a^2*b^2*tan(c)^4 - 15*b^4*tan(c)^4 + 80*a^3*b*tan(d*x)^3 - 80*a*b^
3*tan(d*x)^3 - 180*a^4*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) + 1080*a^2*b^2*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) - 180*b^4*
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) + 1440*a^3*b*tan(d*x)^2*tan(c) - 1440*a*b^3*tan(d*x)^2*tan(c) + 1440*a^3*b*tan(
d*x)*tan(c)^2 - 1440*a*b^3*tan(d*x)*tan(c)^2 + 80*a^3*b*tan(c)^3 - 80*a*b^3*tan(c)^3 + 240*a^3*b*d*x - 240*a*b
^3*d*x + 30*a^4*tan(d*x)^2 - 180*a^2*b^2*tan(d*x)^2 + 30*b^4*tan(d*x)^2 - 120*a^4*tan(d*x)*tan(c) + 1260*a^2*b
^2*tan(d*x)*tan(c) - 270*b^4*tan(d*x)*tan(c) + 30*a^4*tan(c)^2 - 180*a^2*b^2*tan(c)^2 + 30*b^4*tan(c)^2 + 30*a
^4*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)) - 180*a^2*b^2*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*t
an(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)) + 30*b^4*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)) - 240*a^3*b*tan(d*x) + 240*a*b^3*tan(d*
x) - 240*a^3*b*tan(c) + 240*a*b^3*tan(c) + 30*a^4 - 270*a^2*b^2 + 55*b^4)/(d*tan(d*x)^6*tan(c)^6 - 6*d*tan(d*x
)^5*tan(c)^5 + 15*d*tan(d*x)^4*tan(c)^4 - 20*d*tan(d*x)^3*tan(c)^3 + 15*d*tan(d*x)^2*tan(c)^2 - 6*d*tan(d*x)*t
an(c) + d)