3.1197 \(\int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2 \, dx\)

Optimal. Leaf size=215 \[ -\frac{\left (3 a^2 b \left (c^2-d^2\right )+2 a^3 c d-6 a b^2 c d-b^3 \left (c^2-d^2\right )\right ) \log (\cos (e+f x))}{f}-x \left (6 a^2 b c d+a^3 \left (-\left (c^2-d^2\right )\right )+3 a b^2 \left (c^2-d^2\right )-2 b^3 c d\right )+\frac{\left (2 a c d+b \left (c^2-d^2\right )\right ) (a+b \tan (e+f x))^2}{2 f}+\frac{2 c d (a+b \tan (e+f x))^3}{3 f}+\frac{2 b (a d+b c) (a c-b d) \tan (e+f x)}{f}+\frac{d^2 (a+b \tan (e+f x))^4}{4 b f} \]

[Out]

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

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Rubi [A]  time = 0.265793, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3543, 3528, 3525, 3475} \[ -\frac{\left (3 a^2 b \left (c^2-d^2\right )+2 a^3 c d-6 a b^2 c d-b^3 \left (c^2-d^2\right )\right ) \log (\cos (e+f x))}{f}-x \left (6 a^2 b c d+a^3 \left (-\left (c^2-d^2\right )\right )+3 a b^2 \left (c^2-d^2\right )-2 b^3 c d\right )+\frac{\left (2 a c d+b \left (c^2-d^2\right )\right ) (a+b \tan (e+f x))^2}{2 f}+\frac{2 c d (a+b \tan (e+f x))^3}{3 f}+\frac{2 b (a d+b c) (a c-b d) \tan (e+f x)}{f}+\frac{d^2 (a+b \tan (e+f x))^4}{4 b f} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^2,x]

[Out]

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

Rule 3543

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

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

Mathematica [C]  time = 2.44156, size = 221, normalized size = 1.03 \[ \frac{-4 c d \left (6 b^2 \left (b^2-6 a^2\right ) \tan (e+f x)-12 a b^3 \tan ^2(e+f x)-3 i (a-i b)^4 \log (\tan (e+f x)+i)+3 i (a+i b)^4 \log (-\tan (e+f x)+i)-2 b^4 \tan ^3(e+f x)\right )-6 \left (2 a c d+b \left (d^2-c^2\right )\right ) \left (6 a b^2 \tan (e+f x)+(-b+i a)^3 \log (-\tan (e+f x)+i)-(b+i a)^3 \log (\tan (e+f x)+i)+b^3 \tan ^2(e+f x)\right )+3 d^2 (a+b \tan (e+f x))^4}{12 b f} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^2,x]

[Out]

(3*d^2*(a + b*Tan[e + f*x])^4 - 6*(2*a*c*d + b*(-c^2 + d^2))*((I*a - b)^3*Log[I - Tan[e + f*x]] - (I*a + b)^3*
Log[I + Tan[e + f*x]] + 6*a*b^2*Tan[e + f*x] + b^3*Tan[e + f*x]^2) - 4*c*d*((3*I)*(a + I*b)^4*Log[I - Tan[e +
f*x]] - (3*I)*(a - I*b)^4*Log[I + Tan[e + f*x]] + 6*b^2*(-6*a^2 + b^2)*Tan[e + f*x] - 12*a*b^3*Tan[e + f*x]^2
- 2*b^4*Tan[e + f*x]^3))/(12*b*f)

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Maple [B]  time = 0.006, size = 460, normalized size = 2.1 \begin{align*}{\frac{{b}^{3}{d}^{2} \left ( \tan \left ( fx+e \right ) \right ) ^{4}}{4\,f}}+{\frac{ \left ( \tan \left ( fx+e \right ) \right ) ^{3}a{b}^{2}{d}^{2}}{f}}+{\frac{2\, \left ( \tan \left ( fx+e \right ) \right ) ^{3}{b}^{3}cd}{3\,f}}+{\frac{3\, \left ( \tan \left ( fx+e \right ) \right ) ^{2}{a}^{2}b{d}^{2}}{2\,f}}+3\,{\frac{ \left ( \tan \left ( fx+e \right ) \right ) ^{2}a{b}^{2}cd}{f}}+{\frac{ \left ( \tan \left ( fx+e \right ) \right ) ^{2}{b}^{3}{c}^{2}}{2\,f}}-{\frac{ \left ( \tan \left ( fx+e \right ) \right ) ^{2}{b}^{3}{d}^{2}}{2\,f}}+{\frac{{a}^{3}\tan \left ( fx+e \right ){d}^{2}}{f}}+6\,{\frac{{a}^{2}bcd\tan \left ( fx+e \right ) }{f}}+3\,{\frac{a{b}^{2}{c}^{2}\tan \left ( fx+e \right ) }{f}}-3\,{\frac{a{b}^{2}{d}^{2}\tan \left ( fx+e \right ) }{f}}-2\,{\frac{{b}^{3}cd\tan \left ( fx+e \right ) }{f}}+{\frac{{a}^{3}\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) cd}{f}}+{\frac{3\,\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ){a}^{2}b{c}^{2}}{2\,f}}-{\frac{3\,\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ){a}^{2}b{d}^{2}}{2\,f}}-3\,{\frac{\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) a{b}^{2}cd}{f}}-{\frac{\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ){b}^{3}{c}^{2}}{2\,f}}+{\frac{\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ){b}^{3}{d}^{2}}{2\,f}}+{\frac{{a}^{3}\arctan \left ( \tan \left ( fx+e \right ) \right ){c}^{2}}{f}}-{\frac{{a}^{3}\arctan \left ( \tan \left ( fx+e \right ) \right ){d}^{2}}{f}}-6\,{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ){a}^{2}bcd}{f}}-3\,{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ) a{b}^{2}{c}^{2}}{f}}+3\,{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ) a{b}^{2}{d}^{2}}{f}}+2\,{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ){b}^{3}cd}{f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*tan(f*x+e))^3*(c+d*tan(f*x+e))^2,x)

[Out]

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

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Maxima [A]  time = 1.79999, size = 342, normalized size = 1.59 \begin{align*} \frac{3 \, b^{3} d^{2} \tan \left (f x + e\right )^{4} + 4 \,{\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} \tan \left (f x + e\right )^{3} + 6 \,{\left (b^{3} c^{2} + 6 \, a b^{2} c d +{\left (3 \, a^{2} b - b^{3}\right )} d^{2}\right )} \tan \left (f x + e\right )^{2} + 12 \,{\left ({\left (a^{3} - 3 \, a b^{2}\right )} c^{2} - 2 \,{\left (3 \, a^{2} b - b^{3}\right )} c d -{\left (a^{3} - 3 \, a b^{2}\right )} d^{2}\right )}{\left (f x + e\right )} + 6 \,{\left ({\left (3 \, a^{2} b - b^{3}\right )} c^{2} + 2 \,{\left (a^{3} - 3 \, a b^{2}\right )} c d -{\left (3 \, a^{2} b - b^{3}\right )} d^{2}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right ) + 12 \,{\left (3 \, a b^{2} c^{2} + 2 \,{\left (3 \, a^{2} b - b^{3}\right )} c d +{\left (a^{3} - 3 \, a b^{2}\right )} d^{2}\right )} \tan \left (f x + e\right )}{12 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))^3*(c+d*tan(f*x+e))^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.48685, size = 540, normalized size = 2.51 \begin{align*} \frac{3 \, b^{3} d^{2} \tan \left (f x + e\right )^{4} + 4 \,{\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} \tan \left (f x + e\right )^{3} + 12 \,{\left ({\left (a^{3} - 3 \, a b^{2}\right )} c^{2} - 2 \,{\left (3 \, a^{2} b - b^{3}\right )} c d -{\left (a^{3} - 3 \, a b^{2}\right )} d^{2}\right )} f x + 6 \,{\left (b^{3} c^{2} + 6 \, a b^{2} c d +{\left (3 \, a^{2} b - b^{3}\right )} d^{2}\right )} \tan \left (f x + e\right )^{2} - 6 \,{\left ({\left (3 \, a^{2} b - b^{3}\right )} c^{2} + 2 \,{\left (a^{3} - 3 \, a b^{2}\right )} c d -{\left (3 \, a^{2} b - b^{3}\right )} d^{2}\right )} \log \left (\frac{1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 12 \,{\left (3 \, a b^{2} c^{2} + 2 \,{\left (3 \, a^{2} b - b^{3}\right )} c d +{\left (a^{3} - 3 \, a b^{2}\right )} d^{2}\right )} \tan \left (f x + e\right )}{12 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))^3*(c+d*tan(f*x+e))^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.07894, size = 445, normalized size = 2.07 \begin{align*} \begin{cases} a^{3} c^{2} x + \frac{a^{3} c d \log{\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{f} - a^{3} d^{2} x + \frac{a^{3} d^{2} \tan{\left (e + f x \right )}}{f} + \frac{3 a^{2} b c^{2} \log{\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} - 6 a^{2} b c d x + \frac{6 a^{2} b c d \tan{\left (e + f x \right )}}{f} - \frac{3 a^{2} b d^{2} \log{\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac{3 a^{2} b d^{2} \tan ^{2}{\left (e + f x \right )}}{2 f} - 3 a b^{2} c^{2} x + \frac{3 a b^{2} c^{2} \tan{\left (e + f x \right )}}{f} - \frac{3 a b^{2} c d \log{\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{f} + \frac{3 a b^{2} c d \tan ^{2}{\left (e + f x \right )}}{f} + 3 a b^{2} d^{2} x + \frac{a b^{2} d^{2} \tan ^{3}{\left (e + f x \right )}}{f} - \frac{3 a b^{2} d^{2} \tan{\left (e + f x \right )}}{f} - \frac{b^{3} c^{2} \log{\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac{b^{3} c^{2} \tan ^{2}{\left (e + f x \right )}}{2 f} + 2 b^{3} c d x + \frac{2 b^{3} c d \tan ^{3}{\left (e + f x \right )}}{3 f} - \frac{2 b^{3} c d \tan{\left (e + f x \right )}}{f} + \frac{b^{3} d^{2} \log{\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac{b^{3} d^{2} \tan ^{4}{\left (e + f x \right )}}{4 f} - \frac{b^{3} d^{2} \tan ^{2}{\left (e + f x \right )}}{2 f} & \text{for}\: f \neq 0 \\x \left (a + b \tan{\left (e \right )}\right )^{3} \left (c + d \tan{\left (e \right )}\right )^{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))**3*(c+d*tan(f*x+e))**2,x)

[Out]

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

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Giac [B]  time = 6.30377, size = 6152, normalized size = 28.61 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))^3*(c+d*tan(f*x+e))^2,x, algorithm="giac")

[Out]

1/12*(12*a^3*c^2*f*x*tan(f*x)^4*tan(e)^4 - 36*a*b^2*c^2*f*x*tan(f*x)^4*tan(e)^4 - 72*a^2*b*c*d*f*x*tan(f*x)^4*
tan(e)^4 + 24*b^3*c*d*f*x*tan(f*x)^4*tan(e)^4 - 12*a^3*d^2*f*x*tan(f*x)^4*tan(e)^4 + 36*a*b^2*d^2*f*x*tan(f*x)
^4*tan(e)^4 - 18*a^2*b*c^2*log(4*(tan(e)^2 + 1)/(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)
^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1))*tan(f*x)^4*tan(e)^4 + 6*b^3*c^2*log(4*(tan(e)^2 + 1)/(tan(f*x)^4*tan
(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1))*tan(f*x)^4*tan(e)^4 -
12*a^3*c*d*log(4*(tan(e)^2 + 1)/(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2
- 2*tan(f*x)*tan(e) + 1))*tan(f*x)^4*tan(e)^4 + 36*a*b^2*c*d*log(4*(tan(e)^2 + 1)/(tan(f*x)^4*tan(e)^2 - 2*ta
n(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1))*tan(f*x)^4*tan(e)^4 + 18*a^2*b*d^
2*log(4*(tan(e)^2 + 1)/(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f
*x)*tan(e) + 1))*tan(f*x)^4*tan(e)^4 - 6*b^3*d^2*log(4*(tan(e)^2 + 1)/(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(
e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1))*tan(f*x)^4*tan(e)^4 - 48*a^3*c^2*f*x*tan(f*x)^
3*tan(e)^3 + 144*a*b^2*c^2*f*x*tan(f*x)^3*tan(e)^3 + 288*a^2*b*c*d*f*x*tan(f*x)^3*tan(e)^3 - 96*b^3*c*d*f*x*ta
n(f*x)^3*tan(e)^3 + 48*a^3*d^2*f*x*tan(f*x)^3*tan(e)^3 - 144*a*b^2*d^2*f*x*tan(f*x)^3*tan(e)^3 + 6*b^3*c^2*tan
(f*x)^4*tan(e)^4 + 36*a*b^2*c*d*tan(f*x)^4*tan(e)^4 + 18*a^2*b*d^2*tan(f*x)^4*tan(e)^4 - 9*b^3*d^2*tan(f*x)^4*
tan(e)^4 + 72*a^2*b*c^2*log(4*(tan(e)^2 + 1)/(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2
+ tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1))*tan(f*x)^3*tan(e)^3 - 24*b^3*c^2*log(4*(tan(e)^2 + 1)/(tan(f*x)^4*tan(e
)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1))*tan(f*x)^3*tan(e)^3 + 4
8*a^3*c*d*log(4*(tan(e)^2 + 1)/(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 -
2*tan(f*x)*tan(e) + 1))*tan(f*x)^3*tan(e)^3 - 144*a*b^2*c*d*log(4*(tan(e)^2 + 1)/(tan(f*x)^4*tan(e)^2 - 2*tan
(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1))*tan(f*x)^3*tan(e)^3 - 72*a^2*b*d^2
*log(4*(tan(e)^2 + 1)/(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*
x)*tan(e) + 1))*tan(f*x)^3*tan(e)^3 + 24*b^3*d^2*log(4*(tan(e)^2 + 1)/(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(
e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1))*tan(f*x)^3*tan(e)^3 - 36*a*b^2*c^2*tan(f*x)^4*
tan(e)^3 - 72*a^2*b*c*d*tan(f*x)^4*tan(e)^3 + 24*b^3*c*d*tan(f*x)^4*tan(e)^3 - 12*a^3*d^2*tan(f*x)^4*tan(e)^3
+ 36*a*b^2*d^2*tan(f*x)^4*tan(e)^3 - 36*a*b^2*c^2*tan(f*x)^3*tan(e)^4 - 72*a^2*b*c*d*tan(f*x)^3*tan(e)^4 + 24*
b^3*c*d*tan(f*x)^3*tan(e)^4 - 12*a^3*d^2*tan(f*x)^3*tan(e)^4 + 36*a*b^2*d^2*tan(f*x)^3*tan(e)^4 + 72*a^3*c^2*f
*x*tan(f*x)^2*tan(e)^2 - 216*a*b^2*c^2*f*x*tan(f*x)^2*tan(e)^2 - 432*a^2*b*c*d*f*x*tan(f*x)^2*tan(e)^2 + 144*b
^3*c*d*f*x*tan(f*x)^2*tan(e)^2 - 72*a^3*d^2*f*x*tan(f*x)^2*tan(e)^2 + 216*a*b^2*d^2*f*x*tan(f*x)^2*tan(e)^2 +
6*b^3*c^2*tan(f*x)^4*tan(e)^2 + 36*a*b^2*c*d*tan(f*x)^4*tan(e)^2 + 18*a^2*b*d^2*tan(f*x)^4*tan(e)^2 - 6*b^3*d^
2*tan(f*x)^4*tan(e)^2 - 12*b^3*c^2*tan(f*x)^3*tan(e)^3 - 72*a*b^2*c*d*tan(f*x)^3*tan(e)^3 - 36*a^2*b*d^2*tan(f
*x)^3*tan(e)^3 + 24*b^3*d^2*tan(f*x)^3*tan(e)^3 + 6*b^3*c^2*tan(f*x)^2*tan(e)^4 + 36*a*b^2*c*d*tan(f*x)^2*tan(
e)^4 + 18*a^2*b*d^2*tan(f*x)^2*tan(e)^4 - 6*b^3*d^2*tan(f*x)^2*tan(e)^4 - 8*b^3*c*d*tan(f*x)^4*tan(e) - 12*a*b
^2*d^2*tan(f*x)^4*tan(e) - 108*a^2*b*c^2*log(4*(tan(e)^2 + 1)/(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan
(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1))*tan(f*x)^2*tan(e)^2 + 36*b^3*c^2*log(4*(tan(e)^2 + 1)/
(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1))*tan(f*
x)^2*tan(e)^2 - 72*a^3*c*d*log(4*(tan(e)^2 + 1)/(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)
^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1))*tan(f*x)^2*tan(e)^2 + 216*a*b^2*c*d*log(4*(tan(e)^2 + 1)/(tan(f*x)^4
*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1))*tan(f*x)^2*tan(e)
^2 + 108*a^2*b*d^2*log(4*(tan(e)^2 + 1)/(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan
(f*x)^2 - 2*tan(f*x)*tan(e) + 1))*tan(f*x)^2*tan(e)^2 - 36*b^3*d^2*log(4*(tan(e)^2 + 1)/(tan(f*x)^4*tan(e)^2 -
2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1))*tan(f*x)^2*tan(e)^2 + 108*a*
b^2*c^2*tan(f*x)^3*tan(e)^2 + 216*a^2*b*c*d*tan(f*x)^3*tan(e)^2 - 96*b^3*c*d*tan(f*x)^3*tan(e)^2 + 36*a^3*d^2*
tan(f*x)^3*tan(e)^2 - 144*a*b^2*d^2*tan(f*x)^3*tan(e)^2 + 108*a*b^2*c^2*tan(f*x)^2*tan(e)^3 + 216*a^2*b*c*d*ta
n(f*x)^2*tan(e)^3 - 96*b^3*c*d*tan(f*x)^2*tan(e)^3 + 36*a^3*d^2*tan(f*x)^2*tan(e)^3 - 144*a*b^2*d^2*tan(f*x)^2
*tan(e)^3 - 8*b^3*c*d*tan(f*x)*tan(e)^4 - 12*a*b^2*d^2*tan(f*x)*tan(e)^4 + 3*b^3*d^2*tan(f*x)^4 - 48*a^3*c^2*f
*x*tan(f*x)*tan(e) + 144*a*b^2*c^2*f*x*tan(f*x)*tan(e) + 288*a^2*b*c*d*f*x*tan(f*x)*tan(e) - 96*b^3*c*d*f*x*ta
n(f*x)*tan(e) + 48*a^3*d^2*f*x*tan(f*x)*tan(e) - 144*a*b^2*d^2*f*x*tan(f*x)*tan(e) - 12*b^3*c^2*tan(f*x)^3*tan
(e) - 72*a*b^2*c*d*tan(f*x)^3*tan(e) - 36*a^2*b*d^2*tan(f*x)^3*tan(e) + 24*b^3*d^2*tan(f*x)^3*tan(e) + 12*b^3*
c^2*tan(f*x)^2*tan(e)^2 + 72*a*b^2*c*d*tan(f*x)^2*tan(e)^2 + 36*a^2*b*d^2*tan(f*x)^2*tan(e)^2 - 12*b^3*d^2*tan
(f*x)^2*tan(e)^2 - 12*b^3*c^2*tan(f*x)*tan(e)^3 - 72*a*b^2*c*d*tan(f*x)*tan(e)^3 - 36*a^2*b*d^2*tan(f*x)*tan(e
)^3 + 24*b^3*d^2*tan(f*x)*tan(e)^3 + 3*b^3*d^2*tan(e)^4 + 8*b^3*c*d*tan(f*x)^3 + 12*a*b^2*d^2*tan(f*x)^3 + 72*
a^2*b*c^2*log(4*(tan(e)^2 + 1)/(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 -
2*tan(f*x)*tan(e) + 1))*tan(f*x)*tan(e) - 24*b^3*c^2*log(4*(tan(e)^2 + 1)/(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3
*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1))*tan(f*x)*tan(e) + 48*a^3*c*d*log(4*(tan(e
)^2 + 1)/(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1
))*tan(f*x)*tan(e) - 144*a*b^2*c*d*log(4*(tan(e)^2 + 1)/(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^
2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1))*tan(f*x)*tan(e) - 72*a^2*b*d^2*log(4*(tan(e)^2 + 1)/(tan(f*x
)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1))*tan(f*x)*tan(e
) + 24*b^3*d^2*log(4*(tan(e)^2 + 1)/(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x
)^2 - 2*tan(f*x)*tan(e) + 1))*tan(f*x)*tan(e) - 108*a*b^2*c^2*tan(f*x)^2*tan(e) - 216*a^2*b*c*d*tan(f*x)^2*tan
(e) + 96*b^3*c*d*tan(f*x)^2*tan(e) - 36*a^3*d^2*tan(f*x)^2*tan(e) + 144*a*b^2*d^2*tan(f*x)^2*tan(e) - 108*a*b^
2*c^2*tan(f*x)*tan(e)^2 - 216*a^2*b*c*d*tan(f*x)*tan(e)^2 + 96*b^3*c*d*tan(f*x)*tan(e)^2 - 36*a^3*d^2*tan(f*x)
*tan(e)^2 + 144*a*b^2*d^2*tan(f*x)*tan(e)^2 + 8*b^3*c*d*tan(e)^3 + 12*a*b^2*d^2*tan(e)^3 + 12*a^3*c^2*f*x - 36
*a*b^2*c^2*f*x - 72*a^2*b*c*d*f*x + 24*b^3*c*d*f*x - 12*a^3*d^2*f*x + 36*a*b^2*d^2*f*x + 6*b^3*c^2*tan(f*x)^2
+ 36*a*b^2*c*d*tan(f*x)^2 + 18*a^2*b*d^2*tan(f*x)^2 - 6*b^3*d^2*tan(f*x)^2 - 12*b^3*c^2*tan(f*x)*tan(e) - 72*a
*b^2*c*d*tan(f*x)*tan(e) - 36*a^2*b*d^2*tan(f*x)*tan(e) + 24*b^3*d^2*tan(f*x)*tan(e) + 6*b^3*c^2*tan(e)^2 + 36
*a*b^2*c*d*tan(e)^2 + 18*a^2*b*d^2*tan(e)^2 - 6*b^3*d^2*tan(e)^2 - 18*a^2*b*c^2*log(4*(tan(e)^2 + 1)/(tan(f*x)
^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)) + 6*b^3*c^2*log
(4*(tan(e)^2 + 1)/(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*t
an(e) + 1)) - 12*a^3*c*d*log(4*(tan(e)^2 + 1)/(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2
+ tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)) + 36*a*b^2*c*d*log(4*(tan(e)^2 + 1)/(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^
3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)) + 18*a^2*b*d^2*log(4*(tan(e)^2 + 1)/(tan
(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)) - 6*b^3*d^
2*log(4*(tan(e)^2 + 1)/(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f
*x)*tan(e) + 1)) + 36*a*b^2*c^2*tan(f*x) + 72*a^2*b*c*d*tan(f*x) - 24*b^3*c*d*tan(f*x) + 12*a^3*d^2*tan(f*x) -
36*a*b^2*d^2*tan(f*x) + 36*a*b^2*c^2*tan(e) + 72*a^2*b*c*d*tan(e) - 24*b^3*c*d*tan(e) + 12*a^3*d^2*tan(e) - 3
6*a*b^2*d^2*tan(e) + 6*b^3*c^2 + 36*a*b^2*c*d + 18*a^2*b*d^2 - 9*b^3*d^2)/(f*tan(f*x)^4*tan(e)^4 - 4*f*tan(f*x
)^3*tan(e)^3 + 6*f*tan(f*x)^2*tan(e)^2 - 4*f*tan(f*x)*tan(e) + f)