3.1.50 \(\int (a+b \tan (e+f x))^3 (c+d \tan (e+f x)) (A+B \tan (e+f x)+C \tan ^2(e+f x)) \, dx\) [50]

Optimal. Leaf size=353 \[ \left (a^3 (A c-c C-B d)-3 a b^2 (A c-c C-B d)-3 a^2 b (B c+(A-C) d)+b^3 (B c+(A-C) d)\right ) x-\frac {\left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a b^2 (B c+(A-C) d)\right ) \log (\cos (e+f x))}{f}+\frac {b \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) \tan (e+f x)}{f}+\frac {(A b c+a B c-b c C+a A d-b B d-a C d) (a+b \tan (e+f x))^2}{2 f}+\frac {(B c+(A-C) d) (a+b \tan (e+f x))^3}{3 f}-\frac {(a C d-5 b (c C+B d)) (a+b \tan (e+f x))^4}{20 b^2 f}+\frac {C d \tan (e+f x) (a+b \tan (e+f x))^4}{5 b f} \]

[Out]

(a^3*(A*c-B*d-C*c)-3*a*b^2*(A*c-B*d-C*c)-3*a^2*b*(B*c+(A-C)*d)+b^3*(B*c+(A-C)*d))*x-(3*a^2*b*(A*c-B*d-C*c)-b^3
*(A*c-B*d-C*c)+a^3*(B*c+(A-C)*d)-3*a*b^2*(B*c+(A-C)*d))*ln(cos(f*x+e))/f+b*(2*a*b*(A*c-B*d-C*c)+a^2*(B*c+(A-C)
*d)-b^2*(B*c+(A-C)*d))*tan(f*x+e)/f+1/2*(A*a*d+A*b*c+B*a*c-B*b*d-C*a*d-C*b*c)*(a+b*tan(f*x+e))^2/f+1/3*(B*c+(A
-C)*d)*(a+b*tan(f*x+e))^3/f-1/20*(a*C*d-5*b*(B*d+C*c))*(a+b*tan(f*x+e))^4/b^2/f+1/5*C*d*tan(f*x+e)*(a+b*tan(f*
x+e))^4/b/f

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Rubi [A]
time = 0.55, antiderivative size = 353, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {3718, 3711, 3609, 3606, 3556} \begin {gather*} \frac {b \tan (e+f x) \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{f}-\frac {\log (\cos (e+f x)) \left (a^3 (d (A-C)+B c)+3 a^2 b (A c-B d-c C)-3 a b^2 (d (A-C)+B c)-b^3 (A c-B d-c C)\right )}{f}+x \left (a^3 (A c-B d-c C)-3 a^2 b (d (A-C)+B c)-3 a b^2 (A c-B d-c C)+b^3 (d (A-C)+B c)\right )+\frac {(d (A-C)+B c) (a+b \tan (e+f x))^3}{3 f}+\frac {(a+b \tan (e+f x))^2 (a A d+a B c-a C d+A b c-b B d-b c C)}{2 f}-\frac {(a C d-5 b (B d+c C)) (a+b \tan (e+f x))^4}{20 b^2 f}+\frac {C d \tan (e+f x) (a+b \tan (e+f x))^4}{5 b f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(A*c - c*C - B*d) - 3*a*b^2*(A*c - c*C - B*d) - 3*a^2*b*(B*c + (A - C)*d) + b^3*(B*c + (A - C)*d))*x - ((
3*a^2*b*(A*c - c*C - B*d) - b^3*(A*c - c*C - B*d) + a^3*(B*c + (A - C)*d) - 3*a*b^2*(B*c + (A - C)*d))*Log[Cos
[e + f*x]])/f + (b*(2*a*b*(A*c - c*C - B*d) + a^2*(B*c + (A - C)*d) - b^2*(B*c + (A - C)*d))*Tan[e + f*x])/f +
((A*b*c + a*B*c - b*c*C + a*A*d - b*B*d - a*C*d)*(a + b*Tan[e + f*x])^2)/(2*f) + ((B*c + (A - C)*d)*(a + b*Ta
n[e + f*x])^3)/(3*f) - ((a*C*d - 5*b*(c*C + B*d))*(a + b*Tan[e + f*x])^4)/(20*b^2*f) + (C*d*Tan[e + f*x]*(a +
b*Tan[e + f*x])^4)/(5*b*f)

Rule 3556

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

Rule 3606

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 3609

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 3711

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 3718

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*tan[(e
_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[b*C*Tan[e + f*x]*((c + d*Tan[e + f*x])
^(n + 1)/(d*f*(n + 2))), x] - Dist[1/(d*(n + 2)), Int[(c + d*Tan[e + f*x])^n*Simp[b*c*C - a*A*d*(n + 2) - (A*b
+ a*B - b*C)*d*(n + 2)*Tan[e + f*x] - (a*C*d*(n + 2) - b*(c*C - B*d*(n + 2)))*Tan[e + f*x]^2, x], x], x] /; F
reeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] &&  !LtQ[n, -1]

Rubi steps

\begin {align*} \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x)) \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx &=\frac {C d \tan (e+f x) (a+b \tan (e+f x))^4}{5 b f}-\frac {\int (a+b \tan (e+f x))^3 \left (-5 A b c+a C d-5 b (B c+(A-C) d) \tan (e+f x)+(a C d-5 b (c C+B d)) \tan ^2(e+f x)\right ) \, dx}{5 b}\\ &=-\frac {(a C d-5 b (c C+B d)) (a+b \tan (e+f x))^4}{20 b^2 f}+\frac {C d \tan (e+f x) (a+b \tan (e+f x))^4}{5 b f}-\frac {\int (a+b \tan (e+f x))^3 (-5 b (A c-c C-B d)-5 b (B c+(A-C) d) \tan (e+f x)) \, dx}{5 b}\\ &=\frac {(B c+(A-C) d) (a+b \tan (e+f x))^3}{3 f}-\frac {(a C d-5 b (c C+B d)) (a+b \tan (e+f x))^4}{20 b^2 f}+\frac {C d \tan (e+f x) (a+b \tan (e+f x))^4}{5 b f}-\frac {\int (a+b \tan (e+f x))^2 (5 b (b B c+b (A-C) d-a (A c-c C-B d))-5 b (A b c+a B c-b c C+a A d-b B d-a C d) \tan (e+f x)) \, dx}{5 b}\\ &=\frac {(A b c+a B c-b c C+a A d-b B d-a C d) (a+b \tan (e+f x))^2}{2 f}+\frac {(B c+(A-C) d) (a+b \tan (e+f x))^3}{3 f}-\frac {(a C d-5 b (c C+B d)) (a+b \tan (e+f x))^4}{20 b^2 f}+\frac {C d \tan (e+f x) (a+b \tan (e+f x))^4}{5 b f}-\frac {\int (a+b \tan (e+f x)) \left (-5 b \left (a^2 (A c-c C-B d)-b^2 (A c-c C-B d)-2 a b (B c+(A-C) d)\right )-5 b \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) \tan (e+f x)\right ) \, dx}{5 b}\\ &=\left (a^3 (A c-c C-B d)-3 a b^2 (A c-c C-B d)-3 a^2 b (B c+(A-C) d)+b^3 (B c+(A-C) d)\right ) x+\frac {b \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) \tan (e+f x)}{f}+\frac {(A b c+a B c-b c C+a A d-b B d-a C d) (a+b \tan (e+f x))^2}{2 f}+\frac {(B c+(A-C) d) (a+b \tan (e+f x))^3}{3 f}-\frac {(a C d-5 b (c C+B d)) (a+b \tan (e+f x))^4}{20 b^2 f}+\frac {C d \tan (e+f x) (a+b \tan (e+f x))^4}{5 b f}-\left (-3 a^2 b (A c-c C-B d)+b^3 (A c-c C-B d)-a^3 (B c+(A-C) d)+3 a b^2 (B c+(A-C) d)\right ) \int \tan (e+f x) \, dx\\ &=\left (a^3 (A c-c C-B d)-3 a b^2 (A c-c C-B d)-3 a^2 b (B c+(A-C) d)+b^3 (B c+(A-C) d)\right ) x-\frac {\left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)+a^3 (B c+(A-C) d)-3 a b^2 (B c+(A-C) d)\right ) \log (\cos (e+f x))}{f}+\frac {b \left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) \tan (e+f x)}{f}+\frac {(A b c+a B c-b c C+a A d-b B d-a C d) (a+b \tan (e+f x))^2}{2 f}+\frac {(B c+(A-C) d) (a+b \tan (e+f x))^3}{3 f}-\frac {(a C d-5 b (c C+B d)) (a+b \tan (e+f x))^4}{20 b^2 f}+\frac {C d \tan (e+f x) (a+b \tan (e+f x))^4}{5 b f}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 5.85, size = 280, normalized size = 0.79 \begin {gather*} \frac {\frac {3 (-a C d+5 b (c C+B d)) (a+b \tan (e+f x))^4}{b}+12 C d \tan (e+f x) (a+b \tan (e+f x))^4+30 (A b c-a B c-b c C-a A d-b B d+a C d) \left ((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 ^2(e+f x)\right )-10 (B c+(A-C) d) \left (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 \left (-6 a^2+b^2\right ) \tan (e+f x)-12 a b^3 \tan ^2(e+f x)-2 b^4 \tan ^3(e+f x)\right )}{60 b f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((3*(-(a*C*d) + 5*b*(c*C + B*d))*(a + b*Tan[e + f*x])^4)/b + 12*C*d*Tan[e + f*x]*(a + b*Tan[e + f*x])^4 + 30*(
A*b*c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d)*((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) - 10*(B*c + (A - 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))/(60*b*f)

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Maple [A]
time = 0.21, size = 639, normalized size = 1.81 Too large to display

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))*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x,method=_RETURNVERBOSE)

[Out]

1/f*(1/2*(A*a^3*d+3*A*a^2*b*c-3*A*a*b^2*d-A*b^3*c+B*a^3*c-3*B*a^2*b*d-3*B*a*b^2*c+B*b^3*d-C*a^3*d-3*C*a^2*b*c+
3*C*a*b^2*d+C*b^3*c)*ln(1+tan(f*x+e)^2)+(A*a^3*c-3*A*a^2*b*d-3*A*a*b^2*c+A*b^3*d-B*a^3*d-3*B*a^2*b*c+3*B*a*b^2
*d+B*b^3*c-C*a^3*c+3*C*a^2*b*d+3*C*a*b^2*c-C*b^3*d)*arctan(tan(f*x+e))+C*b^3*d*tan(f*x+e)-1/2*C*b^3*c*tan(f*x+
e)^2+1/5*C*b^3*d*tan(f*x+e)^5+1/4*B*b^3*d*tan(f*x+e)^4+1/4*C*b^3*c*tan(f*x+e)^4+1/3*A*b^3*d*tan(f*x+e)^3+1/3*B
*b^3*c*tan(f*x+e)^3-1/3*C*b^3*d*tan(f*x+e)^3+1/2*A*b^3*c*tan(f*x+e)^2-1/2*B*b^3*d*tan(f*x+e)^2+B*a^3*d*tan(f*x
+e)-B*b^3*c*tan(f*x+e)+C*a^3*c*tan(f*x+e)-A*b^3*d*tan(f*x+e)+1/2*C*a^3*d*tan(f*x+e)^2-3/2*C*a*b^2*d*tan(f*x+e)
^2+B*a*b^2*d*tan(f*x+e)^3+C*a^2*b*d*tan(f*x+e)^3+C*a*b^2*c*tan(f*x+e)^3+3*A*a^2*b*d*tan(f*x+e)+3*A*a*b^2*c*tan
(f*x+e)+3*B*a^2*b*c*tan(f*x+e)-3*B*a*b^2*d*tan(f*x+e)-3*C*a^2*b*d*tan(f*x+e)-3*C*a*b^2*c*tan(f*x+e)+3/4*C*a*b^
2*d*tan(f*x+e)^4+3/2*A*a*b^2*d*tan(f*x+e)^2+3/2*B*a^2*b*d*tan(f*x+e)^2+3/2*B*a*b^2*c*tan(f*x+e)^2+3/2*C*a^2*b*
c*tan(f*x+e)^2)

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Maxima [A]
time = 0.51, size = 423, normalized size = 1.20 \begin {gather*} \frac {12 \, C b^{3} d \tan \left (f x + e\right )^{5} + 15 \, {\left (C b^{3} c + {\left (3 \, C a b^{2} + B b^{3}\right )} d\right )} \tan \left (f x + e\right )^{4} + 20 \, {\left ({\left (3 \, C a b^{2} + B b^{3}\right )} c + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + {\left (A - C\right )} b^{3}\right )} d\right )} \tan \left (f x + e\right )^{3} + 30 \, {\left ({\left (3 \, C a^{2} b + 3 \, B a b^{2} + {\left (A - C\right )} b^{3}\right )} c + {\left (C a^{3} + 3 \, B a^{2} b + 3 \, {\left (A - C\right )} a b^{2} - B b^{3}\right )} d\right )} \tan \left (f x + e\right )^{2} + 60 \, {\left ({\left ({\left (A - C\right )} a^{3} - 3 \, B a^{2} b - 3 \, {\left (A - C\right )} a b^{2} + B b^{3}\right )} c - {\left (B a^{3} + 3 \, {\left (A - C\right )} a^{2} b - 3 \, B a b^{2} - {\left (A - C\right )} b^{3}\right )} d\right )} {\left (f x + e\right )} + 30 \, {\left ({\left (B a^{3} + 3 \, {\left (A - C\right )} a^{2} b - 3 \, B a b^{2} - {\left (A - C\right )} b^{3}\right )} c + {\left ({\left (A - C\right )} a^{3} - 3 \, B a^{2} b - 3 \, {\left (A - C\right )} a b^{2} + B b^{3}\right )} d\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right ) + 60 \, {\left ({\left (C a^{3} + 3 \, B a^{2} b + 3 \, {\left (A - C\right )} a b^{2} - B b^{3}\right )} c + {\left (B a^{3} + 3 \, {\left (A - C\right )} a^{2} b - 3 \, B a b^{2} - {\left (A - C\right )} b^{3}\right )} d\right )} \tan \left (f x + e\right )}{60 \, f} \end {gather*}

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))*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x, algorithm="maxima")

[Out]

1/60*(12*C*b^3*d*tan(f*x + e)^5 + 15*(C*b^3*c + (3*C*a*b^2 + B*b^3)*d)*tan(f*x + e)^4 + 20*((3*C*a*b^2 + B*b^3
)*c + (3*C*a^2*b + 3*B*a*b^2 + (A - C)*b^3)*d)*tan(f*x + e)^3 + 30*((3*C*a^2*b + 3*B*a*b^2 + (A - C)*b^3)*c +
(C*a^3 + 3*B*a^2*b + 3*(A - C)*a*b^2 - B*b^3)*d)*tan(f*x + e)^2 + 60*(((A - C)*a^3 - 3*B*a^2*b - 3*(A - C)*a*b
^2 + B*b^3)*c - (B*a^3 + 3*(A - C)*a^2*b - 3*B*a*b^2 - (A - C)*b^3)*d)*(f*x + e) + 30*((B*a^3 + 3*(A - C)*a^2*
b - 3*B*a*b^2 - (A - C)*b^3)*c + ((A - C)*a^3 - 3*B*a^2*b - 3*(A - C)*a*b^2 + B*b^3)*d)*log(tan(f*x + e)^2 + 1
) + 60*((C*a^3 + 3*B*a^2*b + 3*(A - C)*a*b^2 - B*b^3)*c + (B*a^3 + 3*(A - C)*a^2*b - 3*B*a*b^2 - (A - C)*b^3)*
d)*tan(f*x + e))/f

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Fricas [A]
time = 1.48, size = 421, normalized size = 1.19 \begin {gather*} \frac {12 \, C b^{3} d \tan \left (f x + e\right )^{5} + 15 \, {\left (C b^{3} c + {\left (3 \, C a b^{2} + B b^{3}\right )} d\right )} \tan \left (f x + e\right )^{4} + 20 \, {\left ({\left (3 \, C a b^{2} + B b^{3}\right )} c + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + {\left (A - C\right )} b^{3}\right )} d\right )} \tan \left (f x + e\right )^{3} + 60 \, {\left ({\left ({\left (A - C\right )} a^{3} - 3 \, B a^{2} b - 3 \, {\left (A - C\right )} a b^{2} + B b^{3}\right )} c - {\left (B a^{3} + 3 \, {\left (A - C\right )} a^{2} b - 3 \, B a b^{2} - {\left (A - C\right )} b^{3}\right )} d\right )} f x + 30 \, {\left ({\left (3 \, C a^{2} b + 3 \, B a b^{2} + {\left (A - C\right )} b^{3}\right )} c + {\left (C a^{3} + 3 \, B a^{2} b + 3 \, {\left (A - C\right )} a b^{2} - B b^{3}\right )} d\right )} \tan \left (f x + e\right )^{2} - 30 \, {\left ({\left (B a^{3} + 3 \, {\left (A - C\right )} a^{2} b - 3 \, B a b^{2} - {\left (A - C\right )} b^{3}\right )} c + {\left ({\left (A - C\right )} a^{3} - 3 \, B a^{2} b - 3 \, {\left (A - C\right )} a b^{2} + B b^{3}\right )} d\right )} \log \left (\frac {1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 60 \, {\left ({\left (C a^{3} + 3 \, B a^{2} b + 3 \, {\left (A - C\right )} a b^{2} - B b^{3}\right )} c + {\left (B a^{3} + 3 \, {\left (A - C\right )} a^{2} b - 3 \, B a b^{2} - {\left (A - C\right )} b^{3}\right )} d\right )} \tan \left (f x + e\right )}{60 \, f} \end {gather*}

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))*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x, algorithm="fricas")

[Out]

1/60*(12*C*b^3*d*tan(f*x + e)^5 + 15*(C*b^3*c + (3*C*a*b^2 + B*b^3)*d)*tan(f*x + e)^4 + 20*((3*C*a*b^2 + B*b^3
)*c + (3*C*a^2*b + 3*B*a*b^2 + (A - C)*b^3)*d)*tan(f*x + e)^3 + 60*(((A - C)*a^3 - 3*B*a^2*b - 3*(A - C)*a*b^2
+ B*b^3)*c - (B*a^3 + 3*(A - C)*a^2*b - 3*B*a*b^2 - (A - C)*b^3)*d)*f*x + 30*((3*C*a^2*b + 3*B*a*b^2 + (A - C
)*b^3)*c + (C*a^3 + 3*B*a^2*b + 3*(A - C)*a*b^2 - B*b^3)*d)*tan(f*x + e)^2 - 30*((B*a^3 + 3*(A - C)*a^2*b - 3*
B*a*b^2 - (A - C)*b^3)*c + ((A - C)*a^3 - 3*B*a^2*b - 3*(A - C)*a*b^2 + B*b^3)*d)*log(1/(tan(f*x + e)^2 + 1))
+ 60*((C*a^3 + 3*B*a^2*b + 3*(A - C)*a*b^2 - B*b^3)*c + (B*a^3 + 3*(A - C)*a^2*b - 3*B*a*b^2 - (A - C)*b^3)*d)
*tan(f*x + e))/f

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1001 vs. \(2 (316) = 632\).
time = 0.35, size = 1001, normalized size = 2.84 \begin {gather*} \begin {cases} A a^{3} c x + \frac {A a^{3} d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {3 A a^{2} b c \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} - 3 A a^{2} b d x + \frac {3 A a^{2} b d \tan {\left (e + f x \right )}}{f} - 3 A a b^{2} c x + \frac {3 A a b^{2} c \tan {\left (e + f x \right )}}{f} - \frac {3 A a b^{2} d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {3 A a b^{2} d \tan ^{2}{\left (e + f x \right )}}{2 f} - \frac {A b^{3} c \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {A b^{3} c \tan ^{2}{\left (e + f x \right )}}{2 f} + A b^{3} d x + \frac {A b^{3} d \tan ^{3}{\left (e + f x \right )}}{3 f} - \frac {A b^{3} d \tan {\left (e + f x \right )}}{f} + \frac {B a^{3} c \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} - B a^{3} d x + \frac {B a^{3} d \tan {\left (e + f x \right )}}{f} - 3 B a^{2} b c x + \frac {3 B a^{2} b c \tan {\left (e + f x \right )}}{f} - \frac {3 B a^{2} b d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {3 B a^{2} b d \tan ^{2}{\left (e + f x \right )}}{2 f} - \frac {3 B a b^{2} c \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {3 B a b^{2} c \tan ^{2}{\left (e + f x \right )}}{2 f} + 3 B a b^{2} d x + \frac {B a b^{2} d \tan ^{3}{\left (e + f x \right )}}{f} - \frac {3 B a b^{2} d \tan {\left (e + f x \right )}}{f} + B b^{3} c x + \frac {B b^{3} c \tan ^{3}{\left (e + f x \right )}}{3 f} - \frac {B b^{3} c \tan {\left (e + f x \right )}}{f} + \frac {B b^{3} d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {B b^{3} d \tan ^{4}{\left (e + f x \right )}}{4 f} - \frac {B b^{3} d \tan ^{2}{\left (e + f x \right )}}{2 f} - C a^{3} c x + \frac {C a^{3} c \tan {\left (e + f x \right )}}{f} - \frac {C a^{3} d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {C a^{3} d \tan ^{2}{\left (e + f x \right )}}{2 f} - \frac {3 C a^{2} b c \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {3 C a^{2} b c \tan ^{2}{\left (e + f x \right )}}{2 f} + 3 C a^{2} b d x + \frac {C a^{2} b d \tan ^{3}{\left (e + f x \right )}}{f} - \frac {3 C a^{2} b d \tan {\left (e + f x \right )}}{f} + 3 C a b^{2} c x + \frac {C a b^{2} c \tan ^{3}{\left (e + f x \right )}}{f} - \frac {3 C a b^{2} c \tan {\left (e + f x \right )}}{f} + \frac {3 C a b^{2} d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {3 C a b^{2} d \tan ^{4}{\left (e + f x \right )}}{4 f} - \frac {3 C a b^{2} d \tan ^{2}{\left (e + f x \right )}}{2 f} + \frac {C b^{3} c \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {C b^{3} c \tan ^{4}{\left (e + f x \right )}}{4 f} - \frac {C b^{3} c \tan ^{2}{\left (e + f x \right )}}{2 f} - C b^{3} d x + \frac {C b^{3} d \tan ^{5}{\left (e + f x \right )}}{5 f} - \frac {C b^{3} d \tan ^{3}{\left (e + f x \right )}}{3 f} + \frac {C b^{3} d \tan {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (a + b \tan {\left (e \right )}\right )^{3} \left (c + d \tan {\left (e \right )}\right ) \left (A + B \tan {\left (e \right )} + C \tan ^{2}{\left (e \right )}\right ) & \text {otherwise} \end {cases} \end {gather*}

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))*(A+B*tan(f*x+e)+C*tan(f*x+e)**2),x)

[Out]

Piecewise((A*a**3*c*x + A*a**3*d*log(tan(e + f*x)**2 + 1)/(2*f) + 3*A*a**2*b*c*log(tan(e + f*x)**2 + 1)/(2*f)
- 3*A*a**2*b*d*x + 3*A*a**2*b*d*tan(e + f*x)/f - 3*A*a*b**2*c*x + 3*A*a*b**2*c*tan(e + f*x)/f - 3*A*a*b**2*d*l
og(tan(e + f*x)**2 + 1)/(2*f) + 3*A*a*b**2*d*tan(e + f*x)**2/(2*f) - A*b**3*c*log(tan(e + f*x)**2 + 1)/(2*f) +
A*b**3*c*tan(e + f*x)**2/(2*f) + A*b**3*d*x + A*b**3*d*tan(e + f*x)**3/(3*f) - A*b**3*d*tan(e + f*x)/f + B*a*
*3*c*log(tan(e + f*x)**2 + 1)/(2*f) - B*a**3*d*x + B*a**3*d*tan(e + f*x)/f - 3*B*a**2*b*c*x + 3*B*a**2*b*c*tan
(e + f*x)/f - 3*B*a**2*b*d*log(tan(e + f*x)**2 + 1)/(2*f) + 3*B*a**2*b*d*tan(e + f*x)**2/(2*f) - 3*B*a*b**2*c*
log(tan(e + f*x)**2 + 1)/(2*f) + 3*B*a*b**2*c*tan(e + f*x)**2/(2*f) + 3*B*a*b**2*d*x + B*a*b**2*d*tan(e + f*x)
**3/f - 3*B*a*b**2*d*tan(e + f*x)/f + B*b**3*c*x + B*b**3*c*tan(e + f*x)**3/(3*f) - B*b**3*c*tan(e + f*x)/f +
B*b**3*d*log(tan(e + f*x)**2 + 1)/(2*f) + B*b**3*d*tan(e + f*x)**4/(4*f) - B*b**3*d*tan(e + f*x)**2/(2*f) - C*
a**3*c*x + C*a**3*c*tan(e + f*x)/f - C*a**3*d*log(tan(e + f*x)**2 + 1)/(2*f) + C*a**3*d*tan(e + f*x)**2/(2*f)
- 3*C*a**2*b*c*log(tan(e + f*x)**2 + 1)/(2*f) + 3*C*a**2*b*c*tan(e + f*x)**2/(2*f) + 3*C*a**2*b*d*x + C*a**2*b
*d*tan(e + f*x)**3/f - 3*C*a**2*b*d*tan(e + f*x)/f + 3*C*a*b**2*c*x + C*a*b**2*c*tan(e + f*x)**3/f - 3*C*a*b**
2*c*tan(e + f*x)/f + 3*C*a*b**2*d*log(tan(e + f*x)**2 + 1)/(2*f) + 3*C*a*b**2*d*tan(e + f*x)**4/(4*f) - 3*C*a*
b**2*d*tan(e + f*x)**2/(2*f) + C*b**3*c*log(tan(e + f*x)**2 + 1)/(2*f) + C*b**3*c*tan(e + f*x)**4/(4*f) - C*b*
*3*c*tan(e + f*x)**2/(2*f) - C*b**3*d*x + C*b**3*d*tan(e + f*x)**5/(5*f) - C*b**3*d*tan(e + f*x)**3/(3*f) + C*
b**3*d*tan(e + f*x)/f, Ne(f, 0)), (x*(a + b*tan(e))**3*(c + d*tan(e))*(A + B*tan(e) + C*tan(e)**2), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 11805 vs. \(2 (352) = 704\).
time = 8.91, size = 11805, normalized size = 33.44 \begin {gather*} \text {Too large to display} \end {gather*}

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))*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x, algorithm="giac")

[Out]

1/60*(60*A*a^3*c*f*x*tan(f*x)^5*tan(e)^5 - 60*C*a^3*c*f*x*tan(f*x)^5*tan(e)^5 - 180*B*a^2*b*c*f*x*tan(f*x)^5*t
an(e)^5 - 180*A*a*b^2*c*f*x*tan(f*x)^5*tan(e)^5 + 180*C*a*b^2*c*f*x*tan(f*x)^5*tan(e)^5 + 60*B*b^3*c*f*x*tan(f
*x)^5*tan(e)^5 - 60*B*a^3*d*f*x*tan(f*x)^5*tan(e)^5 - 180*A*a^2*b*d*f*x*tan(f*x)^5*tan(e)^5 + 180*C*a^2*b*d*f*
x*tan(f*x)^5*tan(e)^5 + 180*B*a*b^2*d*f*x*tan(f*x)^5*tan(e)^5 + 60*A*b^3*d*f*x*tan(f*x)^5*tan(e)^5 - 60*C*b^3*
d*f*x*tan(f*x)^5*tan(e)^5 - 30*B*a^3*c*log(4*(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(e)^2 + 1))*tan(f*x)^5*tan(e)^5 - 90*A*a^2*b*c*log(4*(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(e)^2 + 1))*tan(f*
x)^5*tan(e)^5 + 90*C*a^2*b*c*log(4*(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(e)^2 + 1))*tan(f*x)^5*tan(e)^5 + 90*B*a*b^2*c*log(4*(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(e)^2 + 1))*tan(f*x)^5*tan(e
)^5 + 30*A*b^3*c*log(4*(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(e)^2 + 1))*tan(f*x)^5*tan(e)^5 - 30*C*b^3*c*log(4*(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(e)^2 + 1))*tan(f*x)^5*tan(e)^5 - 30*A*a^3
*d*log(4*(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(e)^2 + 1))*tan(f*x)^5*tan(e)^5 + 30*C*a^3*d*log(4*(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(e)^2 + 1))*tan(f*x)^5*tan(e)^5 + 90*B*a^2*b*d*log(4*(ta
n(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(e)^2
+ 1))*tan(f*x)^5*tan(e)^5 + 90*A*a*b^2*d*log(4*(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(e)^2 + 1))*tan(f*x)^5*tan(e)^5 - 90*C*a*b^2*d*log(4*(tan(f*x)^4*t
an(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(e)^2 + 1))*tan(
f*x)^5*tan(e)^5 - 30*B*b^3*d*log(4*(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(e)^2 + 1))*tan(f*x)^5*tan(e)^5 - 300*A*a^3*c*f*x*tan(f*x)^4*tan(e)^4 + 300*C*
a^3*c*f*x*tan(f*x)^4*tan(e)^4 + 900*B*a^2*b*c*f*x*tan(f*x)^4*tan(e)^4 + 900*A*a*b^2*c*f*x*tan(f*x)^4*tan(e)^4
- 900*C*a*b^2*c*f*x*tan(f*x)^4*tan(e)^4 - 300*B*b^3*c*f*x*tan(f*x)^4*tan(e)^4 + 300*B*a^3*d*f*x*tan(f*x)^4*tan
(e)^4 + 900*A*a^2*b*d*f*x*tan(f*x)^4*tan(e)^4 - 900*C*a^2*b*d*f*x*tan(f*x)^4*tan(e)^4 - 900*B*a*b^2*d*f*x*tan(
f*x)^4*tan(e)^4 - 300*A*b^3*d*f*x*tan(f*x)^4*tan(e)^4 + 300*C*b^3*d*f*x*tan(f*x)^4*tan(e)^4 + 90*C*a^2*b*c*tan
(f*x)^5*tan(e)^5 + 90*B*a*b^2*c*tan(f*x)^5*tan(e)^5 + 30*A*b^3*c*tan(f*x)^5*tan(e)^5 - 45*C*b^3*c*tan(f*x)^5*t
an(e)^5 + 30*C*a^3*d*tan(f*x)^5*tan(e)^5 + 90*B*a^2*b*d*tan(f*x)^5*tan(e)^5 + 90*A*a*b^2*d*tan(f*x)^5*tan(e)^5
- 135*C*a*b^2*d*tan(f*x)^5*tan(e)^5 - 45*B*b^3*d*tan(f*x)^5*tan(e)^5 + 150*B*a^3*c*log(4*(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(e)^2 + 1))*tan(f*x)^4*
tan(e)^4 + 450*A*a^2*b*c*log(4*(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(e)^2 + 1))*tan(f*x)^4*tan(e)^4 - 450*C*a^2*b*c*log(4*(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(e)^2 + 1))*tan(f*x)^4*tan(e)^4
- 450*B*a*b^2*c*log(4*(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(e)^2 + 1))*tan(f*x)^4*tan(e)^4 - 150*A*b^3*c*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*ta
n(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^4*tan(e)^4 + 150*C*b
^3*c*log(4*(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(e)^2 + 1))*tan(f*x)^4*tan(e)^4 + 150*A*a^3*d*log(4*(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(e)^2 + 1))*tan(f*x)^4*tan(e)^4 - 150*C*a^3*d*log(4*(
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(e)^
2 + 1))*tan(f*x)^4*tan(e)^4 - 450*B*a^2*b*d*log(4*(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(e)^2 + 1))*tan(f*x)^4*tan(e)^4 - 450*A*a*b^2*d*log(4*(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(e)^2 + 1))*
tan(f*x)^4*tan(e)^4 + 450*C*a*b^2*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + t
an(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^4*tan(e)^4 + 150*B*b^3*d*log(4*(tan(f*x)^4*tan(e)^
2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 +...

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Mupad [B]
time = 9.00, size = 477, normalized size = 1.35 \begin {gather*} x\,\left (A\,a^3\,c+A\,b^3\,d-B\,a^3\,d+B\,b^3\,c-C\,a^3\,c-C\,b^3\,d-3\,A\,a\,b^2\,c-3\,A\,a^2\,b\,d-3\,B\,a^2\,b\,c+3\,B\,a\,b^2\,d+3\,C\,a\,b^2\,c+3\,C\,a^2\,b\,d\right )+\frac {{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (\frac {B\,b^3\,d}{4}+\frac {C\,b^3\,c}{4}+\frac {3\,C\,a\,b^2\,d}{4}\right )}{f}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (\frac {A\,b^3\,d}{3}+\frac {B\,b^3\,c}{3}-\frac {C\,b^3\,d}{3}+B\,a\,b^2\,d+C\,a\,b^2\,c+C\,a^2\,b\,d\right )}{f}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {A\,b^3\,c}{2}-\frac {B\,b^3\,d}{2}+\frac {C\,a^3\,d}{2}-\frac {C\,b^3\,c}{2}+\frac {3\,A\,a\,b^2\,d}{2}+\frac {3\,B\,a\,b^2\,c}{2}+\frac {3\,B\,a^2\,b\,d}{2}+\frac {3\,C\,a^2\,b\,c}{2}-\frac {3\,C\,a\,b^2\,d}{2}\right )}{f}+\frac {\ln \left ({\mathrm {tan}\left (e+f\,x\right )}^2+1\right )\,\left (\frac {A\,a^3\,d}{2}-\frac {A\,b^3\,c}{2}+\frac {B\,a^3\,c}{2}+\frac {B\,b^3\,d}{2}-\frac {C\,a^3\,d}{2}+\frac {C\,b^3\,c}{2}+\frac {3\,A\,a^2\,b\,c}{2}-\frac {3\,A\,a\,b^2\,d}{2}-\frac {3\,B\,a\,b^2\,c}{2}-\frac {3\,B\,a^2\,b\,d}{2}-\frac {3\,C\,a^2\,b\,c}{2}+\frac {3\,C\,a\,b^2\,d}{2}\right )}{f}+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (B\,a^3\,d-A\,b^3\,d-B\,b^3\,c+C\,a^3\,c+C\,b^3\,d+3\,A\,a\,b^2\,c+3\,A\,a^2\,b\,d+3\,B\,a^2\,b\,c-3\,B\,a\,b^2\,d-3\,C\,a\,b^2\,c-3\,C\,a^2\,b\,d\right )}{f}+\frac {C\,b^3\,d\,{\mathrm {tan}\left (e+f\,x\right )}^5}{5\,f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(A*a^3*c + A*b^3*d - B*a^3*d + B*b^3*c - C*a^3*c - C*b^3*d - 3*A*a*b^2*c - 3*A*a^2*b*d - 3*B*a^2*b*c + 3*B*a
*b^2*d + 3*C*a*b^2*c + 3*C*a^2*b*d) + (tan(e + f*x)^4*((B*b^3*d)/4 + (C*b^3*c)/4 + (3*C*a*b^2*d)/4))/f + (tan(
e + f*x)^3*((A*b^3*d)/3 + (B*b^3*c)/3 - (C*b^3*d)/3 + B*a*b^2*d + C*a*b^2*c + C*a^2*b*d))/f + (tan(e + f*x)^2*
((A*b^3*c)/2 - (B*b^3*d)/2 + (C*a^3*d)/2 - (C*b^3*c)/2 + (3*A*a*b^2*d)/2 + (3*B*a*b^2*c)/2 + (3*B*a^2*b*d)/2 +
(3*C*a^2*b*c)/2 - (3*C*a*b^2*d)/2))/f + (log(tan(e + f*x)^2 + 1)*((A*a^3*d)/2 - (A*b^3*c)/2 + (B*a^3*c)/2 + (
B*b^3*d)/2 - (C*a^3*d)/2 + (C*b^3*c)/2 + (3*A*a^2*b*c)/2 - (3*A*a*b^2*d)/2 - (3*B*a*b^2*c)/2 - (3*B*a^2*b*d)/2
- (3*C*a^2*b*c)/2 + (3*C*a*b^2*d)/2))/f + (tan(e + f*x)*(B*a^3*d - A*b^3*d - B*b^3*c + C*a^3*c + C*b^3*d + 3*
A*a*b^2*c + 3*A*a^2*b*d + 3*B*a^2*b*c - 3*B*a*b^2*d - 3*C*a*b^2*c - 3*C*a^2*b*d))/f + (C*b^3*d*tan(e + f*x)^5)
/(5*f)

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