3.1.65 \(\int (a+b \tan (e+f x)) (c+d \tan (e+f x))^3 (A+B \tan (e+f x)+C \tan ^2(e+f x)) \, dx\) [65]

Optimal. Leaf size=389 \[ \left (a \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )-b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x-\frac {\left (A \left (b c^3+3 a c^2 d-3 b c d^2-a d^3\right )-b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3\right )+a \left (B c^3-3 c^2 C d-3 B c d^2+C d^3\right )\right ) \log (\cos (e+f x))}{f}+\frac {d \left (a \left (B c^2-2 c C d-B d^2\right )-b \left (c^2 C+2 B c d-C d^2\right )+A \left (2 a c d+b \left (c^2-d^2\right )\right )\right ) \tan (e+f x)}{f}+\frac {(A b c+a B c-b c C+a A d-b B d-a C d) (c+d \tan (e+f x))^2}{2 f}+\frac {(A b+a B-b C) (c+d \tan (e+f x))^3}{3 f}-\frac {(b c C-5 b B d-5 a C d) (c+d \tan (e+f x))^4}{20 d^2 f}+\frac {b C \tan (e+f x) (c+d \tan (e+f x))^4}{5 d f} \]

[Out]

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

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Rubi [A]
time = 0.52, antiderivative size = 387, normalized size of antiderivative = 0.99, 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 {d \tan (e+f x) \left (2 a A c d+a B \left (c^2-d^2\right )-2 a c C d+A b \left (c^2-d^2\right )-b \left (2 B c d+c^2 C-C d^2\right )\right )}{f}-\frac {\log (\cos (e+f x)) \left (A \left (3 a c^2 d-a d^3+b c^3-3 b c d^2\right )+a \left (B c^3-3 B c d^2-3 c^2 C d+C d^3\right )-b \left (3 B c^2 d-B d^3+c^3 C-3 c C d^2\right )\right )}{f}-x \left (-a \left (A c^3-3 A c d^2-3 B c^2 d+B d^3-c^3 C+3 c C d^2\right )+b d (A-C) \left (3 c^2-d^2\right )+b B \left (c^3-3 c d^2\right )\right )+\frac {(a B+A b-b C) (c+d \tan (e+f x))^3}{3 f}+\frac {(c+d \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 {(-5 a C d-5 b B d+b c C) (c+d \tan (e+f x))^4}{20 d^2 f}+\frac {b C \tan (e+f x) (c+d \tan (e+f x))^4}{5 d f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [C] Result contains complex when optimal does not.
time = 6.26, size = 297, normalized size = 0.76 \begin {gather*} \frac {b C \tan (e+f x) (c+d \tan (e+f x))^4}{5 d f}-\frac {\frac {(b c C-5 b B d-5 a C d) (c+d \tan (e+f x))^4}{4 d f}+\frac {5 \left (3 (A b c+a B c-b c C-a A d+b B d+a C d) \left ((i c-d)^3 \log (i-\tan (e+f x))-(i c+d)^3 \log (i+\tan (e+f x))+6 c d^2 \tan (e+f x)+d^3 \tan ^2(e+f x)\right )+(A b+a B-b C) \left (3 i (c+i d)^4 \log (i-\tan (e+f x))-3 i (c-i d)^4 \log (i+\tan (e+f x))-6 d^2 \left (6 c^2-d^2\right ) \tan (e+f x)-12 c d^3 \tan ^2(e+f x)-2 d^4 \tan ^3(e+f x)\right )\right )}{6 f}}{5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*C*Tan[e + f*x]*(c + d*Tan[e + f*x])^4)/(5*d*f) - (((b*c*C - 5*b*B*d - 5*a*C*d)*(c + d*Tan[e + f*x])^4)/(4*d
*f) + (5*(3*(A*b*c + a*B*c - b*c*C - a*A*d + b*B*d + a*C*d)*((I*c - d)^3*Log[I - Tan[e + f*x]] - (I*c + d)^3*L
og[I + Tan[e + f*x]] + 6*c*d^2*Tan[e + f*x] + d^3*Tan[e + f*x]^2) + (A*b + a*B - b*C)*((3*I)*(c + I*d)^4*Log[I
- Tan[e + f*x]] - (3*I)*(c - I*d)^4*Log[I + Tan[e + f*x]] - 6*d^2*(6*c^2 - d^2)*Tan[e + f*x] - 12*c*d^3*Tan[e
+ f*x]^2 - 2*d^4*Tan[e + f*x]^3)))/(6*f))/(5*d)

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Maple [A]
time = 0.16, size = 639, normalized size = 1.64 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.52, size = 394, normalized size = 1.01 \begin {gather*} \frac {12 \, C b d^{3} \tan \left (f x + e\right )^{5} + 15 \, {\left (3 \, C b c d^{2} + {\left (C a + B b\right )} d^{3}\right )} \tan \left (f x + e\right )^{4} + 20 \, {\left (3 \, C b c^{2} d + 3 \, {\left (C a + B b\right )} c d^{2} + {\left (B a + {\left (A - C\right )} b\right )} d^{3}\right )} \tan \left (f x + e\right )^{3} + 30 \, {\left (C b c^{3} + 3 \, {\left (C a + B b\right )} c^{2} d + 3 \, {\left (B a + {\left (A - C\right )} b\right )} c d^{2} + {\left ({\left (A - C\right )} a - B b\right )} d^{3}\right )} \tan \left (f x + e\right )^{2} + 60 \, {\left ({\left ({\left (A - C\right )} a - B b\right )} c^{3} - 3 \, {\left (B a + {\left (A - C\right )} b\right )} c^{2} d - 3 \, {\left ({\left (A - C\right )} a - B b\right )} c d^{2} + {\left (B a + {\left (A - C\right )} b\right )} d^{3}\right )} {\left (f x + e\right )} + 30 \, {\left ({\left (B a + {\left (A - C\right )} b\right )} c^{3} + 3 \, {\left ({\left (A - C\right )} a - B b\right )} c^{2} d - 3 \, {\left (B a + {\left (A - C\right )} b\right )} c d^{2} - {\left ({\left (A - C\right )} a - B b\right )} d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right ) + 60 \, {\left ({\left (C a + B b\right )} c^{3} + 3 \, {\left (B a + {\left (A - C\right )} b\right )} c^{2} d + 3 \, {\left ({\left (A - C\right )} a - B b\right )} c d^{2} - {\left (B a + {\left (A - C\right )} b\right )} d^{3}\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))*(c+d*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.59, size = 392, normalized size = 1.01 \begin {gather*} \frac {12 \, C b d^{3} \tan \left (f x + e\right )^{5} + 15 \, {\left (3 \, C b c d^{2} + {\left (C a + B b\right )} d^{3}\right )} \tan \left (f x + e\right )^{4} + 20 \, {\left (3 \, C b c^{2} d + 3 \, {\left (C a + B b\right )} c d^{2} + {\left (B a + {\left (A - C\right )} b\right )} d^{3}\right )} \tan \left (f x + e\right )^{3} + 60 \, {\left ({\left ({\left (A - C\right )} a - B b\right )} c^{3} - 3 \, {\left (B a + {\left (A - C\right )} b\right )} c^{2} d - 3 \, {\left ({\left (A - C\right )} a - B b\right )} c d^{2} + {\left (B a + {\left (A - C\right )} b\right )} d^{3}\right )} f x + 30 \, {\left (C b c^{3} + 3 \, {\left (C a + B b\right )} c^{2} d + 3 \, {\left (B a + {\left (A - C\right )} b\right )} c d^{2} + {\left ({\left (A - C\right )} a - B b\right )} d^{3}\right )} \tan \left (f x + e\right )^{2} - 30 \, {\left ({\left (B a + {\left (A - C\right )} b\right )} c^{3} + 3 \, {\left ({\left (A - C\right )} a - B b\right )} c^{2} d - 3 \, {\left (B a + {\left (A - C\right )} b\right )} c d^{2} - {\left ({\left (A - C\right )} a - B b\right )} d^{3}\right )} \log \left (\frac {1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 60 \, {\left ({\left (C a + B b\right )} c^{3} + 3 \, {\left (B a + {\left (A - C\right )} b\right )} c^{2} d + 3 \, {\left ({\left (A - C\right )} a - B b\right )} c d^{2} - {\left (B a + {\left (A - C\right )} b\right )} d^{3}\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))*(c+d*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x, algorithm="fricas")

[Out]

1/60*(12*C*b*d^3*tan(f*x + e)^5 + 15*(3*C*b*c*d^2 + (C*a + B*b)*d^3)*tan(f*x + e)^4 + 20*(3*C*b*c^2*d + 3*(C*a
+ B*b)*c*d^2 + (B*a + (A - C)*b)*d^3)*tan(f*x + e)^3 + 60*(((A - C)*a - B*b)*c^3 - 3*(B*a + (A - C)*b)*c^2*d
- 3*((A - C)*a - B*b)*c*d^2 + (B*a + (A - C)*b)*d^3)*f*x + 30*(C*b*c^3 + 3*(C*a + B*b)*c^2*d + 3*(B*a + (A - C
)*b)*c*d^2 + ((A - C)*a - B*b)*d^3)*tan(f*x + e)^2 - 30*((B*a + (A - C)*b)*c^3 + 3*((A - C)*a - B*b)*c^2*d - 3
*(B*a + (A - C)*b)*c*d^2 - ((A - C)*a - B*b)*d^3)*log(1/(tan(f*x + e)^2 + 1)) + 60*((C*a + B*b)*c^3 + 3*(B*a +
(A - C)*b)*c^2*d + 3*((A - C)*a - B*b)*c*d^2 - (B*a + (A - C)*b)*d^3)*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. \(2 (379) = 758\).
time = 0.35, size = 1001, normalized size = 2.57 \begin {gather*} \begin {cases} A a c^{3} x + \frac {3 A a c^{2} d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} - 3 A a c d^{2} x + \frac {3 A a c d^{2} \tan {\left (e + f x \right )}}{f} - \frac {A a d^{3} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {A a d^{3} \tan ^{2}{\left (e + f x \right )}}{2 f} + \frac {A b c^{3} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} - 3 A b c^{2} d x + \frac {3 A b c^{2} d \tan {\left (e + f x \right )}}{f} - \frac {3 A b c d^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {3 A b c d^{2} \tan ^{2}{\left (e + f x \right )}}{2 f} + A b d^{3} x + \frac {A b d^{3} \tan ^{3}{\left (e + f x \right )}}{3 f} - \frac {A b d^{3} \tan {\left (e + f x \right )}}{f} + \frac {B a c^{3} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} - 3 B a c^{2} d x + \frac {3 B a c^{2} d \tan {\left (e + f x \right )}}{f} - \frac {3 B a c d^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {3 B a c d^{2} \tan ^{2}{\left (e + f x \right )}}{2 f} + B a d^{3} x + \frac {B a d^{3} \tan ^{3}{\left (e + f x \right )}}{3 f} - \frac {B a d^{3} \tan {\left (e + f x \right )}}{f} - B b c^{3} x + \frac {B b c^{3} \tan {\left (e + f x \right )}}{f} - \frac {3 B b c^{2} d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {3 B b c^{2} d \tan ^{2}{\left (e + f x \right )}}{2 f} + 3 B b c d^{2} x + \frac {B b c d^{2} \tan ^{3}{\left (e + f x \right )}}{f} - \frac {3 B b c d^{2} \tan {\left (e + f x \right )}}{f} + \frac {B b d^{3} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {B b d^{3} \tan ^{4}{\left (e + f x \right )}}{4 f} - \frac {B b d^{3} \tan ^{2}{\left (e + f x \right )}}{2 f} - C a c^{3} x + \frac {C a c^{3} \tan {\left (e + f x \right )}}{f} - \frac {3 C a c^{2} d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {3 C a c^{2} d \tan ^{2}{\left (e + f x \right )}}{2 f} + 3 C a c d^{2} x + \frac {C a c d^{2} \tan ^{3}{\left (e + f x \right )}}{f} - \frac {3 C a c d^{2} \tan {\left (e + f x \right )}}{f} + \frac {C a d^{3} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {C a d^{3} \tan ^{4}{\left (e + f x \right )}}{4 f} - \frac {C a d^{3} \tan ^{2}{\left (e + f x \right )}}{2 f} - \frac {C b c^{3} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {C b c^{3} \tan ^{2}{\left (e + f x \right )}}{2 f} + 3 C b c^{2} d x + \frac {C b c^{2} d \tan ^{3}{\left (e + f x \right )}}{f} - \frac {3 C b c^{2} d \tan {\left (e + f x \right )}}{f} + \frac {3 C b c d^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {3 C b c d^{2} \tan ^{4}{\left (e + f x \right )}}{4 f} - \frac {3 C b c d^{2} \tan ^{2}{\left (e + f x \right )}}{2 f} - C b d^{3} x + \frac {C b d^{3} \tan ^{5}{\left (e + f x \right )}}{5 f} - \frac {C b d^{3} \tan ^{3}{\left (e + f x \right )}}{3 f} + \frac {C b d^{3} \tan {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (a + b \tan {\left (e \right )}\right ) \left (c + d \tan {\left (e \right )}\right )^{3} \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))*(c+d*tan(f*x+e))**3*(A+B*tan(f*x+e)+C*tan(f*x+e)**2),x)

[Out]

Piecewise((A*a*c**3*x + 3*A*a*c**2*d*log(tan(e + f*x)**2 + 1)/(2*f) - 3*A*a*c*d**2*x + 3*A*a*c*d**2*tan(e + f*
x)/f - A*a*d**3*log(tan(e + f*x)**2 + 1)/(2*f) + A*a*d**3*tan(e + f*x)**2/(2*f) + A*b*c**3*log(tan(e + f*x)**2
+ 1)/(2*f) - 3*A*b*c**2*d*x + 3*A*b*c**2*d*tan(e + f*x)/f - 3*A*b*c*d**2*log(tan(e + f*x)**2 + 1)/(2*f) + 3*A
*b*c*d**2*tan(e + f*x)**2/(2*f) + A*b*d**3*x + A*b*d**3*tan(e + f*x)**3/(3*f) - A*b*d**3*tan(e + f*x)/f + B*a*
c**3*log(tan(e + f*x)**2 + 1)/(2*f) - 3*B*a*c**2*d*x + 3*B*a*c**2*d*tan(e + f*x)/f - 3*B*a*c*d**2*log(tan(e +
f*x)**2 + 1)/(2*f) + 3*B*a*c*d**2*tan(e + f*x)**2/(2*f) + B*a*d**3*x + B*a*d**3*tan(e + f*x)**3/(3*f) - B*a*d*
*3*tan(e + f*x)/f - B*b*c**3*x + B*b*c**3*tan(e + f*x)/f - 3*B*b*c**2*d*log(tan(e + f*x)**2 + 1)/(2*f) + 3*B*b
*c**2*d*tan(e + f*x)**2/(2*f) + 3*B*b*c*d**2*x + B*b*c*d**2*tan(e + f*x)**3/f - 3*B*b*c*d**2*tan(e + f*x)/f +
B*b*d**3*log(tan(e + f*x)**2 + 1)/(2*f) + B*b*d**3*tan(e + f*x)**4/(4*f) - B*b*d**3*tan(e + f*x)**2/(2*f) - C*
a*c**3*x + C*a*c**3*tan(e + f*x)/f - 3*C*a*c**2*d*log(tan(e + f*x)**2 + 1)/(2*f) + 3*C*a*c**2*d*tan(e + f*x)**
2/(2*f) + 3*C*a*c*d**2*x + C*a*c*d**2*tan(e + f*x)**3/f - 3*C*a*c*d**2*tan(e + f*x)/f + C*a*d**3*log(tan(e + f
*x)**2 + 1)/(2*f) + C*a*d**3*tan(e + f*x)**4/(4*f) - C*a*d**3*tan(e + f*x)**2/(2*f) - C*b*c**3*log(tan(e + f*x
)**2 + 1)/(2*f) + C*b*c**3*tan(e + f*x)**2/(2*f) + 3*C*b*c**2*d*x + C*b*c**2*d*tan(e + f*x)**3/f - 3*C*b*c**2*
d*tan(e + f*x)/f + 3*C*b*c*d**2*log(tan(e + f*x)**2 + 1)/(2*f) + 3*C*b*c*d**2*tan(e + f*x)**4/(4*f) - 3*C*b*c*
d**2*tan(e + f*x)**2/(2*f) - C*b*d**3*x + C*b*d**3*tan(e + f*x)**5/(5*f) - C*b*d**3*tan(e + f*x)**3/(3*f) + C*
b*d**3*tan(e + f*x)/f, Ne(f, 0)), (x*(a + b*tan(e))*(c + d*tan(e))**3*(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. \(2 (388) = 776\).
time = 8.91, size = 11805, normalized size = 30.35 \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))*(c+d*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x, algorithm="giac")

[Out]

1/60*(60*A*a*c^3*f*x*tan(f*x)^5*tan(e)^5 - 60*C*a*c^3*f*x*tan(f*x)^5*tan(e)^5 - 60*B*b*c^3*f*x*tan(f*x)^5*tan(
e)^5 - 180*B*a*c^2*d*f*x*tan(f*x)^5*tan(e)^5 - 180*A*b*c^2*d*f*x*tan(f*x)^5*tan(e)^5 + 180*C*b*c^2*d*f*x*tan(f
*x)^5*tan(e)^5 - 180*A*a*c*d^2*f*x*tan(f*x)^5*tan(e)^5 + 180*C*a*c*d^2*f*x*tan(f*x)^5*tan(e)^5 + 180*B*b*c*d^2
*f*x*tan(f*x)^5*tan(e)^5 + 60*B*a*d^3*f*x*tan(f*x)^5*tan(e)^5 + 60*A*b*d^3*f*x*tan(f*x)^5*tan(e)^5 - 60*C*b*d^
3*f*x*tan(f*x)^5*tan(e)^5 - 30*B*a*c^3*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*c^3*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*c^3*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*c^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*c^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*B*b*c^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*B*a*c
*d^2*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*b*c*d^2*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*b*c*d^2*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*d^3*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*d^3*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*d^3*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*c^3*f*x*tan(f*x)^4*tan(e)^4 + 300*C*
a*c^3*f*x*tan(f*x)^4*tan(e)^4 + 300*B*b*c^3*f*x*tan(f*x)^4*tan(e)^4 + 900*B*a*c^2*d*f*x*tan(f*x)^4*tan(e)^4 +
900*A*b*c^2*d*f*x*tan(f*x)^4*tan(e)^4 - 900*C*b*c^2*d*f*x*tan(f*x)^4*tan(e)^4 + 900*A*a*c*d^2*f*x*tan(f*x)^4*t
an(e)^4 - 900*C*a*c*d^2*f*x*tan(f*x)^4*tan(e)^4 - 900*B*b*c*d^2*f*x*tan(f*x)^4*tan(e)^4 - 300*B*a*d^3*f*x*tan(
f*x)^4*tan(e)^4 - 300*A*b*d^3*f*x*tan(f*x)^4*tan(e)^4 + 300*C*b*d^3*f*x*tan(f*x)^4*tan(e)^4 + 30*C*b*c^3*tan(f
*x)^5*tan(e)^5 + 90*C*a*c^2*d*tan(f*x)^5*tan(e)^5 + 90*B*b*c^2*d*tan(f*x)^5*tan(e)^5 + 90*B*a*c*d^2*tan(f*x)^5
*tan(e)^5 + 90*A*b*c*d^2*tan(f*x)^5*tan(e)^5 - 135*C*b*c*d^2*tan(f*x)^5*tan(e)^5 + 30*A*a*d^3*tan(f*x)^5*tan(e
)^5 - 45*C*a*d^3*tan(f*x)^5*tan(e)^5 - 45*B*b*d^3*tan(f*x)^5*tan(e)^5 + 150*B*a*c^3*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*c^3*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*b*c^3*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 + 4
50*A*a*c^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*c^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*B*b*c
^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*B*a*c*d^2*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*b*c*d^2*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*b*c*d^2*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*d^3*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*d^3*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*d^3*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 = 9.04, size = 478, normalized size = 1.23 \begin {gather*} x\,\left (A\,a\,c^3+A\,b\,d^3+B\,a\,d^3-B\,b\,c^3-C\,a\,c^3-C\,b\,d^3-3\,A\,a\,c\,d^2-3\,A\,b\,c^2\,d-3\,B\,a\,c^2\,d+3\,B\,b\,c\,d^2+3\,C\,a\,c\,d^2+3\,C\,b\,c^2\,d\right )+\frac {{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (\frac {B\,b\,d^3}{4}+\frac {C\,a\,d^3}{4}+\frac {3\,C\,b\,c\,d^2}{4}\right )}{f}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (\frac {A\,b\,d^3}{3}+\frac {B\,a\,d^3}{3}-\frac {C\,b\,d^3}{3}+B\,b\,c\,d^2+C\,a\,c\,d^2+C\,b\,c^2\,d\right )}{f}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {A\,a\,d^3}{2}-\frac {B\,b\,d^3}{2}-\frac {C\,a\,d^3}{2}+\frac {C\,b\,c^3}{2}+\frac {3\,A\,b\,c\,d^2}{2}+\frac {3\,B\,a\,c\,d^2}{2}+\frac {3\,B\,b\,c^2\,d}{2}+\frac {3\,C\,a\,c^2\,d}{2}-\frac {3\,C\,b\,c\,d^2}{2}\right )}{f}-\frac {\ln \left ({\mathrm {tan}\left (e+f\,x\right )}^2+1\right )\,\left (\frac {A\,a\,d^3}{2}-\frac {A\,b\,c^3}{2}-\frac {B\,a\,c^3}{2}-\frac {B\,b\,d^3}{2}-\frac {C\,a\,d^3}{2}+\frac {C\,b\,c^3}{2}-\frac {3\,A\,a\,c^2\,d}{2}+\frac {3\,A\,b\,c\,d^2}{2}+\frac {3\,B\,a\,c\,d^2}{2}+\frac {3\,B\,b\,c^2\,d}{2}+\frac {3\,C\,a\,c^2\,d}{2}-\frac {3\,C\,b\,c\,d^2}{2}\right )}{f}+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (B\,b\,c^3-B\,a\,d^3-A\,b\,d^3+C\,a\,c^3+C\,b\,d^3+3\,A\,a\,c\,d^2+3\,A\,b\,c^2\,d+3\,B\,a\,c^2\,d-3\,B\,b\,c\,d^2-3\,C\,a\,c\,d^2-3\,C\,b\,c^2\,d\right )}{f}+\frac {C\,b\,d^3\,{\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))*(c + d*tan(e + f*x))^3*(A + B*tan(e + f*x) + C*tan(e + f*x)^2),x)

[Out]

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

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