∫ (a+b tan(e+fx))3(A+B tan(e+fx)+C tan2(e+fx))___________________________________

3.70 \(\int \frac {(a+b \tan (e+f x))^3 (A+B \tan (e+f x)+C \tan ^2(e+f x))}{c+d \tan (e+f x)} \, dx\)

Optimal. Leaf size=337 \[ -\frac {\log (\cos (e+f x)) \left (a^3 (B c-d (A-C))+3 a^2 b (A c+B d-c C)-3 a b^2 (B c-d (A-C))-b^3 (A c+B d-c C)\right )}{f \left (c^2+d^2\right )}+\frac {x \left (a^3 (A c+B d-c C)-3 a^2 b (B c-d (A-C))-3 a b^2 (A c+B d-c C)+b^3 (B c-d (A-C))\right )}{c^2+d^2}-\frac {(b c-a d)^3 \left (A d^2-B c d+c^2 C\right ) \log (c+d \tan (e+f x))}{d^4 f \left (c^2+d^2\right )}+\frac {b \tan (e+f x) \left (b d^2 (a B+A b-b C)+(b c-a d) (-a C d-b B d+b c C)\right )}{d^3 f}-\frac {(-a C d-b B d+b c C) (a+b \tan (e+f x))^2}{2 d^2 f}+\frac {C (a+b \tan (e+f x))^3}{3 d f} \]

[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/(c^2+d^2)-(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))/(c^2+d^2)/f-(-a*d+b*c)^3*(A*d

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

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

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Rubi [A] time = 1.59, antiderivative size = 337, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3647, 3637, 3626, 3617, 31, 3475} \[ -\frac {\log (\cos (e+f x)) \left (3 a^2 b (A c+B d-c C)+a^3 (B c-d (A-C))-3 a b^2 (B c-d (A-C))-b^3 (A c+B d-c C)\right )}{f \left (c^2+d^2\right )}+\frac {x \left (-3 a^2 b (B c-d (A-C))+a^3 (A c+B d-c C)-3 a b^2 (A c+B d-c C)+b^3 (B c-d (A-C))\right )}{c^2+d^2}-\frac {(b c-a d)^3 \left (A d^2-B c d+c^2 C\right ) \log (c+d \tan (e+f x))}{d^4 f \left (c^2+d^2\right )}+\frac {b \tan (e+f x) \left (b d^2 (a B+A b-b C)+(b c-a d) (-a C d-b B d+b c C)\right )}{d^3 f}-\frac {(-a C d-b B d+b c C) (a+b \tan (e+f x))^2}{2 d^2 f}+\frac {C (a+b \tan (e+f x))^3}{3 d f} \]

Antiderivative was successfully veriﬁed.

[In]

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

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

^4*(c^2 + d^2)*f) + (b*(b*(A*b + a*B - b*C)*d^2 + (b*c - a*d)*(b*c*C - b*B*d - a*C*d))*Tan[e + f*x])/(d^3*f) -

((b*c*C - b*B*d - a*C*d)*(a + b*Tan[e + f*x])^2)/(2*d^2*f) + (C*(a + b*Tan[e + f*x])^3)/(3*d*f)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 3475

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

Rule 3617

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[

A/(b*f), Subst[Int[(a + x)^m, x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A, C]

Rule 3626

Int[((A_) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2)/((a_.) + (b_.)*tan[(e_.) + (f_.)*

(x_)]), x_Symbol] :> Simp[((a*A + b*B - a*C)*x)/(a^2 + b^2), x] + (Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 + b^2), I

nt[(1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x], x] - Dist[(A*b - a*B - b*C)/(a^2 + b^2), Int[Tan[e + f*x], x

], x]) /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && NeQ[a^2 + b^2, 0] && NeQ[A*b - a

*B - b*C, 0]

Rule 3637

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]

Rule 3647

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*

tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(C*(a + b*Tan[e + f*x])^m*(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 - 1)*(c +

d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C*(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f

*x] - (C*m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}

, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !Intege

rQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rubi steps

\begin {align*} \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{c+d \tan (e+f x)} \, dx &=\frac {C (a+b \tan (e+f x))^3}{3 d f}+\frac {\int \frac {(a+b \tan (e+f x))^2 \left (-3 (b c C-a A d)+3 (A b+a B-b C) d \tan (e+f x)-3 (b c C-b B d-a C d) \tan ^2(e+f x)\right )}{c+d \tan (e+f x)} \, dx}{3 d}\\ &=-\frac {(b c C-b B d-a C d) (a+b \tan (e+f x))^2}{2 d^2 f}+\frac {C (a+b \tan (e+f x))^3}{3 d f}+\frac {\int \frac {(a+b \tan (e+f x)) \left (-6 \left (2 a b c C d-a^2 A d^2-b^2 c (c C-B d)\right )+6 \left (a^2 B-b^2 B+2 a b (A-C)\right ) d^2 \tan (e+f x)+6 \left (b (A b+a B-b C) d^2+(b c-a d) (b c C-b B d-a C d)\right ) \tan ^2(e+f x)\right )}{c+d \tan (e+f x)} \, dx}{6 d^2}\\ &=\frac {b \left (b (A b+a B-b C) d^2+(b c-a d) (b c C-b B d-a C d)\right ) \tan (e+f x)}{d^3 f}-\frac {(b c C-b B d-a C d) (a+b \tan (e+f x))^2}{2 d^2 f}+\frac {C (a+b \tan (e+f x))^3}{3 d f}-\frac {\int \frac {6 \left (3 a^2 b c C d^2-a^3 A d^3-3 a b^2 c d (c C-B d)+b^3 c \left (c^2 C-B c d+(A-C) d^2\right )\right )-6 \left (a^3 B-3 a b^2 B+3 a^2 b (A-C)-b^3 (A-C)\right ) d^3 \tan (e+f x)-6 \left (a^3 C d^3-3 a^2 b d^2 (c C-B d)+3 a b^2 d \left (c^2 C-B c d+(A-C) d^2\right )-b^3 \left (c^3 C-B c^2 d+c (A-C) d^2+B d^3\right )\right ) \tan ^2(e+f x)}{c+d \tan (e+f x)} \, dx}{6 d^3}\\ &=\frac {\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}{c^2+d^2}+\frac {b \left (b (A b+a B-b C) d^2+(b c-a d) (b c C-b B d-a C d)\right ) \tan (e+f x)}{d^3 f}-\frac {(b c C-b B d-a C d) (a+b \tan (e+f x))^2}{2 d^2 f}+\frac {C (a+b \tan (e+f x))^3}{3 d f}-\frac {\left ((b c-a d)^3 \left (c^2 C-B c d+A d^2\right )\right ) \int \frac {1+\tan ^2(e+f x)}{c+d \tan (e+f x)} \, dx}{d^3 \left (c^2+d^2\right )}+\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 ) \int \tan (e+f x) \, dx}{c^2+d^2}\\ &=\frac {\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}{c^2+d^2}-\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))}{\left (c^2+d^2\right ) f}+\frac {b \left (b (A b+a B-b C) d^2+(b c-a d) (b c C-b B d-a C d)\right ) \tan (e+f x)}{d^3 f}-\frac {(b c C-b B d-a C d) (a+b \tan (e+f x))^2}{2 d^2 f}+\frac {C (a+b \tan (e+f x))^3}{3 d f}-\frac {\left ((b c-a d)^3 \left (c^2 C-B c d+A d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{c+x} \, dx,x,d \tan (e+f x)\right )}{d^4 \left (c^2+d^2\right ) f}\\ &=\frac {\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}{c^2+d^2}-\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))}{\left (c^2+d^2\right ) f}-\frac {(b c-a d)^3 \left (c^2 C-B c d+A d^2\right ) \log (c+d \tan (e+f x))}{d^4 \left (c^2+d^2\right ) f}+\frac {b \left (b (A b+a B-b C) d^2+(b c-a d) (b c C-b B d-a C d)\right ) \tan (e+f x)}{d^3 f}-\frac {(b c C-b B d-a C d) (a+b \tan (e+f x))^2}{2 d^2 f}+\frac {C (a+b \tan (e+f x))^3}{3 d f}\\ \end {align*}

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Mathematica [C] time = 4.52, size = 258, normalized size = 0.77 \[ \frac {6 b^2 d \tan (e+f x) (a B+A b-b C)+\frac {6 (a d-b c)^3 \left (A d^2-B c d+c^2 C\right ) \log (c+d \tan (e+f x))}{d^2 \left (c^2+d^2\right )}+\frac {3 d^2 (a-i b)^3 (i A+B-i C) \log (\tan (e+f x)+i)}{c-i d}+\frac {3 d^2 (a+i b)^3 (-i A+B+i C) \log (-\tan (e+f x)+i)}{c+i d}-3 (-a C d-b B d+b c C) (a+b \tan (e+f x))^2-\frac {6 b (b c-a d) \tan (e+f x) (a C d+b B d-b c C)}{d}+2 C d (a+b \tan (e+f x))^3}{6 d^2 f} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

^2 + d^2)) + 6*b^2*(A*b + a*B - b*C)*d*Tan[e + f*x] - (6*b*(b*c - a*d)*(-(b*c*C) + b*B*d + a*C*d)*Tan[e + f*x]

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

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fricas [A] time = 2.73, size = 627, normalized size = 1.86 \[ \frac {2 \, {\left (C b^{3} c^{2} d^{3} + C b^{3} d^{5}\right )} \tan \left (f x + e\right )^{3} + 6 \, {\left ({\left ({\left (A - C\right )} a^{3} - 3 \, B a^{2} b - 3 \, {\left (A - C\right )} a b^{2} + B b^{3}\right )} c d^{4} + {\left (B a^{3} + 3 \, {\left (A - C\right )} a^{2} b - 3 \, B a b^{2} - {\left (A - C\right )} b^{3}\right )} d^{5}\right )} f x - 3 \, {\left (C b^{3} c^{3} d^{2} + C b^{3} c d^{4} - {\left (3 \, C a b^{2} + B b^{3}\right )} c^{2} d^{3} - {\left (3 \, C a b^{2} + B b^{3}\right )} d^{5}\right )} \tan \left (f x + e\right )^{2} - 3 \, {\left (C b^{3} c^{5} - A a^{3} d^{5} - {\left (3 \, C a b^{2} + B b^{3}\right )} c^{4} d + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{3} d^{2} - {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c^{2} d^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} c d^{4}\right )} \log \left (\frac {d^{2} \tan \left (f x + e\right )^{2} + 2 \, c d \tan \left (f x + e\right ) + c^{2}}{\tan \left (f x + e\right )^{2} + 1}\right ) + 3 \, {\left (C b^{3} c^{5} - {\left (3 \, C a b^{2} + B b^{3}\right )} c^{4} d + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{3} d^{2} - {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c^{2} d^{3} + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + {\left (A - C\right )} b^{3}\right )} c d^{4} - {\left (C a^{3} + 3 \, B a^{2} b + 3 \, {\left (A - C\right )} a b^{2} - B b^{3}\right )} d^{5}\right )} \log \left (\frac {1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 6 \, {\left (C b^{3} c^{4} d - {\left (3 \, C a b^{2} + B b^{3}\right )} c^{3} d^{2} + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{2} d^{3} - {\left (3 \, C a b^{2} + B b^{3}\right )} c d^{4} + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + {\left (A - C\right )} b^{3}\right )} d^{5}\right )} \tan \left (f x + e\right )}{6 \, {\left (c^{2} d^{4} + d^{6}\right )} f} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*(C*b^3*c^2*d^3 + C*b^3*d^5)*tan(f*x + e)^3 + 6*(((A - C)*a^3 - 3*B*a^2*b - 3*(A - C)*a*b^2 + B*b^3)*c*d

^4 + (B*a^3 + 3*(A - C)*a^2*b - 3*B*a*b^2 - (A - C)*b^3)*d^5)*f*x - 3*(C*b^3*c^3*d^2 + C*b^3*c*d^4 - (3*C*a*b^

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

c^4*d + (3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^3*d^2 - (C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c^2*d^3 + (B*a^3 + 3*A*a^2*b

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

+ B*b^3)*c^4*d + (3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^3*d^2 - (C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c^2*d^3 + (3*C*a^2

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

+ 1)) + 6*(C*b^3*c^4*d - (3*C*a*b^2 + B*b^3)*c^3*d^2 + (3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^2*d^3 - (3*C*a*b^2 +

B*b^3)*c*d^4 + (3*C*a^2*b + 3*B*a*b^2 + (A - C)*b^3)*d^5)*tan(f*x + e))/((c^2*d^4 + d^6)*f)

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giac [A] time = 5.52, size = 573, normalized size = 1.70 \[ \frac {\frac {6 \, {\left (A a^{3} c - C a^{3} c - 3 \, B a^{2} b c - 3 \, A a b^{2} c + 3 \, C a b^{2} c + B b^{3} c + B a^{3} d + 3 \, A a^{2} b d - 3 \, C a^{2} b d - 3 \, B a b^{2} d - A b^{3} d + C b^{3} d\right )} {\left (f x + e\right )}}{c^{2} + d^{2}} + \frac {3 \, {\left (B a^{3} c + 3 \, A a^{2} b c - 3 \, C a^{2} b c - 3 \, B a b^{2} c - A b^{3} c + C b^{3} c - A a^{3} d + C a^{3} d + 3 \, B a^{2} b d + 3 \, A a b^{2} d - 3 \, C a b^{2} d - B b^{3} d\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{c^{2} + d^{2}} - \frac {6 \, {\left (C b^{3} c^{5} - 3 \, C a b^{2} c^{4} d - B b^{3} c^{4} d + 3 \, C a^{2} b c^{3} d^{2} + 3 \, B a b^{2} c^{3} d^{2} + A b^{3} c^{3} d^{2} - C a^{3} c^{2} d^{3} - 3 \, B a^{2} b c^{2} d^{3} - 3 \, A a b^{2} c^{2} d^{3} + B a^{3} c d^{4} + 3 \, A a^{2} b c d^{4} - A a^{3} d^{5}\right )} \log \left ({\left | d \tan \left (f x + e\right ) + c \right |}\right )}{c^{2} d^{4} + d^{6}} + \frac {2 \, C b^{3} d^{2} \tan \left (f x + e\right )^{3} - 3 \, C b^{3} c d \tan \left (f x + e\right )^{2} + 9 \, C a b^{2} d^{2} \tan \left (f x + e\right )^{2} + 3 \, B b^{3} d^{2} \tan \left (f x + e\right )^{2} + 6 \, C b^{3} c^{2} \tan \left (f x + e\right ) - 18 \, C a b^{2} c d \tan \left (f x + e\right ) - 6 \, B b^{3} c d \tan \left (f x + e\right ) + 18 \, C a^{2} b d^{2} \tan \left (f x + e\right ) + 18 \, B a b^{2} d^{2} \tan \left (f x + e\right ) + 6 \, A b^{3} d^{2} \tan \left (f x + e\right ) - 6 \, C b^{3} d^{2} \tan \left (f x + e\right )}{d^{3}}}{6 \, f} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(6*(A*a^3*c - C*a^3*c - 3*B*a^2*b*c - 3*A*a*b^2*c + 3*C*a*b^2*c + B*b^3*c + B*a^3*d + 3*A*a^2*b*d - 3*C*a^

2*b*d - 3*B*a*b^2*d - A*b^3*d + C*b^3*d)*(f*x + e)/(c^2 + d^2) + 3*(B*a^3*c + 3*A*a^2*b*c - 3*C*a^2*b*c - 3*B*

a*b^2*c - A*b^3*c + C*b^3*c - A*a^3*d + C*a^3*d + 3*B*a^2*b*d + 3*A*a*b^2*d - 3*C*a*b^2*d - B*b^3*d)*log(tan(f

*x + e)^2 + 1)/(c^2 + d^2) - 6*(C*b^3*c^5 - 3*C*a*b^2*c^4*d - B*b^3*c^4*d + 3*C*a^2*b*c^3*d^2 + 3*B*a*b^2*c^3*

d^2 + A*b^3*c^3*d^2 - C*a^3*c^2*d^3 - 3*B*a^2*b*c^2*d^3 - 3*A*a*b^2*c^2*d^3 + B*a^3*c*d^4 + 3*A*a^2*b*c*d^4 -

A*a^3*d^5)*log(abs(d*tan(f*x + e) + c))/(c^2*d^4 + d^6) + (2*C*b^3*d^2*tan(f*x + e)^3 - 3*C*b^3*c*d*tan(f*x +

e)^2 + 9*C*a*b^2*d^2*tan(f*x + e)^2 + 3*B*b^3*d^2*tan(f*x + e)^2 + 6*C*b^3*c^2*tan(f*x + e) - 18*C*a*b^2*c*d*t

an(f*x + e) - 6*B*b^3*c*d*tan(f*x + e) + 18*C*a^2*b*d^2*tan(f*x + e) + 18*B*a*b^2*d^2*tan(f*x + e) + 6*A*b^3*d

^2*tan(f*x + e) - 6*C*b^3*d^2*tan(f*x + e))/d^3)/f

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maple [B] time = 0.27, size = 1304, normalized size = 3.87 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

f*x+e)^3

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maxima [A] time = 0.59, size = 445, normalized size = 1.32 \[ \frac {\frac {6 \, {\left ({\left ({\left (A - C\right )} a^{3} - 3 \, B a^{2} b - 3 \, {\left (A - C\right )} a b^{2} + B b^{3}\right )} c + {\left (B a^{3} + 3 \, {\left (A - C\right )} a^{2} b - 3 \, B a b^{2} - {\left (A - C\right )} b^{3}\right )} d\right )} {\left (f x + e\right )}}{c^{2} + d^{2}} - \frac {6 \, {\left (C b^{3} c^{5} - A a^{3} d^{5} - {\left (3 \, C a b^{2} + B b^{3}\right )} c^{4} d + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{3} d^{2} - {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c^{2} d^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} c d^{4}\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{c^{2} d^{4} + d^{6}} + \frac {3 \, {\left ({\left (B a^{3} + 3 \, {\left (A - C\right )} a^{2} b - 3 \, B a b^{2} - {\left (A - C\right )} b^{3}\right )} c - {\left ({\left (A - C\right )} a^{3} - 3 \, B a^{2} b - 3 \, {\left (A - C\right )} a b^{2} + B b^{3}\right )} d\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{c^{2} + d^{2}} + \frac {2 \, C b^{3} d^{2} \tan \left (f x + e\right )^{3} - 3 \, {\left (C b^{3} c d - {\left (3 \, C a b^{2} + B b^{3}\right )} d^{2}\right )} \tan \left (f x + e\right )^{2} + 6 \, {\left (C b^{3} c^{2} - {\left (3 \, C a b^{2} + B b^{3}\right )} c d + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + {\left (A - C\right )} b^{3}\right )} d^{2}\right )} \tan \left (f x + e\right )}{d^{3}}}{6 \, f} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(6*(((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)/(c^2 + d^2) - 6*(C*b^3*c^5 - A*a^3*d^5 - (3*C*a*b^2 + B*b^3)*c^4*d + (3*C*a^2*b + 3*B*a*b^2

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

c)/(c^2*d^4 + d^6) + 3*((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)/(c^2 + d^2) + (2*C*b^3*d^2*tan(f*x + e)^3 - 3*(C*b^3*c*d - (3

*C*a*b^2 + B*b^3)*d^2)*tan(f*x + e)^2 + 6*(C*b^3*c^2 - (3*C*a*b^2 + B*b^3)*c*d + (3*C*a^2*b + 3*B*a*b^2 + (A -

C)*b^3)*d^2)*tan(f*x + e))/d^3)/f

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mupad [B] time = 13.39, size = 508, normalized size = 1.51 \[ \frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {B\,b^3+3\,C\,a\,b^2}{2\,d}-\frac {C\,b^3\,c}{2\,d^2}\right )}{f}-\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (\frac {c\,\left (\frac {B\,b^3+3\,C\,a\,b^2}{d}-\frac {C\,b^3\,c}{d^2}\right )}{d}-\frac {3\,C\,a^2\,b+3\,B\,a\,b^2+A\,b^3}{d}+\frac {C\,b^3}{d}\right )}{f}-\frac {\ln \left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (d^4\,\left (B\,c\,a^3+3\,A\,b\,c\,a^2\right )-d^3\,\left (C\,a^3\,c^2+3\,B\,a^2\,b\,c^2+3\,A\,a\,b^2\,c^2\right )+d^2\,\left (3\,C\,a^2\,b\,c^3+3\,B\,a\,b^2\,c^3+A\,b^3\,c^3\right )-d\,\left (B\,b^3\,c^4+3\,C\,a\,b^2\,c^4\right )-A\,a^3\,d^5+C\,b^3\,c^5\right )}{f\,\left (c^2\,d^4+d^6\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (A\,a^3+A\,b^3\,1{}\mathrm {i}-B\,a^3\,1{}\mathrm {i}+B\,b^3-C\,a^3-C\,b^3\,1{}\mathrm {i}-3\,A\,a\,b^2-A\,a^2\,b\,3{}\mathrm {i}+B\,a\,b^2\,3{}\mathrm {i}-3\,B\,a^2\,b+3\,C\,a\,b^2+C\,a^2\,b\,3{}\mathrm {i}\right )}{2\,f\,\left (d+c\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,\left (A\,b^3-B\,a^3-C\,b^3-3\,A\,a^2\,b+3\,B\,a\,b^2+3\,C\,a^2\,b+A\,a^3\,1{}\mathrm {i}+B\,b^3\,1{}\mathrm {i}-C\,a^3\,1{}\mathrm {i}-A\,a\,b^2\,3{}\mathrm {i}-B\,a^2\,b\,3{}\mathrm {i}+C\,a\,b^2\,3{}\mathrm {i}\right )}{2\,f\,\left (c+d\,1{}\mathrm {i}\right )}+\frac {C\,b^3\,{\mathrm {tan}\left (e+f\,x\right )}^3}{3\,d\,f} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(tan(e + f*x)^2*((B*b^3 + 3*C*a*b^2)/(2*d) - (C*b^3*c)/(2*d^2)))/f - (tan(e + f*x)*((c*((B*b^3 + 3*C*a*b^2)/d

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

*c + 3*A*a^2*b*c) - d^3*(C*a^3*c^2 + 3*A*a*b^2*c^2 + 3*B*a^2*b*c^2) + d^2*(A*b^3*c^3 + 3*B*a*b^2*c^3 + 3*C*a^2

*b*c^3) - d*(B*b^3*c^4 + 3*C*a*b^2*c^4) - A*a^3*d^5 + C*b^3*c^5))/(f*(d^6 + c^2*d^4)) - (log(tan(e + f*x) + 1i

)*(A*a^3 + A*b^3*1i - B*a^3*1i + B*b^3 - C*a^3 - C*b^3*1i - 3*A*a*b^2 - A*a^2*b*3i + B*a*b^2*3i - 3*B*a^2*b +

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

^3*1i - C*b^3 - A*a*b^2*3i - 3*A*a^2*b + 3*B*a*b^2 - B*a^2*b*3i + C*a*b^2*3i + 3*C*a^2*b))/(2*f*(c + d*1i)) +

(C*b^3*tan(e + f*x)^3)/(3*d*f)

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sympy [A] time = 115.15, size = 7205, normalized size = 21.38 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x*(a + b*tan(e))**3*(A + B*tan(e) + C*tan(e)**2)/tan(e), Eq(c, 0) & Eq(d, 0) & Eq(f, 0)), (-3*I

*A*a**3*f*x*tan(e + f*x)/(-6*d*f*tan(e + f*x) + 6*I*d*f) - 3*A*a**3*f*x/(-6*d*f*tan(e + f*x) + 6*I*d*f) - 3*I*

A*a**3/(-6*d*f*tan(e + f*x) + 6*I*d*f) - 9*A*a**2*b*f*x*tan(e + f*x)/(-6*d*f*tan(e + f*x) + 6*I*d*f) + 9*I*A*a

**2*b*f*x/(-6*d*f*tan(e + f*x) + 6*I*d*f) + 9*A*a**2*b/(-6*d*f*tan(e + f*x) + 6*I*d*f) - 9*I*A*a*b**2*f*x*tan(

e + f*x)/(-6*d*f*tan(e + f*x) + 6*I*d*f) - 9*A*a*b**2*f*x/(-6*d*f*tan(e + f*x) + 6*I*d*f) - 9*A*a*b**2*log(tan

(e + f*x)**2 + 1)*tan(e + f*x)/(-6*d*f*tan(e + f*x) + 6*I*d*f) + 9*I*A*a*b**2*log(tan(e + f*x)**2 + 1)/(-6*d*f

*tan(e + f*x) + 6*I*d*f) + 9*I*A*a*b**2/(-6*d*f*tan(e + f*x) + 6*I*d*f) + 9*A*b**3*f*x*tan(e + f*x)/(-6*d*f*ta

n(e + f*x) + 6*I*d*f) - 9*I*A*b**3*f*x/(-6*d*f*tan(e + f*x) + 6*I*d*f) - 3*I*A*b**3*log(tan(e + f*x)**2 + 1)*t

an(e + f*x)/(-6*d*f*tan(e + f*x) + 6*I*d*f) - 3*A*b**3*log(tan(e + f*x)**2 + 1)/(-6*d*f*tan(e + f*x) + 6*I*d*f

) - 6*A*b**3*tan(e + f*x)**2/(-6*d*f*tan(e + f*x) + 6*I*d*f) - 9*A*b**3/(-6*d*f*tan(e + f*x) + 6*I*d*f) - 3*B*

a**3*f*x*tan(e + f*x)/(-6*d*f*tan(e + f*x) + 6*I*d*f) + 3*I*B*a**3*f*x/(-6*d*f*tan(e + f*x) + 6*I*d*f) + 3*B*a

**3/(-6*d*f*tan(e + f*x) + 6*I*d*f) - 9*I*B*a**2*b*f*x*tan(e + f*x)/(-6*d*f*tan(e + f*x) + 6*I*d*f) - 9*B*a**2

*b*f*x/(-6*d*f*tan(e + f*x) + 6*I*d*f) - 9*B*a**2*b*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(-6*d*f*tan(e + f*x)

+ 6*I*d*f) + 9*I*B*a**2*b*log(tan(e + f*x)**2 + 1)/(-6*d*f*tan(e + f*x) + 6*I*d*f) + 9*I*B*a**2*b/(-6*d*f*tan

(e + f*x) + 6*I*d*f) + 27*B*a*b**2*f*x*tan(e + f*x)/(-6*d*f*tan(e + f*x) + 6*I*d*f) - 27*I*B*a*b**2*f*x/(-6*d*

f*tan(e + f*x) + 6*I*d*f) - 9*I*B*a*b**2*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(-6*d*f*tan(e + f*x) + 6*I*d*f)

- 9*B*a*b**2*log(tan(e + f*x)**2 + 1)/(-6*d*f*tan(e + f*x) + 6*I*d*f) - 18*B*a*b**2*tan(e + f*x)**2/(-6*d*f*t

an(e + f*x) + 6*I*d*f) - 27*B*a*b**2/(-6*d*f*tan(e + f*x) + 6*I*d*f) + 9*I*B*b**3*f*x*tan(e + f*x)/(-6*d*f*tan

(e + f*x) + 6*I*d*f) + 9*B*b**3*f*x/(-6*d*f*tan(e + f*x) + 6*I*d*f) + 6*B*b**3*log(tan(e + f*x)**2 + 1)*tan(e

+ f*x)/(-6*d*f*tan(e + f*x) + 6*I*d*f) - 6*I*B*b**3*log(tan(e + f*x)**2 + 1)/(-6*d*f*tan(e + f*x) + 6*I*d*f) -

3*B*b**3*tan(e + f*x)**3/(-6*d*f*tan(e + f*x) + 6*I*d*f) - 3*I*B*b**3*tan(e + f*x)**2/(-6*d*f*tan(e + f*x) +

6*I*d*f) - 9*I*B*b**3/(-6*d*f*tan(e + f*x) + 6*I*d*f) - 3*I*C*a**3*f*x*tan(e + f*x)/(-6*d*f*tan(e + f*x) + 6*I

*d*f) - 3*C*a**3*f*x/(-6*d*f*tan(e + f*x) + 6*I*d*f) - 3*C*a**3*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(-6*d*f*

tan(e + f*x) + 6*I*d*f) + 3*I*C*a**3*log(tan(e + f*x)**2 + 1)/(-6*d*f*tan(e + f*x) + 6*I*d*f) + 3*I*C*a**3/(-6

*d*f*tan(e + f*x) + 6*I*d*f) + 27*C*a**2*b*f*x*tan(e + f*x)/(-6*d*f*tan(e + f*x) + 6*I*d*f) - 27*I*C*a**2*b*f*

x/(-6*d*f*tan(e + f*x) + 6*I*d*f) - 9*I*C*a**2*b*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(-6*d*f*tan(e + f*x) +

6*I*d*f) - 9*C*a**2*b*log(tan(e + f*x)**2 + 1)/(-6*d*f*tan(e + f*x) + 6*I*d*f) - 18*C*a**2*b*tan(e + f*x)**2/(

-6*d*f*tan(e + f*x) + 6*I*d*f) - 27*C*a**2*b/(-6*d*f*tan(e + f*x) + 6*I*d*f) + 27*I*C*a*b**2*f*x*tan(e + f*x)/

(-6*d*f*tan(e + f*x) + 6*I*d*f) + 27*C*a*b**2*f*x/(-6*d*f*tan(e + f*x) + 6*I*d*f) + 18*C*a*b**2*log(tan(e + f*

x)**2 + 1)*tan(e + f*x)/(-6*d*f*tan(e + f*x) + 6*I*d*f) - 18*I*C*a*b**2*log(tan(e + f*x)**2 + 1)/(-6*d*f*tan(e

+ f*x) + 6*I*d*f) - 9*C*a*b**2*tan(e + f*x)**3/(-6*d*f*tan(e + f*x) + 6*I*d*f) - 9*I*C*a*b**2*tan(e + f*x)**2

/(-6*d*f*tan(e + f*x) + 6*I*d*f) - 27*I*C*a*b**2/(-6*d*f*tan(e + f*x) + 6*I*d*f) - 15*C*b**3*f*x*tan(e + f*x)/

(-6*d*f*tan(e + f*x) + 6*I*d*f) + 15*I*C*b**3*f*x/(-6*d*f*tan(e + f*x) + 6*I*d*f) + 6*I*C*b**3*log(tan(e + f*x

)**2 + 1)*tan(e + f*x)/(-6*d*f*tan(e + f*x) + 6*I*d*f) + 6*C*b**3*log(tan(e + f*x)**2 + 1)/(-6*d*f*tan(e + f*x

) + 6*I*d*f) - 2*C*b**3*tan(e + f*x)**4/(-6*d*f*tan(e + f*x) + 6*I*d*f) - I*C*b**3*tan(e + f*x)**3/(-6*d*f*tan

(e + f*x) + 6*I*d*f) + 9*C*b**3*tan(e + f*x)**2/(-6*d*f*tan(e + f*x) + 6*I*d*f) + 15*C*b**3/(-6*d*f*tan(e + f*

x) + 6*I*d*f), Eq(c, -I*d)), (3*I*A*a**3*f*x*tan(e + f*x)/(-6*d*f*tan(e + f*x) - 6*I*d*f) - 3*A*a**3*f*x/(-6*d

*f*tan(e + f*x) - 6*I*d*f) + 3*I*A*a**3/(-6*d*f*tan(e + f*x) - 6*I*d*f) - 9*A*a**2*b*f*x*tan(e + f*x)/(-6*d*f*

tan(e + f*x) - 6*I*d*f) - 9*I*A*a**2*b*f*x/(-6*d*f*tan(e + f*x) - 6*I*d*f) + 9*A*a**2*b/(-6*d*f*tan(e + f*x) -

6*I*d*f) + 9*I*A*a*b**2*f*x*tan(e + f*x)/(-6*d*f*tan(e + f*x) - 6*I*d*f) - 9*A*a*b**2*f*x/(-6*d*f*tan(e + f*x

) - 6*I*d*f) - 9*A*a*b**2*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(-6*d*f*tan(e + f*x) - 6*I*d*f) - 9*I*A*a*b**2

*log(tan(e + f*x)**2 + 1)/(-6*d*f*tan(e + f*x) - 6*I*d*f) - 9*I*A*a*b**2/(-6*d*f*tan(e + f*x) - 6*I*d*f) + 9*A

*b**3*f*x*tan(e + f*x)/(-6*d*f*tan(e + f*x) - 6*I*d*f) + 9*I*A*b**3*f*x/(-6*d*f*tan(e + f*x) - 6*I*d*f) + 3*I*

A*b**3*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(-6*d*f*tan(e + f*x) - 6*I*d*f) - 3*A*b**3*log(tan(e + f*x)**2 +

1)/(-6*d*f*tan(e + f*x) - 6*I*d*f) - 6*A*b**3*tan(e + f*x)**2/(-6*d*f*tan(e + f*x) - 6*I*d*f) - 9*A*b**3/(-6*d

*f*tan(e + f*x) - 6*I*d*f) - 3*B*a**3*f*x*tan(e + f*x)/(-6*d*f*tan(e + f*x) - 6*I*d*f) - 3*I*B*a**3*f*x/(-6*d*

f*tan(e + f*x) - 6*I*d*f) + 3*B*a**3/(-6*d*f*tan(e + f*x) - 6*I*d*f) + 9*I*B*a**2*b*f*x*tan(e + f*x)/(-6*d*f*t

an(e + f*x) - 6*I*d*f) - 9*B*a**2*b*f*x/(-6*d*f*tan(e + f*x) - 6*I*d*f) - 9*B*a**2*b*log(tan(e + f*x)**2 + 1)*

tan(e + f*x)/(-6*d*f*tan(e + f*x) - 6*I*d*f) - 9*I*B*a**2*b*log(tan(e + f*x)**2 + 1)/(-6*d*f*tan(e + f*x) - 6*

I*d*f) - 9*I*B*a**2*b/(-6*d*f*tan(e + f*x) - 6*I*d*f) + 27*B*a*b**2*f*x*tan(e + f*x)/(-6*d*f*tan(e + f*x) - 6*

I*d*f) + 27*I*B*a*b**2*f*x/(-6*d*f*tan(e + f*x) - 6*I*d*f) + 9*I*B*a*b**2*log(tan(e + f*x)**2 + 1)*tan(e + f*x

)/(-6*d*f*tan(e + f*x) - 6*I*d*f) - 9*B*a*b**2*log(tan(e + f*x)**2 + 1)/(-6*d*f*tan(e + f*x) - 6*I*d*f) - 18*B

*a*b**2*tan(e + f*x)**2/(-6*d*f*tan(e + f*x) - 6*I*d*f) - 27*B*a*b**2/(-6*d*f*tan(e + f*x) - 6*I*d*f) - 9*I*B*

b**3*f*x*tan(e + f*x)/(-6*d*f*tan(e + f*x) - 6*I*d*f) + 9*B*b**3*f*x/(-6*d*f*tan(e + f*x) - 6*I*d*f) + 6*B*b**

3*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(-6*d*f*tan(e + f*x) - 6*I*d*f) + 6*I*B*b**3*log(tan(e + f*x)**2 + 1)/

(-6*d*f*tan(e + f*x) - 6*I*d*f) - 3*B*b**3*tan(e + f*x)**3/(-6*d*f*tan(e + f*x) - 6*I*d*f) + 3*I*B*b**3*tan(e

+ f*x)**2/(-6*d*f*tan(e + f*x) - 6*I*d*f) + 9*I*B*b**3/(-6*d*f*tan(e + f*x) - 6*I*d*f) + 3*I*C*a**3*f*x*tan(e

+ f*x)/(-6*d*f*tan(e + f*x) - 6*I*d*f) - 3*C*a**3*f*x/(-6*d*f*tan(e + f*x) - 6*I*d*f) - 3*C*a**3*log(tan(e + f

*x)**2 + 1)*tan(e + f*x)/(-6*d*f*tan(e + f*x) - 6*I*d*f) - 3*I*C*a**3*log(tan(e + f*x)**2 + 1)/(-6*d*f*tan(e +

f*x) - 6*I*d*f) - 3*I*C*a**3/(-6*d*f*tan(e + f*x) - 6*I*d*f) + 27*C*a**2*b*f*x*tan(e + f*x)/(-6*d*f*tan(e + f

*x) - 6*I*d*f) + 27*I*C*a**2*b*f*x/(-6*d*f*tan(e + f*x) - 6*I*d*f) + 9*I*C*a**2*b*log(tan(e + f*x)**2 + 1)*tan

(e + f*x)/(-6*d*f*tan(e + f*x) - 6*I*d*f) - 9*C*a**2*b*log(tan(e + f*x)**2 + 1)/(-6*d*f*tan(e + f*x) - 6*I*d*f

) - 18*C*a**2*b*tan(e + f*x)**2/(-6*d*f*tan(e + f*x) - 6*I*d*f) - 27*C*a**2*b/(-6*d*f*tan(e + f*x) - 6*I*d*f)

- 27*I*C*a*b**2*f*x*tan(e + f*x)/(-6*d*f*tan(e + f*x) - 6*I*d*f) + 27*C*a*b**2*f*x/(-6*d*f*tan(e + f*x) - 6*I*

d*f) + 18*C*a*b**2*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(-6*d*f*tan(e + f*x) - 6*I*d*f) + 18*I*C*a*b**2*log(t

an(e + f*x)**2 + 1)/(-6*d*f*tan(e + f*x) - 6*I*d*f) - 9*C*a*b**2*tan(e + f*x)**3/(-6*d*f*tan(e + f*x) - 6*I*d*

f) + 9*I*C*a*b**2*tan(e + f*x)**2/(-6*d*f*tan(e + f*x) - 6*I*d*f) + 27*I*C*a*b**2/(-6*d*f*tan(e + f*x) - 6*I*d

*f) - 15*C*b**3*f*x*tan(e + f*x)/(-6*d*f*tan(e + f*x) - 6*I*d*f) - 15*I*C*b**3*f*x/(-6*d*f*tan(e + f*x) - 6*I*

d*f) - 6*I*C*b**3*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(-6*d*f*tan(e + f*x) - 6*I*d*f) + 6*C*b**3*log(tan(e +

f*x)**2 + 1)/(-6*d*f*tan(e + f*x) - 6*I*d*f) - 2*C*b**3*tan(e + f*x)**4/(-6*d*f*tan(e + f*x) - 6*I*d*f) + I*C

*b**3*tan(e + f*x)**3/(-6*d*f*tan(e + f*x) - 6*I*d*f) + 9*C*b**3*tan(e + f*x)**2/(-6*d*f*tan(e + f*x) - 6*I*d*

f) + 15*C*b**3/(-6*d*f*tan(e + f*x) - 6*I*d*f), Eq(c, I*d)), ((A*a**3*x + 3*A*a**2*b*log(tan(e + f*x)**2 + 1)/

(2*f) - 3*A*a*b**2*x + 3*A*a*b**2*tan(e + f*x)/f - A*b**3*log(tan(e + f*x)**2 + 1)/(2*f) + A*b**3*tan(e + f*x)

**2/(2*f) + B*a**3*log(tan(e + f*x)**2 + 1)/(2*f) - 3*B*a**2*b*x + 3*B*a**2*b*tan(e + f*x)/f - 3*B*a*b**2*log(

tan(e + f*x)**2 + 1)/(2*f) + 3*B*a*b**2*tan(e + f*x)**2/(2*f) + B*b**3*x + B*b**3*tan(e + f*x)**3/(3*f) - B*b*

*3*tan(e + f*x)/f - C*a**3*x + C*a**3*tan(e + f*x)/f - 3*C*a**2*b*log(tan(e + f*x)**2 + 1)/(2*f) + 3*C*a**2*b*

tan(e + f*x)**2/(2*f) + 3*C*a*b**2*x + C*a*b**2*tan(e + f*x)**3/f - 3*C*a*b**2*tan(e + f*x)/f + C*b**3*log(tan

(e + f*x)**2 + 1)/(2*f) + C*b**3*tan(e + f*x)**4/(4*f) - C*b**3*tan(e + f*x)**2/(2*f))/c, Eq(d, 0)), (x*(a + b

*tan(e))**3*(A + B*tan(e) + C*tan(e)**2)/(c + d*tan(e)), Eq(f, 0)), (6*A*a**3*c*d**4*f*x/(6*c**2*d**4*f + 6*d*

*6*f) + 6*A*a**3*d**5*log(c/d + tan(e + f*x))/(6*c**2*d**4*f + 6*d**6*f) - 3*A*a**3*d**5*log(tan(e + f*x)**2 +

1)/(6*c**2*d**4*f + 6*d**6*f) - 18*A*a**2*b*c*d**4*log(c/d + tan(e + f*x))/(6*c**2*d**4*f + 6*d**6*f) + 9*A*a

**2*b*c*d**4*log(tan(e + f*x)**2 + 1)/(6*c**2*d**4*f + 6*d**6*f) + 18*A*a**2*b*d**5*f*x/(6*c**2*d**4*f + 6*d**

6*f) + 18*A*a*b**2*c**2*d**3*log(c/d + tan(e + f*x))/(6*c**2*d**4*f + 6*d**6*f) - 18*A*a*b**2*c*d**4*f*x/(6*c*

*2*d**4*f + 6*d**6*f) + 9*A*a*b**2*d**5*log(tan(e + f*x)**2 + 1)/(6*c**2*d**4*f + 6*d**6*f) - 6*A*b**3*c**3*d*

*2*log(c/d + tan(e + f*x))/(6*c**2*d**4*f + 6*d**6*f) + 6*A*b**3*c**2*d**3*tan(e + f*x)/(6*c**2*d**4*f + 6*d**

6*f) - 3*A*b**3*c*d**4*log(tan(e + f*x)**2 + 1)/(6*c**2*d**4*f + 6*d**6*f) - 6*A*b**3*d**5*f*x/(6*c**2*d**4*f

+ 6*d**6*f) + 6*A*b**3*d**5*tan(e + f*x)/(6*c**2*d**4*f + 6*d**6*f) - 6*B*a**3*c*d**4*log(c/d + tan(e + f*x))/

(6*c**2*d**4*f + 6*d**6*f) + 3*B*a**3*c*d**4*log(tan(e + f*x)**2 + 1)/(6*c**2*d**4*f + 6*d**6*f) + 6*B*a**3*d*

*5*f*x/(6*c**2*d**4*f + 6*d**6*f) + 18*B*a**2*b*c**2*d**3*log(c/d + tan(e + f*x))/(6*c**2*d**4*f + 6*d**6*f) -

18*B*a**2*b*c*d**4*f*x/(6*c**2*d**4*f + 6*d**6*f) + 9*B*a**2*b*d**5*log(tan(e + f*x)**2 + 1)/(6*c**2*d**4*f +

6*d**6*f) - 18*B*a*b**2*c**3*d**2*log(c/d + tan(e + f*x))/(6*c**2*d**4*f + 6*d**6*f) + 18*B*a*b**2*c**2*d**3*

tan(e + f*x)/(6*c**2*d**4*f + 6*d**6*f) - 9*B*a*b**2*c*d**4*log(tan(e + f*x)**2 + 1)/(6*c**2*d**4*f + 6*d**6*f

) - 18*B*a*b**2*d**5*f*x/(6*c**2*d**4*f + 6*d**6*f) + 18*B*a*b**2*d**5*tan(e + f*x)/(6*c**2*d**4*f + 6*d**6*f)

+ 6*B*b**3*c**4*d*log(c/d + tan(e + f*x))/(6*c**2*d**4*f + 6*d**6*f) - 6*B*b**3*c**3*d**2*tan(e + f*x)/(6*c**

2*d**4*f + 6*d**6*f) + 3*B*b**3*c**2*d**3*tan(e + f*x)**2/(6*c**2*d**4*f + 6*d**6*f) + 6*B*b**3*c*d**4*f*x/(6*

c**2*d**4*f + 6*d**6*f) - 6*B*b**3*c*d**4*tan(e + f*x)/(6*c**2*d**4*f + 6*d**6*f) - 3*B*b**3*d**5*log(tan(e +

f*x)**2 + 1)/(6*c**2*d**4*f + 6*d**6*f) + 3*B*b**3*d**5*tan(e + f*x)**2/(6*c**2*d**4*f + 6*d**6*f) + 6*C*a**3*

c**2*d**3*log(c/d + tan(e + f*x))/(6*c**2*d**4*f + 6*d**6*f) - 6*C*a**3*c*d**4*f*x/(6*c**2*d**4*f + 6*d**6*f)

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

f*x))/(6*c**2*d**4*f + 6*d**6*f) + 18*C*a**2*b*c**2*d**3*tan(e + f*x)/(6*c**2*d**4*f + 6*d**6*f) - 9*C*a**2*b*

c*d**4*log(tan(e + f*x)**2 + 1)/(6*c**2*d**4*f + 6*d**6*f) - 18*C*a**2*b*d**5*f*x/(6*c**2*d**4*f + 6*d**6*f) +

18*C*a**2*b*d**5*tan(e + f*x)/(6*c**2*d**4*f + 6*d**6*f) + 18*C*a*b**2*c**4*d*log(c/d + tan(e + f*x))/(6*c**2

*d**4*f + 6*d**6*f) - 18*C*a*b**2*c**3*d**2*tan(e + f*x)/(6*c**2*d**4*f + 6*d**6*f) + 9*C*a*b**2*c**2*d**3*tan

(e + f*x)**2/(6*c**2*d**4*f + 6*d**6*f) + 18*C*a*b**2*c*d**4*f*x/(6*c**2*d**4*f + 6*d**6*f) - 18*C*a*b**2*c*d*

*4*tan(e + f*x)/(6*c**2*d**4*f + 6*d**6*f) - 9*C*a*b**2*d**5*log(tan(e + f*x)**2 + 1)/(6*c**2*d**4*f + 6*d**6*

f) + 9*C*a*b**2*d**5*tan(e + f*x)**2/(6*c**2*d**4*f + 6*d**6*f) - 6*C*b**3*c**5*log(c/d + tan(e + f*x))/(6*c**

2*d**4*f + 6*d**6*f) + 6*C*b**3*c**4*d*tan(e + f*x)/(6*c**2*d**4*f + 6*d**6*f) - 3*C*b**3*c**3*d**2*tan(e + f*

x)**2/(6*c**2*d**4*f + 6*d**6*f) + 2*C*b**3*c**2*d**3*tan(e + f*x)**3/(6*c**2*d**4*f + 6*d**6*f) + 3*C*b**3*c*

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

d**6*f) + 6*C*b**3*d**5*f*x/(6*c**2*d**4*f + 6*d**6*f) + 2*C*b**3*d**5*tan(e + f*x)**3/(6*c**2*d**4*f + 6*d**6

*f) - 6*C*b**3*d**5*tan(e + f*x)/(6*c**2*d**4*f + 6*d**6*f), True))

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