\(\int \frac {(c+d \tan (e+f x)) (A+B \tan (e+f x)+C \tan ^2(e+f x))}{(a+b \tan (e+f x))^3} \, dx\) [56]

Optimal result

Integrand size = 43, antiderivative size = 320 \[ \int \frac {(c+d \tan (e+f x)) \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^3} \, dx=\frac {\left (a^3 (A c-c C-B d)-3 a b^2 (A c-c C-B d)+3 a^2 b (B c+(A-C) d)-b^3 (B c+(A-C) d)\right ) x}{\left (a^2+b^2\right )^3}+\frac {\left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)-a^3 (B c+(A-C) d)+3 a b^2 (B c+(A-C) d)\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{\left (a^2+b^2\right )^3 f}-\frac {\left (A b^2-a (b B-a C)\right ) (b c-a d)}{2 b^2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {a^4 C d+b^4 (B c+A d)+2 a b^3 (A c-c C-B d)-a^2 b^2 (B c+(A-3 C) d)}{b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))} \]

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

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {3716, 3709, 3612, 3611} \[ \int \frac {(c+d \tan (e+f x)) \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^3} \, dx=-\frac {(b c-a d) \left (A b^2-a (b B-a C)\right )}{2 b^2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}-\frac {a^4 C d-a^2 b^2 (d (A-3 C)+B c)+2 a b^3 (A c-B d-c C)+b^4 (A d+B c)}{b^2 f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))}+\frac {\left (-\left (a^3 (d (A-C)+B c)\right )+3 a^2 b (A c-B d-c C)+3 a b^2 (d (A-C)+B c)-b^3 (A c-B d-c C)\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right )^3}+\frac {x \left (a^3 (A c-B d-c C)+3 a^2 b (d (A-C)+B c)-3 a b^2 (A c-B d-c C)-b^3 (d (A-C)+B c)\right )}{\left (a^2+b^2\right )^3} \]

[In]

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

Rule 3611

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c/(b*f))
*Log[RemoveContent[a*Cos[e + f*x] + b*Sin[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] && EqQ[a*c + b*d, 0]

Rule 3612

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

Rule 3709

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

Rule 3716

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 - a*d))*(c^2*C - B*c*d + A*d^2)
*((c + d*Tan[e + f*x])^(n + 1)/(d^2*f*(n + 1)*(c^2 + d^2))), x] + Dist[1/(d*(c^2 + d^2)), Int[(c + d*Tan[e + f
*x])^(n + 1)*Simp[a*d*(A*c - c*C + B*d) + b*(c^2*C - B*c*d + A*d^2) + d*(A*b*c + a*B*c - b*c*C - a*A*d + b*B*d
+ a*C*d)*Tan[e + f*x] + b*C*(c^2 + d^2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] &
& NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] && LtQ[n, -1]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.34 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.18 \[ \int \frac {(c+d \tan (e+f x)) \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^3} \, dx=-\frac {C (c+d \tan (e+f x))}{b f (a+b \tan (e+f x))^2}-\frac {-\frac {b c C-b B d-a C d}{2 b f (a+b \tan (e+f x))^2}+\frac {\frac {\left (-2 b^3 (A c-c C-B d)+2 a b^2 (B c+(A-C) d)\right ) \left (-\frac {\log (i-\tan (e+f x))}{2 (i a-b)^3}+\frac {\log (i+\tan (e+f x))}{2 (i a+b)^3}+\frac {b \left (3 a^2-b^2\right ) \log (a+b \tan (e+f x))}{\left (a^2+b^2\right )^3}-\frac {b}{2 \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}-\frac {2 a b}{\left (a^2+b^2\right )^2 (a+b \tan (e+f x))}\right )}{b}-2 b (B c+(A-C) d) \left (-\frac {i \log (i-\tan (e+f x))}{2 (a+i b)^2}+\frac {i \log (i+\tan (e+f x))}{2 (a-i b)^2}+\frac {2 a b \log (a+b \tan (e+f x))}{\left (a^2+b^2\right )^2}-\frac {b}{\left (a^2+b^2\right ) (a+b \tan (e+f x))}\right )}{2 b f}}{b} \]

[In]

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

[Out]

-((C*(c + d*Tan[e + f*x]))/(b*f*(a + b*Tan[e + f*x])^2)) - (-1/2*(b*c*C - b*B*d - a*C*d)/(b*f*(a + b*Tan[e + f
*x])^2) + (((-2*b^3*(A*c - c*C - B*d) + 2*a*b^2*(B*c + (A - C)*d))*(-1/2*Log[I - Tan[e + f*x]]/(I*a - b)^3 + L
og[I + Tan[e + f*x]]/(2*(I*a + b)^3) + (b*(3*a^2 - b^2)*Log[a + b*Tan[e + f*x]])/(a^2 + b^2)^3 - b/(2*(a^2 + b
^2)*(a + b*Tan[e + f*x])^2) - (2*a*b)/((a^2 + b^2)^2*(a + b*Tan[e + f*x]))))/b - 2*b*(B*c + (A - C)*d)*(((-1/2
*I)*Log[I - Tan[e + f*x]])/(a + I*b)^2 + ((I/2)*Log[I + Tan[e + f*x]])/(a - I*b)^2 + (2*a*b*Log[a + b*Tan[e +
f*x]])/(a^2 + b^2)^2 - b/((a^2 + b^2)*(a + b*Tan[e + f*x]))))/(2*b*f))/b

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.54

method	result	size
derivativedivides	\(\frac {-\frac {-A a \,b^{2} d +A \,b^{3} c +B \,a^{2} b d -B a \,b^{2} c -a^{3} C d +C \,a^{2} b c}{2 b^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (f x +e \right )\right )^{2}}-\frac {-A \,a^{2} b^{2} d +2 A a \,b^{3} c +A \,b^{4} d -B \,a^{2} b^{2} c -2 B a \,b^{3} d +B \,b^{4} c +a^{4} C d +3 C \,a^{2} b^{2} d -2 C a \,b^{3} c}{\left (a^{2}+b^{2}\right )^{2} b^{2} \left (a +b \tan \left (f x +e \right )\right )}-\frac {\left (A \,a^{3} d -3 A \,a^{2} b c -3 A a \,b^{2} d +A \,b^{3} c +B \,a^{3} c +3 B \,a^{2} b d -3 B a \,b^{2} c -B \,b^{3} d -a^{3} C d +3 C \,a^{2} b c +3 C a \,b^{2} d -C \,b^{3} c \right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {\left (A \,a^{3} d -3 A \,a^{2} b c -3 A a \,b^{2} d +A \,b^{3} c +B \,a^{3} c +3 B \,a^{2} b d -3 B a \,b^{2} c -B \,b^{3} d -a^{3} C d +3 C \,a^{2} b c +3 C a \,b^{2} d -C \,b^{3} c \right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}+\left (A \,a^{3} c +3 A \,a^{2} b d -3 A a \,b^{2} c -A \,b^{3} d -B \,a^{3} d +3 B \,a^{2} b c +3 B a \,b^{2} d -B \,b^{3} c -C \,a^{3} c -3 C \,a^{2} b d +3 C a \,b^{2} c +C \,b^{3} d \right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{f}\)	\(494\)
default	\(\frac {-\frac {-A a \,b^{2} d +A \,b^{3} c +B \,a^{2} b d -B a \,b^{2} c -a^{3} C d +C \,a^{2} b c}{2 b^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (f x +e \right )\right )^{2}}-\frac {-A \,a^{2} b^{2} d +2 A a \,b^{3} c +A \,b^{4} d -B \,a^{2} b^{2} c -2 B a \,b^{3} d +B \,b^{4} c +a^{4} C d +3 C \,a^{2} b^{2} d -2 C a \,b^{3} c}{\left (a^{2}+b^{2}\right )^{2} b^{2} \left (a +b \tan \left (f x +e \right )\right )}-\frac {\left (A \,a^{3} d -3 A \,a^{2} b c -3 A a \,b^{2} d +A \,b^{3} c +B \,a^{3} c +3 B \,a^{2} b d -3 B a \,b^{2} c -B \,b^{3} d -a^{3} C d +3 C \,a^{2} b c +3 C a \,b^{2} d -C \,b^{3} c \right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {\left (A \,a^{3} d -3 A \,a^{2} b c -3 A a \,b^{2} d +A \,b^{3} c +B \,a^{3} c +3 B \,a^{2} b d -3 B a \,b^{2} c -B \,b^{3} d -a^{3} C d +3 C \,a^{2} b c +3 C a \,b^{2} d -C \,b^{3} c \right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}+\left (A \,a^{3} c +3 A \,a^{2} b d -3 A a \,b^{2} c -A \,b^{3} d -B \,a^{3} d +3 B \,a^{2} b c +3 B a \,b^{2} d -B \,b^{3} c -C \,a^{3} c -3 C \,a^{2} b d +3 C a \,b^{2} c +C \,b^{3} d \right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{f}\)	\(494\)
norman	\(\frac {\frac {\left (A \,a^{2} b^{2} d -2 A a \,b^{3} c -A \,b^{4} d +B \,a^{2} b^{2} c +2 B a \,b^{3} d -B \,b^{4} c -a^{4} C d -3 C \,a^{2} b^{2} d +2 C a \,b^{3} c \right ) \tan \left (f x +e \right )}{f b \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {\left (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 \right ) a^{2} x}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}+\frac {b^{2} \left (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 \right ) x \tan \left (f x +e \right )^{2}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}+\frac {3 A \,a^{3} b^{2} d -5 A \,a^{2} b^{3} c -A \,b^{4} d a -A \,b^{5} c -B \,a^{4} d b +3 B \,a^{3} b^{2} c +3 B \,a^{2} b^{3} d -B \,b^{4} c a -a^{5} C d -C \,a^{4} c b -5 C \,a^{3} b^{2} d +3 C \,a^{2} b^{3} c}{2 f \,b^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 b \left (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 \right ) a x \tan \left (f x +e \right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (f x +e \right )\right )^{2}}+\frac {\left (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 -a^{3} C d +3 C \,a^{2} b c +3 C a \,b^{2} d -C \,b^{3} c \right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2 f \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {\left (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 -a^{3} C d +3 C \,a^{2} b c +3 C a \,b^{2} d -C \,b^{3} c \right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{f \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\)	\(884\)
risch	\(\text {Expression too large to display}\)	\(2383\)
parallelrisch	\(\text {Expression too large to display}\)	\(2787\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 987 vs. \(2 (316) = 632\).

Time = 0.30 (sec) , antiderivative size = 987, normalized size of antiderivative = 3.08 \[ \int \frac {(c+d \tan (e+f x)) \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^3} \, dx=\frac {2 \, {\left ({\left ({\left (A - C\right )} a^{5} + 3 \, B a^{4} b - 3 \, {\left (A - C\right )} a^{3} b^{2} - B a^{2} b^{3}\right )} c - {\left (B a^{5} - 3 \, {\left (A - C\right )} a^{4} b - 3 \, B a^{3} b^{2} + {\left (A - C\right )} a^{2} b^{3}\right )} d\right )} f x + {\left (2 \, {\left ({\left ({\left (A - C\right )} a^{3} b^{2} + 3 \, B a^{2} b^{3} - 3 \, {\left (A - C\right )} a b^{4} - B b^{5}\right )} c - {\left (B a^{3} b^{2} - 3 \, {\left (A - C\right )} a^{2} b^{3} - 3 \, B a b^{4} + {\left (A - C\right )} b^{5}\right )} d\right )} f x + {\left (C a^{4} b - 3 \, B a^{3} b^{2} + 5 \, {\left (A - C\right )} a^{2} b^{3} + 3 \, B a b^{4} - A b^{5}\right )} c + {\left (C a^{5} + B a^{4} b - {\left (3 \, A - 7 \, C\right )} a^{3} b^{2} - 5 \, B a^{2} b^{3} + 3 \, A a b^{4}\right )} d\right )} \tan \left (f x + e\right )^{2} - {\left (3 \, C a^{4} b - 5 \, B a^{3} b^{2} + {\left (7 \, A - 3 \, C\right )} a^{2} b^{3} + B a b^{4} + A b^{5}\right )} c + {\left (C a^{5} - 3 \, B a^{4} b + 5 \, {\left (A - C\right )} a^{3} b^{2} + 3 \, B a^{2} b^{3} - A a b^{4}\right )} d - {\left ({\left ({\left (B a^{3} b^{2} - 3 \, {\left (A - C\right )} a^{2} b^{3} - 3 \, B a b^{4} + {\left (A - C\right )} b^{5}\right )} c + {\left ({\left (A - C\right )} a^{3} b^{2} + 3 \, B a^{2} b^{3} - 3 \, {\left (A - C\right )} a b^{4} - B b^{5}\right )} d\right )} \tan \left (f x + e\right )^{2} + {\left (B a^{5} - 3 \, {\left (A - C\right )} a^{4} b - 3 \, B a^{3} b^{2} + {\left (A - C\right )} a^{2} b^{3}\right )} c + {\left ({\left (A - C\right )} a^{5} + 3 \, B a^{4} b - 3 \, {\left (A - C\right )} a^{3} b^{2} - B a^{2} b^{3}\right )} d + 2 \, {\left ({\left (B a^{4} b - 3 \, {\left (A - C\right )} a^{3} b^{2} - 3 \, B a^{2} b^{3} + {\left (A - C\right )} a b^{4}\right )} c + {\left ({\left (A - C\right )} a^{4} b + 3 \, B a^{3} b^{2} - 3 \, {\left (A - C\right )} a^{2} b^{3} - B a b^{4}\right )} d\right )} \tan \left (f x + e\right )\right )} \log \left (\frac {b^{2} \tan \left (f x + e\right )^{2} + 2 \, a b \tan \left (f x + e\right ) + a^{2}}{\tan \left (f x + e\right )^{2} + 1}\right ) + 2 \, {\left (2 \, {\left ({\left ({\left (A - C\right )} a^{4} b + 3 \, B a^{3} b^{2} - 3 \, {\left (A - C\right )} a^{2} b^{3} - B a b^{4}\right )} c - {\left (B a^{4} b - 3 \, {\left (A - C\right )} a^{3} b^{2} - 3 \, B a^{2} b^{3} + {\left (A - C\right )} a b^{4}\right )} d\right )} f x + {\left (C a^{5} - 2 \, B a^{4} b + 3 \, {\left (A - C\right )} a^{3} b^{2} + 3 \, B a^{2} b^{3} - {\left (3 \, A - 2 \, C\right )} a b^{4} - B b^{5}\right )} c + {\left (B a^{5} - {\left (2 \, A - 3 \, C\right )} a^{4} b - 3 \, B a^{3} b^{2} + 3 \, {\left (A - C\right )} a^{2} b^{3} + 2 \, B a b^{4} - A b^{5}\right )} d\right )} \tan \left (f x + e\right )}{2 \, {\left ({\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} f \tan \left (f x + e\right )^{2} + 2 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} f \tan \left (f x + e\right ) + {\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} f\right )}} \]

[In]

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

[Out]

1/2*(2*(((A - C)*a^5 + 3*B*a^4*b - 3*(A - C)*a^3*b^2 - B*a^2*b^3)*c - (B*a^5 - 3*(A - C)*a^4*b - 3*B*a^3*b^2 +
(A - C)*a^2*b^3)*d)*f*x + (2*(((A - C)*a^3*b^2 + 3*B*a^2*b^3 - 3*(A - C)*a*b^4 - B*b^5)*c - (B*a^3*b^2 - 3*(A
- C)*a^2*b^3 - 3*B*a*b^4 + (A - C)*b^5)*d)*f*x + (C*a^4*b - 3*B*a^3*b^2 + 5*(A - C)*a^2*b^3 + 3*B*a*b^4 - A*b
^5)*c + (C*a^5 + B*a^4*b - (3*A - 7*C)*a^3*b^2 - 5*B*a^2*b^3 + 3*A*a*b^4)*d)*tan(f*x + e)^2 - (3*C*a^4*b - 5*B
*a^3*b^2 + (7*A - 3*C)*a^2*b^3 + B*a*b^4 + A*b^5)*c + (C*a^5 - 3*B*a^4*b + 5*(A - C)*a^3*b^2 + 3*B*a^2*b^3 - A
*a*b^4)*d - (((B*a^3*b^2 - 3*(A - C)*a^2*b^3 - 3*B*a*b^4 + (A - C)*b^5)*c + ((A - C)*a^3*b^2 + 3*B*a^2*b^3 - 3
*(A - C)*a*b^4 - B*b^5)*d)*tan(f*x + e)^2 + (B*a^5 - 3*(A - C)*a^4*b - 3*B*a^3*b^2 + (A - C)*a^2*b^3)*c + ((A
- C)*a^5 + 3*B*a^4*b - 3*(A - C)*a^3*b^2 - B*a^2*b^3)*d + 2*((B*a^4*b - 3*(A - C)*a^3*b^2 - 3*B*a^2*b^3 + (A -
C)*a*b^4)*c + ((A - C)*a^4*b + 3*B*a^3*b^2 - 3*(A - C)*a^2*b^3 - B*a*b^4)*d)*tan(f*x + e))*log((b^2*tan(f*x +
e)^2 + 2*a*b*tan(f*x + e) + a^2)/(tan(f*x + e)^2 + 1)) + 2*(2*(((A - C)*a^4*b + 3*B*a^3*b^2 - 3*(A - C)*a^2*b
^3 - B*a*b^4)*c - (B*a^4*b - 3*(A - C)*a^3*b^2 - 3*B*a^2*b^3 + (A - C)*a*b^4)*d)*f*x + (C*a^5 - 2*B*a^4*b + 3*
(A - C)*a^3*b^2 + 3*B*a^2*b^3 - (3*A - 2*C)*a*b^4 - B*b^5)*c + (B*a^5 - (2*A - 3*C)*a^4*b - 3*B*a^3*b^2 + 3*(A
- C)*a^2*b^3 + 2*B*a*b^4 - A*b^5)*d)*tan(f*x + e))/((a^6*b^2 + 3*a^4*b^4 + 3*a^2*b^6 + b^8)*f*tan(f*x + e)^2
+ 2*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*f*tan(f*x + e) + (a^8 + 3*a^6*b^2 + 3*a^4*b^4 + a^2*b^6)*f)

Sympy [F(-2)]

Exception generated. \[ \int \frac {(c+d \tan (e+f x)) \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^3} \, dx=\text {Exception raised: AttributeError} \]

[In]

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

[Out]

Exception raised: AttributeError >> 'NoneType' object has no attribute 'primitive'

Maxima [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 574, normalized size of antiderivative = 1.79 \[ \int \frac {(c+d \tan (e+f x)) \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^3} \, dx=\frac {\frac {2 \, {\left ({\left ({\left (A - C\right )} a^{3} + 3 \, B a^{2} b - 3 \, {\left (A - C\right )} a b^{2} - B b^{3}\right )} c - {\left (B a^{3} - 3 \, {\left (A - C\right )} a^{2} b - 3 \, B a b^{2} + {\left (A - C\right )} b^{3}\right )} d\right )} {\left (f x + e\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left ({\left (B a^{3} - 3 \, {\left (A - C\right )} a^{2} b - 3 \, B a b^{2} + {\left (A - C\right )} b^{3}\right )} c + {\left ({\left (A - C\right )} a^{3} + 3 \, B a^{2} b - 3 \, {\left (A - C\right )} a b^{2} - B b^{3}\right )} d\right )} \log \left (b \tan \left (f x + e\right ) + a\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left ({\left (B a^{3} - 3 \, {\left (A - C\right )} a^{2} b - 3 \, B a b^{2} + {\left (A - C\right )} b^{3}\right )} c + {\left ({\left (A - C\right )} a^{3} + 3 \, B a^{2} b - 3 \, {\left (A - C\right )} a b^{2} - B b^{3}\right )} d\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (C a^{4} b - 3 \, B a^{3} b^{2} + {\left (5 \, A - 3 \, C\right )} a^{2} b^{3} + B a b^{4} + A b^{5}\right )} c + {\left (C a^{5} + B a^{4} b - {\left (3 \, A - 5 \, C\right )} a^{3} b^{2} - 3 \, B a^{2} b^{3} + A a b^{4}\right )} d - 2 \, {\left ({\left (B a^{2} b^{3} - 2 \, {\left (A - C\right )} a b^{4} - B b^{5}\right )} c - {\left (C a^{4} b - {\left (A - 3 \, C\right )} a^{2} b^{3} - 2 \, B a b^{4} + A b^{5}\right )} d\right )} \tan \left (f x + e\right )}{a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6} + {\left (a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8}\right )} \tan \left (f x + e\right )^{2} + 2 \, {\left (a^{5} b^{3} + 2 \, a^{3} b^{5} + a b^{7}\right )} \tan \left (f x + e\right )}}{2 \, f} \]

[In]

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

[Out]

1/2*(2*(((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)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 2*((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(b*tan(f*x + e) + a)/(a^6 + 3*a^4*b^2 + 3*a^2*
b^4 + b^6) + ((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)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - ((C*a^4*b - 3*B*a^3*b^2 + (5*A -
3*C)*a^2*b^3 + B*a*b^4 + A*b^5)*c + (C*a^5 + B*a^4*b - (3*A - 5*C)*a^3*b^2 - 3*B*a^2*b^3 + A*a*b^4)*d - 2*((B*
a^2*b^3 - 2*(A - C)*a*b^4 - B*b^5)*c - (C*a^4*b - (A - 3*C)*a^2*b^3 - 2*B*a*b^4 + A*b^5)*d)*tan(f*x + e))/(a^6
*b^2 + 2*a^4*b^4 + a^2*b^6 + (a^4*b^4 + 2*a^2*b^6 + b^8)*tan(f*x + e)^2 + 2*(a^5*b^3 + 2*a^3*b^5 + a*b^7)*tan(
f*x + e)))/f

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1006 vs. \(2 (316) = 632\).

Time = 0.81 (sec) , antiderivative size = 1006, normalized size of antiderivative = 3.14 \[ \int \frac {(c+d \tan (e+f x)) \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^3} \, dx=\frac {\frac {2 \, {\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 )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\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 )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (B a^{3} b c - 3 \, A a^{2} b^{2} c + 3 \, C a^{2} b^{2} c - 3 \, B a b^{3} c + A b^{4} c - C b^{4} c + A a^{3} b d - C a^{3} b d + 3 \, B a^{2} b^{2} d - 3 \, A a b^{3} d + 3 \, C a b^{3} d - B b^{4} d\right )} \log \left ({\left | b \tan \left (f x + e\right ) + a \right |}\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}} + \frac {3 \, B a^{3} b^{4} c \tan \left (f x + e\right )^{2} - 9 \, A a^{2} b^{5} c \tan \left (f x + e\right )^{2} + 9 \, C a^{2} b^{5} c \tan \left (f x + e\right )^{2} - 9 \, B a b^{6} c \tan \left (f x + e\right )^{2} + 3 \, A b^{7} c \tan \left (f x + e\right )^{2} - 3 \, C b^{7} c \tan \left (f x + e\right )^{2} + 3 \, A a^{3} b^{4} d \tan \left (f x + e\right )^{2} - 3 \, C a^{3} b^{4} d \tan \left (f x + e\right )^{2} + 9 \, B a^{2} b^{5} d \tan \left (f x + e\right )^{2} - 9 \, A a b^{6} d \tan \left (f x + e\right )^{2} + 9 \, C a b^{6} d \tan \left (f x + e\right )^{2} - 3 \, B b^{7} d \tan \left (f x + e\right )^{2} + 8 \, B a^{4} b^{3} c \tan \left (f x + e\right ) - 22 \, A a^{3} b^{4} c \tan \left (f x + e\right ) + 22 \, C a^{3} b^{4} c \tan \left (f x + e\right ) - 18 \, B a^{2} b^{5} c \tan \left (f x + e\right ) + 2 \, A a b^{6} c \tan \left (f x + e\right ) - 2 \, C a b^{6} c \tan \left (f x + e\right ) - 2 \, B b^{7} c \tan \left (f x + e\right ) - 2 \, C a^{6} b d \tan \left (f x + e\right ) + 8 \, A a^{4} b^{3} d \tan \left (f x + e\right ) - 14 \, C a^{4} b^{3} d \tan \left (f x + e\right ) + 22 \, B a^{3} b^{4} d \tan \left (f x + e\right ) - 18 \, A a^{2} b^{5} d \tan \left (f x + e\right ) + 12 \, C a^{2} b^{5} d \tan \left (f x + e\right ) - 2 \, B a b^{6} d \tan \left (f x + e\right ) - 2 \, A b^{7} d \tan \left (f x + e\right ) - C a^{6} b c + 6 \, B a^{5} b^{2} c - 14 \, A a^{4} b^{3} c + 11 \, C a^{4} b^{3} c - 7 \, B a^{3} b^{4} c - 3 \, A a^{2} b^{5} c - B a b^{6} c - A b^{7} c - C a^{7} d - B a^{6} b d + 6 \, A a^{5} b^{2} d - 9 \, C a^{5} b^{2} d + 11 \, B a^{4} b^{3} d - 7 \, A a^{3} b^{4} d + 4 \, C a^{3} b^{4} d - A a b^{6} d}{{\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} {\left (b \tan \left (f x + e\right ) + a\right )}^{2}}}{2 \, f} \]

[In]

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

[Out]

1/2*(2*(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)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + (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)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 2*(B*a^3*b*c - 3*A*a^2*b^2*c + 3*C*
a^2*b^2*c - 3*B*a*b^3*c + A*b^4*c - C*b^4*c + A*a^3*b*d - C*a^3*b*d + 3*B*a^2*b^2*d - 3*A*a*b^3*d + 3*C*a*b^3*
d - B*b^4*d)*log(abs(b*tan(f*x + e) + a))/(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7) + (3*B*a^3*b^4*c*tan(f*x + e)^
2 - 9*A*a^2*b^5*c*tan(f*x + e)^2 + 9*C*a^2*b^5*c*tan(f*x + e)^2 - 9*B*a*b^6*c*tan(f*x + e)^2 + 3*A*b^7*c*tan(f
*x + e)^2 - 3*C*b^7*c*tan(f*x + e)^2 + 3*A*a^3*b^4*d*tan(f*x + e)^2 - 3*C*a^3*b^4*d*tan(f*x + e)^2 + 9*B*a^2*b
^5*d*tan(f*x + e)^2 - 9*A*a*b^6*d*tan(f*x + e)^2 + 9*C*a*b^6*d*tan(f*x + e)^2 - 3*B*b^7*d*tan(f*x + e)^2 + 8*B
*a^4*b^3*c*tan(f*x + e) - 22*A*a^3*b^4*c*tan(f*x + e) + 22*C*a^3*b^4*c*tan(f*x + e) - 18*B*a^2*b^5*c*tan(f*x +
e) + 2*A*a*b^6*c*tan(f*x + e) - 2*C*a*b^6*c*tan(f*x + e) - 2*B*b^7*c*tan(f*x + e) - 2*C*a^6*b*d*tan(f*x + e)
+ 8*A*a^4*b^3*d*tan(f*x + e) - 14*C*a^4*b^3*d*tan(f*x + e) + 22*B*a^3*b^4*d*tan(f*x + e) - 18*A*a^2*b^5*d*tan(
f*x + e) + 12*C*a^2*b^5*d*tan(f*x + e) - 2*B*a*b^6*d*tan(f*x + e) - 2*A*b^7*d*tan(f*x + e) - C*a^6*b*c + 6*B*a
^5*b^2*c - 14*A*a^4*b^3*c + 11*C*a^4*b^3*c - 7*B*a^3*b^4*c - 3*A*a^2*b^5*c - B*a*b^6*c - A*b^7*c - C*a^7*d - B
*a^6*b*d + 6*A*a^5*b^2*d - 9*C*a^5*b^2*d + 11*B*a^4*b^3*d - 7*A*a^3*b^4*d + 4*C*a^3*b^4*d - A*a*b^6*d)/((a^6*b
^2 + 3*a^4*b^4 + 3*a^2*b^6 + b^8)*(b*tan(f*x + e) + a)^2))/f

Mupad [B] (verification not implemented)

Time = 15.53 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.57 \[ \int \frac {(c+d \tan (e+f x)) \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^3} \, dx=-\frac {\frac {A\,b^5\,c+C\,a^5\,d+A\,a\,b^4\,d+B\,a\,b^4\,c+B\,a^4\,b\,d+C\,a^4\,b\,c+5\,A\,a^2\,b^3\,c-3\,A\,a^3\,b^2\,d-3\,B\,a^3\,b^2\,c-3\,B\,a^2\,b^3\,d-3\,C\,a^2\,b^3\,c+5\,C\,a^3\,b^2\,d}{2\,b^2\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (A\,b^4\,d+B\,b^4\,c+C\,a^4\,d+2\,A\,a\,b^3\,c-2\,B\,a\,b^3\,d-2\,C\,a\,b^3\,c-A\,a^2\,b^2\,d-B\,a^2\,b^2\,c+3\,C\,a^2\,b^2\,d\right )}{b\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{f\,\left (a^2+2\,a\,b\,\mathrm {tan}\left (e+f\,x\right )+b^2\,{\mathrm {tan}\left (e+f\,x\right )}^2\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (B\,d+A\,d\,1{}\mathrm {i}+B\,c\,1{}\mathrm {i}-A\,c+C\,c-C\,d\,1{}\mathrm {i}\right )}{2\,f\,\left (-a^3\,1{}\mathrm {i}-3\,a^2\,b+a\,b^2\,3{}\mathrm {i}+b^3\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,\left (A\,d+B\,c-C\,d-A\,c\,1{}\mathrm {i}+B\,d\,1{}\mathrm {i}+C\,c\,1{}\mathrm {i}\right )}{2\,f\,\left (-a^3-a^2\,b\,3{}\mathrm {i}+3\,a\,b^2+b^3\,1{}\mathrm {i}\right )}-\frac {\ln \left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (\left (A\,d+B\,c-C\,d\right )\,a^3+\left (3\,B\,d-3\,A\,c+3\,C\,c\right )\,a^2\,b+\left (3\,C\,d-3\,B\,c-3\,A\,d\right )\,a\,b^2+\left (A\,c-B\,d-C\,c\right )\,b^3\right )}{f\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )} \]

[In]

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

[Out]

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