\(\int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^2} \, dx\) [81]

Optimal result

Integrand size = 45, antiderivative size = 293 \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^2} \, dx=-\frac {\left (a \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )+b \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) x}{\left (a^2+b^2\right ) \left (c^2+d^2\right )^2}+\frac {b \left (A b^2-a (b B-a C)\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{\left (a^2+b^2\right ) (b c-a d)^2 f}-\frac {\left (b \left (c^4 C-2 B c^3 d+c^2 (3 A-C) d^2+A d^4\right )-a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{(b c-a d)^2 \left (c^2+d^2\right )^2 f}+\frac {c^2 C-B c d+A d^2}{(b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3730, 3732, 3611} \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^2} \, dx=-\frac {x \left (a \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )+b \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )\right )}{\left (a^2+b^2\right ) \left (c^2+d^2\right )^2}+\frac {b \left (A b^2-a (b B-a C)\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right ) (b c-a d)^2}+\frac {A d^2-B c d+c^2 C}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}-\frac {\left (b \left (c^2 d^2 (3 A-C)+A d^4-2 B c^3 d+c^4 C\right )-a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right )^2 (b c-a d)^2} \]

[In]

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

[Out]

-(((a*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + b*(2*c*(A - C)*d - B*(c^2 - d^2)))*x)/((a^2 + b^2)*(c^2 + d^
2)^2)) + (b*(A*b^2 - a*(b*B - a*C))*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)*(b*c - a*d)^2*f) - ((b*
(c^4*C - 2*B*c^3*d + c^2*(3*A - C)*d^2 + A*d^4) - a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2)))*Log[c*Cos[e + f*x] +
d*Sin[e + f*x]])/((b*c - a*d)^2*(c^2 + d^2)^2*f) + (c^2*C - B*c*d + A*d^2)/((b*c - a*d)*(c^2 + d^2)*f*(c + d*T
an[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 3730

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Ta
n[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*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] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3732

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

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(592\) vs. \(2(293)=586\).

Time = 7.59 (sec) , antiderivative size = 592, normalized size of antiderivative = 2.02 \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^2} \, dx=-\frac {-\frac {b (b c-a d) \left (A b c^2-a B c^2-b c^2 C+2 a A c d+2 b B c d-2 a c C d-A b d^2+a B d^2+b C d^2-\frac {\sqrt {-b^2} \left (a \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )+b \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )}{b}\right ) \log \left (\sqrt {-b^2}-b \tan (e+f x)\right )}{2 \left (a^2+b^2\right ) \left (c^2+d^2\right )}+\frac {b^2 \left (A b^2-a (b B-a C)\right ) \left (c^2+d^2\right ) \log (a+b \tan (e+f x))}{\left (a^2+b^2\right ) (b c-a d)}-\frac {b (b c-a d) \left (A b c^2-a B c^2-b c^2 C+2 a A c d+2 b B c d-2 a c C d-A b d^2+a B d^2+b C d^2+\frac {\sqrt {-b^2} \left (a \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )+b \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )}{b}\right ) \log \left (\sqrt {-b^2}+b \tan (e+f x)\right )}{2 \left (a^2+b^2\right ) \left (c^2+d^2\right )}-\frac {b \left (b \left (c^4 C-2 B c^3 d+c^2 (3 A-C) d^2+A d^4\right )-a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right ) \log (c+d \tan (e+f x))}{(b c-a d) \left (c^2+d^2\right )}}{b (-b c+a d) \left (c^2+d^2\right ) f}-\frac {A d^2-c (-c C+B d)}{(-b c+a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))} \]

[In]

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

[Out]

-((-1/2*(b*(b*c - a*d)*(A*b*c^2 - a*B*c^2 - b*c^2*C + 2*a*A*c*d + 2*b*B*c*d - 2*a*c*C*d - A*b*d^2 + a*B*d^2 +
b*C*d^2 - (Sqrt[-b^2]*(a*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + b*(2*c*(A - C)*d - B*(c^2 - d^2))))/b)*Lo
g[Sqrt[-b^2] - b*Tan[e + f*x]])/((a^2 + b^2)*(c^2 + d^2)) + (b^2*(A*b^2 - a*(b*B - a*C))*(c^2 + d^2)*Log[a + b
*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)) - (b*(b*c - a*d)*(A*b*c^2 - a*B*c^2 - b*c^2*C + 2*a*A*c*d + 2*b*B*c*
d - 2*a*c*C*d - A*b*d^2 + a*B*d^2 + b*C*d^2 + (Sqrt[-b^2]*(a*(c^2*C - 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + b*(2*
c*(A - C)*d - B*(c^2 - d^2))))/b)*Log[Sqrt[-b^2] + b*Tan[e + f*x]])/(2*(a^2 + b^2)*(c^2 + d^2)) - (b*(b*(c^4*C
- 2*B*c^3*d + c^2*(3*A - C)*d^2 + A*d^4) - a*d^2*(2*c*(A - C)*d - B*(c^2 - d^2)))*Log[c + d*Tan[e + f*x]])/((
b*c - a*d)*(c^2 + d^2)))/(b*(-(b*c) + a*d)*(c^2 + d^2)*f)) - (A*d^2 - c*(-(c*C) + B*d))/((-(b*c) + a*d)*(c^2 +
d^2)*f*(c + d*Tan[e + f*x]))

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.25

method	result	size
derivativedivides	\(\frac {\frac {\left (A \,b^{2}-B a b +C \,a^{2}\right ) b \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a d -b c \right )^{2} \left (a^{2}+b^{2}\right )}+\frac {\frac {\left (-2 A a c d -A b \,c^{2}+A b \,d^{2}+B a \,c^{2}-B a \,d^{2}-2 B b c d +2 C a c d +C b \,c^{2}-C b \,d^{2}\right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}+\left (A a \,c^{2}-A a \,d^{2}-2 A b c d +2 B a c d +B b \,c^{2}-B b \,d^{2}-C a \,c^{2}+C a \,d^{2}+2 C b c d \right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right ) \left (c^{2}+d^{2}\right )^{2}}+\frac {\left (2 A a c \,d^{3}-3 A b \,c^{2} d^{2}-A b \,d^{4}-B a \,c^{2} d^{2}+B a \,d^{4}+2 B b \,c^{3} d -2 C a c \,d^{3}-C b \,c^{4}+C b \,c^{2} d^{2}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (a d -b c \right )^{2} \left (c^{2}+d^{2}\right )^{2}}-\frac {A \,d^{2}-B c d +c^{2} C}{\left (a d -b c \right ) \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )}}{f}\)	\(365\)
default	\(\frac {\frac {\left (A \,b^{2}-B a b +C \,a^{2}\right ) b \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a d -b c \right )^{2} \left (a^{2}+b^{2}\right )}+\frac {\frac {\left (-2 A a c d -A b \,c^{2}+A b \,d^{2}+B a \,c^{2}-B a \,d^{2}-2 B b c d +2 C a c d +C b \,c^{2}-C b \,d^{2}\right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}+\left (A a \,c^{2}-A a \,d^{2}-2 A b c d +2 B a c d +B b \,c^{2}-B b \,d^{2}-C a \,c^{2}+C a \,d^{2}+2 C b c d \right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right ) \left (c^{2}+d^{2}\right )^{2}}+\frac {\left (2 A a c \,d^{3}-3 A b \,c^{2} d^{2}-A b \,d^{4}-B a \,c^{2} d^{2}+B a \,d^{4}+2 B b \,c^{3} d -2 C a c \,d^{3}-C b \,c^{4}+C b \,c^{2} d^{2}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (a d -b c \right )^{2} \left (c^{2}+d^{2}\right )^{2}}-\frac {A \,d^{2}-B c d +c^{2} C}{\left (a d -b c \right ) \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )}}{f}\)	\(365\)
norman	\(\frac {\frac {\left (A a \,c^{2}-A a \,d^{2}-2 A b c d +2 B a c d +B b \,c^{2}-B b \,d^{2}-C a \,c^{2}+C a \,d^{2}+2 C b c d \right ) c x}{\left (a^{2}+b^{2}\right ) \left (c^{4}+2 c^{2} d^{2}+d^{4}\right )}+\frac {\left (A a \,c^{2}-A a \,d^{2}-2 A b c d +2 B a c d +B b \,c^{2}-B b \,d^{2}-C a \,c^{2}+C a \,d^{2}+2 C b c d \right ) d x \tan \left (f x +e \right )}{\left (a^{2}+b^{2}\right ) \left (c^{4}+2 c^{2} d^{2}+d^{4}\right )}-\frac {A \,d^{3}-B c \,d^{2}+C \,c^{2} d}{d f \left (a \,c^{2} d +a \,d^{3}-b \,c^{3}-b c \,d^{2}\right )}}{c +d \tan \left (f x +e \right )}+\frac {\left (2 A a c \,d^{3}-3 A b \,c^{2} d^{2}-A b \,d^{4}-B a \,c^{2} d^{2}+B a \,d^{4}+2 B b \,c^{3} d -2 C a c \,d^{3}-C b \,c^{4}+C b \,c^{2} d^{2}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{f \left (a^{2} c^{4} d^{2}+2 a^{2} c^{2} d^{4}+a^{2} d^{6}-2 a b \,c^{5} d -4 a b \,c^{3} d^{3}-2 a b c \,d^{5}+b^{2} c^{6}+2 b^{2} c^{4} d^{2}+b^{2} c^{2} d^{4}\right )}+\frac {b \left (A \,b^{2}-B a b +C \,a^{2}\right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) f \left (a^{2}+b^{2}\right )}-\frac {\left (2 A a c d +A b \,c^{2}-A b \,d^{2}-B a \,c^{2}+B a \,d^{2}+2 B b c d -2 C a c d -C b \,c^{2}+C b \,d^{2}\right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2 f \left (a^{2}+b^{2}\right ) \left (c^{4}+2 c^{2} d^{2}+d^{4}\right )}\)	\(588\)
parallelrisch	\(\text {Expression too large to display}\)	\(3455\)
risch	\(\text {Expression too large to display}\)	\(3864\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1275 vs. \(2 (291) = 582\).

Time = 1.11 (sec) , antiderivative size = 1275, normalized size of antiderivative = 4.35 \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Sympy [F(-2)]

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

[In]

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

[Out]

Exception raised: NotImplementedError >> no valid subset found

Maxima [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.75 \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^2} \, dx=\frac {\frac {2 \, {\left ({\left ({\left (A - C\right )} a + B b\right )} c^{2} + 2 \, {\left (B a - {\left (A - C\right )} b\right )} c d - {\left ({\left (A - C\right )} a + B b\right )} d^{2}\right )} {\left (f x + e\right )}}{{\left (a^{2} + b^{2}\right )} c^{4} + 2 \, {\left (a^{2} + b^{2}\right )} c^{2} d^{2} + {\left (a^{2} + b^{2}\right )} d^{4}} + \frac {2 \, {\left (C a^{2} b - B a b^{2} + A b^{3}\right )} \log \left (b \tan \left (f x + e\right ) + a\right )}{{\left (a^{2} b^{2} + b^{4}\right )} c^{2} - 2 \, {\left (a^{3} b + a b^{3}\right )} c d + {\left (a^{4} + a^{2} b^{2}\right )} d^{2}} - \frac {2 \, {\left (C b c^{4} - 2 \, B b c^{3} d - 2 \, {\left (A - C\right )} a c d^{3} + {\left (B a + {\left (3 \, A - C\right )} b\right )} c^{2} d^{2} - {\left (B a - A b\right )} d^{4}\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{b^{2} c^{6} - 2 \, a b c^{5} d - 4 \, a b c^{3} d^{3} - 2 \, a b c d^{5} + a^{2} d^{6} + {\left (a^{2} + 2 \, b^{2}\right )} c^{4} d^{2} + {\left (2 \, a^{2} + b^{2}\right )} c^{2} d^{4}} + \frac {{\left ({\left (B a - {\left (A - C\right )} b\right )} c^{2} - 2 \, {\left ({\left (A - C\right )} a + B b\right )} c d - {\left (B a - {\left (A - C\right )} b\right )} d^{2}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{{\left (a^{2} + b^{2}\right )} c^{4} + 2 \, {\left (a^{2} + b^{2}\right )} c^{2} d^{2} + {\left (a^{2} + b^{2}\right )} d^{4}} + \frac {2 \, {\left (C c^{2} - B c d + A d^{2}\right )}}{b c^{4} - a c^{3} d + b c^{2} d^{2} - a c d^{3} + {\left (b c^{3} d - a c^{2} d^{2} + b c d^{3} - a d^{4}\right )} \tan \left (f x + e\right )}}{2 \, f} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 832 vs. \(2 (291) = 582\).

Time = 0.77 (sec) , antiderivative size = 832, normalized size of antiderivative = 2.84 \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^2} \, dx=\frac {\frac {2 \, {\left (A a c^{2} - C a c^{2} + B b c^{2} + 2 \, B a c d - 2 \, A b c d + 2 \, C b c d - A a d^{2} + C a d^{2} - B b d^{2}\right )} {\left (f x + e\right )}}{a^{2} c^{4} + b^{2} c^{4} + 2 \, a^{2} c^{2} d^{2} + 2 \, b^{2} c^{2} d^{2} + a^{2} d^{4} + b^{2} d^{4}} + \frac {{\left (B a c^{2} - A b c^{2} + C b c^{2} - 2 \, A a c d + 2 \, C a c d - 2 \, B b c d - B a d^{2} + A b d^{2} - C b d^{2}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{2} c^{4} + b^{2} c^{4} + 2 \, a^{2} c^{2} d^{2} + 2 \, b^{2} c^{2} d^{2} + a^{2} d^{4} + b^{2} d^{4}} + \frac {2 \, {\left (C a^{2} b^{2} - B a b^{3} + A b^{4}\right )} \log \left ({\left | b \tan \left (f x + e\right ) + a \right |}\right )}{a^{2} b^{3} c^{2} + b^{5} c^{2} - 2 \, a^{3} b^{2} c d - 2 \, a b^{4} c d + a^{4} b d^{2} + a^{2} b^{3} d^{2}} - \frac {2 \, {\left (C b c^{4} d - 2 \, B b c^{3} d^{2} + B a c^{2} d^{3} + 3 \, A b c^{2} d^{3} - C b c^{2} d^{3} - 2 \, A a c d^{4} + 2 \, C a c d^{4} - B a d^{5} + A b d^{5}\right )} \log \left ({\left | d \tan \left (f x + e\right ) + c \right |}\right )}{b^{2} c^{6} d - 2 \, a b c^{5} d^{2} + a^{2} c^{4} d^{3} + 2 \, b^{2} c^{4} d^{3} - 4 \, a b c^{3} d^{4} + 2 \, a^{2} c^{2} d^{5} + b^{2} c^{2} d^{5} - 2 \, a b c d^{6} + a^{2} d^{7}} + \frac {2 \, {\left (C b c^{4} d \tan \left (f x + e\right ) - 2 \, B b c^{3} d^{2} \tan \left (f x + e\right ) + B a c^{2} d^{3} \tan \left (f x + e\right ) + 3 \, A b c^{2} d^{3} \tan \left (f x + e\right ) - C b c^{2} d^{3} \tan \left (f x + e\right ) - 2 \, A a c d^{4} \tan \left (f x + e\right ) + 2 \, C a c d^{4} \tan \left (f x + e\right ) - B a d^{5} \tan \left (f x + e\right ) + A b d^{5} \tan \left (f x + e\right ) + 2 \, C b c^{5} - C a c^{4} d - 3 \, B b c^{4} d + 2 \, B a c^{3} d^{2} + 4 \, A b c^{3} d^{2} - 3 \, A a c^{2} d^{3} + C a c^{2} d^{3} - B b c^{2} d^{3} + 2 \, A b c d^{4} - A a d^{5}\right )}}{{\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2} + 2 \, b^{2} c^{4} d^{2} - 4 \, a b c^{3} d^{3} + 2 \, a^{2} c^{2} d^{4} + b^{2} c^{2} d^{4} - 2 \, a b c d^{5} + a^{2} d^{6}\right )} {\left (d \tan \left (f x + e\right ) + c\right )}}}{2 \, f} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 65.17 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.47 \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^2} \, dx=\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,\left (B-A\,1{}\mathrm {i}+C\,1{}\mathrm {i}\right )}{2\,f\,\left (a\,c^2-a\,d^2-2\,b\,c\,d+b\,c^2\,1{}\mathrm {i}-b\,d^2\,1{}\mathrm {i}+a\,c\,d\,2{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (A\,1{}\mathrm {i}+B-C\,1{}\mathrm {i}\right )}{2\,f\,\left (a\,d^2-a\,c^2+2\,b\,c\,d+b\,c^2\,1{}\mathrm {i}-b\,d^2\,1{}\mathrm {i}+a\,c\,d\,2{}\mathrm {i}\right )}+\frac {\ln \left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (C\,a^2\,b-B\,a\,b^2+A\,b^3\right )}{f\,\left (a^4\,d^2-2\,a^3\,b\,c\,d+a^2\,b^2\,c^2+a^2\,b^2\,d^2-2\,a\,b^3\,c\,d+b^4\,c^2\right )}-\frac {\ln \left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (C\,b\,c^4-2\,B\,b\,c^3\,d+\left (3\,A\,b+B\,a-C\,b\right )\,c^2\,d^2+\left (2\,C\,a-2\,A\,a\right )\,c\,d^3+\left (A\,b-B\,a\right )\,d^4\right )}{f\,\left (a^2\,c^4\,d^2+2\,a^2\,c^2\,d^4+a^2\,d^6-2\,a\,b\,c^5\,d-4\,a\,b\,c^3\,d^3-2\,a\,b\,c\,d^5+b^2\,c^6+2\,b^2\,c^4\,d^2+b^2\,c^2\,d^4\right )}-\frac {C\,c^2-B\,c\,d+A\,d^2}{f\,\left (a\,d-b\,c\right )\,\left (c^2+d^2\right )\,\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )} \]

[In]

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

[Out]

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