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

Optimal. Leaf size=281 \[ \frac {x \left (a^2 (A c+B d-c C)+2 a b (B c-d (A-C))-b^2 (A c+B d-c C)\right )}{\left (a^2+b^2\right )^2 \left (c^2+d^2\right )}-\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))}+\frac {\left (a^4 (-C) d+2 a^3 b B d-a^2 b^2 (3 A d+B c-C d)+2 a b^3 c (A-C)+b^4 (B c-A d)\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right )^2 (b c-a d)^2}+\frac {d \left (A d^2-B c d+c^2 C\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right ) (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.80, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3649, 3651, 3530} \[ \frac {x \left (a^2 (A c+B d-c C)+2 a b (B c-d (A-C))-b^2 (A c+B d-c C)\right )}{\left (a^2+b^2\right )^2 \left (c^2+d^2\right )}-\frac {A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))}+\frac {\left (-a^2 b^2 (3 A d+B c-C d)+2 a^3 b B d+a^4 (-C) d+2 a b^3 c (A-C)+b^4 (B c-A d)\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right )^2 (b c-a d)^2}+\frac {d \left (A d^2-B c d+c^2 C\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right ) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3530

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c*Log[Re
moveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]])/(b*f), 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 3649

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*T
an[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 3651

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

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Mathematica [A]  time = 6.96, size = 543, normalized size = 1.93 \[ \frac {-\frac {(b c-a d) \log \left (\sqrt {-b^2}-b \tan (e+f x)\right ) \left (\frac {\sqrt {-b^2} \left (a^2 (A c+B d-c C)+2 a b (d (C-A)+B c)-b^2 (A c+B d-c C)\right )}{b}+a^2 A d+a^2 (-B) c-a^2 C d+2 a A b c+2 a b B d-2 a b c C-A b^2 d+b^2 B c+b^2 C d\right )}{2 \left (a^2+b^2\right ) \left (c^2+d^2\right )}-\frac {(b c-a d) \log \left (\sqrt {-b^2}+b \tan (e+f x)\right ) \left (\frac {\sqrt {-b^2} \left (-\left (a^2 (A c+B d-c C)\right )-2 a b (d (C-A)+B c)+b^2 (A c+B d-c C)\right )}{b}+a^2 A d+a^2 (-B) c-a^2 C d+2 a A b c+2 a b B d-2 a b c C-A b^2 d+b^2 B c+b^2 C d\right )}{2 \left (a^2+b^2\right ) \left (c^2+d^2\right )}+\frac {d \left (a^2+b^2\right ) \left (A d^2-B c d+c^2 C\right ) \log (c+d \tan (e+f x))}{\left (c^2+d^2\right ) (b c-a d)}+\frac {\left (a^4 C d-2 a^3 b B d+a^2 b^2 (3 A d+B c-C d)+2 a b^3 c (C-A)+b^4 (A d-B c)\right ) \log (a+b \tan (e+f x))}{\left (a^2+b^2\right ) (a d-b c)}-\frac {A b^2}{a+b \tan (e+f x)}+\frac {a (b B-a C)}{a+b \tan (e+f x)}}{f \left (a^2+b^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 2.54, size = 1345, normalized size = 4.79 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 6.62, size = 846, normalized size = 3.01 \[ \frac {\frac {2 \, {\left (A a^{2} c - C a^{2} c + 2 \, B a b c - A b^{2} c + C b^{2} c + B a^{2} d - 2 \, A a b d + 2 \, C a b d - B b^{2} d\right )} {\left (f x + e\right )}}{a^{4} c^{2} + 2 \, a^{2} b^{2} c^{2} + b^{4} c^{2} + a^{4} d^{2} + 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}} + \frac {{\left (B a^{2} c - 2 \, A a b c + 2 \, C a b c - B b^{2} c - A a^{2} d + C a^{2} d - 2 \, B a b d + A b^{2} d - C b^{2} d\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{4} c^{2} + 2 \, a^{2} b^{2} c^{2} + b^{4} c^{2} + a^{4} d^{2} + 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}} - \frac {2 \, {\left (B a^{2} b^{3} c - 2 \, A a b^{4} c + 2 \, C a b^{4} c - B b^{5} c + C a^{4} b d - 2 \, B a^{3} b^{2} d + 3 \, A a^{2} b^{3} d - C a^{2} b^{3} d + A b^{5} d\right )} \log \left ({\left | b \tan \left (f x + e\right ) + a \right |}\right )}{a^{4} b^{3} c^{2} + 2 \, a^{2} b^{5} c^{2} + b^{7} c^{2} - 2 \, a^{5} b^{2} c d - 4 \, a^{3} b^{4} c d - 2 \, a b^{6} c d + a^{6} b d^{2} + 2 \, a^{4} b^{3} d^{2} + a^{2} b^{5} d^{2}} + \frac {2 \, {\left (C c^{2} d^{2} - B c d^{3} + A d^{4}\right )} \log \left ({\left | d \tan \left (f x + e\right ) + c \right |}\right )}{b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3} + b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}} + \frac {2 \, {\left (B a^{2} b^{3} c \tan \left (f x + e\right ) - 2 \, A a b^{4} c \tan \left (f x + e\right ) + 2 \, C a b^{4} c \tan \left (f x + e\right ) - B b^{5} c \tan \left (f x + e\right ) + C a^{4} b d \tan \left (f x + e\right ) - 2 \, B a^{3} b^{2} d \tan \left (f x + e\right ) + 3 \, A a^{2} b^{3} d \tan \left (f x + e\right ) - C a^{2} b^{3} d \tan \left (f x + e\right ) + A b^{5} d \tan \left (f x + e\right ) - C a^{4} b c + 2 \, B a^{3} b^{2} c - 3 \, A a^{2} b^{3} c + C a^{2} b^{3} c - A b^{5} c + 2 \, C a^{5} d - 3 \, B a^{4} b d + 4 \, A a^{3} b^{2} d - B a^{2} b^{3} d + 2 \, A a b^{4} d\right )}}{{\left (a^{4} b^{2} c^{2} + 2 \, a^{2} b^{4} c^{2} + b^{6} c^{2} - 2 \, a^{5} b c d - 4 \, a^{3} b^{3} c d - 2 \, a b^{5} c d + a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right )} {\left (b \tan \left (f x + e\right ) + a\right )}}}{2 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.55, size = 1262, normalized size = 4.49 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.51, size = 520, normalized size = 1.85 \[ \frac {\frac {2 \, {\left ({\left ({\left (A - C\right )} a^{2} + 2 \, B a b - {\left (A - C\right )} b^{2}\right )} c + {\left (B a^{2} - 2 \, {\left (A - C\right )} a b - B b^{2}\right )} d\right )} {\left (f x + e\right )}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} c^{2} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{2}} - \frac {2 \, {\left ({\left (B a^{2} b^{2} - 2 \, {\left (A - C\right )} a b^{3} - B b^{4}\right )} c + {\left (C a^{4} - 2 \, B a^{3} b + {\left (3 \, A - C\right )} a^{2} b^{2} + A b^{4}\right )} d\right )} \log \left (b \tan \left (f x + e\right ) + a\right )}{{\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} c^{2} - 2 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} c d + {\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} d^{2}} + \frac {2 \, {\left (C c^{2} d - B c d^{2} + A d^{3}\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{b^{2} c^{4} - 2 \, a b c^{3} d - 2 \, a b c d^{3} + a^{2} d^{4} + {\left (a^{2} + b^{2}\right )} c^{2} d^{2}} + \frac {{\left ({\left (B a^{2} - 2 \, {\left (A - C\right )} a b - B b^{2}\right )} c - {\left ({\left (A - C\right )} a^{2} + 2 \, B a b - {\left (A - C\right )} b^{2}\right )} d\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} c^{2} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{2}} - \frac {2 \, {\left (C a^{2} - B a b + A b^{2}\right )}}{{\left (a^{3} b + a b^{3}\right )} c - {\left (a^{4} + a^{2} b^{2}\right )} d + {\left ({\left (a^{2} b^{2} + b^{4}\right )} c - {\left (a^{3} b + a b^{3}\right )} d\right )} \tan \left (f x + e\right )}}{2 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 63.66, size = 393, normalized size = 1.40 \[ \frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,\left (B-A\,1{}\mathrm {i}+C\,1{}\mathrm {i}\right )}{2\,f\,\left (a^2\,c-b^2\,c-2\,a\,b\,d+a^2\,d\,1{}\mathrm {i}-b^2\,d\,1{}\mathrm {i}+a\,b\,c\,2{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (A\,1{}\mathrm {i}+B-C\,1{}\mathrm {i}\right )}{2\,f\,\left (b^2\,c-a^2\,c+2\,a\,b\,d+a^2\,d\,1{}\mathrm {i}-b^2\,d\,1{}\mathrm {i}+a\,b\,c\,2{}\mathrm {i}\right )}-\frac {\ln \left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (C\,d\,a^4-2\,B\,d\,a^3\,b+\left (3\,A\,d+B\,c-C\,d\right )\,a^2\,b^2+\left (2\,C\,c-2\,A\,c\right )\,a\,b^3+\left (A\,d-B\,c\right )\,b^4\right )}{f\,\left (a^6\,d^2-2\,a^5\,b\,c\,d+a^4\,b^2\,c^2+2\,a^4\,b^2\,d^2-4\,a^3\,b^3\,c\,d+2\,a^2\,b^4\,c^2+a^2\,b^4\,d^2-2\,a\,b^5\,c\,d+b^6\,c^2\right )}+\frac {C\,a^2-B\,a\,b+A\,b^2}{f\,\left (a\,d-b\,c\right )\,\left (a^2+b^2\right )\,\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}+\frac {d\,\ln \left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (C\,c^2-B\,c\,d+A\,d^2\right )}{f\,{\left (a\,d-b\,c\right )}^2\,\left (c^2+d^2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError

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