3.1.88 \(\int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^3} \, dx\) [88]

Optimal. Leaf size=487 \[ -\frac {\left (a \left (c^3 C-3 B c^2 d-3 c C d^2+B d^3-A \left (c^3-3 c d^2\right )\right )+b \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )\right ) x}{\left (a^2+b^2\right ) \left (c^2+d^2\right )^3}+\frac {b^2 \left (A b^2-a (b B-a C)\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{\left (a^2+b^2\right ) (b c-a d)^3 f}-\frac {\left (b^2 \left (c^6 C-3 B c^5 d+3 c^4 (2 A-C) d^2+B c^3 d^3+3 A c^2 d^4+A d^6\right )+a^2 d^3 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )-a b d^2 \left (8 c^3 (A-C) d-B \left (3 c^4-6 c^2 d^2-d^4\right )\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{(b c-a d)^3 \left (c^2+d^2\right )^3 f}+\frac {c^2 C-B c d+A d^2}{2 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}+\frac {b \left (c^4 C-2 B c^3 d+c^2 (3 A-C) d^2+A d^4\right )-a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )}{(b c-a d)^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))} \]

[Out]

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

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Rubi [A]
time = 1.23, antiderivative size = 487, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3730, 3732, 3611} \begin {gather*} -\frac {x \left (a \left (-A \left (c^3-3 c d^2\right )-3 B c^2 d+B d^3+c^3 C-3 c C d^2\right )+b \left (d (A-C) \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )\right )}{\left (a^2+b^2\right ) \left (c^2+d^2\right )^3}-\frac {\left (a^2 d^3 \left (d (A-C) \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )-a b d^2 \left (8 c^3 d (A-C)-B \left (3 c^4-6 c^2 d^2-d^4\right )\right )+b^2 \left (3 c^4 d^2 (2 A-C)+3 A c^2 d^4+A d^6-3 B c^5 d+B c^3 d^3+c^6 C\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right )^3 (b c-a d)^3}+\frac {b^2 \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)^3}+\frac {A d^2-B c d+c^2 C}{2 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^2}+\frac {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 )}{f \left (c^2+d^2\right )^2 (b c-a d)^2 (c+d \tan (e+f x))} \end {gather*}

Antiderivative was successfully verified.

[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])^3),x]

[Out]

-(((a*(c^3*C - 3*B*c^2*d - 3*c*C*d^2 + B*d^3 - A*(c^3 - 3*c*d^2)) + b*((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*
d^2)))*x)/((a^2 + b^2)*(c^2 + d^2)^3)) + (b^2*(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)^3*f) - ((b^2*(c^6*C - 3*B*c^5*d + 3*c^4*(2*A - C)*d^2 + B*c^3*d^3 + 3*A*c^2*d^4 + A*d^6
) + a^2*d^3*((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*d^2)) - a*b*d^2*(8*c^3*(A - C)*d - B*(3*c^4 - 6*c^2*d^2 -
d^4)))*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((b*c - a*d)^3*(c^2 + d^2)^3*f) + (c^2*C - B*c*d + A*d^2)/(2*(b*c
- a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) + (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)))/((b*c - a*d)^2*(c^2 + d^2)^2*f*(c + d*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 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*} \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))^3} \, dx &=\frac {c^2 C-B c d+A d^2}{2 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}+\frac {\int \frac {-2 \left (a A c d-a d (c C-B d)-A b \left (c^2+d^2\right )\right )+2 (b c-a d) (B c-(A-C) d) \tan (e+f x)+2 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))^2} \, dx}{2 (b c-a d) \left (c^2+d^2\right )}\\ &=\frac {c^2 C-B c d+A d^2}{2 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}+\frac {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 )}{(b c-a d)^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}+\frac {\int \frac {-2 \left (A \left (2 a b c^3 d-a^2 d^2 \left (c^2-d^2\right )-b^2 \left (c^2+d^2\right )^2\right )+a d \left (a d \left (c^2 C-2 B c d-C d^2\right )-b \left (2 c^3 C-3 B c^2 d-B d^3\right )\right )\right )-2 (b c-a d)^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) \tan (e+f x)+2 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 ) \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))} \, dx}{2 (b c-a d)^2 \left (c^2+d^2\right )^2}\\ &=-\frac {\left (b (A-C) d \left (3 c^2-d^2\right )-b B \left (c^3-3 c d^2\right )-a \left (A c^3-c^3 C+3 B c^2 d-3 A c d^2+3 c C d^2-B d^3\right )\right ) x}{\left (a^2+b^2\right ) \left (c^2+d^2\right )^3}+\frac {c^2 C-B c d+A d^2}{2 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}+\frac {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 )}{(b c-a d)^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}+\frac {\left (b^2 \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)^3}-\frac {\left (b^2 \left (c^6 C-3 B c^5 d+3 c^4 (2 A-C) d^2+B c^3 d^3+3 A c^2 d^4+A d^6\right )+a^2 d^3 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )-a b d^2 \left (8 c^3 (A-C) d-B \left (3 c^4-6 c^2 d^2-d^4\right )\right )\right ) \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{(b c-a d)^3 \left (c^2+d^2\right )^3}\\ &=-\frac {\left (b (A-C) d \left (3 c^2-d^2\right )-b B \left (c^3-3 c d^2\right )-a \left (A c^3-c^3 C+3 B c^2 d-3 A c d^2+3 c C d^2-B d^3\right )\right ) x}{\left (a^2+b^2\right ) \left (c^2+d^2\right )^3}+\frac {b^2 \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)^3 f}-\frac {\left (b^2 \left (c^6 C-3 B c^5 d+3 c^4 (2 A-C) d^2+B c^3 d^3+3 A c^2 d^4+A d^6\right )+a^2 d^3 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )-a b d^2 \left (8 c^3 (A-C) d-B \left (3 c^4-6 c^2 d^2-d^4\right )\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{(b c-a d)^3 \left (c^2+d^2\right )^3 f}+\frac {c^2 C-B c d+A d^2}{2 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}+\frac {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 )}{(b c-a d)^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}\\ \end {align*}

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Mathematica [A]
time = 8.12, size = 912, normalized size = 1.87 \begin {gather*} -\frac {A d^2-c (-c C+B d)}{2 (-b c+a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {-\frac {-\frac {b (b c-a d)^2 \left (A b c^3-a B c^3-b c^3 C+3 a A c^2 d+3 b B c^2 d-3 a c^2 C d-3 A b c d^2+3 a B c d^2+3 b c C d^2-a A d^3-b B d^3+a C d^3-\frac {\sqrt {-b^2} \left (a \left (c^3 C-3 B c^2 d-3 c C d^2+B d^3-A \left (c^3-3 c d^2\right )\right )+b \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )\right )}{b}\right ) \log \left (\sqrt {-b^2}-b \tan (e+f x)\right )}{\left (a^2+b^2\right ) \left (c^2+d^2\right )}+\frac {2 b^3 \left (A b^2-a (b B-a C)\right ) \left (c^2+d^2\right )^2 \log (a+b \tan (e+f x))}{\left (a^2+b^2\right ) (b c-a d)}-\frac {b (b c-a d)^2 \left (A b c^3-a B c^3-b c^3 C+3 a A c^2 d+3 b B c^2 d-3 a c^2 C d-3 A b c d^2+3 a B c d^2+3 b c C d^2-a A d^3-b B d^3+a C d^3+\frac {\sqrt {-b^2} \left (b (A-C) d \left (3 c^2-d^2\right )-b B \left (c^3-3 c d^2\right )-a \left (A c^3-c^3 C+3 B c^2 d-3 A c d^2+3 c C d^2-B d^3\right )\right )}{b}\right ) \log \left (\sqrt {-b^2}+b \tan (e+f x)\right )}{\left (a^2+b^2\right ) \left (c^2+d^2\right )}-\frac {2 b \left (b^2 \left (c^6 C-3 B c^5 d+3 c^4 (2 A-C) d^2+B c^3 d^3+3 A c^2 d^4+A d^6\right )+a^2 d^3 \left ((A-C) d \left (3 c^2-d^2\right )-B \left (c^3-3 c d^2\right )\right )-a b d^2 \left (8 c^3 (A-C) d-B \left (3 c^4-6 c^2 d^2-d^4\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 {-2 d^2 \left (a A c d-a d (c C-B d)-A b \left (c^2+d^2\right )\right )-c \left (2 d (b c-a d) (B c-(A-C) d)-2 b c \left (c^2 C-B c d+A d^2\right )\right )}{(-b c+a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))}}{2 (-b c+a d) \left (c^2+d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[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])^3),x]

[Out]

-1/2*(A*d^2 - c*(-(c*C) + B*d))/((-(b*c) + a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^2) - (-((-((b*(b*c - a*d)^2
*(A*b*c^3 - a*B*c^3 - b*c^3*C + 3*a*A*c^2*d + 3*b*B*c^2*d - 3*a*c^2*C*d - 3*A*b*c*d^2 + 3*a*B*c*d^2 + 3*b*c*C*
d^2 - a*A*d^3 - b*B*d^3 + a*C*d^3 - (Sqrt[-b^2]*(a*(c^3*C - 3*B*c^2*d - 3*c*C*d^2 + B*d^3 - A*(c^3 - 3*c*d^2))
+ b*((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*d^2))))/b)*Log[Sqrt[-b^2] - b*Tan[e + f*x]])/((a^2 + b^2)*(c^2 +
d^2))) + (2*b^3*(A*b^2 - a*(b*B - a*C))*(c^2 + d^2)^2*Log[a + b*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)) - (b*
(b*c - a*d)^2*(A*b*c^3 - a*B*c^3 - b*c^3*C + 3*a*A*c^2*d + 3*b*B*c^2*d - 3*a*c^2*C*d - 3*A*b*c*d^2 + 3*a*B*c*d
^2 + 3*b*c*C*d^2 - a*A*d^3 - b*B*d^3 + a*C*d^3 + (Sqrt[-b^2]*(b*(A - C)*d*(3*c^2 - d^2) - b*B*(c^3 - 3*c*d^2)
- a*(A*c^3 - c^3*C + 3*B*c^2*d - 3*A*c*d^2 + 3*c*C*d^2 - B*d^3)))/b)*Log[Sqrt[-b^2] + b*Tan[e + f*x]])/((a^2 +
b^2)*(c^2 + d^2)) - (2*b*(b^2*(c^6*C - 3*B*c^5*d + 3*c^4*(2*A - C)*d^2 + B*c^3*d^3 + 3*A*c^2*d^4 + A*d^6) + a
^2*d^3*((A - C)*d*(3*c^2 - d^2) - B*(c^3 - 3*c*d^2)) - a*b*d^2*(8*c^3*(A - C)*d - B*(3*c^4 - 6*c^2*d^2 - d^4))
)*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)) - (-2*d^2*(a*A*c*d - a
*d*(c*C - B*d) - A*b*(c^2 + d^2)) - c*(2*d*(b*c - a*d)*(B*c - (A - C)*d) - 2*b*c*(c^2*C - B*c*d + A*d^2)))/((-
(b*c) + a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])))/(2*(-(b*c) + a*d)*(c^2 + d^2))

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Maple [A]
time = 1.83, size = 649, normalized size = 1.33 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))/(c+d*tan(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1085 vs. \(2 (491) = 982\).
time = 0.60, size = 1085, normalized size = 2.23 \begin {gather*} \frac {\frac {2 \, {\left ({\left ({\left (A - C\right )} a + B b\right )} c^{3} + 3 \, {\left (B a - {\left (A - C\right )} b\right )} c^{2} d - 3 \, {\left ({\left (A - C\right )} a + B b\right )} c d^{2} - {\left (B a - {\left (A - C\right )} b\right )} d^{3}\right )} {\left (f x + e\right )}}{{\left (a^{2} + b^{2}\right )} c^{6} + 3 \, {\left (a^{2} + b^{2}\right )} c^{4} d^{2} + 3 \, {\left (a^{2} + b^{2}\right )} c^{2} d^{4} + {\left (a^{2} + b^{2}\right )} d^{6}} + \frac {2 \, {\left (C a^{2} b^{2} - B a b^{3} + A b^{4}\right )} \log \left (b \tan \left (f x + e\right ) + a\right )}{{\left (a^{2} b^{3} + b^{5}\right )} c^{3} - 3 \, {\left (a^{3} b^{2} + a b^{4}\right )} c^{2} d + 3 \, {\left (a^{4} b + a^{2} b^{3}\right )} c d^{2} - {\left (a^{5} + a^{3} b^{2}\right )} d^{3}} - \frac {2 \, {\left (C b^{2} c^{6} - 3 \, B b^{2} c^{5} d + 3 \, B a^{2} c d^{5} + 3 \, {\left (B a b + {\left (2 \, A - C\right )} b^{2}\right )} c^{4} d^{2} - {\left (B a^{2} + 8 \, {\left (A - C\right )} a b - B b^{2}\right )} c^{3} d^{3} + 3 \, {\left ({\left (A - C\right )} a^{2} - 2 \, B a b + A b^{2}\right )} c^{2} d^{4} - {\left ({\left (A - C\right )} a^{2} + B a b - A b^{2}\right )} d^{6}\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{b^{3} c^{9} - 3 \, a b^{2} c^{8} d + 3 \, a^{2} b c d^{8} - a^{3} d^{9} + 3 \, {\left (a^{2} b + b^{3}\right )} c^{7} d^{2} - {\left (a^{3} + 9 \, a b^{2}\right )} c^{6} d^{3} + 3 \, {\left (3 \, a^{2} b + b^{3}\right )} c^{5} d^{4} - 3 \, {\left (a^{3} + 3 \, a b^{2}\right )} c^{4} d^{5} + {\left (9 \, a^{2} b + b^{3}\right )} c^{3} d^{6} - 3 \, {\left (a^{3} + a b^{2}\right )} c^{2} d^{7}} + \frac {{\left ({\left (B a - {\left (A - C\right )} b\right )} c^{3} - 3 \, {\left ({\left (A - C\right )} a + B b\right )} c^{2} d - 3 \, {\left (B a - {\left (A - C\right )} b\right )} c d^{2} + {\left ({\left (A - C\right )} a + B b\right )} d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{{\left (a^{2} + b^{2}\right )} c^{6} + 3 \, {\left (a^{2} + b^{2}\right )} c^{4} d^{2} + 3 \, {\left (a^{2} + b^{2}\right )} c^{2} d^{4} + {\left (a^{2} + b^{2}\right )} d^{6}} + \frac {3 \, C b c^{5} - A a d^{5} - {\left (C a + 5 \, B b\right )} c^{4} d + {\left (3 \, B a + {\left (7 \, A - C\right )} b\right )} c^{3} d^{2} - {\left ({\left (5 \, A - 3 \, C\right )} a + B b\right )} c^{2} d^{3} - {\left (B a - 3 \, A b\right )} c d^{4} + 2 \, {\left (C b c^{4} d - 2 \, B b c^{3} d^{2} - 2 \, {\left (A - C\right )} a c d^{4} + {\left (B a + {\left (3 \, A - C\right )} b\right )} c^{2} d^{3} - {\left (B a - A b\right )} d^{5}\right )} \tan \left (f x + e\right )}{b^{2} c^{8} - 2 \, a b c^{7} d - 4 \, a b c^{5} d^{3} - 2 \, a b c^{3} d^{5} + a^{2} c^{2} d^{6} + {\left (a^{2} + 2 \, b^{2}\right )} c^{6} d^{2} + {\left (2 \, a^{2} + b^{2}\right )} c^{4} d^{4} + {\left (b^{2} c^{6} d^{2} - 2 \, a b c^{5} d^{3} - 4 \, a b c^{3} d^{5} - 2 \, a b c d^{7} + a^{2} d^{8} + {\left (a^{2} + 2 \, b^{2}\right )} c^{4} d^{4} + {\left (2 \, a^{2} + b^{2}\right )} c^{2} d^{6}\right )} \tan \left (f x + e\right )^{2} + 2 \, {\left (b^{2} c^{7} d - 2 \, a b c^{6} d^{2} - 4 \, a b c^{4} d^{4} - 2 \, a b c^{2} d^{6} + a^{2} c d^{7} + {\left (a^{2} + 2 \, b^{2}\right )} c^{5} d^{3} + {\left (2 \, a^{2} + b^{2}\right )} c^{3} d^{5}\right )} \tan \left (f x + e\right )}}{2 \, f} \end {gather*}

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))/(c+d*tan(f*x+e))^3,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3510 vs. \(2 (491) = 982\).
time = 48.34, size = 3510, normalized size = 7.21 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(5*(C*a^2*b^2 + C*b^4)*c^6*d^2 - (8*C*a^3*b + 7*B*a^2*b^2 + 8*C*a*b^3 + 7*B*b^4)*c^5*d^3 + (3*C*a^4 + 12*B
*a^3*b + (9*A + 2*C)*a^2*b^2 + 12*B*a*b^3 + (9*A - C)*b^4)*c^4*d^4 - (5*B*a^4 + 4*(4*A - C)*a^3*b + 6*B*a^2*b^
2 + 4*(4*A - C)*a*b^3 + B*b^4)*c^3*d^5 + ((7*A - 3*C)*a^4 + (10*A - 3*C)*a^2*b^2 + 3*A*b^4)*c^2*d^6 + (B*a^4 -
4*A*a^3*b + B*a^2*b^2 - 4*A*a*b^3)*c*d^7 + (A*a^4 + A*a^2*b^2)*d^8 + 2*(((A - C)*a*b^3 + B*b^4)*c^8 - 3*((A -
C)*a^2*b^2 + (A - C)*b^4)*c^7*d + 3*((A - C)*a^3*b - 2*B*a^2*b^2 + 2*(A - C)*a*b^3 - B*b^4)*c^6*d^2 - ((A - C
)*a^4 - 8*B*a^3*b - 8*B*a*b^3 - (A - C)*b^4)*c^5*d^3 - 3*(B*a^4 + 2*(A - C)*a^3*b + 2*B*a^2*b^2 + (A - C)*a*b^
3)*c^4*d^4 + 3*((A - C)*a^4 + (A - C)*a^2*b^2)*c^3*d^5 + (B*a^4 - (A - C)*a^3*b)*c^2*d^6)*f*x - (3*(C*a^2*b^2
+ C*b^4)*c^6*d^2 - (4*C*a^3*b + 5*B*a^2*b^2 + 4*C*a*b^3 + 5*B*b^4)*c^5*d^3 + (C*a^4 + 8*B*a^3*b + (7*A - 2*C)*
a^2*b^2 + 8*B*a*b^3 + (7*A - 3*C)*b^4)*c^4*d^4 - (3*B*a^4 + 4*(3*A - 2*C)*a^3*b + 2*B*a^2*b^2 + 4*(3*A - 2*C)*
a*b^3 - B*b^4)*c^3*d^5 + (5*(A - C)*a^4 - 4*B*a^3*b + (6*A - 5*C)*a^2*b^2 - 4*B*a*b^3 + A*b^4)*c^2*d^6 + 3*(B*
a^4 + B*a^2*b^2)*c*d^7 - (A*a^4 + A*a^2*b^2)*d^8 - 2*(((A - C)*a*b^3 + B*b^4)*c^6*d^2 - 3*((A - C)*a^2*b^2 + (
A - C)*b^4)*c^5*d^3 + 3*((A - C)*a^3*b - 2*B*a^2*b^2 + 2*(A - C)*a*b^3 - B*b^4)*c^4*d^4 - ((A - C)*a^4 - 8*B*a
^3*b - 8*B*a*b^3 - (A - C)*b^4)*c^3*d^5 - 3*(B*a^4 + 2*(A - C)*a^3*b + 2*B*a^2*b^2 + (A - C)*a*b^3)*c^2*d^6 +
3*((A - C)*a^4 + (A - C)*a^2*b^2)*c*d^7 + (B*a^4 - (A - C)*a^3*b)*d^8)*f*x)*tan(f*x + e)^2 + ((C*a^2*b^2 - B*a
*b^3 + A*b^4)*c^8 + 3*(C*a^2*b^2 - B*a*b^3 + A*b^4)*c^6*d^2 + 3*(C*a^2*b^2 - B*a*b^3 + A*b^4)*c^4*d^4 + (C*a^2
*b^2 - B*a*b^3 + A*b^4)*c^2*d^6 + ((C*a^2*b^2 - B*a*b^3 + A*b^4)*c^6*d^2 + 3*(C*a^2*b^2 - B*a*b^3 + A*b^4)*c^4
*d^4 + 3*(C*a^2*b^2 - B*a*b^3 + A*b^4)*c^2*d^6 + (C*a^2*b^2 - B*a*b^3 + A*b^4)*d^8)*tan(f*x + e)^2 + 2*((C*a^2
*b^2 - B*a*b^3 + A*b^4)*c^7*d + 3*(C*a^2*b^2 - B*a*b^3 + A*b^4)*c^5*d^3 + 3*(C*a^2*b^2 - B*a*b^3 + A*b^4)*c^3*
d^5 + (C*a^2*b^2 - B*a*b^3 + A*b^4)*c*d^7)*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^2 + C*b^4)*c^8 - 3*(B*a^2*b^2 + B*b^4)*c^7*d + 3*(B*a^3*b + (2*A - C)*a^2*b^2
+ B*a*b^3 + (2*A - C)*b^4)*c^6*d^2 - (B*a^4 + 8*(A - C)*a^3*b + 8*(A - C)*a*b^3 - B*b^4)*c^5*d^3 + 3*((A - C)
*a^4 - 2*B*a^3*b + (2*A - C)*a^2*b^2 - 2*B*a*b^3 + A*b^4)*c^4*d^4 + 3*(B*a^4 + B*a^2*b^2)*c^3*d^5 - ((A - C)*a
^4 + B*a^3*b - C*a^2*b^2 + B*a*b^3 - A*b^4)*c^2*d^6 + ((C*a^2*b^2 + C*b^4)*c^6*d^2 - 3*(B*a^2*b^2 + B*b^4)*c^5
*d^3 + 3*(B*a^3*b + (2*A - C)*a^2*b^2 + B*a*b^3 + (2*A - C)*b^4)*c^4*d^4 - (B*a^4 + 8*(A - C)*a^3*b + 8*(A - C
)*a*b^3 - B*b^4)*c^3*d^5 + 3*((A - C)*a^4 - 2*B*a^3*b + (2*A - C)*a^2*b^2 - 2*B*a*b^3 + A*b^4)*c^2*d^6 + 3*(B*
a^4 + B*a^2*b^2)*c*d^7 - ((A - C)*a^4 + B*a^3*b - C*a^2*b^2 + B*a*b^3 - A*b^4)*d^8)*tan(f*x + e)^2 + 2*((C*a^2
*b^2 + C*b^4)*c^7*d - 3*(B*a^2*b^2 + B*b^4)*c^6*d^2 + 3*(B*a^3*b + (2*A - C)*a^2*b^2 + B*a*b^3 + (2*A - C)*b^4
)*c^5*d^3 - (B*a^4 + 8*(A - C)*a^3*b + 8*(A - C)*a*b^3 - B*b^4)*c^4*d^4 + 3*((A - C)*a^4 - 2*B*a^3*b + (2*A -
C)*a^2*b^2 - 2*B*a*b^3 + A*b^4)*c^3*d^5 + 3*(B*a^4 + B*a^2*b^2)*c^2*d^6 - ((A - C)*a^4 + B*a^3*b - C*a^2*b^2 +
B*a*b^3 - A*b^4)*c*d^7)*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*(2*(C*a^2*b^2 + C*b^4)*c^7*d - 3*(C*a^3*b + B*a^2*b^2 + C*a*b^3 + B*b^4)*c^6*d^2 + (C*a^4 + 5*B*a^3*b +
2*(2*A - C)*a^2*b^2 + 5*B*a*b^3 + (4*A - 3*C)*b^4)*c^5*d^3 - (2*B*a^4 + (7*A - 6*C)*a^3*b - B*a^2*b^2 + (7*A
- 6*C)*a*b^3 - 3*B*b^4)*c^4*d^4 + (3*(A - C)*a^4 - 6*B*a^3*b - 2*C*a^2*b^2 - 6*B*a*b^3 - (3*A - C)*b^4)*c^3*d^
5 + 3*(B*a^4 + (2*A - C)*a^3*b + B*a^2*b^2 + (2*A - C)*a*b^3)*c^2*d^6 - ((3*A - 2*C)*a^4 - B*a^3*b + 2*(2*A -
C)*a^2*b^2 - B*a*b^3 + A*b^4)*c*d^7 - (B*a^4 - A*a^3*b + B*a^2*b^2 - A*a*b^3)*d^8 - 2*(((A - C)*a*b^3 + B*b^4)
*c^7*d - 3*((A - C)*a^2*b^2 + (A - C)*b^4)*c^6*d^2 + 3*((A - C)*a^3*b - 2*B*a^2*b^2 + 2*(A - C)*a*b^3 - B*b^4)
*c^5*d^3 - ((A - C)*a^4 - 8*B*a^3*b - 8*B*a*b^3 - (A - C)*b^4)*c^4*d^4 - 3*(B*a^4 + 2*(A - C)*a^3*b + 2*B*a^2*
b^2 + (A - C)*a*b^3)*c^3*d^5 + 3*((A - C)*a^4 + (A - C)*a^2*b^2)*c^2*d^6 + (B*a^4 - (A - C)*a^3*b)*c*d^7)*f*x)
*tan(f*x + e))/(((a^2*b^3 + b^5)*c^9*d^2 - 3*(a^3*b^2 + a*b^4)*c^8*d^3 + 3*(a^4*b + 2*a^2*b^3 + b^5)*c^7*d^4 -
(a^5 + 10*a^3*b^2 + 9*a*b^4)*c^6*d^5 + 3*(3*a^4*b + 4*a^2*b^3 + b^5)*c^5*d^6 - 3*(a^5 + 4*a^3*b^2 + 3*a*b^4)*
c^4*d^7 + (9*a^4*b + 10*a^2*b^3 + b^5)*c^3*d^8 - 3*(a^5 + 2*a^3*b^2 + a*b^4)*c^2*d^9 + 3*(a^4*b + a^2*b^3)*c*d
^10 - (a^5 + a^3*b^2)*d^11)*f*tan(f*x + e)^2 + 2*((a^2*b^3 + b^5)*c^10*d - 3*(a^3*b^2 + a*b^4)*c^9*d^2 + 3*(a^
4*b + 2*a^2*b^3 + b^5)*c^8*d^3 - (a^5 + 10*a^3*b^2 + 9*a*b^4)*c^7*d^4 + 3*(3*a^4*b + 4*a^2*b^3 + b^5)*c^6*d^5
- 3*(a^5 + 4*a^3*b^2 + 3*a*b^4)*c^5*d^6 + (9*a^4*b + 10*a^2*b^3 + b^5)*c^4*d^7 - 3*(a^5 + 2*a^3*b^2 + a*b^4)*c
^3*d^8 + 3*(a^4*b + a^2*b^3)*c^2*d^9 - (a^5 + a^3*b^2)*c*d^10)*f*tan(f*x + e) + ((a^2*b^3 + b^5)*c^11 - 3*(a^3
*b^2 + a*b^4)*c^10*d + 3*(a^4*b + 2*a^2*b^3 + b...

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

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))/(c+d*tan(f*x+e))**3,x)

[Out]

Exception raised: NotImplementedError >> no valid subset found

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2125 vs. \(2 (491) = 982\).
time = 1.19, size = 2125, normalized size = 4.36 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*(A*a*c^3 - C*a*c^3 + B*b*c^3 + 3*B*a*c^2*d - 3*A*b*c^2*d + 3*C*b*c^2*d - 3*A*a*c*d^2 + 3*C*a*c*d^2 - 3*
B*b*c*d^2 - B*a*d^3 + A*b*d^3 - C*b*d^3)*(f*x + e)/(a^2*c^6 + b^2*c^6 + 3*a^2*c^4*d^2 + 3*b^2*c^4*d^2 + 3*a^2*
c^2*d^4 + 3*b^2*c^2*d^4 + a^2*d^6 + b^2*d^6) + (B*a*c^3 - A*b*c^3 + C*b*c^3 - 3*A*a*c^2*d + 3*C*a*c^2*d - 3*B*
b*c^2*d - 3*B*a*c*d^2 + 3*A*b*c*d^2 - 3*C*b*c*d^2 + A*a*d^3 - C*a*d^3 + B*b*d^3)*log(tan(f*x + e)^2 + 1)/(a^2*
c^6 + b^2*c^6 + 3*a^2*c^4*d^2 + 3*b^2*c^4*d^2 + 3*a^2*c^2*d^4 + 3*b^2*c^2*d^4 + a^2*d^6 + b^2*d^6) + 2*(C*a^2*
b^3 - B*a*b^4 + A*b^5)*log(abs(b*tan(f*x + e) + a))/(a^2*b^4*c^3 + b^6*c^3 - 3*a^3*b^3*c^2*d - 3*a*b^5*c^2*d +
3*a^4*b^2*c*d^2 + 3*a^2*b^4*c*d^2 - a^5*b*d^3 - a^3*b^3*d^3) - 2*(C*b^2*c^6*d - 3*B*b^2*c^5*d^2 + 3*B*a*b*c^4
*d^3 + 6*A*b^2*c^4*d^3 - 3*C*b^2*c^4*d^3 - B*a^2*c^3*d^4 - 8*A*a*b*c^3*d^4 + 8*C*a*b*c^3*d^4 + B*b^2*c^3*d^4 +
3*A*a^2*c^2*d^5 - 3*C*a^2*c^2*d^5 - 6*B*a*b*c^2*d^5 + 3*A*b^2*c^2*d^5 + 3*B*a^2*c*d^6 - A*a^2*d^7 + C*a^2*d^7
- B*a*b*d^7 + A*b^2*d^7)*log(abs(d*tan(f*x + e) + c))/(b^3*c^9*d - 3*a*b^2*c^8*d^2 + 3*a^2*b*c^7*d^3 + 3*b^3*
c^7*d^3 - a^3*c^6*d^4 - 9*a*b^2*c^6*d^4 + 9*a^2*b*c^5*d^5 + 3*b^3*c^5*d^5 - 3*a^3*c^4*d^6 - 9*a*b^2*c^4*d^6 +
9*a^2*b*c^3*d^7 + b^3*c^3*d^7 - 3*a^3*c^2*d^8 - 3*a*b^2*c^2*d^8 + 3*a^2*b*c*d^9 - a^3*d^10) + (3*C*b^2*c^6*d^2
*tan(f*x + e)^2 - 9*B*b^2*c^5*d^3*tan(f*x + e)^2 + 9*B*a*b*c^4*d^4*tan(f*x + e)^2 + 18*A*b^2*c^4*d^4*tan(f*x +
e)^2 - 9*C*b^2*c^4*d^4*tan(f*x + e)^2 - 3*B*a^2*c^3*d^5*tan(f*x + e)^2 - 24*A*a*b*c^3*d^5*tan(f*x + e)^2 + 24
*C*a*b*c^3*d^5*tan(f*x + e)^2 + 3*B*b^2*c^3*d^5*tan(f*x + e)^2 + 9*A*a^2*c^2*d^6*tan(f*x + e)^2 - 9*C*a^2*c^2*
d^6*tan(f*x + e)^2 - 18*B*a*b*c^2*d^6*tan(f*x + e)^2 + 9*A*b^2*c^2*d^6*tan(f*x + e)^2 + 9*B*a^2*c*d^7*tan(f*x
+ e)^2 - 3*A*a^2*d^8*tan(f*x + e)^2 + 3*C*a^2*d^8*tan(f*x + e)^2 - 3*B*a*b*d^8*tan(f*x + e)^2 + 3*A*b^2*d^8*ta
n(f*x + e)^2 + 8*C*b^2*c^7*d*tan(f*x + e) - 2*C*a*b*c^6*d^2*tan(f*x + e) - 22*B*b^2*c^6*d^2*tan(f*x + e) + 24*
B*a*b*c^5*d^3*tan(f*x + e) + 42*A*b^2*c^5*d^3*tan(f*x + e) - 18*C*b^2*c^5*d^3*tan(f*x + e) - 8*B*a^2*c^4*d^4*t
an(f*x + e) - 58*A*a*b*c^4*d^4*tan(f*x + e) + 52*C*a*b*c^4*d^4*tan(f*x + e) + 2*B*b^2*c^4*d^4*tan(f*x + e) + 2
2*A*a^2*c^3*d^5*tan(f*x + e) - 22*C*a^2*c^3*d^5*tan(f*x + e) - 32*B*a*b*c^3*d^5*tan(f*x + e) + 26*A*b^2*c^3*d^
5*tan(f*x + e) - 2*C*b^2*c^3*d^5*tan(f*x + e) + 18*B*a^2*c^2*d^6*tan(f*x + e) - 12*A*a*b*c^2*d^6*tan(f*x + e)
+ 6*C*a*b*c^2*d^6*tan(f*x + e) - 2*A*a^2*c*d^7*tan(f*x + e) + 2*C*a^2*c*d^7*tan(f*x + e) - 8*B*a*b*c*d^7*tan(f
*x + e) + 8*A*b^2*c*d^7*tan(f*x + e) + 2*B*a^2*d^8*tan(f*x + e) - 2*A*a*b*d^8*tan(f*x + e) + 6*C*b^2*c^8 - 4*C
*a*b*c^7*d - 14*B*b^2*c^7*d + C*a^2*c^6*d^2 + 17*B*a*b*c^6*d^2 + 25*A*b^2*c^6*d^2 - 7*C*b^2*c^6*d^2 - 6*B*a^2*
c^5*d^3 - 36*A*a*b*c^5*d^3 + 24*C*a*b*c^5*d^3 - 3*B*b^2*c^5*d^3 + 14*A*a^2*c^4*d^4 - 11*C*a^2*c^4*d^4 - 10*B*a
*b*c^4*d^4 + 19*A*b^2*c^4*d^4 - C*b^2*c^4*d^4 + 7*B*a^2*c^3*d^5 - 16*A*a*b*c^3*d^5 + 4*C*a*b*c^3*d^5 - B*b^2*c
^3*d^5 + 3*A*a^2*c^2*d^6 - 3*B*a*b*c^2*d^6 + 6*A*b^2*c^2*d^6 + B*a^2*c*d^7 - 4*A*a*b*c*d^7 + A*a^2*d^8)/((b^3*
c^9 - 3*a*b^2*c^8*d + 3*a^2*b*c^7*d^2 + 3*b^3*c^7*d^2 - a^3*c^6*d^3 - 9*a*b^2*c^6*d^3 + 9*a^2*b*c^5*d^4 + 3*b^
3*c^5*d^4 - 3*a^3*c^4*d^5 - 9*a*b^2*c^4*d^5 + 9*a^2*b*c^3*d^6 + b^3*c^3*d^6 - 3*a^3*c^2*d^7 - 3*a*b^2*c^2*d^7
+ 3*a^2*b*c*d^8 - a^3*d^9)*(d*tan(f*x + e) + c)^2))/f

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Mupad [B]
time = 24.61, size = 2500, normalized size = 5.13 \begin {gather*} \text {Too large to display} \end {gather*}

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))*(c + d*tan(e + f*x))^3),x)

[Out]

(symsum(log(- root(480*a^9*b*c^7*d^11*f^4 + 480*a*b^9*c^11*d^7*f^4 + 360*a^9*b*c^9*d^9*f^4 + 360*a^9*b*c^5*d^1
3*f^4 + 360*a*b^9*c^13*d^5*f^4 + 360*a*b^9*c^9*d^9*f^4 + 144*a^9*b*c^11*d^7*f^4 + 144*a^9*b*c^3*d^15*f^4 + 144
*a*b^9*c^15*d^3*f^4 + 144*a*b^9*c^7*d^11*f^4 + 48*a^7*b^3*c*d^17*f^4 + 48*a^3*b^7*c^17*d*f^4 + 24*a^9*b*c^13*d
^5*f^4 + 24*a^5*b^5*c^17*d*f^4 + 24*a^5*b^5*c*d^17*f^4 + 24*a*b^9*c^5*d^13*f^4 + 24*a^9*b*c*d^17*f^4 + 24*a*b^
9*c^17*d*f^4 + 3920*a^5*b^5*c^9*d^9*f^4 - 3360*a^6*b^4*c^8*d^10*f^4 - 3360*a^4*b^6*c^10*d^8*f^4 - 3024*a^6*b^4
*c^10*d^8*f^4 + 3024*a^5*b^5*c^11*d^7*f^4 + 3024*a^5*b^5*c^7*d^11*f^4 - 3024*a^4*b^6*c^8*d^10*f^4 + 2320*a^7*b
^3*c^9*d^9*f^4 + 2320*a^3*b^7*c^9*d^9*f^4 - 2240*a^6*b^4*c^6*d^12*f^4 - 2240*a^4*b^6*c^12*d^6*f^4 + 2160*a^7*b
^3*c^7*d^11*f^4 + 2160*a^3*b^7*c^11*d^7*f^4 - 1624*a^6*b^4*c^12*d^6*f^4 - 1624*a^4*b^6*c^6*d^12*f^4 + 1488*a^7
*b^3*c^11*d^7*f^4 + 1488*a^3*b^7*c^7*d^11*f^4 + 1344*a^5*b^5*c^13*d^5*f^4 + 1344*a^5*b^5*c^5*d^13*f^4 - 1320*a
^8*b^2*c^8*d^10*f^4 - 1320*a^2*b^8*c^10*d^8*f^4 + 1200*a^7*b^3*c^5*d^13*f^4 + 1200*a^3*b^7*c^13*d^5*f^4 - 1060
*a^8*b^2*c^6*d^12*f^4 - 1060*a^2*b^8*c^12*d^6*f^4 - 948*a^8*b^2*c^10*d^8*f^4 - 948*a^2*b^8*c^8*d^10*f^4 - 840*
a^6*b^4*c^4*d^14*f^4 - 840*a^4*b^6*c^14*d^4*f^4 + 528*a^7*b^3*c^13*d^5*f^4 + 528*a^3*b^7*c^5*d^13*f^4 - 480*a^
8*b^2*c^4*d^14*f^4 - 480*a^6*b^4*c^14*d^4*f^4 - 480*a^4*b^6*c^4*d^14*f^4 - 480*a^2*b^8*c^14*d^4*f^4 - 368*a^8*
b^2*c^12*d^6*f^4 + 368*a^7*b^3*c^3*d^15*f^4 + 368*a^3*b^7*c^15*d^3*f^4 - 368*a^2*b^8*c^6*d^12*f^4 + 304*a^5*b^
5*c^15*d^3*f^4 + 304*a^5*b^5*c^3*d^15*f^4 - 144*a^6*b^4*c^2*d^16*f^4 - 144*a^4*b^6*c^16*d^2*f^4 - 108*a^8*b^2*
c^2*d^16*f^4 - 108*a^2*b^8*c^16*d^2*f^4 + 80*a^7*b^3*c^15*d^3*f^4 + 80*a^3*b^7*c^3*d^15*f^4 - 60*a^8*b^2*c^14*
d^4*f^4 - 60*a^6*b^4*c^16*d^2*f^4 - 60*a^4*b^6*c^2*d^16*f^4 - 60*a^2*b^8*c^4*d^14*f^4 - 80*b^10*c^12*d^6*f^4 -
60*b^10*c^14*d^4*f^4 - 60*b^10*c^10*d^8*f^4 - 24*b^10*c^16*d^2*f^4 - 24*b^10*c^8*d^10*f^4 - 4*b^10*c^6*d^12*f
^4 - 80*a^10*c^6*d^12*f^4 - 60*a^10*c^8*d^10*f^4 - 60*a^10*c^4*d^14*f^4 - 24*a^10*c^10*d^8*f^4 - 24*a^10*c^2*d
^16*f^4 - 4*a^10*c^12*d^6*f^4 - 8*a^8*b^2*d^18*f^4 - 4*a^6*b^4*d^18*f^4 - 8*a^2*b^8*c^18*f^4 - 4*a^4*b^6*c^18*
f^4 - 4*b^10*c^18*f^4 - 4*a^10*d^18*f^4 - 12*A*C*a^7*b*c*d^11*f^2 - 12*A*C*a*b^7*c^11*d*f^2 - 912*B*C*a^4*b^4*
c^5*d^7*f^2 + 792*B*C*a^5*b^3*c^4*d^8*f^2 - 792*B*C*a^3*b^5*c^8*d^4*f^2 + 720*B*C*a^4*b^4*c^7*d^5*f^2 - 480*B*
C*a^6*b^2*c^5*d^7*f^2 - 408*B*C*a^2*b^6*c^5*d^7*f^2 + 384*B*C*a^2*b^6*c^7*d^5*f^2 - 336*B*C*a^5*b^3*c^8*d^4*f^
2 + 324*B*C*a^3*b^5*c^4*d^8*f^2 + 312*B*C*a^6*b^2*c^7*d^5*f^2 - 248*B*C*a^6*b^2*c^3*d^9*f^2 + 216*B*C*a^2*b^6*
c^9*d^3*f^2 - 196*B*C*a^4*b^4*c^3*d^9*f^2 + 132*B*C*a^4*b^4*c^9*d^3*f^2 + 80*B*C*a^3*b^5*c^6*d^6*f^2 - 64*B*C*
a^5*b^3*c^6*d^6*f^2 - 36*B*C*a^3*b^5*c^2*d^10*f^2 - 28*B*C*a^2*b^6*c^3*d^9*f^2 + 12*B*C*a^5*b^3*c^10*d^2*f^2 -
12*B*C*a^5*b^3*c^2*d^10*f^2 - 12*B*C*a^3*b^5*c^10*d^2*f^2 - 4*B*C*a^6*b^2*c^9*d^3*f^2 - 1468*A*C*a^4*b^4*c^6*
d^6*f^2 + 996*A*C*a^3*b^5*c^7*d^5*f^2 + 900*A*C*a^5*b^3*c^5*d^7*f^2 - 676*A*C*a^6*b^2*c^6*d^6*f^2 - 660*A*C*a^
2*b^6*c^6*d^6*f^2 + 636*A*C*a^3*b^5*c^5*d^7*f^2 + 540*A*C*a^5*b^3*c^7*d^5*f^2 - 236*A*C*a^5*b^3*c^3*d^9*f^2 -
204*A*C*a^3*b^5*c^9*d^3*f^2 + 156*A*C*a^2*b^6*c^10*d^2*f^2 + 132*A*C*a^6*b^2*c^2*d^10*f^2 - 72*A*C*a^6*b^2*c^4
*d^8*f^2 - 72*A*C*a^5*b^3*c^9*d^3*f^2 + 66*A*C*a^2*b^6*c^4*d^8*f^2 + 54*A*C*a^4*b^4*c^10*d^2*f^2 + 54*A*C*a^4*
b^4*c^2*d^10*f^2 - 48*A*C*a^4*b^4*c^4*d^8*f^2 - 48*A*C*a^2*b^6*c^8*d^4*f^2 + 42*A*C*a^6*b^2*c^8*d^4*f^2 - 40*A
*C*a^3*b^5*c^3*d^9*f^2 - 36*A*C*a^4*b^4*c^8*d^4*f^2 + 24*A*C*a^2*b^6*c^2*d^10*f^2 + 960*A*B*a^4*b^4*c^5*d^7*f^
2 - 864*A*B*a^5*b^3*c^4*d^8*f^2 + 756*A*B*a^3*b^5*c^8*d^4*f^2 - 744*A*B*a^4*b^4*c^7*d^5*f^2 - 528*A*B*a^3*b^5*
c^4*d^8*f^2 + 504*A*B*a^6*b^2*c^5*d^7*f^2 - 432*A*B*a^2*b^6*c^7*d^5*f^2 + 432*A*B*a^2*b^6*c^5*d^7*f^2 + 348*A*
B*a^5*b^3*c^8*d^4*f^2 - 312*A*B*a^6*b^2*c^7*d^5*f^2 - 284*A*B*a^2*b^6*c^9*d^3*f^2 + 280*A*B*a^6*b^2*c^3*d^9*f^
2 + 264*A*B*a^4*b^4*c^3*d^9*f^2 - 240*A*B*a^3*b^5*c^6*d^6*f^2 - 172*A*B*a^4*b^4*c^9*d^3*f^2 + 68*A*B*a^2*b^6*c
^3*d^9*f^2 - 60*A*B*a^3*b^5*c^2*d^10*f^2 + 24*A*B*a^5*b^3*c^6*d^6*f^2 - 24*A*B*a^5*b^3*c^2*d^10*f^2 + 12*A*B*a
^3*b^5*c^10*d^2*f^2 + 360*B*C*a^7*b*c^4*d^8*f^2 - 336*B*C*a*b^7*c^8*d^4*f^2 + 168*B*C*a*b^7*c^6*d^6*f^2 - 136*
B*C*a^7*b*c^6*d^6*f^2 + 36*B*C*a^6*b^2*c*d^11*f^2 - 36*B*C*a^2*b^6*c^11*d*f^2 - 24*B*C*a^7*b*c^2*d^10*f^2 + 24
*B*C*a*b^7*c^10*d^2*f^2 - 12*B*C*a^4*b^4*c^11*d*f^2 + 12*B*C*a^4*b^4*c*d^11*f^2 + 12*B*C*a*b^7*c^4*d^8*f^2 + 4
44*A*C*a*b^7*c^7*d^5*f^2 + 348*A*C*a^7*b*c^5*d^7*f^2 - 164*A*C*a^7*b*c^3*d^9*f^2 - 132*A*C*a*b^7*c^9*d^3*f^2 +
84*A*C*a*b^7*c^5*d^7*f^2 + 32*A*C*a*b^7*c^3*d^9*f^2 - 12*A*C*a^7*b*c^7*d^5*f^2 - 12*A*C*a^5*b^3*c*d^11*f^2 -
12*A*C*a^3*b^5*c^11*d*f^2 - 360*A*B*a^7*b*c^4*d^8*f^2 + 288*A*B*a*b^7*c^8*d^4*f^2 - 288*A*B*a*b^7*c^6*d^6*f^2
- 144*A*B*a*b^7*c^4*d^8*f^2 + 136*A*B*a^7*b*c^6*d^6*f^2 - 60*A*B*a*b^7*c^2*d^10*f^2 - 36*A*B*a*b^7*c^10*d^2*f^
2 + 24*A*B*a^7*b*c^2*d^10*f^2 - 24*A*B*a^6*b^2*...

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