3.1.76 \(\int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx\) [76]

Optimal. Leaf size=477 \[ \frac {\left (a^3 (A c-c C+B d)-3 a b^2 (A c-c C+B d)+3 a^2 b (B c-(A-C) d)-b^3 (B c-(A-C) d)\right ) x}{\left (a^2+b^2\right )^3 \left (c^2+d^2\right )}+\frac {\left (3 a b^5 B c^2-3 a^5 b B d^2+a^6 C d^2+3 a^4 b^2 d (B c+2 A d-C d)+b^6 \left (c (c C-B d)-A \left (c^2-d^2\right )\right )-a^3 b^3 \left (8 c (A-C) d+B \left (c^2-d^2\right )\right )-3 a^2 b^4 \left (c (c C+2 B d)-A \left (c^2+d^2\right )\right )\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{\left (a^2+b^2\right )^3 (b c-a d)^3 f}-\frac {d^2 \left (c^2 C-B c d+A d^2\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{(b c-a d)^3 \left (c^2+d^2\right ) f}-\frac {A b^2-a (b B-a C)}{2 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^2}-\frac {2 a b^3 c (A-C)+2 a^3 b B d-a^4 C d+b^4 (B c-A d)-a^2 b^2 (B c+3 A d-C d)}{\left (a^2+b^2\right )^2 (b c-a d)^2 f (a+b \tan (e+f x))} \]

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

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Rubi [A]
time = 1.17, antiderivative size = 477, 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 {A b^2-a (b B-a C)}{2 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^2}+\frac {x \left (a^3 (A c+B d-c C)+3 a^2 b (B c-d (A-C))-3 a b^2 (A c+B d-c C)-b^3 (B c-d (A-C))\right )}{\left (a^2+b^2\right )^3 \left (c^2+d^2\right )}-\frac {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)}{f \left (a^2+b^2\right )^2 (b c-a d)^2 (a+b \tan (e+f x))}+\frac {\left (a^6 C d^2-3 a^5 b B d^2+3 a^4 b^2 d (2 A d+B c-C d)-a^3 b^3 \left (8 c d (A-C)+B \left (c^2-d^2\right )\right )-3 a^2 b^4 \left (c (2 B d+c C)-A \left (c^2+d^2\right )\right )+3 a b^5 B c^2+b^6 \left (c (c C-B d)-A \left (c^2-d^2\right )\right )\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right )^3 (b c-a d)^3}-\frac {d^2 \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)^3} \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])^3*(c + d*Tan[e + f*x])),x]

[Out]

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

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Mathematica [A]
time = 7.95, size = 898, normalized size = 1.88 \begin {gather*} -\frac {A b^2-a (b B-a C)}{2 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^2}-\frac {-\frac {-\frac {b (b c-a d)^2 \left (3 a^2 A b c-A b^3 c-a^3 B c+3 a b^2 B c-3 a^2 b c C+b^3 c C+a^3 A d-3 a A b^2 d+3 a^2 b B d-b^3 B d-a^3 C d+3 a b^2 C d+\frac {\sqrt {-b^2} \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 )}{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 (3 a b^5 B c^2-3 a^5 b B d^2+a^6 C d^2+3 a^4 b^2 d (B c+2 A d-C d)+b^6 \left (c (c C-B d)-A \left (c^2-d^2\right )\right )-a^3 b^3 \left (8 c (A-C) d+B \left (c^2-d^2\right )\right )-3 a^2 b^4 \left (c (c C+2 B d)-A \left (c^2+d^2\right )\right )\right ) \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 (3 a^2 A b c-A b^3 c-a^3 B c+3 a b^2 B c-3 a^2 b c C+b^3 c C+a^3 A d-3 a A b^2 d+3 a^2 b B d-b^3 B d-a^3 C d+3 a b^2 C d-\frac {\sqrt {-b^2} \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 )}{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 (a^2+b^2\right )^2 d^2 \left (c^2 C-B c d+A d^2\right ) \log (c+d \tan (e+f x))}{(b c-a d) \left (c^2+d^2\right )}}{b \left (a^2+b^2\right ) (b c-a d) f}-\frac {-a \left (-2 a \left (A b^2-a (b B-a C)\right ) d+2 b (A b-a B-b C) (b c-a d)\right )-2 b^2 \left (a b c (A-C)-a^2 A d+b^2 (B c-A d)\right )}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))}}{2 \left (a^2+b^2\right ) (b c-a d)} \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])^3*(c + d*Tan[e + f*x])),x]

[Out]

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

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Maple [A]
time = 1.60, size = 647, normalized size = 1.36 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))^3/(c+d*tan(f*x+e)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1103 vs. \(2 (481) = 962\).
time = 0.63, size = 1103, normalized size = 2.31 \begin {gather*} \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 )}}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} c^{2} + {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} d^{2}} - \frac {2 \, {\left ({\left (B a^{3} b^{3} - 3 \, {\left (A - C\right )} a^{2} b^{4} - 3 \, B a b^{5} + {\left (A - C\right )} b^{6}\right )} c^{2} - {\left (3 \, B a^{4} b^{2} - 8 \, {\left (A - C\right )} a^{3} b^{3} - 6 \, B a^{2} b^{4} - B b^{6}\right )} c d - {\left (C a^{6} - 3 \, B a^{5} b + 3 \, {\left (2 \, A - C\right )} a^{4} b^{2} + B a^{3} b^{3} + 3 \, A a^{2} b^{4} + A b^{6}\right )} d^{2}\right )} \log \left (b \tan \left (f x + e\right ) + a\right )}{{\left (a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}\right )} c^{3} - 3 \, {\left (a^{7} b^{2} + 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} + a b^{8}\right )} c^{2} d + 3 \, {\left (a^{8} b + 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} + a^{2} b^{7}\right )} c d^{2} - {\left (a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6}\right )} d^{3}} - \frac {2 \, {\left (C c^{2} d^{2} - B c d^{3} + A d^{4}\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c d^{4} - a^{3} d^{5} + {\left (3 \, a^{2} b + b^{3}\right )} c^{3} d^{2} - {\left (a^{3} + 3 \, a b^{2}\right )} c^{2} d^{3}} + \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 )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} c^{2} + {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} d^{2}} - \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 (3 \, C a^{5} - 5 \, B a^{4} b + {\left (7 \, A - C\right )} a^{3} b^{2} - B a^{2} b^{3} + 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 - 2 \, B a^{3} b^{2} + {\left (3 \, A - C\right )} a^{2} b^{3} + A b^{5}\right )} d\right )} \tan \left (f x + e\right )}{{\left (a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6}\right )} c^{2} - 2 \, {\left (a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} c d + {\left (a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} d^{2} + {\left ({\left (a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8}\right )} c^{2} - 2 \, {\left (a^{5} b^{3} + 2 \, a^{3} b^{5} + a b^{7}\right )} c d + {\left (a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6}\right )} d^{2}\right )} \tan \left (f x + e\right )^{2} + 2 \, {\left ({\left (a^{5} b^{3} + 2 \, a^{3} b^{5} + a b^{7}\right )} c^{2} - 2 \, {\left (a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6}\right )} c d + {\left (a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} d^{2}\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))^3/(c+d*tan(f*x+e)),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)*c^2 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2) - 2*((B
*a^3*b^3 - 3*(A - C)*a^2*b^4 - 3*B*a*b^5 + (A - C)*b^6)*c^2 - (3*B*a^4*b^2 - 8*(A - C)*a^3*b^3 - 6*B*a^2*b^4 -
B*b^6)*c*d - (C*a^6 - 3*B*a^5*b + 3*(2*A - C)*a^4*b^2 + B*a^3*b^3 + 3*A*a^2*b^4 + A*b^6)*d^2)*log(b*tan(f*x +
e) + a)/((a^6*b^3 + 3*a^4*b^5 + 3*a^2*b^7 + b^9)*c^3 - 3*(a^7*b^2 + 3*a^5*b^4 + 3*a^3*b^6 + a*b^8)*c^2*d + 3*
(a^8*b + 3*a^6*b^3 + 3*a^4*b^5 + a^2*b^7)*c*d^2 - (a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a^3*b^6)*d^3) - 2*(C*c^2*d^2
- B*c*d^3 + A*d^4)*log(d*tan(f*x + e) + c)/(b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c*d^4 - a^3*d^5 + (3*a^2*b + b^3
)*c^3*d^2 - (a^3 + 3*a*b^2)*c^2*d^3) + ((B*a^3 - 3*(A - C)*a^2*b - 3*B*a*b^2 + (A - C)*b^3)*c - ((A - C)*a^3 +
3*B*a^2*b - 3*(A - C)*a*b^2 - B*b^3)*d)*log(tan(f*x + e)^2 + 1)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c^2 + (a
^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2) - ((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 -
(3*C*a^5 - 5*B*a^4*b + (7*A - C)*a^3*b^2 - B*a^2*b^3 + 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 - 2*B*a^3*b^2 + (3*A - C)*a^2*b^3 + A*b^5)*d)*tan(f*x + e))/((a^6*b^2 + 2*a^4*b^4 + a^2*b^6)*c^2
- 2*(a^7*b + 2*a^5*b^3 + a^3*b^5)*c*d + (a^8 + 2*a^6*b^2 + a^4*b^4)*d^2 + ((a^4*b^4 + 2*a^2*b^6 + b^8)*c^2 -
2*(a^5*b^3 + 2*a^3*b^5 + a*b^7)*c*d + (a^6*b^2 + 2*a^4*b^4 + a^2*b^6)*d^2)*tan(f*x + e)^2 + 2*((a^5*b^3 + 2*a^
3*b^5 + a*b^7)*c^2 - 2*(a^6*b^2 + 2*a^4*b^4 + a^2*b^6)*c*d + (a^7*b + 2*a^5*b^3 + a^3*b^5)*d^2)*tan(f*x + e)))
/f

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3657 vs. \(2 (481) = 962\).
time = 12.07, size = 3657, normalized size = 7.67 \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))^3/(c+d*tan(f*x+e)),x, algorithm="fricas")

[Out]

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

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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))**3/(c+d*tan(f*x+e)),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. 2127 vs. \(2 (481) = 962\).
time = 1.22, size = 2127, normalized size = 4.46 \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))^3/(c+d*tan(f*x+e)),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*c^2 + 3*a^4*b^2*c^2 + 3*a^2*b^4*c^2 + b^6*c^2 + a^6*d^
2 + 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 + b^6*d^2) + (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*
c^2 + 3*a^4*b^2*c^2 + 3*a^2*b^4*c^2 + b^6*c^2 + a^6*d^2 + 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 + b^6*d^2) - 2*(B*a^3*
b^4*c^2 - 3*A*a^2*b^5*c^2 + 3*C*a^2*b^5*c^2 - 3*B*a*b^6*c^2 + A*b^7*c^2 - C*b^7*c^2 - 3*B*a^4*b^3*c*d + 8*A*a^
3*b^4*c*d - 8*C*a^3*b^4*c*d + 6*B*a^2*b^5*c*d + B*b^7*c*d - C*a^6*b*d^2 + 3*B*a^5*b^2*d^2 - 6*A*a^4*b^3*d^2 +
3*C*a^4*b^3*d^2 - B*a^3*b^4*d^2 - 3*A*a^2*b^5*d^2 - A*b^7*d^2)*log(abs(b*tan(f*x + e) + a))/(a^6*b^4*c^3 + 3*a
^4*b^6*c^3 + 3*a^2*b^8*c^3 + b^10*c^3 - 3*a^7*b^3*c^2*d - 9*a^5*b^5*c^2*d - 9*a^3*b^7*c^2*d - 3*a*b^9*c^2*d +
3*a^8*b^2*c*d^2 + 9*a^6*b^4*c*d^2 + 9*a^4*b^6*c*d^2 + 3*a^2*b^8*c*d^2 - a^9*b*d^3 - 3*a^7*b^3*d^3 - 3*a^5*b^5*
d^3 - a^3*b^7*d^3) - 2*(C*c^2*d^3 - B*c*d^4 + A*d^5)*log(abs(d*tan(f*x + e) + c))/(b^3*c^5*d - 3*a*b^2*c^4*d^2
+ 3*a^2*b*c^3*d^3 + b^3*c^3*d^3 - a^3*c^2*d^4 - 3*a*b^2*c^2*d^4 + 3*a^2*b*c*d^5 - a^3*d^6) + (3*B*a^3*b^5*c^2
*tan(f*x + e)^2 - 9*A*a^2*b^6*c^2*tan(f*x + e)^2 + 9*C*a^2*b^6*c^2*tan(f*x + e)^2 - 9*B*a*b^7*c^2*tan(f*x + e)
^2 + 3*A*b^8*c^2*tan(f*x + e)^2 - 3*C*b^8*c^2*tan(f*x + e)^2 - 9*B*a^4*b^4*c*d*tan(f*x + e)^2 + 24*A*a^3*b^5*c
*d*tan(f*x + e)^2 - 24*C*a^3*b^5*c*d*tan(f*x + e)^2 + 18*B*a^2*b^6*c*d*tan(f*x + e)^2 + 3*B*b^8*c*d*tan(f*x +
e)^2 - 3*C*a^6*b^2*d^2*tan(f*x + e)^2 + 9*B*a^5*b^3*d^2*tan(f*x + e)^2 - 18*A*a^4*b^4*d^2*tan(f*x + e)^2 + 9*C
*a^4*b^4*d^2*tan(f*x + e)^2 - 3*B*a^3*b^5*d^2*tan(f*x + e)^2 - 9*A*a^2*b^6*d^2*tan(f*x + e)^2 - 3*A*b^8*d^2*ta
n(f*x + e)^2 + 8*B*a^4*b^4*c^2*tan(f*x + e) - 22*A*a^3*b^5*c^2*tan(f*x + e) + 22*C*a^3*b^5*c^2*tan(f*x + e) -
18*B*a^2*b^6*c^2*tan(f*x + e) + 2*A*a*b^7*c^2*tan(f*x + e) - 2*C*a*b^7*c^2*tan(f*x + e) - 2*B*b^8*c^2*tan(f*x
+ e) + 2*C*a^6*b^2*c*d*tan(f*x + e) - 24*B*a^5*b^3*c*d*tan(f*x + e) + 58*A*a^4*b^4*c*d*tan(f*x + e) - 52*C*a^4
*b^4*c*d*tan(f*x + e) + 32*B*a^3*b^5*c*d*tan(f*x + e) + 12*A*a^2*b^6*c*d*tan(f*x + e) - 6*C*a^2*b^6*c*d*tan(f*
x + e) + 8*B*a*b^7*c*d*tan(f*x + e) + 2*A*b^8*c*d*tan(f*x + e) - 8*C*a^7*b*d^2*tan(f*x + e) + 22*B*a^6*b^2*d^2
*tan(f*x + e) - 42*A*a^5*b^3*d^2*tan(f*x + e) + 18*C*a^5*b^3*d^2*tan(f*x + e) - 2*B*a^4*b^4*d^2*tan(f*x + e) -
26*A*a^3*b^5*d^2*tan(f*x + e) + 2*C*a^3*b^5*d^2*tan(f*x + e) - 8*A*a*b^7*d^2*tan(f*x + e) - C*a^6*b^2*c^2 + 6
*B*a^5*b^3*c^2 - 14*A*a^4*b^4*c^2 + 11*C*a^4*b^4*c^2 - 7*B*a^3*b^5*c^2 - 3*A*a^2*b^6*c^2 - B*a*b^7*c^2 - A*b^8
*c^2 + 4*C*a^7*b*c*d - 17*B*a^6*b^2*c*d + 36*A*a^5*b^3*c*d - 24*C*a^5*b^3*c*d + 10*B*a^4*b^4*c*d + 16*A*a^3*b^
5*c*d - 4*C*a^3*b^5*c*d + 3*B*a^2*b^6*c*d + 4*A*a*b^7*c*d - 6*C*a^8*d^2 + 14*B*a^7*b*d^2 - 25*A*a^6*b^2*d^2 +
7*C*a^6*b^2*d^2 + 3*B*a^5*b^3*d^2 - 19*A*a^4*b^4*d^2 + C*a^4*b^4*d^2 + B*a^3*b^5*d^2 - 6*A*a^2*b^6*d^2)/((a^6*
b^3*c^3 + 3*a^4*b^5*c^3 + 3*a^2*b^7*c^3 + b^9*c^3 - 3*a^7*b^2*c^2*d - 9*a^5*b^4*c^2*d - 9*a^3*b^6*c^2*d - 3*a*
b^8*c^2*d + 3*a^8*b*c*d^2 + 9*a^6*b^3*c*d^2 + 9*a^4*b^5*c*d^2 + 3*a^2*b^7*c*d^2 - a^9*d^3 - 3*a^7*b^2*d^3 - 3*
a^5*b^4*d^3 - a^3*b^6*d^3)*(b*tan(f*x + e) + a)^2))/f

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Mupad [B]
time = 24.03, size = 2500, normalized size = 5.24 \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))^3*(c + d*tan(e + f*x))),x)

[Out]

-(((A*b^5*c - 3*C*a^5*d - 3*A*a*b^4*d + B*a*b^4*c + 5*B*a^4*b*d + C*a^4*b*c + 5*A*a^2*b^3*c - 7*A*a^3*b^2*d -
3*B*a^3*b^2*c + B*a^2*b^3*d - 3*C*a^2*b^3*c + C*a^3*b^2*d)/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(a^4 + b^4 + 2*a
^2*b^2)) - (tan(e + f*x)*(A*b^5*d - B*b^5*c - 2*A*a*b^4*c + 2*C*a*b^4*c + C*a^4*b*d + 3*A*a^2*b^3*d + B*a^2*b^
3*c - 2*B*a^3*b^2*d - C*a^2*b^3*d))/((a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(a^4 + b^4 + 2*a^2*b^2)))/(a^2 + b^2*tan(
e + f*x)^2 + 2*a*b*tan(e + f*x)) - symsum(log(- (A^3*b^8*c^2*d^4 - 4*A^3*a^2*b^6*d^6 - 7*A^3*a^4*b^4*d^6 - A^3
*b^8*d^6 + A^2*C*b^8*d^6 - 3*A^3*a^2*b^6*c^2*d^4 - B^3*a^3*b^5*c^2*d^4 - C^3*a^2*b^6*c^2*d^4 - 2*C^3*a^3*b^5*c
^3*d^3 + 7*C^3*a^4*b^4*c^2*d^4 + A^2*B*a*b^7*d^6 + A^2*B*b^8*c*d^5 + A^3*a*b^7*c*d^5 + C^3*a^7*b*c*d^5 - A*B^2
*a^2*b^6*d^6 - 3*A*B^2*a^6*b^2*d^6 + 2*A^2*B*a^3*b^5*d^6 + 9*A^2*B*a^5*b^3*d^6 - A*C^2*a^2*b^6*d^6 - 4*A*C^2*a
^4*b^4*d^6 + A*C^2*a^6*b^2*d^6 + 5*A^2*C*a^2*b^6*d^6 + 11*A^2*C*a^4*b^4*d^6 - A^2*C*a^6*b^2*d^6 + A*C^2*b^8*c^
2*d^4 - 2*A^2*C*b^8*c^2*d^4 - B*C^2*b^8*c^3*d^3 + B^2*C*b^8*c^2*d^4 + 9*A^3*a^3*b^5*c*d^5 - B^3*a*b^7*c^2*d^4
+ B^3*a^2*b^6*c*d^5 + B^3*a^4*b^4*c*d^5 + 2*C^3*a*b^7*c^3*d^3 - 3*C^3*a^5*b^3*c*d^5 + A*B*C*a^7*b*d^6 - 2*A*B*
C*b^8*c*d^5 + 3*A*B^2*a^2*b^6*c^2*d^4 - A*B^2*a^4*b^4*c^2*d^4 + 3*A^2*B*a^3*b^5*c^2*d^4 - A*C^2*a^2*b^6*c^2*d^
4 + 4*A*C^2*a^3*b^5*c^3*d^3 - 14*A*C^2*a^4*b^4*c^2*d^4 + 5*A^2*C*a^2*b^6*c^2*d^4 - 2*A^2*C*a^3*b^5*c^3*d^3 + 7
*A^2*C*a^4*b^4*c^2*d^4 + 6*B*C^2*a^2*b^6*c^3*d^3 - 15*B*C^2*a^3*b^5*c^2*d^4 - B*C^2*a^4*b^4*c^3*d^3 + 3*B*C^2*
a^5*b^3*c^2*d^4 + 5*B^2*C*a^2*b^6*c^2*d^4 + 2*B^2*C*a^3*b^5*c^3*d^3 - 4*B^2*C*a^4*b^4*c^2*d^4 + A*B*C*a^3*b^5*
d^6 - 6*A*B*C*a^5*b^3*d^6 + A*B*C*b^8*c^3*d^3 + 2*A*C^2*a*b^7*c*d^5 - A*C^2*a^7*b*c*d^5 - 3*A^2*C*a*b^7*c*d^5
- 5*A*B^2*a^3*b^5*c*d^5 + 3*A*B^2*a^5*b^3*c*d^5 - 5*A^2*B*a*b^7*c^2*d^4 + 7*A^2*B*a^2*b^6*c*d^5 - 10*A^2*B*a^4
*b^4*c*d^5 - 4*A*C^2*a*b^7*c^3*d^3 + 12*A*C^2*a^3*b^5*c*d^5 + 9*A*C^2*a^5*b^3*c*d^5 + 2*A^2*C*a*b^7*c^3*d^3 -
21*A^2*C*a^3*b^5*c*d^5 - 6*A^2*C*a^5*b^3*c*d^5 - 2*B*C^2*a*b^7*c^2*d^4 + B*C^2*a^2*b^6*c*d^5 + 5*B*C^2*a^4*b^4
*c*d^5 - 4*B*C^2*a^6*b^2*c*d^5 - 2*B^2*C*a*b^7*c^3*d^3 - B^2*C*a^3*b^5*c*d^5 + 3*B^2*C*a^5*b^3*c*d^5 - 6*A*B*C
*a^2*b^6*c^3*d^3 + 12*A*B*C*a^3*b^5*c^2*d^4 + A*B*C*a^4*b^4*c^3*d^3 - 3*A*B*C*a^5*b^3*c^2*d^4 + 7*A*B*C*a*b^7*
c^2*d^4 - 11*A*B*C*a^2*b^6*c*d^5 + 2*A*B*C*a^4*b^4*c*d^5 + 3*A*B*C*a^6*b^2*c*d^5)/(a^12*d^4 + b^12*c^4 + 4*a^2
*b^10*c^4 + 6*a^4*b^8*c^4 + 4*a^6*b^6*c^4 + a^8*b^4*c^4 + a^4*b^8*d^4 + 4*a^6*b^6*d^4 + 6*a^8*b^4*d^4 + 4*a^10
*b^2*d^4 - 4*a^3*b^9*c*d^3 - 16*a^3*b^9*c^3*d - 16*a^5*b^7*c*d^3 - 24*a^5*b^7*c^3*d - 24*a^7*b^5*c*d^3 - 16*a^
7*b^5*c^3*d - 16*a^9*b^3*c*d^3 - 4*a^9*b^3*c^3*d + 6*a^2*b^10*c^2*d^2 + 24*a^4*b^8*c^2*d^2 + 36*a^6*b^6*c^2*d^
2 + 24*a^8*b^4*c^2*d^2 + 6*a^10*b^2*c^2*d^2 - 4*a*b^11*c^3*d - 4*a^11*b*c*d^3) - root(480*a^11*b^7*c*d^9*f^4 +
480*a^7*b^11*c^9*d*f^4 + 360*a^13*b^5*c*d^9*f^4 + 360*a^9*b^9*c^9*d*f^4 + 360*a^9*b^9*c*d^9*f^4 + 360*a^5*b^1
3*c^9*d*f^4 + 144*a^15*b^3*c*d^9*f^4 + 144*a^11*b^7*c^9*d*f^4 + 144*a^7*b^11*c*d^9*f^4 + 144*a^3*b^15*c^9*d*f^
4 + 48*a^17*b*c^3*d^7*f^4 + 48*a*b^17*c^7*d^3*f^4 + 24*a^17*b*c^5*d^5*f^4 + 24*a^13*b^5*c^9*d*f^4 + 24*a^5*b^1
3*c*d^9*f^4 + 24*a*b^17*c^5*d^5*f^4 + 24*a^17*b*c*d^9*f^4 + 24*a*b^17*c^9*d*f^4 + 3920*a^9*b^9*c^5*d^5*f^4 - 3
360*a^10*b^8*c^4*d^6*f^4 - 3360*a^8*b^10*c^6*d^4*f^4 + 3024*a^11*b^7*c^5*d^5*f^4 - 3024*a^10*b^8*c^6*d^4*f^4 -
3024*a^8*b^10*c^4*d^6*f^4 + 3024*a^7*b^11*c^5*d^5*f^4 + 2320*a^9*b^9*c^7*d^3*f^4 + 2320*a^9*b^9*c^3*d^7*f^4 -
2240*a^12*b^6*c^4*d^6*f^4 - 2240*a^6*b^12*c^6*d^4*f^4 + 2160*a^11*b^7*c^3*d^7*f^4 + 2160*a^7*b^11*c^7*d^3*f^4
- 1624*a^12*b^6*c^6*d^4*f^4 - 1624*a^6*b^12*c^4*d^6*f^4 + 1488*a^11*b^7*c^7*d^3*f^4 + 1488*a^7*b^11*c^3*d^7*f
^4 + 1344*a^13*b^5*c^5*d^5*f^4 + 1344*a^5*b^13*c^5*d^5*f^4 - 1320*a^10*b^8*c^2*d^8*f^4 - 1320*a^8*b^10*c^8*d^2
*f^4 + 1200*a^13*b^5*c^3*d^7*f^4 + 1200*a^5*b^13*c^7*d^3*f^4 - 1060*a^12*b^6*c^2*d^8*f^4 - 1060*a^6*b^12*c^8*d
^2*f^4 - 948*a^10*b^8*c^8*d^2*f^4 - 948*a^8*b^10*c^2*d^8*f^4 - 840*a^14*b^4*c^4*d^6*f^4 - 840*a^4*b^14*c^6*d^4
*f^4 + 528*a^13*b^5*c^7*d^3*f^4 + 528*a^5*b^13*c^3*d^7*f^4 - 480*a^14*b^4*c^6*d^4*f^4 - 480*a^14*b^4*c^2*d^8*f
^4 - 480*a^4*b^14*c^8*d^2*f^4 - 480*a^4*b^14*c^4*d^6*f^4 + 368*a^15*b^3*c^3*d^7*f^4 - 368*a^12*b^6*c^8*d^2*f^4
- 368*a^6*b^12*c^2*d^8*f^4 + 368*a^3*b^15*c^7*d^3*f^4 + 304*a^15*b^3*c^5*d^5*f^4 + 304*a^3*b^15*c^5*d^5*f^4 -
144*a^16*b^2*c^4*d^6*f^4 - 144*a^2*b^16*c^6*d^4*f^4 - 108*a^16*b^2*c^2*d^8*f^4 - 108*a^2*b^16*c^8*d^2*f^4 + 8
0*a^15*b^3*c^7*d^3*f^4 + 80*a^3*b^15*c^3*d^7*f^4 - 60*a^16*b^2*c^6*d^4*f^4 - 60*a^14*b^4*c^8*d^2*f^4 - 60*a^4*
b^14*c^2*d^8*f^4 - 60*a^2*b^16*c^4*d^6*f^4 - 8*b^18*c^8*d^2*f^4 - 4*b^18*c^6*d^4*f^4 - 8*a^18*c^2*d^8*f^4 - 4*
a^18*c^4*d^6*f^4 - 80*a^12*b^6*d^10*f^4 - 60*a^14*b^4*d^10*f^4 - 60*a^10*b^8*d^10*f^4 - 24*a^16*b^2*d^10*f^4 -
24*a^8*b^10*d^10*f^4 - 4*a^6*b^12*d^10*f^4 - 80*a^6*b^12*c^10*f^4 - 60*a^8*b^10*c^10*f^4 - 60*a^4*b^14*c^10*f
^4 - 24*a^10*b^8*c^10*f^4 - 24*a^2*b^16*c^10*f^...

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