3.13.22 \(\int \frac {1}{(a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2} \, dx\) [1222]

Optimal. Leaf size=457 \[ -\frac {\left (6 a^2 b c d-2 b^3 c d-a^3 \left (c^2-d^2\right )+3 a b^2 \left (c^2-d^2\right )\right ) x}{\left (a^2+b^2\right )^3 \left (c^2+d^2\right )^2}-\frac {b^3 \left (10 a^3 b c d+2 a b^3 c d-10 a^4 d^2+b^4 \left (c^2-3 d^2\right )-3 a^2 b^2 \left (c^2+3 d^2\right )\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{\left (a^2+b^2\right )^3 (b c-a d)^4 f}-\frac {d^4 \left (5 b c^2-2 a c d+3 b d^2\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{(b c-a d)^4 \left (c^2+d^2\right )^2 f}+\frac {d \left (a^4 d^3-2 a b^3 c \left (c^2+d^2\right )+2 a^2 b^2 d \left (2 c^2+3 d^2\right )+b^4 d \left (2 c^2+3 d^2\right )\right )}{\left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right ) f (c+d \tan (e+f x))}-\frac {b^2}{2 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^2 (c+d \tan (e+f x))}-\frac {b^2 \left (4 a b c-7 a^2 d-3 b^2 d\right )}{2 \left (a^2+b^2\right )^2 (b c-a d)^2 f (a+b \tan (e+f x)) (c+d \tan (e+f x))} \]

[Out]

-(6*a^2*b*c*d-2*b^3*c*d-a^3*(c^2-d^2)+3*a*b^2*(c^2-d^2))*x/(a^2+b^2)^3/(c^2+d^2)^2-b^3*(10*a^3*b*c*d+2*a*b^3*c
*d-10*a^4*d^2+b^4*(c^2-3*d^2)-3*a^2*b^2*(c^2+3*d^2))*ln(a*cos(f*x+e)+b*sin(f*x+e))/(a^2+b^2)^3/(-a*d+b*c)^4/f-
d^4*(-2*a*c*d+5*b*c^2+3*b*d^2)*ln(c*cos(f*x+e)+d*sin(f*x+e))/(-a*d+b*c)^4/(c^2+d^2)^2/f+d*(a^4*d^3-2*a*b^3*c*(
c^2+d^2)+2*a^2*b^2*d*(2*c^2+3*d^2)+b^4*d*(2*c^2+3*d^2))/(a^2+b^2)^2/(-a*d+b*c)^3/(c^2+d^2)/f/(c+d*tan(f*x+e))-
1/2*b^2/(a^2+b^2)/(-a*d+b*c)/f/(a+b*tan(f*x+e))^2/(c+d*tan(f*x+e))-1/2*b^2*(-7*a^2*d+4*a*b*c-3*b^2*d)/(a^2+b^2
)^2/(-a*d+b*c)^2/f/(a+b*tan(f*x+e))/(c+d*tan(f*x+e))

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Rubi [A]
time = 1.24, antiderivative size = 457, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3650, 3730, 3732, 3611} \begin {gather*} -\frac {b^2 \left (-7 a^2 d+4 a b c-3 b^2 d\right )}{2 f \left (a^2+b^2\right )^2 (b c-a d)^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))}-\frac {b^2}{2 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^2 (c+d \tan (e+f x))}+\frac {d \left (a^4 d^3+2 a^2 b^2 d \left (2 c^2+3 d^2\right )-2 a b^3 c \left (c^2+d^2\right )+b^4 d \left (2 c^2+3 d^2\right )\right )}{f \left (a^2+b^2\right )^2 \left (c^2+d^2\right ) (b c-a d)^3 (c+d \tan (e+f x))}-\frac {x \left (-\left (a^3 \left (c^2-d^2\right )\right )+6 a^2 b c d+3 a b^2 \left (c^2-d^2\right )-2 b^3 c d\right )}{\left (a^2+b^2\right )^3 \left (c^2+d^2\right )^2}-\frac {b^3 \left (-10 a^4 d^2+10 a^3 b c d-3 a^2 b^2 \left (c^2+3 d^2\right )+2 a b^3 c d+b^4 \left (c^2-3 d^2\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)^4}-\frac {d^4 \left (-2 a c d+5 b c^2+3 b d^2\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right )^2 (b c-a d)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^2),x]

[Out]

-(((6*a^2*b*c*d - 2*b^3*c*d - a^3*(c^2 - d^2) + 3*a*b^2*(c^2 - d^2))*x)/((a^2 + b^2)^3*(c^2 + d^2)^2)) - (b^3*
(10*a^3*b*c*d + 2*a*b^3*c*d - 10*a^4*d^2 + b^4*(c^2 - 3*d^2) - 3*a^2*b^2*(c^2 + 3*d^2))*Log[a*Cos[e + f*x] + b
*Sin[e + f*x]])/((a^2 + b^2)^3*(b*c - a*d)^4*f) - (d^4*(5*b*c^2 - 2*a*c*d + 3*b*d^2)*Log[c*Cos[e + f*x] + d*Si
n[e + f*x]])/((b*c - a*d)^4*(c^2 + d^2)^2*f) + (d*(a^4*d^3 - 2*a*b^3*c*(c^2 + d^2) + 2*a^2*b^2*d*(2*c^2 + 3*d^
2) + b^4*d*(2*c^2 + 3*d^2)))/((a^2 + b^2)^2*(b*c - a*d)^3*(c^2 + d^2)*f*(c + d*Tan[e + f*x])) - b^2/(2*(a^2 +
b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])) - (b^2*(4*a*b*c - 7*a^2*d - 3*b^2*d))/(2*(a^2
+ b^2)^2*(b*c - a*d)^2*f*(a + b*Tan[e + f*x])*(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 3650

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[b^2*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d))), x] + D
ist[1/((m + 1)*(a^2 + b^2)*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c -
a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && I
ntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] &&
NeQ[a, 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 {1}{(a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2} \, dx &=-\frac {b^2}{2 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^2 (c+d \tan (e+f x))}-\frac {\int \frac {-2 a b c+2 a^2 d+3 b^2 d+2 b (b c-a d) \tan (e+f x)+3 b^2 d \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx}{2 \left (a^2+b^2\right ) (b c-a d)}\\ &=-\frac {b^2}{2 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^2 (c+d \tan (e+f x))}-\frac {b^2 \left (4 a b c-7 a^2 d-3 b^2 d\right )}{2 \left (a^2+b^2\right )^2 (b c-a d)^2 f (a+b \tan (e+f x)) (c+d \tan (e+f x))}+\frac {\int \frac {-2 \left (2 a^3 b c d+2 a b^3 c d-a^4 d^2+b^4 \left (c^2-3 d^2\right )-a^2 b^2 \left (c^2+6 d^2\right )\right )-4 a b (b c-a d)^2 \tan (e+f x)-2 b^2 d \left (4 a b c-7 a^2 d-3 b^2 d\right ) \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^2} \, dx}{2 \left (a^2+b^2\right )^2 (b c-a d)^2}\\ &=\frac {d \left (a^4 d^3-2 a b^3 c \left (c^2+d^2\right )+2 a^2 b^2 d \left (2 c^2+3 d^2\right )+b^4 d \left (2 c^2+3 d^2\right )\right )}{\left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right ) f (c+d \tan (e+f x))}-\frac {b^2}{2 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^2 (c+d \tan (e+f x))}-\frac {b^2 \left (4 a b c-7 a^2 d-3 b^2 d\right )}{2 \left (a^2+b^2\right )^2 (b c-a d)^2 f (a+b \tan (e+f x)) (c+d \tan (e+f x))}+\frac {\int \frac {-2 \left (a^5 c d^3-3 a^4 b d^2 \left (c^2+d^2\right )+a b^4 c d \left (c^2+2 d^2\right )+a^3 b^2 c d \left (3 c^2+5 d^2\right )+b^5 \left (c^4-2 c^2 d^2-3 d^4\right )-a^2 b^3 \left (c^4+7 c^2 d^2+6 d^4\right )\right )-2 (b c-a d)^3 \left (2 a b c+a^2 d-b^2 d\right ) \tan (e+f x)+2 b d \left (a^4 d^3-2 a b^3 c \left (c^2+d^2\right )+2 a^2 b^2 d \left (2 c^2+3 d^2\right )+b^4 d \left (2 c^2+3 d^2\right )\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)^3 \left (c^2+d^2\right )}\\ &=-\frac {\left (6 a^2 b c d-2 b^3 c d-a^3 \left (c^2-d^2\right )+3 a b^2 \left (c^2-d^2\right )\right ) x}{\left (a^2+b^2\right )^3 \left (c^2+d^2\right )^2}+\frac {d \left (a^4 d^3-2 a b^3 c \left (c^2+d^2\right )+2 a^2 b^2 d \left (2 c^2+3 d^2\right )+b^4 d \left (2 c^2+3 d^2\right )\right )}{\left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right ) f (c+d \tan (e+f x))}-\frac {b^2}{2 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^2 (c+d \tan (e+f x))}-\frac {b^2 \left (4 a b c-7 a^2 d-3 b^2 d\right )}{2 \left (a^2+b^2\right )^2 (b c-a d)^2 f (a+b \tan (e+f x)) (c+d \tan (e+f x))}-\frac {\left (d^4 \left (5 b c^2-2 a c d+3 b d^2\right )\right ) \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{(b c-a d)^4 \left (c^2+d^2\right )^2}-\frac {\left (b^3 \left (10 a^3 b c d+2 a b^3 c d-10 a^4 d^2+b^4 \left (c^2-3 d^2\right )-3 a^2 b^2 \left (c^2+3 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)^4}\\ &=-\frac {\left (6 a^2 b c d-2 b^3 c d-a^3 \left (c^2-d^2\right )+3 a b^2 \left (c^2-d^2\right )\right ) x}{\left (a^2+b^2\right )^3 \left (c^2+d^2\right )^2}-\frac {b^3 \left (10 a^3 b c d+2 a b^3 c d-10 a^4 d^2+b^4 \left (c^2-3 d^2\right )-3 a^2 b^2 \left (c^2+3 d^2\right )\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{\left (a^2+b^2\right )^3 (b c-a d)^4 f}-\frac {d^4 \left (5 b c^2-2 a c d+3 b d^2\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{(b c-a d)^4 \left (c^2+d^2\right )^2 f}+\frac {d \left (a^4 d^3-2 a b^3 c \left (c^2+d^2\right )+2 a^2 b^2 d \left (2 c^2+3 d^2\right )+b^4 d \left (2 c^2+3 d^2\right )\right )}{\left (a^2+b^2\right )^2 (b c-a d)^3 \left (c^2+d^2\right ) f (c+d \tan (e+f x))}-\frac {b^2}{2 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^2 (c+d \tan (e+f x))}-\frac {b^2 \left (4 a b c-7 a^2 d-3 b^2 d\right )}{2 \left (a^2+b^2\right )^2 (b c-a d)^2 f (a+b \tan (e+f x)) (c+d \tan (e+f x))}\\ \end {align*}

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Mathematica [A]
time = 7.30, size = 840, normalized size = 1.84 \begin {gather*} -\frac {b^2}{2 \left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))^2 (c+d \tan (e+f x))}-\frac {-\frac {b^2 \left (-2 a b c+2 a^2 d+3 b^2 d\right )-a \left (-3 a b^2 d+2 b^2 (b c-a d)\right )}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))}-\frac {-\frac {-\frac {b (b c-a d)^3 \left (3 a^2 b c^2-b^3 c^2+2 a^3 c d-6 a b^2 c d-3 a^2 b d^2+b^3 d^2-\frac {\sqrt {-b^2} \left (6 a^2 b c d-2 b^3 c d-a^3 \left (c^2-d^2\right )+3 a b^2 \left (c^2-d^2\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^4 \left (c^2+d^2\right ) \left (10 a^3 b c d+2 a b^3 c d-10 a^4 d^2+b^4 \left (c^2-3 d^2\right )-3 a^2 b^2 \left (c^2+3 d^2\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)^3 \left (3 a^2 b c^2-b^3 c^2+2 a^3 c d-6 a b^2 c d-3 a^2 b d^2+b^3 d^2+\frac {\sqrt {-b^2} \left (6 a^2 b c d-2 b^3 c d-a^3 \left (c^2-d^2\right )+3 a b^2 \left (c^2-d^2\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 (a^2+b^2\right )^2 d^4 \left (5 b c^2-2 a c d+3 b d^2\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 {-c \left (-4 a b d (b c-a d)^2+2 b^2 c d \left (4 a b c-7 a^2 d-3 b^2 d\right )\right )-2 d^2 \left (2 a^3 b c d+2 a b^3 c d-a^4 d^2+b^4 \left (c^2-3 d^2\right )-a^2 b^2 \left (c^2+6 d^2\right )\right )}{(-b c+a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))}}{\left (a^2+b^2\right ) (b c-a d)}}{2 \left (a^2+b^2\right ) (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^2),x]

[Out]

-1/2*b^2/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])) - (-((b^2*(-2*a*b*c + 2*a^2*d
+ 3*b^2*d) - a*(-3*a*b^2*d + 2*b^2*(b*c - a*d)))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])*(c + d*Tan[e
+ f*x]))) - (-((-((b*(b*c - a*d)^3*(3*a^2*b*c^2 - b^3*c^2 + 2*a^3*c*d - 6*a*b^2*c*d - 3*a^2*b*d^2 + b^3*d^2 -
(Sqrt[-b^2]*(6*a^2*b*c*d - 2*b^3*c*d - a^3*(c^2 - d^2) + 3*a*b^2*(c^2 - d^2)))/b)*Log[Sqrt[-b^2] - b*Tan[e +
f*x]])/((a^2 + b^2)*(c^2 + d^2))) - (2*b^4*(c^2 + d^2)*(10*a^3*b*c*d + 2*a*b^3*c*d - 10*a^4*d^2 + b^4*(c^2 - 3
*d^2) - 3*a^2*b^2*(c^2 + 3*d^2))*Log[a + b*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)) - (b*(b*c - a*d)^3*(3*a^2*
b*c^2 - b^3*c^2 + 2*a^3*c*d - 6*a*b^2*c*d - 3*a^2*b*d^2 + b^3*d^2 + (Sqrt[-b^2]*(6*a^2*b*c*d - 2*b^3*c*d - a^3
*(c^2 - d^2) + 3*a*b^2*(c^2 - d^2)))/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^4*(5*b*c^2 - 2*a*c*d + 3*b*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)) - (-(c*(-4*a*b*d*(b*c - a*d)^2 + 2*b^2*c*d*(4*a*b*c - 7*a^2*d - 3*b^2*d))) - 2*d^2*(2*a^3
*b*c*d + 2*a*b^3*c*d - a^4*d^2 + b^4*(c^2 - 3*d^2) - a^2*b^2*(c^2 + 6*d^2)))/((-(b*c) + a*d)*(c^2 + d^2)*f*(c
+ d*Tan[e + f*x])))/((a^2 + b^2)*(b*c - a*d)))/(2*(a^2 + b^2)*(b*c - a*d))

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Maple [A]
time = 1.39, size = 419, normalized size = 0.92

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (-2 a^{3} c d -3 a^{2} b \,c^{2}+3 a^{2} b \,d^{2}+6 a \,b^{2} c d +b^{3} c^{2}-b^{3} d^{2}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (a^{3} c^{2}-a^{3} d^{2}-6 a^{2} b c d -3 a \,b^{2} c^{2}+3 a \,b^{2} d^{2}+2 b^{3} c d \right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right )^{3} \left (c^{2}+d^{2}\right )^{2}}-\frac {b^{3}}{2 \left (a d -b c \right )^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (f x +e \right )\right )^{2}}+\frac {b^{3} \left (10 a^{4} d^{2}-10 a^{3} b c d +3 a^{2} b^{2} c^{2}+9 a^{2} b^{2} d^{2}-2 a \,b^{3} c d -b^{4} c^{2}+3 b^{4} d^{2}\right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a d -b c \right )^{4} \left (a^{2}+b^{2}\right )^{3}}-\frac {2 b^{3} \left (2 a^{2} d -a b c +b^{2} d \right )}{\left (a d -b c \right )^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (f x +e \right )\right )}-\frac {d^{4}}{\left (c^{2}+d^{2}\right ) \left (a d -b c \right )^{3} \left (c +d \tan \left (f x +e \right )\right )}+\frac {d^{4} \left (2 a c d -5 b \,c^{2}-3 b \,d^{2}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{2} \left (a d -b c \right )^{4}}}{f}\)	\(419\)
default	\(\frac {\frac {\frac {\left (-2 a^{3} c d -3 a^{2} b \,c^{2}+3 a^{2} b \,d^{2}+6 a \,b^{2} c d +b^{3} c^{2}-b^{3} d^{2}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (a^{3} c^{2}-a^{3} d^{2}-6 a^{2} b c d -3 a \,b^{2} c^{2}+3 a \,b^{2} d^{2}+2 b^{3} c d \right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right )^{3} \left (c^{2}+d^{2}\right )^{2}}-\frac {b^{3}}{2 \left (a d -b c \right )^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (f x +e \right )\right )^{2}}+\frac {b^{3} \left (10 a^{4} d^{2}-10 a^{3} b c d +3 a^{2} b^{2} c^{2}+9 a^{2} b^{2} d^{2}-2 a \,b^{3} c d -b^{4} c^{2}+3 b^{4} d^{2}\right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a d -b c \right )^{4} \left (a^{2}+b^{2}\right )^{3}}-\frac {2 b^{3} \left (2 a^{2} d -a b c +b^{2} d \right )}{\left (a d -b c \right )^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (f x +e \right )\right )}-\frac {d^{4}}{\left (c^{2}+d^{2}\right ) \left (a d -b c \right )^{3} \left (c +d \tan \left (f x +e \right )\right )}+\frac {d^{4} \left (2 a c d -5 b \,c^{2}-3 b \,d^{2}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (c^{2}+d^{2}\right )^{2} \left (a d -b c \right )^{4}}}{f}\)	\(419\)
norman	\(\text {Expression too large to display}\)	\(1676\)
risch	\(\text {Expression too large to display}\)	\(8456\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*tan(f*x+e))^3/(c+d*tan(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

1/f*(1/(a^2+b^2)^3/(c^2+d^2)^2*(1/2*(-2*a^3*c*d-3*a^2*b*c^2+3*a^2*b*d^2+6*a*b^2*c*d+b^3*c^2-b^3*d^2)*ln(1+tan(
f*x+e)^2)+(a^3*c^2-a^3*d^2-6*a^2*b*c*d-3*a*b^2*c^2+3*a*b^2*d^2+2*b^3*c*d)*arctan(tan(f*x+e)))-1/2*b^3/(a*d-b*c
)^2/(a^2+b^2)/(a+b*tan(f*x+e))^2+b^3*(10*a^4*d^2-10*a^3*b*c*d+3*a^2*b^2*c^2+9*a^2*b^2*d^2-2*a*b^3*c*d-b^4*c^2+
3*b^4*d^2)/(a*d-b*c)^4/(a^2+b^2)^3*ln(a+b*tan(f*x+e))-2*b^3*(2*a^2*d-a*b*c+b^2*d)/(a*d-b*c)^3/(a^2+b^2)^2/(a+b
*tan(f*x+e))-d^4/(c^2+d^2)/(a*d-b*c)^3/(c+d*tan(f*x+e))+d^4*(2*a*c*d-5*b*c^2-3*b*d^2)/(c^2+d^2)^2/(a*d-b*c)^4*
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. 1842 vs. \(2 (462) = 924\).
time = 0.69, size = 1842, normalized size = 4.03 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(f*x+e))^3/(c+d*tan(f*x+e))^2,x, algorithm="maxima")

[Out]

1/2*(2*((a^3 - 3*a*b^2)*c^2 - 2*(3*a^2*b - b^3)*c*d - (a^3 - 3*a*b^2)*d^2)*(f*x + e)/((a^6 + 3*a^4*b^2 + 3*a^2
*b^4 + b^6)*c^4 + 2*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c^2*d^2 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4) + 2
*((3*a^2*b^5 - b^7)*c^2 - 2*(5*a^3*b^4 + a*b^6)*c*d + (10*a^4*b^3 + 9*a^2*b^5 + 3*b^7)*d^2)*log(b*tan(f*x + e)
+ a)/((a^6*b^4 + 3*a^4*b^6 + 3*a^2*b^8 + b^10)*c^4 - 4*(a^7*b^3 + 3*a^5*b^5 + 3*a^3*b^7 + a*b^9)*c^3*d + 6*(a
^8*b^2 + 3*a^6*b^4 + 3*a^4*b^6 + a^2*b^8)*c^2*d^2 - 4*(a^9*b + 3*a^7*b^3 + 3*a^5*b^5 + a^3*b^7)*c*d^3 + (a^10
+ 3*a^8*b^2 + 3*a^6*b^4 + a^4*b^6)*d^4) - 2*(5*b*c^2*d^4 - 2*a*c*d^5 + 3*b*d^6)*log(d*tan(f*x + e) + c)/(b^4*c
^8 - 4*a*b^3*c^7*d - 4*a^3*b*c*d^7 + a^4*d^8 + 2*(3*a^2*b^2 + b^4)*c^6*d^2 - 4*(a^3*b + 2*a*b^3)*c^5*d^3 + (a^
4 + 12*a^2*b^2 + b^4)*c^4*d^4 - 4*(2*a^3*b + a*b^3)*c^3*d^5 + 2*(a^4 + 3*a^2*b^2)*c^2*d^6) - ((3*a^2*b - b^3)*
c^2 + 2*(a^3 - 3*a*b^2)*c*d - (3*a^2*b - b^3)*d^2)*log(tan(f*x + e)^2 + 1)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6
)*c^4 + 2*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c^2*d^2 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4) - ((5*a^2*b^4
+ b^6)*c^4 - (9*a^3*b^3 + 5*a*b^5)*c^3*d + (5*a^2*b^4 + b^6)*c^2*d^2 - (9*a^3*b^3 + 5*a*b^5)*c*d^3 - 2*(a^6 +
2*a^4*b^2 + a^2*b^4)*d^4 + 2*(2*a*b^5*c^3*d + 2*a*b^5*c*d^3 - 2*(2*a^2*b^4 + b^6)*c^2*d^2 - (a^4*b^2 + 6*a^2*
b^4 + 3*b^6)*d^4)*tan(f*x + e)^2 + (4*a*b^5*c^4 - 3*(a^2*b^4 + b^6)*c^3*d - (9*a^3*b^3 + a*b^5)*c^2*d^2 - 3*(a
^2*b^4 + b^6)*c*d^3 - (4*a^5*b + 17*a^3*b^3 + 9*a*b^5)*d^4)*tan(f*x + e))/((a^6*b^3 + 2*a^4*b^5 + a^2*b^7)*c^6
- 3*(a^7*b^2 + 2*a^5*b^4 + a^3*b^6)*c^5*d + (3*a^8*b + 7*a^6*b^3 + 5*a^4*b^5 + a^2*b^7)*c^4*d^2 - (a^9 + 5*a^
7*b^2 + 7*a^5*b^4 + 3*a^3*b^6)*c^3*d^3 + 3*(a^8*b + 2*a^6*b^3 + a^4*b^5)*c^2*d^4 - (a^9 + 2*a^7*b^2 + a^5*b^4)
*c*d^5 + ((a^4*b^5 + 2*a^2*b^7 + b^9)*c^5*d - 3*(a^5*b^4 + 2*a^3*b^6 + a*b^8)*c^4*d^2 + (3*a^6*b^3 + 7*a^4*b^5
+ 5*a^2*b^7 + b^9)*c^3*d^3 - (a^7*b^2 + 5*a^5*b^4 + 7*a^3*b^6 + 3*a*b^8)*c^2*d^4 + 3*(a^6*b^3 + 2*a^4*b^5 + a
^2*b^7)*c*d^5 - (a^7*b^2 + 2*a^5*b^4 + a^3*b^6)*d^6)*tan(f*x + e)^3 + ((a^4*b^5 + 2*a^2*b^7 + b^9)*c^6 - (a^5*
b^4 + 2*a^3*b^6 + a*b^8)*c^5*d - (3*a^6*b^3 + 5*a^4*b^5 + a^2*b^7 - b^9)*c^4*d^2 + (5*a^7*b^2 + 9*a^5*b^4 + 3*
a^3*b^6 - a*b^8)*c^3*d^3 - (2*a^8*b + 7*a^6*b^3 + 8*a^4*b^5 + 3*a^2*b^7)*c^2*d^4 + 5*(a^7*b^2 + 2*a^5*b^4 + a^
3*b^6)*c*d^5 - 2*(a^8*b + 2*a^6*b^3 + a^4*b^5)*d^6)*tan(f*x + e)^2 + (2*(a^5*b^4 + 2*a^3*b^6 + a*b^8)*c^6 - 5*
(a^6*b^3 + 2*a^4*b^5 + a^2*b^7)*c^5*d + (3*a^7*b^2 + 8*a^5*b^4 + 7*a^3*b^6 + 2*a*b^8)*c^4*d^2 + (a^8*b - 3*a^6
*b^3 - 9*a^4*b^5 - 5*a^2*b^7)*c^3*d^3 - (a^9 - a^7*b^2 - 5*a^5*b^4 - 3*a^3*b^6)*c^2*d^4 + (a^8*b + 2*a^6*b^3 +
a^4*b^5)*c*d^5 - (a^9 + 2*a^7*b^2 + a^5*b^4)*d^6)*tan(f*x + e)))/f

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4767 vs. \(2 (462) = 924\).
time = 4.39, size = 4767, normalized size = 10.43 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(f*x+e))^3/(c+d*tan(f*x+e))^2,x, algorithm="fricas")

[Out]

-1/2*((7*a^2*b^7 + b^9)*c^7 - 6*(3*a^3*b^6 + a*b^8)*c^6*d + (11*a^4*b^5 + 19*a^2*b^7 + 2*b^9)*c^5*d^2 - 12*(3*
a^3*b^6 + a*b^8)*c^4*d^3 + (22*a^4*b^5 + 17*a^2*b^7 + b^9)*c^3*d^4 - 6*(3*a^3*b^6 + a*b^8)*c^2*d^5 - (2*a^8*b
+ 6*a^6*b^3 - 5*a^4*b^5 - 3*a^2*b^7)*c*d^6 + 2*(a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a^3*b^6)*d^7 - ((5*a^2*b^7 - b^9
)*c^6*d - 2*(7*a^3*b^6 + a*b^8)*c^5*d^2 + (9*a^4*b^5 + 13*a^2*b^7 - 2*b^9)*c^4*d^3 - 4*(7*a^3*b^6 + a*b^8)*c^3
*d^4 - (2*a^6*b^3 - 12*a^4*b^5 - 5*a^2*b^7 + 3*b^9)*c^2*d^5 + 2*(a^7*b^2 + 3*a^5*b^4 - 4*a^3*b^6)*c*d^6 + 3*(3
*a^4*b^5 + a^2*b^7)*d^7 + 2*((a^3*b^6 - 3*a*b^8)*c^6*d - 2*(2*a^4*b^5 - 3*a^2*b^7 - b^9)*c^5*d^2 + (6*a^5*b^4
+ 5*a^3*b^6 - 5*a*b^8)*c^4*d^3 - 4*(a^6*b^3 + 5*a^4*b^5)*c^3*d^4 + (a^7*b^2 + 15*a^5*b^4 + 10*a^3*b^6)*c^2*d^5
- 2*(a^6*b^3 + 5*a^4*b^5)*c*d^6 - (a^7*b^2 - 3*a^5*b^4)*d^7)*f*x)*tan(f*x + e)^3 - 2*((a^5*b^4 - 3*a^3*b^6)*c
^7 - 2*(2*a^6*b^3 - 3*a^4*b^5 - a^2*b^7)*c^6*d + (6*a^7*b^2 + 5*a^5*b^4 - 5*a^3*b^6)*c^5*d^2 - 4*(a^8*b + 5*a^
6*b^3)*c^4*d^3 + (a^9 + 15*a^7*b^2 + 10*a^5*b^4)*c^3*d^4 - 2*(a^8*b + 5*a^6*b^3)*c^2*d^5 - (a^9 - 3*a^7*b^2)*c
*d^6)*f*x - ((5*a^2*b^7 - b^9)*c^7 - 8*(a^3*b^6 + a*b^8)*c^6*d - (7*a^4*b^5 - 25*a^2*b^7 - 2*b^9)*c^5*d^2 + 2*
(5*a^5*b^4 - 11*a^3*b^6 - 10*a*b^8)*c^4*d^3 - 7*(2*a^4*b^5 - 5*a^2*b^7 - b^9)*c^3*d^4 - 4*(a^7*b^2 - 2*a^5*b^4
+ 8*a^3*b^6 + 5*a*b^8)*c^2*d^5 + (4*a^8*b + 14*a^6*b^3 + 11*a^4*b^5 + 25*a^2*b^7 + 6*b^9)*c*d^6 - 2*(a^7*b^2
- 2*a^5*b^4 + 6*a^3*b^6 + 3*a*b^8)*d^7 + 2*((a^3*b^6 - 3*a*b^8)*c^7 - 2*(a^4*b^5 - b^9)*c^6*d - (2*a^5*b^4 - 1
7*a^3*b^6 + a*b^8)*c^5*d^2 + 2*(4*a^6*b^3 - 5*a^4*b^5 - 5*a^2*b^7)*c^4*d^3 - (7*a^7*b^2 + 25*a^5*b^4 - 10*a^3*
b^6)*c^3*d^4 + 2*(a^8*b + 14*a^6*b^3 + 5*a^4*b^5)*c^2*d^5 - (5*a^7*b^2 + 17*a^5*b^4)*c*d^6 - 2*(a^8*b - 3*a^6*
b^3)*d^7)*f*x)*tan(f*x + e)^2 - ((3*a^4*b^5 - a^2*b^7)*c^7 - 2*(5*a^5*b^4 + a^3*b^6)*c^6*d + (10*a^6*b^3 + 15*
a^4*b^5 + a^2*b^7)*c^5*d^2 - 4*(5*a^5*b^4 + a^3*b^6)*c^4*d^3 + (20*a^6*b^3 + 21*a^4*b^5 + 5*a^2*b^7)*c^3*d^4 -
2*(5*a^5*b^4 + a^3*b^6)*c^2*d^5 + (10*a^6*b^3 + 9*a^4*b^5 + 3*a^2*b^7)*c*d^6 + ((3*a^2*b^7 - b^9)*c^6*d - 2*(
5*a^3*b^6 + a*b^8)*c^5*d^2 + (10*a^4*b^5 + 15*a^2*b^7 + b^9)*c^4*d^3 - 4*(5*a^3*b^6 + a*b^8)*c^3*d^4 + (20*a^4
*b^5 + 21*a^2*b^7 + 5*b^9)*c^2*d^5 - 2*(5*a^3*b^6 + a*b^8)*c*d^6 + (10*a^4*b^5 + 9*a^2*b^7 + 3*b^9)*d^7)*tan(f
*x + e)^3 + ((3*a^2*b^7 - b^9)*c^7 - 4*(a^3*b^6 + a*b^8)*c^6*d - (10*a^4*b^5 - 11*a^2*b^7 - b^9)*c^5*d^2 + 2*(
10*a^5*b^4 + 5*a^3*b^6 - a*b^8)*c^4*d^3 - (20*a^4*b^5 - 13*a^2*b^7 - 5*b^9)*c^3*d^4 + 8*(5*a^5*b^4 + 4*a^3*b^6
+ a*b^8)*c^2*d^5 - (10*a^4*b^5 - 5*a^2*b^7 - 3*b^9)*c*d^6 + 2*(10*a^5*b^4 + 9*a^3*b^6 + 3*a*b^8)*d^7)*tan(f*x
+ e)^2 + (2*(3*a^3*b^6 - a*b^8)*c^7 - (17*a^4*b^5 + 5*a^2*b^7)*c^6*d + 2*(5*a^5*b^4 + 14*a^3*b^6 + a*b^8)*c^5
*d^2 + (10*a^6*b^3 - 25*a^4*b^5 - 7*a^2*b^7)*c^4*d^3 + 2*(10*a^5*b^4 + 19*a^3*b^6 + 5*a*b^8)*c^3*d^4 + (20*a^6
*b^3 + a^4*b^5 + a^2*b^7)*c^2*d^5 + 2*(5*a^5*b^4 + 8*a^3*b^6 + 3*a*b^8)*c*d^6 + (10*a^6*b^3 + 9*a^4*b^5 + 3*a^
2*b^7)*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)) + (5*(a^8*
b + 3*a^6*b^3 + 3*a^4*b^5 + a^2*b^7)*c^3*d^4 - 2*(a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a^3*b^6)*c^2*d^5 + 3*(a^8*b +
3*a^6*b^3 + 3*a^4*b^5 + a^2*b^7)*c*d^6 + (5*(a^6*b^3 + 3*a^4*b^5 + 3*a^2*b^7 + b^9)*c^2*d^5 - 2*(a^7*b^2 + 3*a
^5*b^4 + 3*a^3*b^6 + a*b^8)*c*d^6 + 3*(a^6*b^3 + 3*a^4*b^5 + 3*a^2*b^7 + b^9)*d^7)*tan(f*x + e)^3 + (5*(a^6*b^
3 + 3*a^4*b^5 + 3*a^2*b^7 + b^9)*c^3*d^4 + 8*(a^7*b^2 + 3*a^5*b^4 + 3*a^3*b^6 + a*b^8)*c^2*d^5 - (4*a^8*b + 9*
a^6*b^3 + 3*a^4*b^5 - 5*a^2*b^7 - 3*b^9)*c*d^6 + 6*(a^7*b^2 + 3*a^5*b^4 + 3*a^3*b^6 + a*b^8)*d^7)*tan(f*x + e)
^2 + (10*(a^7*b^2 + 3*a^5*b^4 + 3*a^3*b^6 + a*b^8)*c^3*d^4 + (a^8*b + 3*a^6*b^3 + 3*a^4*b^5 + a^2*b^7)*c^2*d^5
- 2*(a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*c*d^6 + 3*(a^8*b + 3*a^6*b^3 + 3*a^4*b^5 + a^2*b^7)*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)) - (6*(a^3*b^6 - a*b^8)*c^7 - (
16*a^4*b^5 - 5*a^2*b^7 - 3*b^9)*c^6*d + 2*(5*a^5*b^4 + 12*a^3*b^6 - 5*a*b^8)*c^5*d^2 - (43*a^4*b^5 - 5*a^2*b^7
- 6*b^9)*c^4*d^3 + 2*(10*a^5*b^4 + 15*a^3*b^6 - a*b^8)*c^3*d^4 - (2*a^8*b + 6*a^6*b^3 + 44*a^4*b^5 + 7*a^2*b^
7 - 3*b^9)*c^2*d^5 + 2*(a^9 + 5*a^7*b^2 + 14*a^5*b^4 + 13*a^3*b^6 + 3*a*b^8)*c*d^6 - (4*a^8*b + 12*a^6*b^3 + 2
3*a^4*b^5 + 9*a^2*b^7)*d^7 + 2*(2*(a^4*b^5 - 3*a^2*b^7)*c^7 - (7*a^5*b^4 - 9*a^3*b^6 - 4*a*b^8)*c^6*d + 8*(a^6
*b^3 + 2*a^4*b^5 - a^2*b^7)*c^5*d^2 - (2*a^7*b^2 + 35*a^5*b^4 + 5*a^3*b^6)*c^4*d^3 - 2*(a^8*b - 5*a^6*b^3 - 10
*a^4*b^5)*c^3*d^4 + (a^9 + 11*a^7*b^2 - 10*a^5*b^4)*c^2*d^5 - 4*(a^8*b + a^6*b^3)*c*d^6 - (a^9 - 3*a^7*b^2)*d^
7)*f*x)*tan(f*x + e))/(((a^6*b^6 + 3*a^4*b^8 + 3*a^2*b^10 + b^12)*c^8*d - 4*(a^7*b^5 + 3*a^5*b^7 + 3*a^3*b^9 +
a*b^11)*c^7*d^2 + 2*(3*a^8*b^4 + 10*a^6*b^6 + 12*a^4*b^8 + 6*a^2*b^10 + b^12)*c^6*d^3 - 4*(a^9*b^3 + 5*a^7*b^
5 + 9*a^5*b^7 + 7*a^3*b^9 + 2*a*b^11)*c^5*d^4 + (a^10*b^2 + 15*a^8*b^4 + 40*a^6*b^6 + 40*a^4*b^8 + 15*a^2*b^10
+ b^12)*c^4*d^5 - 4*(2*a^9*b^3 + 7*a^7*b^5 + 9...

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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(1/(a+b*tan(f*x+e))**3/(c+d*tan(f*x+e))**2,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. 1759 vs. \(2 (462) = 924\).
time = 0.87, size = 1759, normalized size = 3.85 \begin {gather*} \frac {\frac {2 \, {\left (a^{3} c^{2} - 3 \, a b^{2} c^{2} - 6 \, a^{2} b c d + 2 \, b^{3} c d - a^{3} d^{2} + 3 \, a b^{2} d^{2}\right )} {\left (f x + e\right )}}{a^{6} c^{4} + 3 \, a^{4} b^{2} c^{4} + 3 \, a^{2} b^{4} c^{4} + b^{6} c^{4} + 2 \, a^{6} c^{2} d^{2} + 6 \, a^{4} b^{2} c^{2} d^{2} + 6 \, a^{2} b^{4} c^{2} d^{2} + 2 \, b^{6} c^{2} d^{2} + a^{6} d^{4} + 3 \, a^{4} b^{2} d^{4} + 3 \, a^{2} b^{4} d^{4} + b^{6} d^{4}} - \frac {{\left (3 \, a^{2} b c^{2} - b^{3} c^{2} + 2 \, a^{3} c d - 6 \, a b^{2} c d - 3 \, a^{2} b d^{2} + b^{3} d^{2}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{6} c^{4} + 3 \, a^{4} b^{2} c^{4} + 3 \, a^{2} b^{4} c^{4} + b^{6} c^{4} + 2 \, a^{6} c^{2} d^{2} + 6 \, a^{4} b^{2} c^{2} d^{2} + 6 \, a^{2} b^{4} c^{2} d^{2} + 2 \, b^{6} c^{2} d^{2} + a^{6} d^{4} + 3 \, a^{4} b^{2} d^{4} + 3 \, a^{2} b^{4} d^{4} + b^{6} d^{4}} + \frac {2 \, {\left (3 \, a^{2} b^{6} c^{2} - b^{8} c^{2} - 10 \, a^{3} b^{5} c d - 2 \, a b^{7} c d + 10 \, a^{4} b^{4} d^{2} + 9 \, a^{2} b^{6} d^{2} + 3 \, b^{8} d^{2}\right )} \log \left ({\left | b \tan \left (f x + e\right ) + a \right |}\right )}{a^{6} b^{5} c^{4} + 3 \, a^{4} b^{7} c^{4} + 3 \, a^{2} b^{9} c^{4} + b^{11} c^{4} - 4 \, a^{7} b^{4} c^{3} d - 12 \, a^{5} b^{6} c^{3} d - 12 \, a^{3} b^{8} c^{3} d - 4 \, a b^{10} c^{3} d + 6 \, a^{8} b^{3} c^{2} d^{2} + 18 \, a^{6} b^{5} c^{2} d^{2} + 18 \, a^{4} b^{7} c^{2} d^{2} + 6 \, a^{2} b^{9} c^{2} d^{2} - 4 \, a^{9} b^{2} c d^{3} - 12 \, a^{7} b^{4} c d^{3} - 12 \, a^{5} b^{6} c d^{3} - 4 \, a^{3} b^{8} c d^{3} + a^{10} b d^{4} + 3 \, a^{8} b^{3} d^{4} + 3 \, a^{6} b^{5} d^{4} + a^{4} b^{7} d^{4}} - \frac {2 \, {\left (5 \, b c^{2} d^{5} - 2 \, a c d^{6} + 3 \, b d^{7}\right )} \log \left ({\left | d \tan \left (f x + e\right ) + c \right |}\right )}{b^{4} c^{8} d - 4 \, a b^{3} c^{7} d^{2} + 6 \, a^{2} b^{2} c^{6} d^{3} + 2 \, b^{4} c^{6} d^{3} - 4 \, a^{3} b c^{5} d^{4} - 8 \, a b^{3} c^{5} d^{4} + a^{4} c^{4} d^{5} + 12 \, a^{2} b^{2} c^{4} d^{5} + b^{4} c^{4} d^{5} - 8 \, a^{3} b c^{3} d^{6} - 4 \, a b^{3} c^{3} d^{6} + 2 \, a^{4} c^{2} d^{7} + 6 \, a^{2} b^{2} c^{2} d^{7} - 4 \, a^{3} b c d^{8} + a^{4} d^{9}} + \frac {2 \, {\left (5 \, b c^{2} d^{5} \tan \left (f x + e\right ) - 2 \, a c d^{6} \tan \left (f x + e\right ) + 3 \, b d^{7} \tan \left (f x + e\right ) + 6 \, b c^{3} d^{4} - 3 \, a c^{2} d^{5} + 4 \, b c d^{6} - a d^{7}\right )}}{{\left (b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} + 2 \, b^{4} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} - 8 \, a b^{3} c^{5} d^{3} + a^{4} c^{4} d^{4} + 12 \, a^{2} b^{2} c^{4} d^{4} + b^{4} c^{4} d^{4} - 8 \, a^{3} b c^{3} d^{5} - 4 \, a b^{3} c^{3} d^{5} + 2 \, a^{4} c^{2} d^{6} + 6 \, a^{2} b^{2} c^{2} d^{6} - 4 \, a^{3} b c d^{7} + a^{4} d^{8}\right )} {\left (d \tan \left (f x + e\right ) + c\right )}} - \frac {9 \, a^{2} b^{7} c^{2} \tan \left (f x + e\right )^{2} - 3 \, b^{9} c^{2} \tan \left (f x + e\right )^{2} - 30 \, a^{3} b^{6} c d \tan \left (f x + e\right )^{2} - 6 \, a b^{8} c d \tan \left (f x + e\right )^{2} + 30 \, a^{4} b^{5} d^{2} \tan \left (f x + e\right )^{2} + 27 \, a^{2} b^{7} d^{2} \tan \left (f x + e\right )^{2} + 9 \, b^{9} d^{2} \tan \left (f x + e\right )^{2} + 22 \, a^{3} b^{6} c^{2} \tan \left (f x + e\right ) - 2 \, a b^{8} c^{2} \tan \left (f x + e\right ) - 72 \, a^{4} b^{5} c d \tan \left (f x + e\right ) - 28 \, a^{2} b^{7} c d \tan \left (f x + e\right ) - 4 \, b^{9} c d \tan \left (f x + e\right ) + 68 \, a^{5} b^{4} d^{2} \tan \left (f x + e\right ) + 66 \, a^{3} b^{6} d^{2} \tan \left (f x + e\right ) + 22 \, a b^{8} d^{2} \tan \left (f x + e\right ) + 14 \, a^{4} b^{5} c^{2} + 3 \, a^{2} b^{7} c^{2} + b^{9} c^{2} - 44 \, a^{5} b^{4} c d - 26 \, a^{3} b^{6} c d - 6 \, a b^{8} c d + 39 \, a^{6} b^{3} d^{2} + 41 \, a^{4} b^{5} d^{2} + 14 \, a^{2} b^{7} d^{2}}{{\left (a^{6} b^{4} c^{4} + 3 \, a^{4} b^{6} c^{4} + 3 \, a^{2} b^{8} c^{4} + b^{10} c^{4} - 4 \, a^{7} b^{3} c^{3} d - 12 \, a^{5} b^{5} c^{3} d - 12 \, a^{3} b^{7} c^{3} d - 4 \, a b^{9} c^{3} d + 6 \, a^{8} b^{2} c^{2} d^{2} + 18 \, a^{6} b^{4} c^{2} d^{2} + 18 \, a^{4} b^{6} c^{2} d^{2} + 6 \, a^{2} b^{8} c^{2} d^{2} - 4 \, a^{9} b c d^{3} - 12 \, a^{7} b^{3} c d^{3} - 12 \, a^{5} b^{5} c d^{3} - 4 \, a^{3} b^{7} c d^{3} + a^{10} d^{4} + 3 \, a^{8} b^{2} d^{4} + 3 \, a^{6} b^{4} d^{4} + a^{4} b^{6} d^{4}\right )} {\left (b \tan \left (f x + e\right ) + a\right )}^{2}}}{2 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(f*x+e))^3/(c+d*tan(f*x+e))^2,x, algorithm="giac")

[Out]

1/2*(2*(a^3*c^2 - 3*a*b^2*c^2 - 6*a^2*b*c*d + 2*b^3*c*d - a^3*d^2 + 3*a*b^2*d^2)*(f*x + e)/(a^6*c^4 + 3*a^4*b^
2*c^4 + 3*a^2*b^4*c^4 + b^6*c^4 + 2*a^6*c^2*d^2 + 6*a^4*b^2*c^2*d^2 + 6*a^2*b^4*c^2*d^2 + 2*b^6*c^2*d^2 + a^6*
d^4 + 3*a^4*b^2*d^4 + 3*a^2*b^4*d^4 + b^6*d^4) - (3*a^2*b*c^2 - b^3*c^2 + 2*a^3*c*d - 6*a*b^2*c*d - 3*a^2*b*d^
2 + b^3*d^2)*log(tan(f*x + e)^2 + 1)/(a^6*c^4 + 3*a^4*b^2*c^4 + 3*a^2*b^4*c^4 + b^6*c^4 + 2*a^6*c^2*d^2 + 6*a^
4*b^2*c^2*d^2 + 6*a^2*b^4*c^2*d^2 + 2*b^6*c^2*d^2 + a^6*d^4 + 3*a^4*b^2*d^4 + 3*a^2*b^4*d^4 + b^6*d^4) + 2*(3*
a^2*b^6*c^2 - b^8*c^2 - 10*a^3*b^5*c*d - 2*a*b^7*c*d + 10*a^4*b^4*d^2 + 9*a^2*b^6*d^2 + 3*b^8*d^2)*log(abs(b*t
an(f*x + e) + a))/(a^6*b^5*c^4 + 3*a^4*b^7*c^4 + 3*a^2*b^9*c^4 + b^11*c^4 - 4*a^7*b^4*c^3*d - 12*a^5*b^6*c^3*d
- 12*a^3*b^8*c^3*d - 4*a*b^10*c^3*d + 6*a^8*b^3*c^2*d^2 + 18*a^6*b^5*c^2*d^2 + 18*a^4*b^7*c^2*d^2 + 6*a^2*b^9
*c^2*d^2 - 4*a^9*b^2*c*d^3 - 12*a^7*b^4*c*d^3 - 12*a^5*b^6*c*d^3 - 4*a^3*b^8*c*d^3 + a^10*b*d^4 + 3*a^8*b^3*d^
4 + 3*a^6*b^5*d^4 + a^4*b^7*d^4) - 2*(5*b*c^2*d^5 - 2*a*c*d^6 + 3*b*d^7)*log(abs(d*tan(f*x + e) + c))/(b^4*c^8
*d - 4*a*b^3*c^7*d^2 + 6*a^2*b^2*c^6*d^3 + 2*b^4*c^6*d^3 - 4*a^3*b*c^5*d^4 - 8*a*b^3*c^5*d^4 + a^4*c^4*d^5 + 1
2*a^2*b^2*c^4*d^5 + b^4*c^4*d^5 - 8*a^3*b*c^3*d^6 - 4*a*b^3*c^3*d^6 + 2*a^4*c^2*d^7 + 6*a^2*b^2*c^2*d^7 - 4*a^
3*b*c*d^8 + a^4*d^9) + 2*(5*b*c^2*d^5*tan(f*x + e) - 2*a*c*d^6*tan(f*x + e) + 3*b*d^7*tan(f*x + e) + 6*b*c^3*d
^4 - 3*a*c^2*d^5 + 4*b*c*d^6 - a*d^7)/((b^4*c^8 - 4*a*b^3*c^7*d + 6*a^2*b^2*c^6*d^2 + 2*b^4*c^6*d^2 - 4*a^3*b*
c^5*d^3 - 8*a*b^3*c^5*d^3 + a^4*c^4*d^4 + 12*a^2*b^2*c^4*d^4 + b^4*c^4*d^4 - 8*a^3*b*c^3*d^5 - 4*a*b^3*c^3*d^5
+ 2*a^4*c^2*d^6 + 6*a^2*b^2*c^2*d^6 - 4*a^3*b*c*d^7 + a^4*d^8)*(d*tan(f*x + e) + c)) - (9*a^2*b^7*c^2*tan(f*x
+ e)^2 - 3*b^9*c^2*tan(f*x + e)^2 - 30*a^3*b^6*c*d*tan(f*x + e)^2 - 6*a*b^8*c*d*tan(f*x + e)^2 + 30*a^4*b^5*d
^2*tan(f*x + e)^2 + 27*a^2*b^7*d^2*tan(f*x + e)^2 + 9*b^9*d^2*tan(f*x + e)^2 + 22*a^3*b^6*c^2*tan(f*x + e) - 2
*a*b^8*c^2*tan(f*x + e) - 72*a^4*b^5*c*d*tan(f*x + e) - 28*a^2*b^7*c*d*tan(f*x + e) - 4*b^9*c*d*tan(f*x + e) +
68*a^5*b^4*d^2*tan(f*x + e) + 66*a^3*b^6*d^2*tan(f*x + e) + 22*a*b^8*d^2*tan(f*x + e) + 14*a^4*b^5*c^2 + 3*a^
2*b^7*c^2 + b^9*c^2 - 44*a^5*b^4*c*d - 26*a^3*b^6*c*d - 6*a*b^8*c*d + 39*a^6*b^3*d^2 + 41*a^4*b^5*d^2 + 14*a^2
*b^7*d^2)/((a^6*b^4*c^4 + 3*a^4*b^6*c^4 + 3*a^2*b^8*c^4 + b^10*c^4 - 4*a^7*b^3*c^3*d - 12*a^5*b^5*c^3*d - 12*a
^3*b^7*c^3*d - 4*a*b^9*c^3*d + 6*a^8*b^2*c^2*d^2 + 18*a^6*b^4*c^2*d^2 + 18*a^4*b^6*c^2*d^2 + 6*a^2*b^8*c^2*d^2
- 4*a^9*b*c*d^3 - 12*a^7*b^3*c*d^3 - 12*a^5*b^5*c*d^3 - 4*a^3*b^7*c*d^3 + a^10*d^4 + 3*a^8*b^2*d^4 + 3*a^6*b^
4*d^4 + a^4*b^6*d^4)*(b*tan(f*x + e) + a)^2))/f

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Mupad [B]
time = 34.61, size = 1421, normalized size = 3.11 \begin {gather*} -\frac {\frac {2\,a^6\,d^4+4\,a^4\,b^2\,d^4+9\,a^3\,b^3\,c^3\,d+9\,a^3\,b^3\,c\,d^3-5\,a^2\,b^4\,c^4-5\,a^2\,b^4\,c^2\,d^2+2\,a^2\,b^4\,d^4+5\,a\,b^5\,c^3\,d+5\,a\,b^5\,c\,d^3-b^6\,c^4-b^6\,c^2\,d^2}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )\,\left (a^4\,c^2+a^4\,d^2+2\,a^2\,b^2\,c^2+2\,a^2\,b^2\,d^2+b^4\,c^2+b^4\,d^2\right )}+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (4\,a^5\,b\,d^4+9\,a^3\,b^3\,c^2\,d^2+17\,a^3\,b^3\,d^4+3\,a^2\,b^4\,c^3\,d+3\,a^2\,b^4\,c\,d^3-4\,a\,b^5\,c^4+a\,b^5\,c^2\,d^2+9\,a\,b^5\,d^4+3\,b^6\,c^3\,d+3\,b^6\,c\,d^3\right )}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )\,\left (a^4\,c^2+a^4\,d^2+2\,a^2\,b^2\,c^2+2\,a^2\,b^2\,d^2+b^4\,c^2+b^4\,d^2\right )}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (a^4\,b^2\,d^4+4\,a^2\,b^4\,c^2\,d^2+6\,a^2\,b^4\,d^4-2\,a\,b^5\,c^3\,d-2\,a\,b^5\,c\,d^3+2\,b^6\,c^2\,d^2+3\,b^6\,d^4\right )}{\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )\,\left (a^4\,c^2+a^4\,d^2+2\,a^2\,b^2\,c^2+2\,a^2\,b^2\,d^2+b^4\,c^2+b^4\,d^2\right )}}{f\,\left (\mathrm {tan}\left (e+f\,x\right )\,\left (d\,a^2+2\,b\,c\,a\right )+a^2\,c+{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (c\,b^2+2\,a\,d\,b\right )+b^2\,d\,{\mathrm {tan}\left (e+f\,x\right )}^3\right )}-\frac {\ln \left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (d\,\left (10\,c\,a^3\,b^4+2\,c\,a\,b^6\right )-d^2\,\left (10\,a^4\,b^3+9\,a^2\,b^5+3\,b^7\right )+b^7\,c^2-3\,a^2\,b^5\,c^2\right )}{f\,\left (a^{10}\,d^4-4\,a^9\,b\,c\,d^3+6\,a^8\,b^2\,c^2\,d^2+3\,a^8\,b^2\,d^4-4\,a^7\,b^3\,c^3\,d-12\,a^7\,b^3\,c\,d^3+a^6\,b^4\,c^4+18\,a^6\,b^4\,c^2\,d^2+3\,a^6\,b^4\,d^4-12\,a^5\,b^5\,c^3\,d-12\,a^5\,b^5\,c\,d^3+3\,a^4\,b^6\,c^4+18\,a^4\,b^6\,c^2\,d^2+a^4\,b^6\,d^4-12\,a^3\,b^7\,c^3\,d-4\,a^3\,b^7\,c\,d^3+3\,a^2\,b^8\,c^4+6\,a^2\,b^8\,c^2\,d^2-4\,a\,b^9\,c^3\,d+b^{10}\,c^4\right )}-\frac {\ln \left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (b\,\left (5\,c^2\,d^4+3\,d^6\right )-2\,a\,c\,d^5\right )}{f\,\left (a^4\,c^4\,d^4+2\,a^4\,c^2\,d^6+a^4\,d^8-4\,a^3\,b\,c^5\,d^3-8\,a^3\,b\,c^3\,d^5-4\,a^3\,b\,c\,d^7+6\,a^2\,b^2\,c^6\,d^2+12\,a^2\,b^2\,c^4\,d^4+6\,a^2\,b^2\,c^2\,d^6-4\,a\,b^3\,c^7\,d-8\,a\,b^3\,c^5\,d^3-4\,a\,b^3\,c^3\,d^5+b^4\,c^8+2\,b^4\,c^6\,d^2+b^4\,c^4\,d^4\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,f\,\left (a^3\,c^2+a^3\,c\,d\,2{}\mathrm {i}-a^3\,d^2+a^2\,b\,c^2\,3{}\mathrm {i}-6\,a^2\,b\,c\,d-a^2\,b\,d^2\,3{}\mathrm {i}-3\,a\,b^2\,c^2-a\,b^2\,c\,d\,6{}\mathrm {i}+3\,a\,b^2\,d^2-b^3\,c^2\,1{}\mathrm {i}+2\,b^3\,c\,d+b^3\,d^2\,1{}\mathrm {i}\right )}+\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,f\,\left (a^3\,c^2-a^3\,c\,d\,2{}\mathrm {i}-a^3\,d^2-a^2\,b\,c^2\,3{}\mathrm {i}-6\,a^2\,b\,c\,d+a^2\,b\,d^2\,3{}\mathrm {i}-3\,a\,b^2\,c^2+a\,b^2\,c\,d\,6{}\mathrm {i}+3\,a\,b^2\,d^2+b^3\,c^2\,1{}\mathrm {i}+2\,b^3\,c\,d-b^3\,d^2\,1{}\mathrm {i}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*tan(e + f*x))^3*(c + d*tan(e + f*x))^2),x)

[Out]

(log(tan(e + f*x) + 1i)*1i)/(2*f*(a^3*c^2 - a^3*d^2 + b^3*c^2*1i - b^3*d^2*1i - 3*a*b^2*c^2 - a^2*b*c^2*3i + 3
*a*b^2*d^2 + a^2*b*d^2*3i - a^3*c*d*2i + 2*b^3*c*d + a*b^2*c*d*6i - 6*a^2*b*c*d)) - (log(tan(e + f*x) - 1i)*1i
)/(2*f*(a^3*c^2 - a^3*d^2 - b^3*c^2*1i + b^3*d^2*1i - 3*a*b^2*c^2 + a^2*b*c^2*3i + 3*a*b^2*d^2 - a^2*b*d^2*3i
+ a^3*c*d*2i + 2*b^3*c*d - a*b^2*c*d*6i - 6*a^2*b*c*d)) - ((2*a^6*d^4 - b^6*c^4 - 5*a^2*b^4*c^4 + 2*a^2*b^4*d^
4 + 4*a^4*b^2*d^4 - b^6*c^2*d^2 + 9*a^3*b^3*c*d^3 + 9*a^3*b^3*c^3*d - 5*a^2*b^4*c^2*d^2 + 5*a*b^5*c*d^3 + 5*a*
b^5*c^3*d)/(2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)*(a^4*c^2 + a^4*d^2 + b^4*c^2 + b^4*d^2 + 2*a
^2*b^2*c^2 + 2*a^2*b^2*d^2)) + (tan(e + f*x)*(9*a*b^5*d^4 - 4*a*b^5*c^4 + 4*a^5*b*d^4 + 3*b^6*c*d^3 + 3*b^6*c^
3*d + 17*a^3*b^3*d^4 + a*b^5*c^2*d^2 + 3*a^2*b^4*c*d^3 + 3*a^2*b^4*c^3*d + 9*a^3*b^3*c^2*d^2))/(2*(a^3*d^3 - b
^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)*(a^4*c^2 + a^4*d^2 + b^4*c^2 + b^4*d^2 + 2*a^2*b^2*c^2 + 2*a^2*b^2*d^2
)) + (tan(e + f*x)^2*(3*b^6*d^4 + 6*a^2*b^4*d^4 + a^4*b^2*d^4 + 2*b^6*c^2*d^2 + 4*a^2*b^4*c^2*d^2 - 2*a*b^5*c*
d^3 - 2*a*b^5*c^3*d))/((a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)*(a^4*c^2 + a^4*d^2 + b^4*c^2 + b^4*
d^2 + 2*a^2*b^2*c^2 + 2*a^2*b^2*d^2)))/(f*(tan(e + f*x)*(a^2*d + 2*a*b*c) + a^2*c + tan(e + f*x)^2*(b^2*c + 2*
a*b*d) + b^2*d*tan(e + f*x)^3)) - (log(a + b*tan(e + f*x))*(d*(10*a^3*b^4*c + 2*a*b^6*c) - d^2*(3*b^7 + 9*a^2*
b^5 + 10*a^4*b^3) + b^7*c^2 - 3*a^2*b^5*c^2))/(f*(a^10*d^4 + b^10*c^4 + 3*a^2*b^8*c^4 + 3*a^4*b^6*c^4 + a^6*b^
4*c^4 + a^4*b^6*d^4 + 3*a^6*b^4*d^4 + 3*a^8*b^2*d^4 - 4*a^3*b^7*c*d^3 - 12*a^3*b^7*c^3*d - 12*a^5*b^5*c*d^3 -
12*a^5*b^5*c^3*d - 12*a^7*b^3*c*d^3 - 4*a^7*b^3*c^3*d + 6*a^2*b^8*c^2*d^2 + 18*a^4*b^6*c^2*d^2 + 18*a^6*b^4*c^
2*d^2 + 6*a^8*b^2*c^2*d^2 - 4*a*b^9*c^3*d - 4*a^9*b*c*d^3)) - (log(c + d*tan(e + f*x))*(b*(3*d^6 + 5*c^2*d^4)
- 2*a*c*d^5))/(f*(a^4*d^8 + b^4*c^8 + 2*a^4*c^2*d^6 + a^4*c^4*d^4 + b^4*c^4*d^4 + 2*b^4*c^6*d^2 - 4*a*b^3*c^3*
d^5 - 8*a*b^3*c^5*d^3 - 8*a^3*b*c^3*d^5 - 4*a^3*b*c^5*d^3 + 6*a^2*b^2*c^2*d^6 + 12*a^2*b^2*c^4*d^4 + 6*a^2*b^2
*c^6*d^2 - 4*a*b^3*c^7*d - 4*a^3*b*c*d^7))

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