3.1221 \(\int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx\)

Optimal. Leaf size=290 \[ -\frac {d \left (a^2 d^2+b^2 \left (c^2+2 d^2\right )\right )}{f \left (a^2+b^2\right ) \left (c^2+d^2\right ) (b c-a d)^2 (c+d \tan (e+f x))}+\frac {x (a (c+d)+b (c-d)) (a (c-d)-b (c+d))}{\left (a^2+b^2\right )^2 \left (c^2+d^2\right )^2}-\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))}+\frac {2 b^3 \left (-2 a^2 d+a b c-b^2 d\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right )^2 (b c-a d)^3}-\frac {2 d^3 \left (a c d-b \left (2 c^2+d^2\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right )^2 (b c-a d)^3} \]

[Out]

(b*(c-d)+a*(c+d))*(a*(c-d)-b*(c+d))*x/(a^2+b^2)^2/(c^2+d^2)^2+2*b^3*(-2*a^2*d+a*b*c-b^2*d)*ln(a*cos(f*x+e)+b*s
in(f*x+e))/(a^2+b^2)^2/(-a*d+b*c)^3/f-2*d^3*(a*c*d-b*(2*c^2+d^2))*ln(c*cos(f*x+e)+d*sin(f*x+e))/(-a*d+b*c)^3/(
c^2+d^2)^2/f-d*(a^2*d^2+b^2*(c^2+2*d^2))/(a^2+b^2)/(-a*d+b*c)^2/(c^2+d^2)/f/(c+d*tan(f*x+e))-b^2/(a^2+b^2)/(-a
*d+b*c)/f/(a+b*tan(f*x+e))/(c+d*tan(f*x+e))

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Rubi [A]  time = 1.03, antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3569, 3649, 3651, 3530} \[ -\frac {d \left (a^2 d^2+b^2 \left (c^2+2 d^2\right )\right )}{f \left (a^2+b^2\right ) \left (c^2+d^2\right ) (b c-a d)^2 (c+d \tan (e+f x))}+\frac {x (a (c+d)+b (c-d)) (a (c-d)-b (c+d))}{\left (a^2+b^2\right )^2 \left (c^2+d^2\right )^2}-\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))}+\frac {2 b^3 \left (-2 a^2 d+a b c-b^2 d\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right )^2 (b c-a d)^3}-\frac {2 d^3 \left (a c d-b \left (2 c^2+d^2\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right )^2 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3530

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c*Log[Re
moveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]])/(b*f), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d,
0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]

Rule 3569

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 3649

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

Rule 3651

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

Rubi steps

\begin {align*} \int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx &=-\frac {b^2}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))}-\frac {\int \frac {-a b c+a^2 d+2 b^2 d+b (b c-a d) \tan (e+f x)+2 b^2 d \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^2} \, dx}{\left (a^2+b^2\right ) (b c-a d)}\\ &=-\frac {d \left (a^2 d^2+b^2 \left (c^2+2 d^2\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))}-\frac {b^2}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))}-\frac {\int \frac {a b d^2 (b c+a d)+\left (a b c-a^2 d-2 b^2 d\right ) \left (a c d-b \left (c^2+d^2\right )\right )+(b c-a d)^2 (b c+a d) \tan (e+f x)+b d \left (a^2 d^2+b^2 \left (c^2+2 d^2\right )\right ) \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))} \, dx}{\left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right )}\\ &=\frac {(b (c-d)+a (c+d)) (a (c-d)-b (c+d)) x}{\left (a^2+b^2\right )^2 \left (c^2+d^2\right )^2}-\frac {d \left (a^2 d^2+b^2 \left (c^2+2 d^2\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))}-\frac {b^2}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))}+\frac {\left (2 b^3 \left (a b c-2 a^2 d-b^2 d\right )\right ) \int \frac {b-a \tan (e+f x)}{a+b \tan (e+f x)} \, dx}{\left (a^2+b^2\right )^2 (b c-a d)^3}-\frac {\left (2 d^3 \left (a c d-b \left (2 c^2+d^2\right )\right )\right ) \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{(b c-a d)^3 \left (c^2+d^2\right )^2}\\ &=\frac {(b (c-d)+a (c+d)) (a (c-d)-b (c+d)) x}{\left (a^2+b^2\right )^2 \left (c^2+d^2\right )^2}+\frac {2 b^3 \left (a b c-2 a^2 d-b^2 d\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{\left (a^2+b^2\right )^2 (b c-a d)^3 f}-\frac {2 d^3 \left (a c d-b \left (2 c^2+d^2\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{(b c-a d)^3 \left (c^2+d^2\right )^2 f}-\frac {d \left (a^2 d^2+b^2 \left (c^2+2 d^2\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))}-\frac {b^2}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) (c+d \tan (e+f x))}\\ \end {align*}

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Mathematica [A]  time = 6.94, size = 556, normalized size = 1.92 \[ -\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))}-\frac {-\frac {d^2 \left (a^2 d-a b c+2 b^2 d\right )-c \left (b d (b c-a d)-2 b^2 c d\right )}{f \left (c^2+d^2\right ) (a d-b c) (c+d \tan (e+f x))}-\frac {\frac {b (b c-a d)^2 \left (\frac {b \left (-\left (a^2 \left (c^2-d^2\right )\right )+4 a b c d+b^2 \left (c^2-d^2\right )\right )}{\sqrt {-b^2}}+2 a^2 c d+2 a b c^2-2 a b d^2-2 b^2 c d\right ) \log \left (\sqrt {-b^2}-b \tan (e+f x)\right )}{2 \left (a^2+b^2\right ) \left (c^2+d^2\right )}+\frac {b (b c-a d)^2 \left (\frac {\sqrt {-b^2} \left (-\left (a^2 \left (c^2-d^2\right )\right )+4 a b c d+b^2 \left (c^2-d^2\right )\right )}{b}+2 a^2 c d+2 a b c^2-2 a b d^2-2 b^2 c d\right ) \log \left (\sqrt {-b^2}+b \tan (e+f x)\right )}{2 \left (a^2+b^2\right ) \left (c^2+d^2\right )}+\frac {2 b d^3 \left (a^2+b^2\right ) \left (a c d-b \left (2 c^2+d^2\right )\right ) \log (c+d \tan (e+f x))}{\left (c^2+d^2\right ) (b c-a d)}-\frac {2 b^4 \left (c^2+d^2\right ) \left (-2 a^2 d+a b c-b^2 d\right ) \log (a+b \tan (e+f x))}{\left (a^2+b^2\right ) (b c-a d)}}{b f \left (c^2+d^2\right ) (a d-b c)}}{\left (a^2+b^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^2/((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)^2*(2*a*b*c^2
+ 2*a^2*c*d - 2*b^2*c*d - 2*a*b*d^2 + (b*(4*a*b*c*d - a^2*(c^2 - d^2) + b^2*(c^2 - d^2)))/Sqrt[-b^2])*Log[Sqr
t[-b^2] - b*Tan[e + f*x]])/(2*(a^2 + b^2)*(c^2 + d^2)) - (2*b^4*(a*b*c - 2*a^2*d - b^2*d)*(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*(2*a*b*c^2 + 2*a^2*c*d - 2*b^2*c*d - 2*a*b*d^2 +
(Sqrt[-b^2]*(4*a*b*c*d - a^2*(c^2 - d^2) + b^2*(c^2 - d^2)))/b)*Log[Sqrt[-b^2] + b*Tan[e + f*x]])/(2*(a^2 + b
^2)*(c^2 + d^2)) + (2*b*(a^2 + b^2)*d^3*(a*c*d - b*(2*c^2 + d^2))*Log[c + d*Tan[e + f*x]])/((b*c - a*d)*(c^2 +
d^2)))/(b*(-(b*c) + a*d)*(c^2 + d^2)*f)) - (d^2*(-(a*b*c) + a^2*d + 2*b^2*d) - c*(-2*b^2*c*d + b*d*(b*c - a*d
)))/((-(b*c) + a*d)*(c^2 + d^2)*f*(c + d*Tan[e + f*x])))/((a^2 + b^2)*(b*c - a*d))

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fricas [B]  time = 2.07, size = 2217, normalized size = 7.64 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b^6*c^6 - a*b^5*c^5*d + 2*b^6*c^4*d^2 - 2*a*b^5*c^3*d^3 + b^6*c^2*d^4 + (a^5*b + 2*a^3*b^3)*c*d^5 - (a^6 + 2
*a^4*b^2 + a^2*b^4)*d^6 - ((a^3*b^3 - a*b^5)*c^6 - (3*a^4*b^2 + a^2*b^4)*c^5*d + (3*a^5*b + 8*a^3*b^3 + a*b^5)
*c^4*d^2 - (a^6 + 8*a^4*b^2 + 3*a^2*b^4)*c^3*d^3 + (a^5*b + 3*a^3*b^3)*c^2*d^4 + (a^6 - a^4*b^2)*c*d^5)*f*x -
(a*b^5*c^5*d - a^2*b^4*c^4*d^2 + 2*a*b^5*c^3*d^3 - a^2*b^4*d^6 + (a^4*b^2 + b^6)*c^2*d^4 - (a^5*b + 2*a^3*b^3)
*c*d^5 + ((a^2*b^4 - b^6)*c^5*d - (3*a^3*b^3 + a*b^5)*c^4*d^2 + (3*a^4*b^2 + 8*a^2*b^4 + b^6)*c^3*d^3 - (a^5*b
+ 8*a^3*b^3 + 3*a*b^5)*c^2*d^4 + (a^4*b^2 + 3*a^2*b^4)*c*d^5 + (a^5*b - a^3*b^3)*d^6)*f*x)*tan(f*x + e)^2 - (
a^2*b^4*c^6 + 2*a^2*b^4*c^4*d^2 + a^2*b^4*c^2*d^4 - (2*a^3*b^3 + a*b^5)*c^5*d - 2*(2*a^3*b^3 + a*b^5)*c^3*d^3
- (2*a^3*b^3 + a*b^5)*c*d^5 + (a*b^5*c^5*d + 2*a*b^5*c^3*d^3 + a*b^5*c*d^5 - (2*a^2*b^4 + b^6)*c^4*d^2 - 2*(2*
a^2*b^4 + b^6)*c^2*d^4 - (2*a^2*b^4 + b^6)*d^6)*tan(f*x + e)^2 + (a*b^5*c^6 - (a^2*b^4 + b^6)*c^5*d - (2*a^3*b
^3 - a*b^5)*c^4*d^2 - 2*(a^2*b^4 + b^6)*c^3*d^3 - (4*a^3*b^3 + a*b^5)*c^2*d^4 - (a^2*b^4 + b^6)*c*d^5 - (2*a^3
*b^3 + a*b^5)*d^6)*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)) - (
2*(a^5*b + 2*a^3*b^3 + a*b^5)*c^3*d^3 - (a^6 + 2*a^4*b^2 + a^2*b^4)*c^2*d^4 + (a^5*b + 2*a^3*b^3 + a*b^5)*c*d^
5 + (2*(a^4*b^2 + 2*a^2*b^4 + b^6)*c^2*d^4 - (a^5*b + 2*a^3*b^3 + a*b^5)*c*d^5 + (a^4*b^2 + 2*a^2*b^4 + b^6)*d
^6)*tan(f*x + e)^2 + (2*(a^4*b^2 + 2*a^2*b^4 + b^6)*c^3*d^3 + (a^5*b + 2*a^3*b^3 + a*b^5)*c^2*d^4 - (a^6 + a^4
*b^2 - a^2*b^4 - b^6)*c*d^5 + (a^5*b + 2*a^3*b^3 + a*b^5)*d^6)*tan(f*x + e))*log((d^2*tan(f*x + e)^2 + 2*c*d*t
an(f*x + e) + c^2)/(tan(f*x + e)^2 + 1)) - (a*b^5*c^6 + 3*a*b^5*c^4*d^2 - (a^2*b^4 + b^6)*c^5*d - 2*(a^2*b^4 +
b^6)*c^3*d^3 + (a^5*b + 2*a^3*b^3 + 4*a*b^5)*c^2*d^4 - (a^6 + 3*a^4*b^2 + 4*a^2*b^4 + 2*b^6)*c*d^5 + (a^5*b +
2*a^3*b^3 + 2*a*b^5)*d^6 + ((a^2*b^4 - b^6)*c^6 - 2*(a^3*b^3 + a*b^5)*c^5*d + (7*a^2*b^4 + b^6)*c^4*d^2 + 2*(
a^5*b - a*b^5)*c^3*d^3 - (a^6 + 7*a^4*b^2)*c^2*d^4 + 2*(a^5*b + a^3*b^3)*c*d^5 + (a^6 - a^4*b^2)*d^6)*f*x)*tan
(f*x + e))/(((a^4*b^4 + 2*a^2*b^6 + b^8)*c^7*d - 3*(a^5*b^3 + 2*a^3*b^5 + a*b^7)*c^6*d^2 + (3*a^6*b^2 + 8*a^4*
b^4 + 7*a^2*b^6 + 2*b^8)*c^5*d^3 - (a^7*b + 8*a^5*b^3 + 13*a^3*b^5 + 6*a*b^7)*c^4*d^4 + (6*a^6*b^2 + 13*a^4*b^
4 + 8*a^2*b^6 + b^8)*c^3*d^5 - (2*a^7*b + 7*a^5*b^3 + 8*a^3*b^5 + 3*a*b^7)*c^2*d^6 + 3*(a^6*b^2 + 2*a^4*b^4 +
a^2*b^6)*c*d^7 - (a^7*b + 2*a^5*b^3 + a^3*b^5)*d^8)*f*tan(f*x + e)^2 + ((a^4*b^4 + 2*a^2*b^6 + b^8)*c^8 - 2*(a
^5*b^3 + 2*a^3*b^5 + a*b^7)*c^7*d + 2*(a^4*b^4 + 2*a^2*b^6 + b^8)*c^6*d^2 + 2*(a^7*b - 3*a^3*b^5 - 2*a*b^7)*c^
5*d^3 - (a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*c^4*d^4 + 2*(2*a^7*b + 3*a^5*b^3 - a*b^7)*c^3*d^5 - 2*(a^8 + 2*a^6
*b^2 + a^4*b^4)*c^2*d^6 + 2*(a^7*b + 2*a^5*b^3 + a^3*b^5)*c*d^7 - (a^8 + 2*a^6*b^2 + a^4*b^4)*d^8)*f*tan(f*x +
e) + ((a^5*b^3 + 2*a^3*b^5 + a*b^7)*c^8 - 3*(a^6*b^2 + 2*a^4*b^4 + a^2*b^6)*c^7*d + (3*a^7*b + 8*a^5*b^3 + 7*
a^3*b^5 + 2*a*b^7)*c^6*d^2 - (a^8 + 8*a^6*b^2 + 13*a^4*b^4 + 6*a^2*b^6)*c^5*d^3 + (6*a^7*b + 13*a^5*b^3 + 8*a^
3*b^5 + a*b^7)*c^4*d^4 - (2*a^8 + 7*a^6*b^2 + 8*a^4*b^4 + 3*a^2*b^6)*c^3*d^5 + 3*(a^7*b + 2*a^5*b^3 + a^3*b^5)
*c^2*d^6 - (a^8 + 2*a^6*b^2 + a^4*b^4)*c*d^7)*f)

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giac [B]  time = 1.85, size = 1395, normalized size = 4.81 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a^2*c^2 - b^2*c^2 - 4*a*b*c*d - a^2*d^2 + b^2*d^2)*(f*x + e)/(a^4*c^4 + 2*a^2*b^2*c^4 + b^4*c^4 + 2*a^4*c^2*
d^2 + 4*a^2*b^2*c^2*d^2 + 2*b^4*c^2*d^2 + a^4*d^4 + 2*a^2*b^2*d^4 + b^4*d^4) - (a*b*c^2 + a^2*c*d - b^2*c*d -
a*b*d^2)*log(tan(f*x + e)^2 + 1)/(a^4*c^4 + 2*a^2*b^2*c^4 + b^4*c^4 + 2*a^4*c^2*d^2 + 4*a^2*b^2*c^2*d^2 + 2*b^
4*c^2*d^2 + a^4*d^4 + 2*a^2*b^2*d^4 + b^4*d^4) + 2*(a*b^5*c - 2*a^2*b^4*d - b^6*d)*log(abs(b*tan(f*x + e) + a)
)/(a^4*b^4*c^3 + 2*a^2*b^6*c^3 + b^8*c^3 - 3*a^5*b^3*c^2*d - 6*a^3*b^5*c^2*d - 3*a*b^7*c^2*d + 3*a^6*b^2*c*d^2
+ 6*a^4*b^4*c*d^2 + 3*a^2*b^6*c*d^2 - a^7*b*d^3 - 2*a^5*b^3*d^3 - a^3*b^5*d^3) + 2*(2*b*c^2*d^4 - a*c*d^5 + b
*d^6)*log(abs(d*tan(f*x + e) + c))/(b^3*c^7*d - 3*a*b^2*c^6*d^2 + 3*a^2*b*c^5*d^3 + 2*b^3*c^5*d^3 - a^3*c^4*d^
4 - 6*a*b^2*c^4*d^4 + 6*a^2*b*c^3*d^5 + b^3*c^3*d^5 - 2*a^3*c^2*d^6 - 3*a*b^2*c^2*d^6 + 3*a^2*b*c*d^7 - a^3*d^
8) - (a*b^4*c^4*d*tan(f*x + e)^2 - a^2*b^3*c^3*d^2*tan(f*x + e)^2 - b^5*c^3*d^2*tan(f*x + e)^2 - a^3*b^2*c^2*d
^3*tan(f*x + e)^2 + a*b^4*c^2*d^3*tan(f*x + e)^2 + a^4*b*c*d^4*tan(f*x + e)^2 + a^2*b^3*c*d^4*tan(f*x + e)^2 -
a^3*b^2*d^5*tan(f*x + e)^2 + a*b^4*c^5*tan(f*x + e) + a^2*b^3*c^4*d*tan(f*x + e) - 2*a^3*b^2*c^3*d^2*tan(f*x
+ e) + a^4*b*c^2*d^3*tan(f*x + e) + 6*a^2*b^3*c^2*d^3*tan(f*x + e) + 3*b^5*c^2*d^3*tan(f*x + e) + a^5*c*d^4*ta
n(f*x + e) + 3*a^2*b^3*d^5*tan(f*x + e) + 2*b^5*d^5*tan(f*x + e) + 2*a^2*b^3*c^5 + b^5*c^5 - a^3*b^2*c^4*d - a
*b^4*c^4*d - a^4*b*c^3*d^2 + 3*a^2*b^3*c^3*d^2 + 2*b^5*c^3*d^2 + 2*a^5*c^2*d^3 + 3*a^3*b^2*c^2*d^3 + a*b^4*c^2
*d^3 - a^4*b*c*d^4 + a^2*b^3*c*d^4 + b^5*c*d^4 + a^5*d^5 + 2*a^3*b^2*d^5 + a*b^4*d^5)/((a^4*b^2*c^6 + 2*a^2*b^
4*c^6 + b^6*c^6 - 2*a^5*b*c^5*d - 4*a^3*b^3*c^5*d - 2*a*b^5*c^5*d + a^6*c^4*d^2 + 4*a^4*b^2*c^4*d^2 + 5*a^2*b^
4*c^4*d^2 + 2*b^6*c^4*d^2 - 4*a^5*b*c^3*d^3 - 8*a^3*b^3*c^3*d^3 - 4*a*b^5*c^3*d^3 + 2*a^6*c^2*d^4 + 5*a^4*b^2*
c^2*d^4 + 4*a^2*b^4*c^2*d^4 + b^6*c^2*d^4 - 2*a^5*b*c*d^5 - 4*a^3*b^3*c*d^5 - 2*a*b^5*c*d^5 + a^6*d^6 + 2*a^4*
b^2*d^6 + a^2*b^4*d^6)*(b*d*tan(f*x + e)^2 + b*c*tan(f*x + e) + a*d*tan(f*x + e) + a*c)))/f

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maple [B]  time = 0.41, size = 652, normalized size = 2.25 \[ -\frac {b^{3}}{f \left (d a -c b \right )^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (f x +e \right )\right )}+\frac {4 b^{3} \ln \left (a +b \tan \left (f x +e \right )\right ) a^{2} d}{f \left (d a -c b \right )^{3} \left (a^{2}+b^{2}\right )^{2}}-\frac {2 b^{4} \ln \left (a +b \tan \left (f x +e \right )\right ) a c}{f \left (d a -c b \right )^{3} \left (a^{2}+b^{2}\right )^{2}}+\frac {2 b^{5} \ln \left (a +b \tan \left (f x +e \right )\right ) d}{f \left (d a -c b \right )^{3} \left (a^{2}+b^{2}\right )^{2}}-\frac {d^{3}}{f \left (d a -c b \right )^{2} \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )}+\frac {2 d^{4} \ln \left (c +d \tan \left (f x +e \right )\right ) a c}{f \left (d a -c b \right )^{3} \left (c^{2}+d^{2}\right )^{2}}-\frac {4 d^{3} \ln \left (c +d \tan \left (f x +e \right )\right ) c^{2} b}{f \left (d a -c b \right )^{3} \left (c^{2}+d^{2}\right )^{2}}-\frac {2 d^{5} \ln \left (c +d \tan \left (f x +e \right )\right ) b}{f \left (d a -c b \right )^{3} \left (c^{2}+d^{2}\right )^{2}}-\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right ) a^{2} c d}{f \left (a^{2}+b^{2}\right )^{2} \left (c^{2}+d^{2}\right )^{2}}-\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right ) a b \,c^{2}}{f \left (a^{2}+b^{2}\right )^{2} \left (c^{2}+d^{2}\right )^{2}}+\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right ) a b \,d^{2}}{f \left (a^{2}+b^{2}\right )^{2} \left (c^{2}+d^{2}\right )^{2}}+\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right ) b^{2} c d}{f \left (a^{2}+b^{2}\right )^{2} \left (c^{2}+d^{2}\right )^{2}}+\frac {\arctan \left (\tan \left (f x +e \right )\right ) a^{2} c^{2}}{f \left (a^{2}+b^{2}\right )^{2} \left (c^{2}+d^{2}\right )^{2}}-\frac {\arctan \left (\tan \left (f x +e \right )\right ) a^{2} d^{2}}{f \left (a^{2}+b^{2}\right )^{2} \left (c^{2}+d^{2}\right )^{2}}-\frac {4 \arctan \left (\tan \left (f x +e \right )\right ) a b c d}{f \left (a^{2}+b^{2}\right )^{2} \left (c^{2}+d^{2}\right )^{2}}-\frac {\arctan \left (\tan \left (f x +e \right )\right ) b^{2} c^{2}}{f \left (a^{2}+b^{2}\right )^{2} \left (c^{2}+d^{2}\right )^{2}}+\frac {\arctan \left (\tan \left (f x +e \right )\right ) b^{2} d^{2}}{f \left (a^{2}+b^{2}\right )^{2} \left (c^{2}+d^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.80, size = 878, normalized size = 3.03 \[ -\frac {\frac {{\left (4 \, a b c d - {\left (a^{2} - b^{2}\right )} c^{2} + {\left (a^{2} - b^{2}\right )} d^{2}\right )} {\left (f x + e\right )}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} c^{4} + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} c^{2} d^{2} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}} - \frac {2 \, {\left (a b^{4} c - {\left (2 \, a^{2} b^{3} + b^{5}\right )} d\right )} \log \left (b \tan \left (f x + e\right ) + a\right )}{{\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} c^{3} - 3 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} c^{2} d + 3 \, {\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} c d^{2} - {\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d^{3}} - \frac {2 \, {\left (2 \, b c^{2} d^{3} - a c d^{4} + b d^{5}\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{b^{3} c^{7} - 3 \, a b^{2} c^{6} d + 3 \, a^{2} b c d^{6} - a^{3} d^{7} + {\left (3 \, a^{2} b + 2 \, b^{3}\right )} c^{5} d^{2} - {\left (a^{3} + 6 \, a b^{2}\right )} c^{4} d^{3} + {\left (6 \, a^{2} b + b^{3}\right )} c^{3} d^{4} - {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c^{2} d^{5}} + \frac {{\left (a b c^{2} - a b d^{2} + {\left (a^{2} - b^{2}\right )} c d\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} c^{4} + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} c^{2} d^{2} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{4}} + \frac {b^{3} c^{3} + b^{3} c d^{2} + {\left (a^{3} + a b^{2}\right )} d^{3} + {\left (b^{3} c^{2} d + {\left (a^{2} b + 2 \, b^{3}\right )} d^{3}\right )} \tan \left (f x + e\right )}{{\left (a^{3} b^{2} + a b^{4}\right )} c^{5} - 2 \, {\left (a^{4} b + a^{2} b^{3}\right )} c^{4} d + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} c^{3} d^{2} - 2 \, {\left (a^{4} b + a^{2} b^{3}\right )} c^{2} d^{3} + {\left (a^{5} + a^{3} b^{2}\right )} c d^{4} + {\left ({\left (a^{2} b^{3} + b^{5}\right )} c^{4} d - 2 \, {\left (a^{3} b^{2} + a b^{4}\right )} c^{3} d^{2} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} c^{2} d^{3} - 2 \, {\left (a^{3} b^{2} + a b^{4}\right )} c d^{4} + {\left (a^{4} b + a^{2} b^{3}\right )} d^{5}\right )} \tan \left (f x + e\right )^{2} + {\left ({\left (a^{2} b^{3} + b^{5}\right )} c^{5} - {\left (a^{3} b^{2} + a b^{4}\right )} c^{4} d - {\left (a^{4} b - b^{5}\right )} c^{3} d^{2} + {\left (a^{5} - a b^{4}\right )} c^{2} d^{3} - {\left (a^{4} b + a^{2} b^{3}\right )} c d^{4} + {\left (a^{5} + a^{3} b^{2}\right )} d^{5}\right )} \tan \left (f x + e\right )}}{f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 10.38, size = 725, normalized size = 2.50 \[ \frac {\ln \left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (d\,\left (4\,a^2\,b^3+2\,b^5\right )-2\,a\,b^4\,c\right )}{f\,\left (a^7\,d^3-3\,a^6\,b\,c\,d^2+3\,a^5\,b^2\,c^2\,d+2\,a^5\,b^2\,d^3-a^4\,b^3\,c^3-6\,a^4\,b^3\,c\,d^2+6\,a^3\,b^4\,c^2\,d+a^3\,b^4\,d^3-2\,a^2\,b^5\,c^3-3\,a^2\,b^5\,c\,d^2+3\,a\,b^6\,c^2\,d-b^7\,c^3\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )}{2\,f\,\left (-a^2\,c^2\,1{}\mathrm {i}+2\,a^2\,c\,d+a^2\,d^2\,1{}\mathrm {i}+2\,a\,b\,c^2+a\,b\,c\,d\,4{}\mathrm {i}-2\,a\,b\,d^2+b^2\,c^2\,1{}\mathrm {i}-2\,b^2\,c\,d-b^2\,d^2\,1{}\mathrm {i}\right )}+\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )}{2\,f\,\left (-a^2\,c^2\,1{}\mathrm {i}-2\,a^2\,c\,d+a^2\,d^2\,1{}\mathrm {i}-2\,a\,b\,c^2+a\,b\,c\,d\,4{}\mathrm {i}+2\,a\,b\,d^2+b^2\,c^2\,1{}\mathrm {i}+2\,b^2\,c\,d-b^2\,d^2\,1{}\mathrm {i}\right )}-\frac {\frac {a^3\,d^3+a\,b^2\,d^3+b^3\,c^3+b^3\,c\,d^2}{\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )\,\left (a^2\,c^2+a^2\,d^2+b^2\,c^2+b^2\,d^2\right )}+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (a^2\,b\,d^3+b^3\,c^2\,d+2\,b^3\,d^3\right )}{\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )\,\left (a^2\,c^2+a^2\,d^2+b^2\,c^2+b^2\,d^2\right )}}{f\,\left (b\,d\,{\mathrm {tan}\left (e+f\,x\right )}^2+\left (a\,d+b\,c\right )\,\mathrm {tan}\left (e+f\,x\right )+a\,c\right )}-\frac {\ln \left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (b\,\left (4\,c^2\,d^3+2\,d^5\right )-2\,a\,c\,d^4\right )}{f\,\left (a^3\,c^4\,d^3+2\,a^3\,c^2\,d^5+a^3\,d^7-3\,a^2\,b\,c^5\,d^2-6\,a^2\,b\,c^3\,d^4-3\,a^2\,b\,c\,d^6+3\,a\,b^2\,c^6\,d+6\,a\,b^2\,c^4\,d^3+3\,a\,b^2\,c^2\,d^5-b^3\,c^7-2\,b^3\,c^5\,d^2-b^3\,c^3\,d^4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError

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