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*(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]))

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Rubi [A]  time = 1.03094, antiderivative size = 290, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, 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 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]

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]

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*}

Mathematica [A]  time = 6.91134, size = 556, normalized size = 1.92 \[ -\frac{-\frac{d^2 \left (a^2 d-a b c+2 b^2 d\right )-c \left (b d (b c-a d)-2 b^2 c d\right )}{f \left (c^2+d^2\right ) (a d-b c) (c+d \tan (e+f x))}-\frac{-\frac{2 b^4 \left (c^2+d^2\right ) \left (-2 a^2 d+a b c-b^2 d\right ) \log (a+b \tan (e+f x))}{\left (a^2+b^2\right ) (b c-a d)}+\frac{b (b c-a d)^2 \left (\frac{b \left (a^2 \left (-\left (c^2-d^2\right )\right )+4 a b c d+b^2 \left (c^2-d^2\right )\right )}{\sqrt{-b^2}}+2 a^2 c d+2 a b c^2-2 a b d^2-2 b^2 c d\right ) \log \left (\sqrt{-b^2}-b \tan (e+f x)\right )}{2 \left (a^2+b^2\right ) \left (c^2+d^2\right )}+\frac{b (b c-a d)^2 \left (\frac{\sqrt{-b^2} \left (a^2 \left (-\left (c^2-d^2\right )\right )+4 a b c d+b^2 \left (c^2-d^2\right )\right )}{b}+2 a^2 c d+2 a b c^2-2 a b d^2-2 b^2 c d\right ) \log \left (\sqrt{-b^2}+b \tan (e+f x)\right )}{2 \left (a^2+b^2\right ) \left (c^2+d^2\right )}+\frac{2 b d^3 \left (a^2+b^2\right ) \left (a c d-b \left (2 c^2+d^2\right )\right ) \log (c+d \tan (e+f x))}{\left (c^2+d^2\right ) (b c-a d)}}{b f \left (c^2+d^2\right ) (a d-b c)}}{\left (a^2+b^2\right ) (b c-a d)}-\frac{b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) (c+d \tan (e+f x))} \]

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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Maple [B]  time = 0.062, size = 652, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}

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

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Maxima [B]  time = 1.95439, size = 1185, normalized size = 4.09 \begin{align*} \text{result too large to display} \end{align*}

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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Fricas [B]  time = 8.97832, size = 4374, normalized size = 15.08 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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

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]

Exception raised: NotImplementedError