3.1227 \(\int \frac{1}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^3} \, dx\)

Optimal. Leaf size=286 \[ \frac{d^2 \left (-a^2 d^2 \left (3 c^2-d^2\right )+8 a b c^3 d-b^2 \left (3 c^2 d^2+6 c^4+d^4\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right )^3 (b c-a d)^3}-\frac{x \left (b d \left (3 c^2-d^2\right )-a \left (c^3-3 c d^2\right )\right )}{\left (a^2+b^2\right ) \left (c^2+d^2\right )^3}+\frac{b^4 \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right ) (b c-a d)^3}-\frac{d^2 \left (2 a c d-b \left (3 c^2+d^2\right )\right )}{f \left (c^2+d^2\right )^2 (b c-a d)^2 (c+d \tan (e+f x))}+\frac{d^2}{2 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^2} \]

[Out]

-(((b*d*(3*c^2 - d^2) - a*(c^3 - 3*c*d^2))*x)/((a^2 + b^2)*(c^2 + d^2)^3)) + (b^4*Log[a*Cos[e + f*x] + b*Sin[e
+ f*x]])/((a^2 + b^2)*(b*c - a*d)^3*f) + (d^2*(8*a*b*c^3*d - a^2*d^2*(3*c^2 - d^2) - b^2*(6*c^4 + 3*c^2*d^2 +
d^4))*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((b*c - a*d)^3*(c^2 + d^2)^3*f) + d^2/(2*(b*c - a*d)*(c^2 + d^2)*
f*(c + d*Tan[e + f*x])^2) - (d^2*(2*a*c*d - b*(3*c^2 + d^2)))/((b*c - a*d)^2*(c^2 + d^2)^2*f*(c + d*Tan[e + f*
x]))

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Rubi [A]  time = 0.967697, antiderivative size = 286, 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} \[ \frac{d^2 \left (-a^2 d^2 \left (3 c^2-d^2\right )+8 a b c^3 d-b^2 \left (3 c^2 d^2+6 c^4+d^4\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right )^3 (b c-a d)^3}-\frac{x \left (b d \left (3 c^2-d^2\right )-a \left (c^3-3 c d^2\right )\right )}{\left (a^2+b^2\right ) \left (c^2+d^2\right )^3}+\frac{b^4 \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right ) (b c-a d)^3}-\frac{d^2 \left (2 a c d-b \left (3 c^2+d^2\right )\right )}{f \left (c^2+d^2\right )^2 (b c-a d)^2 (c+d \tan (e+f x))}+\frac{d^2}{2 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 5.49458, size = 409, normalized size = 1.43 \[ \frac{\frac{-2 b d^2 \left (a^2+b^2\right ) \left (a^2 d^2 \left (d^2-3 c^2\right )+8 a b c^3 d-b^2 \left (3 c^2 d^2+6 c^4+d^4\right )\right ) \log (c+d \tan (e+f x))-2 b^5 \left (c^2+d^2\right )^3 \log (a+b \tan (e+f x))+(b c-a d)^3 \left (b d \left (\sqrt{-b^2}-a\right ) \left (d^2-3 c^2\right )+a \sqrt{-b^2} c \left (c^2-3 d^2\right )+b^2 \left (c^3-3 c d^2\right )\right ) \log \left (\sqrt{-b^2}-b \tan (e+f x)\right )-(b c-a d)^3 \left (b d \left (a+\sqrt{-b^2}\right ) \left (d^2-3 c^2\right )+a \sqrt{-b^2} c \left (c^2-3 d^2\right )+b^2 \left (-\left (c^3-3 c d^2\right )\right )\right ) \log \left (\sqrt{-b^2}+b \tan (e+f x)\right )}{b \left (a^2+b^2\right ) \left (c^2+d^2\right )^2 (b c-a d)^2}+\frac{2 d^2 \left (b \left (3 c^2+d^2\right )-2 a c d\right )}{\left (c^2+d^2\right ) (a d-b c) (c+d \tan (e+f x))}-\frac{d^2}{(c+d \tan (e+f x))^2}}{2 f \left (c^2+d^2\right ) (a d-b c)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.057, size = 748, normalized size = 2.6 \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))/(c+d*tan(f*x+e))^3,x)

[Out]

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

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Maxima [B]  time = 1.93845, size = 1079, normalized size = 3.77 \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))/(c+d*tan(f*x+e))^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 12.2314, size = 3745, normalized size = 13.09 \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))/(c+d*tan(f*x+e))^3,x, algorithm="fricas")

[Out]

1/2*(9*(a^2*b^2 + b^4)*c^4*d^4 - 16*(a^3*b + a*b^3)*c^3*d^5 + (7*a^4 + 10*a^2*b^2 + 3*b^4)*c^2*d^6 - 4*(a^3*b
+ a*b^3)*c*d^7 + (a^4 + a^2*b^2)*d^8 + 2*(a*b^3*c^8 - a^3*b*c^2*d^6 - 3*(a^2*b^2 + b^4)*c^7*d + 3*(a^3*b + 2*a
*b^3)*c^6*d^2 - (a^4 - b^4)*c^5*d^3 - 3*(2*a^3*b + a*b^3)*c^4*d^4 + 3*(a^4 + a^2*b^2)*c^3*d^5)*f*x - (7*(a^2*b
^2 + b^4)*c^4*d^4 - 12*(a^3*b + a*b^3)*c^3*d^5 + (5*a^4 + 6*a^2*b^2 + b^4)*c^2*d^6 - (a^4 + a^2*b^2)*d^8 - 2*(
a*b^3*c^6*d^2 - a^3*b*d^8 - 3*(a^2*b^2 + b^4)*c^5*d^3 + 3*(a^3*b + 2*a*b^3)*c^4*d^4 - (a^4 - b^4)*c^3*d^5 - 3*
(2*a^3*b + a*b^3)*c^2*d^6 + 3*(a^4 + a^2*b^2)*c*d^7)*f*x)*tan(f*x + e)^2 + (b^4*c^8 + 3*b^4*c^6*d^2 + 3*b^4*c^
4*d^4 + b^4*c^2*d^6 + (b^4*c^6*d^2 + 3*b^4*c^4*d^4 + 3*b^4*c^2*d^6 + b^4*d^8)*tan(f*x + e)^2 + 2*(b^4*c^7*d +
3*b^4*c^5*d^3 + 3*b^4*c^3*d^5 + b^4*c*d^7)*tan(f*x + e))*log((b^2*tan(f*x + e)^2 + 2*a*b*tan(f*x + e) + a^2)/(
tan(f*x + e)^2 + 1)) - (6*(a^2*b^2 + b^4)*c^6*d^2 - 8*(a^3*b + a*b^3)*c^5*d^3 + 3*(a^4 + 2*a^2*b^2 + b^4)*c^4*
d^4 - (a^4 - b^4)*c^2*d^6 + (6*(a^2*b^2 + b^4)*c^4*d^4 - 8*(a^3*b + a*b^3)*c^3*d^5 + 3*(a^4 + 2*a^2*b^2 + b^4)
*c^2*d^6 - (a^4 - b^4)*d^8)*tan(f*x + e)^2 + 2*(6*(a^2*b^2 + b^4)*c^5*d^3 - 8*(a^3*b + a*b^3)*c^4*d^4 + 3*(a^4
+ 2*a^2*b^2 + b^4)*c^3*d^5 - (a^4 - b^4)*c*d^7)*tan(f*x + e))*log((d^2*tan(f*x + e)^2 + 2*c*d*tan(f*x + e) +
c^2)/(tan(f*x + e)^2 + 1)) - 2*(4*(a^2*b^2 + b^4)*c^5*d^3 - 7*(a^3*b + a*b^3)*c^4*d^4 + 3*(a^4 - b^4)*c^3*d^5
+ 6*(a^3*b + a*b^3)*c^2*d^6 - (3*a^4 + 4*a^2*b^2 + b^4)*c*d^7 + (a^3*b + a*b^3)*d^8 - 2*(a*b^3*c^7*d - a^3*b*c
*d^7 - 3*(a^2*b^2 + b^4)*c^6*d^2 + 3*(a^3*b + 2*a*b^3)*c^5*d^3 - (a^4 - b^4)*c^4*d^4 - 3*(2*a^3*b + a*b^3)*c^3
*d^5 + 3*(a^4 + a^2*b^2)*c^2*d^6)*f*x)*tan(f*x + e))/(((a^2*b^3 + b^5)*c^9*d^2 - 3*(a^3*b^2 + a*b^4)*c^8*d^3 +
3*(a^4*b + 2*a^2*b^3 + b^5)*c^7*d^4 - (a^5 + 10*a^3*b^2 + 9*a*b^4)*c^6*d^5 + 3*(3*a^4*b + 4*a^2*b^3 + b^5)*c^
5*d^6 - 3*(a^5 + 4*a^3*b^2 + 3*a*b^4)*c^4*d^7 + (9*a^4*b + 10*a^2*b^3 + b^5)*c^3*d^8 - 3*(a^5 + 2*a^3*b^2 + a*
b^4)*c^2*d^9 + 3*(a^4*b + a^2*b^3)*c*d^10 - (a^5 + a^3*b^2)*d^11)*f*tan(f*x + e)^2 + 2*((a^2*b^3 + b^5)*c^10*d
- 3*(a^3*b^2 + a*b^4)*c^9*d^2 + 3*(a^4*b + 2*a^2*b^3 + b^5)*c^8*d^3 - (a^5 + 10*a^3*b^2 + 9*a*b^4)*c^7*d^4 +
3*(3*a^4*b + 4*a^2*b^3 + b^5)*c^6*d^5 - 3*(a^5 + 4*a^3*b^2 + 3*a*b^4)*c^5*d^6 + (9*a^4*b + 10*a^2*b^3 + b^5)*c
^4*d^7 - 3*(a^5 + 2*a^3*b^2 + a*b^4)*c^3*d^8 + 3*(a^4*b + a^2*b^3)*c^2*d^9 - (a^5 + a^3*b^2)*c*d^10)*f*tan(f*x
+ e) + ((a^2*b^3 + b^5)*c^11 - 3*(a^3*b^2 + a*b^4)*c^10*d + 3*(a^4*b + 2*a^2*b^3 + b^5)*c^9*d^2 - (a^5 + 10*a
^3*b^2 + 9*a*b^4)*c^8*d^3 + 3*(3*a^4*b + 4*a^2*b^3 + b^5)*c^7*d^4 - 3*(a^5 + 4*a^3*b^2 + 3*a*b^4)*c^6*d^5 + (9
*a^4*b + 10*a^2*b^3 + b^5)*c^5*d^6 - 3*(a^5 + 2*a^3*b^2 + a*b^4)*c^4*d^7 + 3*(a^4*b + a^2*b^3)*c^3*d^8 - (a^5
+ a^3*b^2)*c^2*d^9)*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))/(c+d*tan(f*x+e))**3,x)

[Out]

Timed out

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Giac [B]  time = 1.47956, size = 1501, normalized size = 5.25 \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))/(c+d*tan(f*x+e))^3,x, algorithm="giac")

[Out]

1/2*(2*b^5*log(abs(b*tan(f*x + e) + a))/(a^2*b^4*c^3 + b^6*c^3 - 3*a^3*b^3*c^2*d - 3*a*b^5*c^2*d + 3*a^4*b^2*c
*d^2 + 3*a^2*b^4*c*d^2 - a^5*b*d^3 - a^3*b^3*d^3) + 2*(a*c^3 - 3*b*c^2*d - 3*a*c*d^2 + b*d^3)*(f*x + e)/(a^2*c
^6 + b^2*c^6 + 3*a^2*c^4*d^2 + 3*b^2*c^4*d^2 + 3*a^2*c^2*d^4 + 3*b^2*c^2*d^4 + a^2*d^6 + b^2*d^6) - (b*c^3 + 3
*a*c^2*d - 3*b*c*d^2 - a*d^3)*log(tan(f*x + e)^2 + 1)/(a^2*c^6 + b^2*c^6 + 3*a^2*c^4*d^2 + 3*b^2*c^4*d^2 + 3*a
^2*c^2*d^4 + 3*b^2*c^2*d^4 + a^2*d^6 + b^2*d^6) - 2*(6*b^2*c^4*d^3 - 8*a*b*c^3*d^4 + 3*a^2*c^2*d^5 + 3*b^2*c^2
*d^5 - a^2*d^7 + b^2*d^7)*log(abs(d*tan(f*x + e) + c))/(b^3*c^9*d - 3*a*b^2*c^8*d^2 + 3*a^2*b*c^7*d^3 + 3*b^3*
c^7*d^3 - a^3*c^6*d^4 - 9*a*b^2*c^6*d^4 + 9*a^2*b*c^5*d^5 + 3*b^3*c^5*d^5 - 3*a^3*c^4*d^6 - 9*a*b^2*c^4*d^6 +
9*a^2*b*c^3*d^7 + b^3*c^3*d^7 - 3*a^3*c^2*d^8 - 3*a*b^2*c^2*d^8 + 3*a^2*b*c*d^9 - a^3*d^10) + (18*b^2*c^4*d^4*
tan(f*x + e)^2 - 24*a*b*c^3*d^5*tan(f*x + e)^2 + 9*a^2*c^2*d^6*tan(f*x + e)^2 + 9*b^2*c^2*d^6*tan(f*x + e)^2 -
3*a^2*d^8*tan(f*x + e)^2 + 3*b^2*d^8*tan(f*x + e)^2 + 42*b^2*c^5*d^3*tan(f*x + e) - 58*a*b*c^4*d^4*tan(f*x +
e) + 22*a^2*c^3*d^5*tan(f*x + e) + 26*b^2*c^3*d^5*tan(f*x + e) - 12*a*b*c^2*d^6*tan(f*x + e) - 2*a^2*c*d^7*tan
(f*x + e) + 8*b^2*c*d^7*tan(f*x + e) - 2*a*b*d^8*tan(f*x + e) + 25*b^2*c^6*d^2 - 36*a*b*c^5*d^3 + 14*a^2*c^4*d
^4 + 19*b^2*c^4*d^4 - 16*a*b*c^3*d^5 + 3*a^2*c^2*d^6 + 6*b^2*c^2*d^6 - 4*a*b*c*d^7 + a^2*d^8)/((b^3*c^9 - 3*a*
b^2*c^8*d + 3*a^2*b*c^7*d^2 + 3*b^3*c^7*d^2 - a^3*c^6*d^3 - 9*a*b^2*c^6*d^3 + 9*a^2*b*c^5*d^4 + 3*b^3*c^5*d^4
- 3*a^3*c^4*d^5 - 9*a*b^2*c^4*d^5 + 9*a^2*b*c^3*d^6 + b^3*c^3*d^6 - 3*a^3*c^2*d^7 - 3*a*b^2*c^2*d^7 + 3*a^2*b*
c*d^8 - a^3*d^9)*(d*tan(f*x + e) + c)^2))/f