3.1215 \(\int \frac{1}{(a+b \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx\)

Optimal. Leaf size=279 \[ -\frac{b^2 \left (-3 a^2 b^2 \left (c^2+d^2\right )+8 a^3 b c d-6 a^4 d^2+b^4 \left (c^2-d^2\right )\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right )^3 (b c-a d)^3}+\frac{x \left (-3 a^2 b d+a^3 c-3 a b^2 c+b^3 d\right )}{\left (a^2+b^2\right )^3 \left (c^2+d^2\right )}-\frac{b^2 \left (-3 a^2 d+2 a b c-b^2 d\right )}{f \left (a^2+b^2\right )^2 (b c-a d)^2 (a+b \tan (e+f x))}-\frac{b^2}{2 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^2}-\frac{d^4 \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right ) (b c-a d)^3} \]

[Out]

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

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Rubi [A]  time = 0.925831, antiderivative size = 279, 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} \[ -\frac{b^2 \left (-3 a^2 b^2 \left (c^2+d^2\right )+8 a^3 b c d-6 a^4 d^2+b^4 \left (c^2-d^2\right )\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right )^3 (b c-a d)^3}+\frac{x \left (-3 a^2 b d+a^3 c-3 a b^2 c+b^3 d\right )}{\left (a^2+b^2\right )^3 \left (c^2+d^2\right )}-\frac{b^2 \left (-3 a^2 d+2 a b c-b^2 d\right )}{f \left (a^2+b^2\right )^2 (b c-a d)^2 (a+b \tan (e+f x))}-\frac{b^2}{2 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^2}-\frac{d^4 \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right ) (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 6.79733, size = 529, normalized size = 1.9 \[ -\frac{-\frac{-\frac{2 b d^4 \left (a^2+b^2\right )^2 \log (c+d \tan (e+f x))}{\left (c^2+d^2\right ) (b c-a d)}-\frac{b (b c-a d)^2 \left (\frac{\sqrt{-b^2} \left (-3 a^2 b d+a^3 c-3 a b^2 c+b^3 d\right )}{b}+3 a^2 b c+a^3 d-3 a b^2 d-b^3 c\right ) \log \left (\sqrt{-b^2}-b \tan (e+f x)\right )}{\left (a^2+b^2\right ) \left (c^2+d^2\right )}-\frac{2 b^3 \left (-3 a^2 b^2 \left (c^2+d^2\right )+8 a^3 b c d-6 a^4 d^2+b^4 \left (c^2-d^2\right )\right ) \log (a+b \tan (e+f x))}{\left (a^2+b^2\right ) (b c-a d)}-\frac{b (b c-a d)^2 \left (\frac{b \left (-3 a^2 b d+a^3 c-3 a b^2 c+b^3 d\right )}{\sqrt{-b^2}}+3 a^2 b c+a^3 d-3 a b^2 d-b^3 c\right ) \log \left (\sqrt{-b^2}+b \tan (e+f x)\right )}{\left (a^2+b^2\right ) \left (c^2+d^2\right )}}{b f \left (a^2+b^2\right ) (b c-a d)}-\frac{-2 b^2 \left (a^2 (-d)+a b c-b^2 d\right )-a \left (2 b^2 (b c-a d)-2 a b^2 d\right )}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))}}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac{b^2}{2 f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.059, size = 747, normalized size = 2.7 \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))^3/(c+d*tan(f*x+e)),x)

[Out]

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

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Maxima [B]  time = 1.93082, size = 1081, normalized size = 3.87 \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))^3/(c+d*tan(f*x+e)),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 7.20337, size = 3822, normalized size = 13.7 \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))^3/(c+d*tan(f*x+e)),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [B]  time = 1.42346, size = 1500, normalized size = 5.38 \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))^3/(c+d*tan(f*x+e)),x, algorithm="giac")

[Out]

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