3.296 \(\int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx\)

Optimal. Leaf size=399 \[ -\frac {(4 A b-a B) \log (\sin (c+d x))}{a^5 d}-\frac {b \left (3 a^2 A-a b B+4 A b^2\right )}{3 a^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}-\frac {b \left (2 a^4 A-3 a^3 b B+8 a^2 A b^2-a b^3 B+4 A b^4\right )}{2 a^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac {x \left (a^4 A+4 a^3 b B-6 a^2 A b^2-4 a b^3 B+A b^4\right )}{\left (a^2+b^2\right )^4}-\frac {b \left (a^6 A-6 a^5 b B+13 a^4 A b^2-3 a^3 b^3 B+12 a^2 A b^4-a b^5 B+4 A b^6\right )}{a^4 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac {b^2 \left (-10 a^7 B+20 a^6 A b-5 a^5 b^2 B+24 a^4 A b^3-4 a^3 b^4 B+16 a^2 A b^5-a b^6 B+4 A b^7\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^5 d \left (a^2+b^2\right )^4}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^3} \]

[Out]

-(A*a^4-6*A*a^2*b^2+A*b^4+4*B*a^3*b-4*B*a*b^3)*x/(a^2+b^2)^4-(4*A*b-B*a)*ln(sin(d*x+c))/a^5/d+b^2*(20*A*a^6*b+
24*A*a^4*b^3+16*A*a^2*b^5+4*A*b^7-10*B*a^7-5*B*a^5*b^2-4*B*a^3*b^4-B*a*b^6)*ln(a*cos(d*x+c)+b*sin(d*x+c))/a^5/
(a^2+b^2)^4/d-1/3*b*(3*A*a^2+4*A*b^2-B*a*b)/a^2/(a^2+b^2)/d/(a+b*tan(d*x+c))^3-A*cot(d*x+c)/a/d/(a+b*tan(d*x+c
))^3-1/2*b*(2*A*a^4+8*A*a^2*b^2+4*A*b^4-3*B*a^3*b-B*a*b^3)/a^3/(a^2+b^2)^2/d/(a+b*tan(d*x+c))^2-b*(A*a^6+13*A*
a^4*b^2+12*A*a^2*b^4+4*A*b^6-6*B*a^5*b-3*B*a^3*b^3-B*a*b^5)/a^4/(a^2+b^2)^3/d/(a+b*tan(d*x+c))

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Rubi [A]  time = 1.32, antiderivative size = 399, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3609, 3649, 3651, 3530, 3475} \[ -\frac {b \left (13 a^4 A b^2+12 a^2 A b^4+a^6 A-3 a^3 b^3 B-6 a^5 b B-a b^5 B+4 A b^6\right )}{a^4 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac {b \left (8 a^2 A b^2+2 a^4 A-3 a^3 b B-a b^3 B+4 A b^4\right )}{2 a^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac {b \left (3 a^2 A-a b B+4 A b^2\right )}{3 a^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {b^2 \left (24 a^4 A b^3+16 a^2 A b^5+20 a^6 A b-5 a^5 b^2 B-4 a^3 b^4 B-10 a^7 B-a b^6 B+4 A b^7\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^5 d \left (a^2+b^2\right )^4}-\frac {x \left (-6 a^2 A b^2+a^4 A+4 a^3 b B-4 a b^3 B+A b^4\right )}{\left (a^2+b^2\right )^4}-\frac {(4 A b-a B) \log (\sin (c+d x))}{a^5 d}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

Int[(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4,x]

[Out]

-(((a^4*A - 6*a^2*A*b^2 + A*b^4 + 4*a^3*b*B - 4*a*b^3*B)*x)/(a^2 + b^2)^4) - ((4*A*b - a*B)*Log[Sin[c + d*x]])
/(a^5*d) + (b^2*(20*a^6*A*b + 24*a^4*A*b^3 + 16*a^2*A*b^5 + 4*A*b^7 - 10*a^7*B - 5*a^5*b^2*B - 4*a^3*b^4*B - a
*b^6*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^5*(a^2 + b^2)^4*d) - (b*(3*a^2*A + 4*A*b^2 - a*b*B))/(3*a^2*(
a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) - (A*Cot[c + d*x])/(a*d*(a + b*Tan[c + d*x])^3) - (b*(2*a^4*A + 8*a^2*A*b
^2 + 4*A*b^4 - 3*a^3*b*B - a*b^3*B))/(2*a^3*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) - (b*(a^6*A + 13*a^4*A*b^2
+ 12*a^2*A*b^4 + 4*A*b^6 - 6*a^5*b*B - 3*a^3*b^3*B - a*b^5*B))/(a^4*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, 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 3609

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(A*b - a*B)*(a + b*Tan[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[b*B*(b*c*(m + 1) + a*d*(n + 1)) + A*(a*(b*c - a*d)*(m + 1) - b^2*d*(
m + n + 2)) - (A*b - a*B)*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n + 2)*Tan[e + f*x]^2, x], x
], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& LtQ[m, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&  !(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 {\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx &=-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^3}-\frac {\int \frac {\cot (c+d x) \left (4 A b-a B+a A \tan (c+d x)+4 A b \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^4} \, dx}{a}\\ &=-\frac {b \left (3 a^2 A+4 A b^2-a b B\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^3}-\frac {\int \frac {\cot (c+d x) \left (3 \left (a^2+b^2\right ) (4 A b-a B)+3 a^2 (a A+b B) \tan (c+d x)+3 b \left (3 a^2 A+4 A b^2-a b B\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{3 a^2 \left (a^2+b^2\right )}\\ &=-\frac {b \left (3 a^2 A+4 A b^2-a b B\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^3}-\frac {b \left (2 a^4 A+8 a^2 A b^2+4 A b^4-3 a^3 b B-a b^3 B\right )}{2 a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {\int \frac {\cot (c+d x) \left (6 \left (a^2+b^2\right )^2 (4 A b-a B)+6 a^3 \left (a^2 A-A b^2+2 a b B\right ) \tan (c+d x)+6 b \left (2 a^4 A+8 a^2 A b^2+4 A b^4-3 a^3 b B-a b^3 B\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{6 a^3 \left (a^2+b^2\right )^2}\\ &=-\frac {b \left (3 a^2 A+4 A b^2-a b B\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^3}-\frac {b \left (2 a^4 A+8 a^2 A b^2+4 A b^4-3 a^3 b B-a b^3 B\right )}{2 a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {b \left (a^6 A+13 a^4 A b^2+12 a^2 A b^4+4 A b^6-6 a^5 b B-3 a^3 b^3 B-a b^5 B\right )}{a^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}-\frac {\int \frac {\cot (c+d x) \left (6 \left (a^2+b^2\right )^3 (4 A b-a B)+6 a^4 \left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right ) \tan (c+d x)+6 b \left (a^6 A+13 a^4 A b^2+12 a^2 A b^4+4 A b^6-6 a^5 b B-3 a^3 b^3 B-a b^5 B\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{6 a^4 \left (a^2+b^2\right )^3}\\ &=-\frac {\left (a^4 A-6 a^2 A b^2+A b^4+4 a^3 b B-4 a b^3 B\right ) x}{\left (a^2+b^2\right )^4}-\frac {b \left (3 a^2 A+4 A b^2-a b B\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^3}-\frac {b \left (2 a^4 A+8 a^2 A b^2+4 A b^4-3 a^3 b B-a b^3 B\right )}{2 a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {b \left (a^6 A+13 a^4 A b^2+12 a^2 A b^4+4 A b^6-6 a^5 b B-3 a^3 b^3 B-a b^5 B\right )}{a^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}-\frac {(4 A b-a B) \int \cot (c+d x) \, dx}{a^5}+\frac {\left (b^2 \left (20 a^6 A b+24 a^4 A b^3+16 a^2 A b^5+4 A b^7-10 a^7 B-5 a^5 b^2 B-4 a^3 b^4 B-a b^6 B\right )\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^5 \left (a^2+b^2\right )^4}\\ &=-\frac {\left (a^4 A-6 a^2 A b^2+A b^4+4 a^3 b B-4 a b^3 B\right ) x}{\left (a^2+b^2\right )^4}-\frac {(4 A b-a B) \log (\sin (c+d x))}{a^5 d}+\frac {b^2 \left (20 a^6 A b+24 a^4 A b^3+16 a^2 A b^5+4 A b^7-10 a^7 B-5 a^5 b^2 B-4 a^3 b^4 B-a b^6 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^5 \left (a^2+b^2\right )^4 d}-\frac {b \left (3 a^2 A+4 A b^2-a b B\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^3}-\frac {b \left (2 a^4 A+8 a^2 A b^2+4 A b^4-3 a^3 b B-a b^3 B\right )}{2 a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {b \left (a^6 A+13 a^4 A b^2+12 a^2 A b^4+4 A b^6-6 a^5 b B-3 a^3 b^3 B-a b^5 B\right )}{a^4 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}\\ \end {align*}

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Mathematica [C]  time = 6.27, size = 357, normalized size = 0.89 \[ \frac {\frac {6 (a B-4 A b) \log (\tan (c+d x))}{a^5}-\frac {6 A \cot (c+d x)}{a^4}+\frac {2 b^2 (a B-A b)}{a^2 \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {3 b^2 \left (3 a^3 B-4 a^2 A b+a b^2 B-2 A b^3\right )}{a^3 \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac {6 b^2 \left (6 a^5 B-10 a^4 A b+3 a^3 b^2 B-9 a^2 A b^3+a b^4 B-3 A b^5\right )}{a^4 \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac {6 b^2 \left (10 a^7 B-20 a^6 A b+5 a^5 b^2 B-24 a^4 A b^3+4 a^3 b^4 B-16 a^2 A b^5+a b^6 B-4 A b^7\right ) \log (a+b \tan (c+d x))}{a^5 \left (a^2+b^2\right )^4}+\frac {3 i (A+i B) \log (-\tan (c+d x)+i)}{(a+i b)^4}-\frac {3 (B+i A) \log (\tan (c+d x)+i)}{(a-i b)^4}}{6 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[c + d*x]^2*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4,x]

[Out]

((-6*A*Cot[c + d*x])/a^4 + ((3*I)*(A + I*B)*Log[I - Tan[c + d*x]])/(a + I*b)^4 + (6*(-4*A*b + a*B)*Log[Tan[c +
d*x]])/a^5 - (3*(I*A + B)*Log[I + Tan[c + d*x]])/(a - I*b)^4 - (6*b^2*(-20*a^6*A*b - 24*a^4*A*b^3 - 16*a^2*A*
b^5 - 4*A*b^7 + 10*a^7*B + 5*a^5*b^2*B + 4*a^3*b^4*B + a*b^6*B)*Log[a + b*Tan[c + d*x]])/(a^5*(a^2 + b^2)^4) +
(2*b^2*(-(A*b) + a*B))/(a^2*(a^2 + b^2)*(a + b*Tan[c + d*x])^3) + (3*b^2*(-4*a^2*A*b - 2*A*b^3 + 3*a^3*B + a*
b^2*B))/(a^3*(a^2 + b^2)^2*(a + b*Tan[c + d*x])^2) + (6*b^2*(-10*a^4*A*b - 9*a^2*A*b^3 - 3*A*b^5 + 6*a^5*B + 3
*a^3*b^2*B + a*b^4*B))/(a^4*(a^2 + b^2)^3*(a + b*Tan[c + d*x])))/(6*d)

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fricas [B]  time = 1.30, size = 1510, normalized size = 3.78 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/6*(6*A*a^12 + 24*A*a^10*b^2 + 36*A*a^8*b^4 + 24*A*a^6*b^6 + 6*A*a^4*b^8 + (47*B*a^7*b^5 - 74*A*a^6*b^6 + 6*
B*a^5*b^7 - 42*A*a^4*b^8 + 3*B*a^3*b^9 - 12*A*a^2*b^10 + 6*(A*a^9*b^3 + 4*B*a^8*b^4 - 6*A*a^7*b^5 - 4*B*a^6*b^
6 + A*a^5*b^7)*d*x)*tan(d*x + c)^4 + 3*(2*A*a^9*b^3 + 35*B*a^8*b^4 - 46*A*a^7*b^5 - 12*B*a^6*b^6 + 8*A*a^5*b^7
- 5*B*a^4*b^8 + 20*A*a^3*b^9 - 2*B*a^2*b^10 + 8*A*a*b^11 + 6*(A*a^10*b^2 + 4*B*a^9*b^3 - 6*A*a^8*b^4 - 4*B*a^
7*b^5 + A*a^6*b^6)*d*x)*tan(d*x + c)^3 + 3*(6*A*a^10*b^2 + 20*B*a^9*b^3 - 6*A*a^8*b^4 - 37*B*a^7*b^5 + 80*A*a^
6*b^6 - 18*B*a^5*b^7 + 68*A*a^4*b^8 - 5*B*a^3*b^9 + 20*A*a^2*b^10 + 6*(A*a^11*b + 4*B*a^10*b^2 - 6*A*a^9*b^3 -
4*B*a^8*b^4 + A*a^7*b^5)*d*x)*tan(d*x + c)^2 - 3*((B*a^9*b^3 - 4*A*a^8*b^4 + 4*B*a^7*b^5 - 16*A*a^6*b^6 + 6*B
*a^5*b^7 - 24*A*a^4*b^8 + 4*B*a^3*b^9 - 16*A*a^2*b^10 + B*a*b^11 - 4*A*b^12)*tan(d*x + c)^4 + 3*(B*a^10*b^2 -
4*A*a^9*b^3 + 4*B*a^8*b^4 - 16*A*a^7*b^5 + 6*B*a^6*b^6 - 24*A*a^5*b^7 + 4*B*a^4*b^8 - 16*A*a^3*b^9 + B*a^2*b^1
0 - 4*A*a*b^11)*tan(d*x + c)^3 + 3*(B*a^11*b - 4*A*a^10*b^2 + 4*B*a^9*b^3 - 16*A*a^8*b^4 + 6*B*a^7*b^5 - 24*A*
a^6*b^6 + 4*B*a^5*b^7 - 16*A*a^4*b^8 + B*a^3*b^9 - 4*A*a^2*b^10)*tan(d*x + c)^2 + (B*a^12 - 4*A*a^11*b + 4*B*a
^10*b^2 - 16*A*a^9*b^3 + 6*B*a^8*b^4 - 24*A*a^7*b^5 + 4*B*a^6*b^6 - 16*A*a^5*b^7 + B*a^4*b^8 - 4*A*a^3*b^9)*ta
n(d*x + c))*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1)) + 3*((10*B*a^7*b^5 - 20*A*a^6*b^6 + 5*B*a^5*b^7 - 24*A*a^
4*b^8 + 4*B*a^3*b^9 - 16*A*a^2*b^10 + B*a*b^11 - 4*A*b^12)*tan(d*x + c)^4 + 3*(10*B*a^8*b^4 - 20*A*a^7*b^5 + 5
*B*a^6*b^6 - 24*A*a^5*b^7 + 4*B*a^4*b^8 - 16*A*a^3*b^9 + B*a^2*b^10 - 4*A*a*b^11)*tan(d*x + c)^3 + 3*(10*B*a^9
*b^3 - 20*A*a^8*b^4 + 5*B*a^7*b^5 - 24*A*a^6*b^6 + 4*B*a^5*b^7 - 16*A*a^4*b^8 + B*a^3*b^9 - 4*A*a^2*b^10)*tan(
d*x + c)^2 + (10*B*a^10*b^2 - 20*A*a^9*b^3 + 5*B*a^8*b^4 - 24*A*a^7*b^5 + 4*B*a^6*b^6 - 16*A*a^5*b^7 + B*a^4*b
^8 - 4*A*a^3*b^9)*tan(d*x + c))*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^2)/(tan(d*x + c)^2 + 1)) + (1
8*A*a^11*b + 72*A*a^9*b^3 - 75*B*a^8*b^4 + 216*A*a^7*b^5 - 42*B*a^6*b^6 + 162*A*a^5*b^7 - 11*B*a^4*b^8 + 44*A*
a^3*b^9 + 6*(A*a^12 + 4*B*a^11*b - 6*A*a^10*b^2 - 4*B*a^9*b^3 + A*a^8*b^4)*d*x)*tan(d*x + c))/((a^13*b^3 + 4*a
^11*b^5 + 6*a^9*b^7 + 4*a^7*b^9 + a^5*b^11)*d*tan(d*x + c)^4 + 3*(a^14*b^2 + 4*a^12*b^4 + 6*a^10*b^6 + 4*a^8*b
^8 + a^6*b^10)*d*tan(d*x + c)^3 + 3*(a^15*b + 4*a^13*b^3 + 6*a^11*b^5 + 4*a^9*b^7 + a^7*b^9)*d*tan(d*x + c)^2
+ (a^16 + 4*a^14*b^2 + 6*a^12*b^4 + 4*a^10*b^6 + a^8*b^8)*d*tan(d*x + c))

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giac [B]  time = 3.44, size = 846, normalized size = 2.12 \[ -\frac {\frac {6 \, {\left (A a^{4} + 4 \, B a^{3} b - 6 \, A a^{2} b^{2} - 4 \, B a b^{3} + A b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {3 \, {\left (B a^{4} - 4 \, A a^{3} b - 6 \, B a^{2} b^{2} + 4 \, A a b^{3} + B b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {6 \, {\left (10 \, B a^{7} b^{3} - 20 \, A a^{6} b^{4} + 5 \, B a^{5} b^{5} - 24 \, A a^{4} b^{6} + 4 \, B a^{3} b^{7} - 16 \, A a^{2} b^{8} + B a b^{9} - 4 \, A b^{10}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{13} b + 4 \, a^{11} b^{3} + 6 \, a^{9} b^{5} + 4 \, a^{7} b^{7} + a^{5} b^{9}} - \frac {110 \, B a^{7} b^{5} \tan \left (d x + c\right )^{3} - 220 \, A a^{6} b^{6} \tan \left (d x + c\right )^{3} + 55 \, B a^{5} b^{7} \tan \left (d x + c\right )^{3} - 264 \, A a^{4} b^{8} \tan \left (d x + c\right )^{3} + 44 \, B a^{3} b^{9} \tan \left (d x + c\right )^{3} - 176 \, A a^{2} b^{10} \tan \left (d x + c\right )^{3} + 11 \, B a b^{11} \tan \left (d x + c\right )^{3} - 44 \, A b^{12} \tan \left (d x + c\right )^{3} + 366 \, B a^{8} b^{4} \tan \left (d x + c\right )^{2} - 720 \, A a^{7} b^{5} \tan \left (d x + c\right )^{2} + 219 \, B a^{6} b^{6} \tan \left (d x + c\right )^{2} - 906 \, A a^{5} b^{7} \tan \left (d x + c\right )^{2} + 156 \, B a^{4} b^{8} \tan \left (d x + c\right )^{2} - 600 \, A a^{3} b^{9} \tan \left (d x + c\right )^{2} + 39 \, B a^{2} b^{10} \tan \left (d x + c\right )^{2} - 150 \, A a b^{11} \tan \left (d x + c\right )^{2} + 411 \, B a^{9} b^{3} \tan \left (d x + c\right ) - 792 \, A a^{8} b^{4} \tan \left (d x + c\right ) + 294 \, B a^{7} b^{5} \tan \left (d x + c\right ) - 1050 \, A a^{6} b^{6} \tan \left (d x + c\right ) + 195 \, B a^{5} b^{7} \tan \left (d x + c\right ) - 696 \, A a^{4} b^{8} \tan \left (d x + c\right ) + 48 \, B a^{3} b^{9} \tan \left (d x + c\right ) - 174 \, A a^{2} b^{10} \tan \left (d x + c\right ) + 157 \, B a^{10} b^{2} - 294 \, A a^{9} b^{3} + 136 \, B a^{8} b^{4} - 414 \, A a^{7} b^{5} + 89 \, B a^{6} b^{6} - 278 \, A a^{5} b^{7} + 22 \, B a^{4} b^{8} - 70 \, A a^{3} b^{9}}{{\left (a^{13} + 4 \, a^{11} b^{2} + 6 \, a^{9} b^{4} + 4 \, a^{7} b^{6} + a^{5} b^{8}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3}} - \frac {6 \, {\left (B a - 4 \, A b\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{5}} + \frac {6 \, {\left (B a \tan \left (d x + c\right ) - 4 \, A b \tan \left (d x + c\right ) + A a\right )}}{a^{5} \tan \left (d x + c\right )}}{6 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x, algorithm="giac")

[Out]

-1/6*(6*(A*a^4 + 4*B*a^3*b - 6*A*a^2*b^2 - 4*B*a*b^3 + A*b^4)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b
^6 + b^8) + 3*(B*a^4 - 4*A*a^3*b - 6*B*a^2*b^2 + 4*A*a*b^3 + B*b^4)*log(tan(d*x + c)^2 + 1)/(a^8 + 4*a^6*b^2 +
6*a^4*b^4 + 4*a^2*b^6 + b^8) + 6*(10*B*a^7*b^3 - 20*A*a^6*b^4 + 5*B*a^5*b^5 - 24*A*a^4*b^6 + 4*B*a^3*b^7 - 16
*A*a^2*b^8 + B*a*b^9 - 4*A*b^10)*log(abs(b*tan(d*x + c) + a))/(a^13*b + 4*a^11*b^3 + 6*a^9*b^5 + 4*a^7*b^7 + a
^5*b^9) - (110*B*a^7*b^5*tan(d*x + c)^3 - 220*A*a^6*b^6*tan(d*x + c)^3 + 55*B*a^5*b^7*tan(d*x + c)^3 - 264*A*a
^4*b^8*tan(d*x + c)^3 + 44*B*a^3*b^9*tan(d*x + c)^3 - 176*A*a^2*b^10*tan(d*x + c)^3 + 11*B*a*b^11*tan(d*x + c)
^3 - 44*A*b^12*tan(d*x + c)^3 + 366*B*a^8*b^4*tan(d*x + c)^2 - 720*A*a^7*b^5*tan(d*x + c)^2 + 219*B*a^6*b^6*ta
n(d*x + c)^2 - 906*A*a^5*b^7*tan(d*x + c)^2 + 156*B*a^4*b^8*tan(d*x + c)^2 - 600*A*a^3*b^9*tan(d*x + c)^2 + 39
*B*a^2*b^10*tan(d*x + c)^2 - 150*A*a*b^11*tan(d*x + c)^2 + 411*B*a^9*b^3*tan(d*x + c) - 792*A*a^8*b^4*tan(d*x
+ c) + 294*B*a^7*b^5*tan(d*x + c) - 1050*A*a^6*b^6*tan(d*x + c) + 195*B*a^5*b^7*tan(d*x + c) - 696*A*a^4*b^8*t
an(d*x + c) + 48*B*a^3*b^9*tan(d*x + c) - 174*A*a^2*b^10*tan(d*x + c) + 157*B*a^10*b^2 - 294*A*a^9*b^3 + 136*B
*a^8*b^4 - 414*A*a^7*b^5 + 89*B*a^6*b^6 - 278*A*a^5*b^7 + 22*B*a^4*b^8 - 70*A*a^3*b^9)/((a^13 + 4*a^11*b^2 + 6
*a^9*b^4 + 4*a^7*b^6 + a^5*b^8)*(b*tan(d*x + c) + a)^3) - 6*(B*a - 4*A*b)*log(abs(tan(d*x + c)))/a^5 + 6*(B*a*
tan(d*x + c) - 4*A*b*tan(d*x + c) + A*a)/(a^5*tan(d*x + c)))/d

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maple [B]  time = 0.69, size = 969, normalized size = 2.43 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x)

[Out]

-1/2/d/(a^2+b^2)^4*ln(1+tan(d*x+c)^2)*a^4*B-1/2/d/(a^2+b^2)^4*ln(1+tan(d*x+c)^2)*B*b^4-1/d/(a^2+b^2)^4*A*arcta
n(tan(d*x+c))*a^4-1/d*A/a^4/tan(d*x+c)+1/d/a^4*ln(tan(d*x+c))*B-10/d/(a^2+b^2)^3/(a+b*tan(d*x+c))*A*b^3+3/2/d/
(a^2+b^2)^2/(a+b*tan(d*x+c))^2*b^2*B-1/d/(a^2+b^2)^4*A*arctan(tan(d*x+c))*b^4-5/d/(a^2+b^2)^4*ln(a+b*tan(d*x+c
))*B*b^4-4/d/a^5*ln(tan(d*x+c))*A*b+20/d*a/(a^2+b^2)^4*b^3*ln(a+b*tan(d*x+c))*A+6/d/(a^2+b^2)^4*A*arctan(tan(d
*x+c))*a^2*b^2-4/d/(a^2+b^2)^4*B*arctan(tan(d*x+c))*a^3*b+4/d/(a^2+b^2)^4*B*arctan(tan(d*x+c))*a*b^3-2/d/(a^2+
b^2)^4*ln(1+tan(d*x+c)^2)*a*A*b^3+3/d/(a^2+b^2)^4*ln(1+tan(d*x+c)^2)*a^2*b^2*B+2/d/(a^2+b^2)^4*ln(1+tan(d*x+c)
^2)*A*a^3*b+6/d/(a^2+b^2)^3/(a+b*tan(d*x+c))*B*a*b^2-10/d*a^2/(a^2+b^2)^4*b^2*ln(a+b*tan(d*x+c))*B-1/d*b^8/a^4
/(a^2+b^2)^4*ln(a+b*tan(d*x+c))*B-4/d*b^6/a^2/(a^2+b^2)^4*ln(a+b*tan(d*x+c))*B+1/d*b^6/a^3/(a^2+b^2)^3/(a+b*ta
n(d*x+c))*B+24/d*b^5/a/(a^2+b^2)^4*ln(a+b*tan(d*x+c))*A+16/d*b^7/a^3/(a^2+b^2)^4*ln(a+b*tan(d*x+c))*A+4/d*b^9/
a^5/(a^2+b^2)^4*ln(a+b*tan(d*x+c))*A-1/d*b^5/a^3/(a^2+b^2)^2/(a+b*tan(d*x+c))^2*A+1/2/d*b^4/a^2/(a^2+b^2)^2/(a
+b*tan(d*x+c))^2*B-9/d*b^5/a^2/(a^2+b^2)^3/(a+b*tan(d*x+c))*A-1/3/d*b^3/a^2/(a^2+b^2)/(a+b*tan(d*x+c))^3*A+1/3
/d*b^2/a/(a^2+b^2)/(a+b*tan(d*x+c))^3*B-3/d*b^7/a^4/(a^2+b^2)^3/(a+b*tan(d*x+c))*A+3/d*b^4/a/(a^2+b^2)^3/(a+b*
tan(d*x+c))*B-2/d*b^3/a/(a^2+b^2)^2/(a+b*tan(d*x+c))^2*A

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maxima [A]  time = 0.91, size = 698, normalized size = 1.75 \[ -\frac {\frac {6 \, {\left (A a^{4} + 4 \, B a^{3} b - 6 \, A a^{2} b^{2} - 4 \, B a b^{3} + A b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {6 \, {\left (10 \, B a^{7} b^{2} - 20 \, A a^{6} b^{3} + 5 \, B a^{5} b^{4} - 24 \, A a^{4} b^{5} + 4 \, B a^{3} b^{6} - 16 \, A a^{2} b^{7} + B a b^{8} - 4 \, A b^{9}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{13} + 4 \, a^{11} b^{2} + 6 \, a^{9} b^{4} + 4 \, a^{7} b^{6} + a^{5} b^{8}} + \frac {3 \, {\left (B a^{4} - 4 \, A a^{3} b - 6 \, B a^{2} b^{2} + 4 \, A a b^{3} + B b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {6 \, A a^{9} + 18 \, A a^{7} b^{2} + 18 \, A a^{5} b^{4} + 6 \, A a^{3} b^{6} + 6 \, {\left (A a^{6} b^{3} - 6 \, B a^{5} b^{4} + 13 \, A a^{4} b^{5} - 3 \, B a^{3} b^{6} + 12 \, A a^{2} b^{7} - B a b^{8} + 4 \, A b^{9}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (6 \, A a^{7} b^{2} - 27 \, B a^{6} b^{3} + 62 \, A a^{5} b^{4} - 16 \, B a^{4} b^{5} + 60 \, A a^{3} b^{6} - 5 \, B a^{2} b^{7} + 20 \, A a b^{8}\right )} \tan \left (d x + c\right )^{2} + {\left (18 \, A a^{8} b - 47 \, B a^{7} b^{2} + 128 \, A a^{6} b^{3} - 34 \, B a^{5} b^{4} + 130 \, A a^{4} b^{5} - 11 \, B a^{3} b^{6} + 44 \, A a^{2} b^{7}\right )} \tan \left (d x + c\right )}{{\left (a^{10} b^{3} + 3 \, a^{8} b^{5} + 3 \, a^{6} b^{7} + a^{4} b^{9}\right )} \tan \left (d x + c\right )^{4} + 3 \, {\left (a^{11} b^{2} + 3 \, a^{9} b^{4} + 3 \, a^{7} b^{6} + a^{5} b^{8}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{12} b + 3 \, a^{10} b^{3} + 3 \, a^{8} b^{5} + a^{6} b^{7}\right )} \tan \left (d x + c\right )^{2} + {\left (a^{13} + 3 \, a^{11} b^{2} + 3 \, a^{9} b^{4} + a^{7} b^{6}\right )} \tan \left (d x + c\right )} - \frac {6 \, {\left (B a - 4 \, A b\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{5}}}{6 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

-1/6*(6*(A*a^4 + 4*B*a^3*b - 6*A*a^2*b^2 - 4*B*a*b^3 + A*b^4)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b
^6 + b^8) + 6*(10*B*a^7*b^2 - 20*A*a^6*b^3 + 5*B*a^5*b^4 - 24*A*a^4*b^5 + 4*B*a^3*b^6 - 16*A*a^2*b^7 + B*a*b^8
- 4*A*b^9)*log(b*tan(d*x + c) + a)/(a^13 + 4*a^11*b^2 + 6*a^9*b^4 + 4*a^7*b^6 + a^5*b^8) + 3*(B*a^4 - 4*A*a^3
*b - 6*B*a^2*b^2 + 4*A*a*b^3 + B*b^4)*log(tan(d*x + c)^2 + 1)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)
+ (6*A*a^9 + 18*A*a^7*b^2 + 18*A*a^5*b^4 + 6*A*a^3*b^6 + 6*(A*a^6*b^3 - 6*B*a^5*b^4 + 13*A*a^4*b^5 - 3*B*a^3*b
^6 + 12*A*a^2*b^7 - B*a*b^8 + 4*A*b^9)*tan(d*x + c)^3 + 3*(6*A*a^7*b^2 - 27*B*a^6*b^3 + 62*A*a^5*b^4 - 16*B*a^
4*b^5 + 60*A*a^3*b^6 - 5*B*a^2*b^7 + 20*A*a*b^8)*tan(d*x + c)^2 + (18*A*a^8*b - 47*B*a^7*b^2 + 128*A*a^6*b^3 -
34*B*a^5*b^4 + 130*A*a^4*b^5 - 11*B*a^3*b^6 + 44*A*a^2*b^7)*tan(d*x + c))/((a^10*b^3 + 3*a^8*b^5 + 3*a^6*b^7
+ a^4*b^9)*tan(d*x + c)^4 + 3*(a^11*b^2 + 3*a^9*b^4 + 3*a^7*b^6 + a^5*b^8)*tan(d*x + c)^3 + 3*(a^12*b + 3*a^10
*b^3 + 3*a^8*b^5 + a^6*b^7)*tan(d*x + c)^2 + (a^13 + 3*a^11*b^2 + 3*a^9*b^4 + a^7*b^6)*tan(d*x + c)) - 6*(B*a
- 4*A*b)*log(tan(d*x + c))/a^5)/d

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mupad [B]  time = 11.02, size = 576, normalized size = 1.44 \[ \frac {b^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-10\,B\,a^7+20\,A\,a^6\,b-5\,B\,a^5\,b^2+24\,A\,a^4\,b^3-4\,B\,a^3\,b^4+16\,A\,a^2\,b^5-B\,a\,b^6+4\,A\,b^7\right )}{a^5\,d\,{\left (a^2+b^2\right )}^4}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (4\,A\,b-B\,a\right )}{a^5\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (A+B\,1{}\mathrm {i}\right )}{2\,d\,\left (a^4\,1{}\mathrm {i}-4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}+4\,a\,b^3+b^4\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )}{2\,d\,\left (a^4-a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2+a\,b^3\,4{}\mathrm {i}+b^4\right )}-\frac {\frac {A}{a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (A\,a^6\,b^3-6\,B\,a^5\,b^4+13\,A\,a^4\,b^5-3\,B\,a^3\,b^6+12\,A\,a^2\,b^7-B\,a\,b^8+4\,A\,b^9\right )}{a^4\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (6\,A\,a^6\,b^2-27\,B\,a^5\,b^3+62\,A\,a^4\,b^4-16\,B\,a^3\,b^5+60\,A\,a^2\,b^6-5\,B\,a\,b^7+20\,A\,b^8\right )}{2\,a^3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (18\,A\,a^6\,b-47\,B\,a^5\,b^2+128\,A\,a^4\,b^3-34\,B\,a^3\,b^4+130\,A\,a^2\,b^5-11\,B\,a\,b^6+44\,A\,b^7\right )}{6\,a^2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}}{d\,\left (a^3\,\mathrm {tan}\left (c+d\,x\right )+3\,a^2\,b\,{\mathrm {tan}\left (c+d\,x\right )}^2+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^3+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cot(c + d*x)^2*(A + B*tan(c + d*x)))/(a + b*tan(c + d*x))^4,x)

[Out]

(b^2*log(a + b*tan(c + d*x))*(4*A*b^7 - 10*B*a^7 + 16*A*a^2*b^5 + 24*A*a^4*b^3 - 4*B*a^3*b^4 - 5*B*a^5*b^2 + 2
0*A*a^6*b - B*a*b^6))/(a^5*d*(a^2 + b^2)^4) - (log(tan(c + d*x))*(4*A*b - B*a))/(a^5*d) - (log(tan(c + d*x) -
1i)*(A + B*1i))/(2*d*(4*a*b^3 - 4*a^3*b + a^4*1i + b^4*1i - a^2*b^2*6i)) - (log(tan(c + d*x) + 1i)*(A*1i + B))
/(2*d*(a*b^3*4i - a^3*b*4i + a^4 + b^4 - 6*a^2*b^2)) - (A/a + (tan(c + d*x)^3*(4*A*b^9 + 12*A*a^2*b^7 + 13*A*a
^4*b^5 + A*a^6*b^3 - 3*B*a^3*b^6 - 6*B*a^5*b^4 - B*a*b^8))/(a^4*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (tan(c
+ d*x)^2*(20*A*b^8 + 60*A*a^2*b^6 + 62*A*a^4*b^4 + 6*A*a^6*b^2 - 16*B*a^3*b^5 - 27*B*a^5*b^3 - 5*B*a*b^7))/(2*
a^3*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (tan(c + d*x)*(44*A*b^7 + 130*A*a^2*b^5 + 128*A*a^4*b^3 - 34*B*a^3*
b^4 - 47*B*a^5*b^2 + 18*A*a^6*b - 11*B*a*b^6))/(6*a^2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))/(d*(a^3*tan(c + d*
x) + b^3*tan(c + d*x)^4 + 3*a^2*b*tan(c + d*x)^2 + 3*a*b^2*tan(c + d*x)^3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))**4,x)

[Out]

Exception raised: AttributeError

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