\(\int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx\) [297]

Optimal result

Integrand size = 31, antiderivative size = 477 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\frac {\left (4 a^3 A b-4 a A b^3-a^4 B+6 a^2 b^2 B-b^4 B\right ) x}{\left (a^2+b^2\right )^4}-\frac {\left (a^2 A-10 A b^2+4 a b B\right ) \log (\sin (c+d x))}{a^6 d}-\frac {b^3 \left (35 a^6 A b+56 a^4 A b^3+39 a^2 A b^5+10 A b^7-20 a^7 B-24 a^5 b^2 B-16 a^3 b^4 B-4 a b^6 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^6 \left (a^2+b^2\right )^4 d}+\frac {b \left (9 a^2 A b+10 A b^3-3 a^3 B-4 a b^2 B\right )}{3 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {(5 A b-2 a B) \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))^3}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^3}+\frac {b \left (7 a^4 A b+19 a^2 A b^3+10 A b^5-2 a^5 B-8 a^3 b^2 B-4 a b^4 B\right )}{2 a^4 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b \left (4 a^6 A b+27 a^4 A b^3+29 a^2 A b^5+10 A b^7-a^7 B-13 a^5 b^2 B-12 a^3 b^4 B-4 a b^6 B\right )}{a^5 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \]

[Out]

(4*A*a^3*b-4*A*a*b^3-B*a^4+6*B*a^2*b^2-B*b^4)*x/(a^2+b^2)^4-(A*a^2-10*A*b^2+4*B*a*b)*ln(sin(d*x+c))/a^6/d-b^3*
(35*A*a^6*b+56*A*a^4*b^3+39*A*a^2*b^5+10*A*b^7-20*B*a^7-24*B*a^5*b^2-16*B*a^3*b^4-4*B*a*b^6)*ln(a*cos(d*x+c)+b
*sin(d*x+c))/a^6/(a^2+b^2)^4/d+1/3*b*(9*A*a^2*b+10*A*b^3-3*B*a^3-4*B*a*b^2)/a^3/(a^2+b^2)/d/(a+b*tan(d*x+c))^3
+1/2*(5*A*b-2*B*a)*cot(d*x+c)/a^2/d/(a+b*tan(d*x+c))^3-1/2*A*cot(d*x+c)^2/a/d/(a+b*tan(d*x+c))^3+1/2*b*(7*A*a^
4*b+19*A*a^2*b^3+10*A*b^5-2*B*a^5-8*B*a^3*b^2-4*B*a*b^4)/a^4/(a^2+b^2)^2/d/(a+b*tan(d*x+c))^2+b*(4*A*a^6*b+27*
A*a^4*b^3+29*A*a^2*b^5+10*A*b^7-B*a^7-13*B*a^5*b^2-12*B*a^3*b^4-4*B*a*b^6)/a^5/(a^2+b^2)^3/d/(a+b*tan(d*x+c))

Rubi [A] (verified)

Time = 2.14 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3690, 3730, 3732, 3611, 3556} \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\frac {(5 A b-2 a B) \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))^3}-\frac {\left (a^2 A+4 a b B-10 A b^2\right ) \log (\sin (c+d x))}{a^6 d}+\frac {b \left (-3 a^3 B+9 a^2 A b-4 a b^2 B+10 A b^3\right )}{3 a^3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {x \left (a^4 (-B)+4 a^3 A b+6 a^2 b^2 B-4 a A b^3-b^4 B\right )}{\left (a^2+b^2\right )^4}+\frac {b \left (-2 a^5 B+7 a^4 A b-8 a^3 b^2 B+19 a^2 A b^3-4 a b^4 B+10 A b^5\right )}{2 a^4 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac {b \left (a^7 (-B)+4 a^6 A b-13 a^5 b^2 B+27 a^4 A b^3-12 a^3 b^4 B+29 a^2 A b^5-4 a b^6 B+10 A b^7\right )}{a^5 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac {b^3 \left (-20 a^7 B+35 a^6 A b-24 a^5 b^2 B+56 a^4 A b^3-16 a^3 b^4 B+39 a^2 A b^5-4 a b^6 B+10 A b^7\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^6 d \left (a^2+b^2\right )^4}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^3} \]

[In]

Int[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4,x]

[Out]

((4*a^3*A*b - 4*a*A*b^3 - a^4*B + 6*a^2*b^2*B - b^4*B)*x)/(a^2 + b^2)^4 - ((a^2*A - 10*A*b^2 + 4*a*b*B)*Log[Si
n[c + d*x]])/(a^6*d) - (b^3*(35*a^6*A*b + 56*a^4*A*b^3 + 39*a^2*A*b^5 + 10*A*b^7 - 20*a^7*B - 24*a^5*b^2*B - 1
6*a^3*b^4*B - 4*a*b^6*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^6*(a^2 + b^2)^4*d) + (b*(9*a^2*A*b + 10*A*b^
3 - 3*a^3*B - 4*a*b^2*B))/(3*a^3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + ((5*A*b - 2*a*B)*Cot[c + d*x])/(2*a^2
*d*(a + b*Tan[c + d*x])^3) - (A*Cot[c + d*x]^2)/(2*a*d*(a + b*Tan[c + d*x])^3) + (b*(7*a^4*A*b + 19*a^2*A*b^3
+ 10*A*b^5 - 2*a^5*B - 8*a^3*b^2*B - 4*a*b^4*B))/(2*a^4*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) + (b*(4*a^6*A*
b + 27*a^4*A*b^3 + 29*a^2*A*b^5 + 10*A*b^7 - a^7*B - 13*a^5*b^2*B - 12*a^3*b^4*B - 4*a*b^6*B))/(a^5*(a^2 + b^2
)^3*d*(a + b*Tan[c + d*x]))

Rule 3556

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

Rule 3611

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c/(b*f))
*Log[RemoveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]], 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 3690

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 3730

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*Ta
n[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 3732

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*} \text {integral}& = -\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^3}-\frac {\int \frac {\cot ^2(c+d x) \left (5 A b-2 a B+2 a A \tan (c+d x)+5 A b \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^4} \, dx}{2 a} \\ & = \frac {(5 A b-2 a B) \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))^3}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^3}+\frac {\int \frac {\cot (c+d x) \left (-2 \left (a^2 A-10 A b^2+4 a b B\right )-2 a^2 B \tan (c+d x)+4 b (5 A b-2 a B) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^4} \, dx}{2 a^2} \\ & = \frac {b \left (9 a^2 A b+10 A b^3-3 a^3 B-4 a b^2 B\right )}{3 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {(5 A b-2 a B) \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))^3}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^3}+\frac {\int \frac {\cot (c+d x) \left (-6 \left (a^2+b^2\right ) \left (a^2 A-10 A b^2+4 a b B\right )+6 a^3 (A b-a B) \tan (c+d x)+6 b \left (9 a^2 A b+10 A b^3-3 a^3 B-4 a b^2 B\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{6 a^3 \left (a^2+b^2\right )} \\ & = \frac {b \left (9 a^2 A b+10 A b^3-3 a^3 B-4 a b^2 B\right )}{3 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {(5 A b-2 a B) \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))^3}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^3}+\frac {b \left (7 a^4 A b+19 a^2 A b^3+10 A b^5-2 a^5 B-8 a^3 b^2 B-4 a b^4 B\right )}{2 a^4 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {\int \frac {\cot (c+d x) \left (-12 \left (a^2+b^2\right )^2 \left (a^2 A-10 A b^2+4 a b B\right )+12 a^4 \left (2 a A b-a^2 B+b^2 B\right ) \tan (c+d x)+12 b \left (7 a^4 A b+19 a^2 A b^3+10 A b^5-2 a^5 B-8 a^3 b^2 B-4 a b^4 B\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{12 a^4 \left (a^2+b^2\right )^2} \\ & = \frac {b \left (9 a^2 A b+10 A b^3-3 a^3 B-4 a b^2 B\right )}{3 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {(5 A b-2 a B) \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))^3}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^3}+\frac {b \left (7 a^4 A b+19 a^2 A b^3+10 A b^5-2 a^5 B-8 a^3 b^2 B-4 a b^4 B\right )}{2 a^4 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b \left (4 a^6 A b+27 a^4 A b^3+29 a^2 A b^5+10 A b^7-a^7 B-13 a^5 b^2 B-12 a^3 b^4 B-4 a b^6 B\right )}{a^5 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\int \frac {\cot (c+d x) \left (-12 \left (a^2+b^2\right )^3 \left (a^2 A-10 A b^2+4 a b B\right )+12 a^5 \left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) \tan (c+d x)+12 b \left (4 a^6 A b+27 a^4 A b^3+29 a^2 A b^5+10 A b^7-a^7 B-13 a^5 b^2 B-12 a^3 b^4 B-4 a b^6 B\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{12 a^5 \left (a^2+b^2\right )^3} \\ & = \frac {\left (4 a^3 A b-4 a A b^3-a^4 B+6 a^2 b^2 B-b^4 B\right ) x}{\left (a^2+b^2\right )^4}+\frac {b \left (9 a^2 A b+10 A b^3-3 a^3 B-4 a b^2 B\right )}{3 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {(5 A b-2 a B) \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))^3}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^3}+\frac {b \left (7 a^4 A b+19 a^2 A b^3+10 A b^5-2 a^5 B-8 a^3 b^2 B-4 a b^4 B\right )}{2 a^4 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b \left (4 a^6 A b+27 a^4 A b^3+29 a^2 A b^5+10 A b^7-a^7 B-13 a^5 b^2 B-12 a^3 b^4 B-4 a b^6 B\right )}{a^5 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}-\frac {\left (a^2 A-10 A b^2+4 a b B\right ) \int \cot (c+d x) \, dx}{a^6}-\frac {\left (b^3 \left (35 a^6 A b+56 a^4 A b^3+39 a^2 A b^5+10 A b^7-20 a^7 B-24 a^5 b^2 B-16 a^3 b^4 B-4 a b^6 B\right )\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^6 \left (a^2+b^2\right )^4} \\ & = \frac {\left (4 a^3 A b-4 a A b^3-a^4 B+6 a^2 b^2 B-b^4 B\right ) x}{\left (a^2+b^2\right )^4}-\frac {\left (a^2 A-10 A b^2+4 a b B\right ) \log (\sin (c+d x))}{a^6 d}-\frac {b^3 \left (35 a^6 A b+56 a^4 A b^3+39 a^2 A b^5+10 A b^7-20 a^7 B-24 a^5 b^2 B-16 a^3 b^4 B-4 a b^6 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^6 \left (a^2+b^2\right )^4 d}+\frac {b \left (9 a^2 A b+10 A b^3-3 a^3 B-4 a b^2 B\right )}{3 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {(5 A b-2 a B) \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))^3}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^3}+\frac {b \left (7 a^4 A b+19 a^2 A b^3+10 A b^5-2 a^5 B-8 a^3 b^2 B-4 a b^4 B\right )}{2 a^4 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b \left (4 a^6 A b+27 a^4 A b^3+29 a^2 A b^5+10 A b^7-a^7 B-13 a^5 b^2 B-12 a^3 b^4 B-4 a b^6 B\right )}{a^5 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.73 (sec) , antiderivative size = 417, normalized size of antiderivative = 0.87 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\frac {(4 A b-a B) \cot (c+d x)}{a^5 d}-\frac {A \cot ^2(c+d x)}{2 a^4 d}+\frac {(A+i B) \log (i-\tan (c+d x))}{2 (a+i b)^4 d}-\frac {\left (a^2 A-10 A b^2+4 a b B\right ) \log (\tan (c+d x))}{a^6 d}+\frac {(A-i B) \log (i+\tan (c+d x))}{2 (a-i b)^4 d}-\frac {b^3 \left (35 a^6 A b+56 a^4 A b^3+39 a^2 A b^5+10 A b^7-20 a^7 B-24 a^5 b^2 B-16 a^3 b^4 B-4 a b^6 B\right ) \log (a+b \tan (c+d x))}{a^6 \left (a^2+b^2\right )^4 d}+\frac {b^3 (A b-a B)}{3 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {b^3 \left (5 a^2 A b+3 A b^3-4 a^3 B-2 a b^2 B\right )}{2 a^4 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b^3 \left (15 a^4 A b+17 a^2 A b^3+6 A b^5-10 a^5 B-9 a^3 b^2 B-3 a b^4 B\right )}{a^5 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \]

[In]

Integrate[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^4,x]

[Out]

((4*A*b - a*B)*Cot[c + d*x])/(a^5*d) - (A*Cot[c + d*x]^2)/(2*a^4*d) + ((A + I*B)*Log[I - Tan[c + d*x]])/(2*(a
+ I*b)^4*d) - ((a^2*A - 10*A*b^2 + 4*a*b*B)*Log[Tan[c + d*x]])/(a^6*d) + ((A - I*B)*Log[I + Tan[c + d*x]])/(2*
(a - I*b)^4*d) - (b^3*(35*a^6*A*b + 56*a^4*A*b^3 + 39*a^2*A*b^5 + 10*A*b^7 - 20*a^7*B - 24*a^5*b^2*B - 16*a^3*
b^4*B - 4*a*b^6*B)*Log[a + b*Tan[c + d*x]])/(a^6*(a^2 + b^2)^4*d) + (b^3*(A*b - a*B))/(3*a^3*(a^2 + b^2)*d*(a
+ b*Tan[c + d*x])^3) + (b^3*(5*a^2*A*b + 3*A*b^3 - 4*a^3*B - 2*a*b^2*B))/(2*a^4*(a^2 + b^2)^2*d*(a + b*Tan[c +
d*x])^2) + (b^3*(15*a^4*A*b + 17*a^2*A*b^3 + 6*A*b^5 - 10*a^5*B - 9*a^3*b^2*B - 3*a*b^4*B))/(a^5*(a^2 + b^2)^
3*d*(a + b*Tan[c + d*x]))

Maple [A] (verified)

Time = 1.08 (sec) , antiderivative size = 429, normalized size of antiderivative = 0.90

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {A}{2 a^{4} \tan \left (d x +c \right )^{2}}-\frac {-4 A b +B a}{a^{5} \tan \left (d x +c \right )}+\frac {\left (-A \,a^{2}+10 A \,b^{2}-4 B a b \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{6}}+\frac {b^{3} \left (5 A \,a^{2} b +3 A \,b^{3}-4 B \,a^{3}-2 B a \,b^{2}\right )}{2 \left (a^{2}+b^{2}\right )^{2} a^{4} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b^{3} \left (15 A \,a^{4} b +17 A \,a^{2} b^{3}+6 A \,b^{5}-10 B \,a^{5}-9 B \,a^{3} b^{2}-3 B a \,b^{4}\right )}{\left (a^{2}+b^{2}\right )^{3} a^{5} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{3} \left (35 A \,a^{6} b +56 A \,a^{4} b^{3}+39 A \,a^{2} b^{5}+10 A \,b^{7}-20 B \,a^{7}-24 B \,a^{5} b^{2}-16 B \,a^{3} b^{4}-4 B a \,b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4} a^{6}}+\frac {\left (A b -B a \right ) b^{3}}{3 \left (a^{2}+b^{2}\right ) a^{3} \left (a +b \tan \left (d x +c \right )\right )^{3}}}{d}\)	\(429\)
default	\(\frac {\frac {\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {A}{2 a^{4} \tan \left (d x +c \right )^{2}}-\frac {-4 A b +B a}{a^{5} \tan \left (d x +c \right )}+\frac {\left (-A \,a^{2}+10 A \,b^{2}-4 B a b \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{6}}+\frac {b^{3} \left (5 A \,a^{2} b +3 A \,b^{3}-4 B \,a^{3}-2 B a \,b^{2}\right )}{2 \left (a^{2}+b^{2}\right )^{2} a^{4} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b^{3} \left (15 A \,a^{4} b +17 A \,a^{2} b^{3}+6 A \,b^{5}-10 B \,a^{5}-9 B \,a^{3} b^{2}-3 B a \,b^{4}\right )}{\left (a^{2}+b^{2}\right )^{3} a^{5} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{3} \left (35 A \,a^{6} b +56 A \,a^{4} b^{3}+39 A \,a^{2} b^{5}+10 A \,b^{7}-20 B \,a^{7}-24 B \,a^{5} b^{2}-16 B \,a^{3} b^{4}-4 B a \,b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4} a^{6}}+\frac {\left (A b -B a \right ) b^{3}}{3 \left (a^{2}+b^{2}\right ) a^{3} \left (a +b \tan \left (d x +c \right )\right )^{3}}}{d}\)	\(429\)
parallelrisch	\(\frac {-70 b^{3} \left (a +b \tan \left (d x +c \right )\right )^{3} \left (A \,a^{6} b +\frac {8}{5} A \,a^{4} b^{3}+\frac {39}{35} A \,a^{2} b^{5}+\frac {2}{7} A \,b^{7}-\frac {4}{7} B \,a^{7}-\frac {24}{35} B \,a^{5} b^{2}-\frac {16}{35} B \,a^{3} b^{4}-\frac {4}{35} B a \,b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )+a^{6} \left (a +b \tan \left (d x +c \right )\right )^{3} \left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) \ln \left (\sec ^{2}\left (d x +c \right )\right )-2 \left (a^{2}+b^{2}\right )^{4} \left (a +b \tan \left (d x +c \right )\right )^{3} \left (A \,a^{2}-10 A \,b^{2}+4 B a b \right ) \ln \left (\tan \left (d x +c \right )\right )+8 \left (-\frac {B x \,a^{10} d}{4}+b \left (A d x +\frac {B}{4}\right ) a^{9}-\frac {23 \left (-\frac {36 B d x}{23}+A \right ) b^{2} a^{8}}{24}-\left (A d x -\frac {17 B}{6}\right ) b^{3} a^{7}-\frac {157 b^{4} \left (\frac {6 B d x}{157}+A \right ) a^{6}}{24}+\frac {61 B \,a^{5} b^{5}}{12}-\frac {93 A \,a^{4} b^{6}}{8}+\frac {10 B \,a^{3} b^{7}}{3}-\frac {65 A \,a^{2} b^{8}}{8}+\frac {5 B a \,b^{9}}{6}-\frac {25 A \,b^{10}}{12}\right ) b^{3} \left (\tan ^{3}\left (d x +c \right )\right )+24 a \left (-\frac {B x \,a^{10} d}{4}+b \left (A d x +\frac {B}{6}\right ) a^{9}-\frac {5 b^{2} \left (-\frac {12 B d x}{5}+A \right ) a^{8}}{8}-\left (A d x -\frac {5 B}{3}\right ) b^{3} a^{7}-\frac {95 \left (\frac {6 B d x}{95}+A \right ) b^{4} a^{6}}{24}+3 B \,a^{5} b^{5}-\frac {167 A \,a^{4} b^{6}}{24}+2 B \,a^{3} b^{7}-\frac {39 A \,a^{2} b^{8}}{8}+\frac {B a \,b^{9}}{2}-\frac {5 A \,b^{10}}{4}\right ) b^{2} \left (\tan ^{2}\left (d x +c \right )\right )+24 d \,a^{8} x \left (A \,a^{3} b -A a \,b^{3}-\frac {1}{4} B \,a^{4}+\frac {3}{2} B \,a^{2} b^{2}-\frac {1}{4} B \,b^{4}\right ) b \tan \left (d x +c \right )-a^{3} \left (A \,a^{2} \left (a^{2}+b^{2}\right )^{4} \left (\cot ^{2}\left (d x +c \right )\right )-5 a \left (A b -\frac {2 B a}{5}\right ) \left (a^{2}+b^{2}\right )^{4} \cot \left (d x +c \right )+2 B x \,a^{10} d -8 \left (A d x -\frac {B}{2}\right ) b \,a^{9}-\frac {40 b^{2} \left (\frac {9 B d x}{10}+A \right ) a^{8}}{3}+8 \left (A d x +\frac {13 B}{4}\right ) b^{3} a^{7}-\frac {202 \left (-\frac {3 B d x}{101}+A \right ) b^{4} a^{6}}{3}+\frac {136 B \,a^{5} b^{5}}{3}-112 A \,a^{4} b^{6}+\frac {94 B \,a^{3} b^{7}}{3}-78 A \,a^{2} b^{8}+8 B a \,b^{9}-20 A \,b^{10}\right )}{2 \left (a^{2}+b^{2}\right )^{4} a^{6} d \left (a +b \tan \left (d x +c \right )\right )^{3}}\)	\(708\)
norman	\(\frac {\frac {b^{3} \left (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) x \left (\tan ^{5}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}+\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) a^{3} x \left (\tan ^{2}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}-\frac {A}{2 a d}+\frac {\left (5 A b -2 B a \right ) \tan \left (d x +c \right )}{2 a^{2} d}-\frac {b^{2} \left (55 A \,a^{6} b^{2}+242 A \,a^{4} b^{4}+261 A \,a^{2} b^{6}+90 A \,b^{8}-16 B \,a^{7} b -102 B \,a^{5} b^{3}-106 B \,a^{3} b^{5}-36 B a \,b^{7}\right ) \left (\tan ^{4}\left (d x +c \right )\right )}{2 a^{5} d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {b^{3} \left (63 A \,a^{6} b^{2}+296 A \,a^{4} b^{4}+319 A \,a^{2} b^{6}+110 A \,b^{8}-18 B \,a^{7} b -128 B \,a^{5} b^{3}-130 B \,a^{3} b^{5}-44 B a \,b^{7}\right ) \left (\tan ^{5}\left (d x +c \right )\right )}{6 d \,a^{6} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {b \left (20 A \,a^{6} b^{2}+81 A \,a^{4} b^{4}+87 A \,a^{2} b^{6}+30 A \,b^{8}-6 B \,a^{7} b -33 B \,a^{5} b^{3}-35 B \,a^{3} b^{5}-12 B a \,b^{7}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{d \,a^{4} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {3 b \left (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) a^{2} x \left (\tan ^{3}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}+\frac {3 b^{2} \left (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) a x \left (\tan ^{4}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}}{\tan \left (d x +c \right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {\left (A \,a^{2}-10 A \,b^{2}+4 B a b \right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{6} d}+\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}-\frac {b^{3} \left (35 A \,a^{6} b +56 A \,a^{4} b^{3}+39 A \,a^{2} b^{5}+10 A \,b^{7}-20 B \,a^{7}-24 B \,a^{5} b^{2}-16 B \,a^{3} b^{4}-4 B a \,b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{6} d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}\)	\(953\)
risch	\(\text {Expression too large to display}\)	\(3351\)

[In]

int(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/(a^2+b^2)^4*(1/2*(A*a^4-6*A*a^2*b^2+A*b^4+4*B*a^3*b-4*B*a*b^3)*ln(1+tan(d*x+c)^2)+(4*A*a^3*b-4*A*a*b^3-
B*a^4+6*B*a^2*b^2-B*b^4)*arctan(tan(d*x+c)))-1/2/a^4*A/tan(d*x+c)^2-(-4*A*b+B*a)/a^5/tan(d*x+c)+(-A*a^2+10*A*b
^2-4*B*a*b)/a^6*ln(tan(d*x+c))+1/2*b^3*(5*A*a^2*b+3*A*b^3-4*B*a^3-2*B*a*b^2)/(a^2+b^2)^2/a^4/(a+b*tan(d*x+c))^
2+b^3*(15*A*a^4*b+17*A*a^2*b^3+6*A*b^5-10*B*a^5-9*B*a^3*b^2-3*B*a*b^4)/(a^2+b^2)^3/a^5/(a+b*tan(d*x+c))-b^3*(3
5*A*a^6*b+56*A*a^4*b^3+39*A*a^2*b^5+10*A*b^7-20*B*a^7-24*B*a^5*b^2-16*B*a^3*b^4-4*B*a*b^6)/(a^2+b^2)^4/a^6*ln(
a+b*tan(d*x+c))+1/3*(A*b-B*a)*b^3/(a^2+b^2)/a^3/(a+b*tan(d*x+c))^3)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1732 vs. \(2 (467) = 934\).

Time = 0.53 (sec) , antiderivative size = 1732, normalized size of antiderivative = 3.63 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\text {Too large to display} \]

[In]

integrate(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/6*(3*A*a^13 + 12*A*a^11*b^2 + 18*A*a^9*b^4 + 12*A*a^7*b^6 + 3*A*a^5*b^8 + (3*A*a^10*b^3 + 12*A*a^8*b^5 - 74
*B*a^7*b^6 + 125*A*a^6*b^7 - 42*B*a^5*b^8 + 102*A*a^4*b^9 - 12*B*a^3*b^10 + 30*A*a^2*b^11 + 6*(B*a^10*b^3 - 4*
A*a^9*b^4 - 6*B*a^8*b^5 + 4*A*a^7*b^6 + B*a^6*b^7)*d*x)*tan(d*x + c)^5 + 3*(3*A*a^11*b^2 + 2*B*a^10*b^3 + 4*A*
a^9*b^4 - 46*B*a^8*b^5 + 63*A*a^7*b^6 + 8*B*a^6*b^7 - 10*A*a^5*b^8 + 20*B*a^4*b^9 - 48*A*a^3*b^10 + 8*B*a^2*b^
11 - 20*A*a*b^12 + 6*(B*a^11*b^2 - 4*A*a^10*b^3 - 6*B*a^9*b^4 + 4*A*a^8*b^5 + B*a^7*b^6)*d*x)*tan(d*x + c)^4 +
3*(3*A*a^12*b + 6*B*a^11*b^2 - 11*A*a^10*b^3 - 6*B*a^9*b^4 - 32*A*a^8*b^5 + 80*B*a^7*b^6 - 177*A*a^6*b^7 + 68
*B*a^5*b^8 - 165*A*a^4*b^9 + 20*B*a^3*b^10 - 50*A*a^2*b^11 + 6*(B*a^12*b - 4*A*a^11*b^2 - 6*B*a^10*b^3 + 4*A*a
^9*b^4 + B*a^8*b^5)*d*x)*tan(d*x + c)^3 + (3*A*a^13 + 18*B*a^12*b - 51*A*a^11*b^2 + 72*B*a^10*b^3 - 234*A*a^9*
b^4 + 216*B*a^8*b^5 - 513*A*a^7*b^6 + 162*B*a^6*b^7 - 399*A*a^5*b^8 + 44*B*a^4*b^9 - 110*A*a^3*b^10 + 6*(B*a^1
3 - 4*A*a^12*b - 6*B*a^11*b^2 + 4*A*a^10*b^3 + B*a^9*b^4)*d*x)*tan(d*x + c)^2 + 3*((A*a^10*b^3 + 4*B*a^9*b^4 -
6*A*a^8*b^5 + 16*B*a^7*b^6 - 34*A*a^6*b^7 + 24*B*a^5*b^8 - 56*A*a^4*b^9 + 16*B*a^3*b^10 - 39*A*a^2*b^11 + 4*B
*a*b^12 - 10*A*b^13)*tan(d*x + c)^5 + 3*(A*a^11*b^2 + 4*B*a^10*b^3 - 6*A*a^9*b^4 + 16*B*a^8*b^5 - 34*A*a^7*b^6
+ 24*B*a^6*b^7 - 56*A*a^5*b^8 + 16*B*a^4*b^9 - 39*A*a^3*b^10 + 4*B*a^2*b^11 - 10*A*a*b^12)*tan(d*x + c)^4 + 3
*(A*a^12*b + 4*B*a^11*b^2 - 6*A*a^10*b^3 + 16*B*a^9*b^4 - 34*A*a^8*b^5 + 24*B*a^7*b^6 - 56*A*a^6*b^7 + 16*B*a^
5*b^8 - 39*A*a^4*b^9 + 4*B*a^3*b^10 - 10*A*a^2*b^11)*tan(d*x + c)^3 + (A*a^13 + 4*B*a^12*b - 6*A*a^11*b^2 + 16
*B*a^10*b^3 - 34*A*a^9*b^4 + 24*B*a^8*b^5 - 56*A*a^7*b^6 + 16*B*a^6*b^7 - 39*A*a^5*b^8 + 4*B*a^4*b^9 - 10*A*a^
3*b^10)*tan(d*x + c)^2)*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1)) - 3*((20*B*a^7*b^6 - 35*A*a^6*b^7 + 24*B*a^5*
b^8 - 56*A*a^4*b^9 + 16*B*a^3*b^10 - 39*A*a^2*b^11 + 4*B*a*b^12 - 10*A*b^13)*tan(d*x + c)^5 + 3*(20*B*a^8*b^5
- 35*A*a^7*b^6 + 24*B*a^6*b^7 - 56*A*a^5*b^8 + 16*B*a^4*b^9 - 39*A*a^3*b^10 + 4*B*a^2*b^11 - 10*A*a*b^12)*tan(
d*x + c)^4 + 3*(20*B*a^9*b^4 - 35*A*a^8*b^5 + 24*B*a^7*b^6 - 56*A*a^6*b^7 + 16*B*a^5*b^8 - 39*A*a^4*b^9 + 4*B*
a^3*b^10 - 10*A*a^2*b^11)*tan(d*x + c)^3 + (20*B*a^10*b^3 - 35*A*a^9*b^4 + 24*B*a^8*b^5 - 56*A*a^7*b^6 + 16*B*
a^6*b^7 - 39*A*a^5*b^8 + 4*B*a^4*b^9 - 10*A*a^3*b^10)*tan(d*x + c)^2)*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x
+ c) + a^2)/(tan(d*x + c)^2 + 1)) + 3*(2*B*a^13 - 5*A*a^12*b + 8*B*a^11*b^2 - 20*A*a^10*b^3 + 12*B*a^9*b^4 - 3
0*A*a^8*b^5 + 8*B*a^7*b^6 - 20*A*a^6*b^7 + 2*B*a^5*b^8 - 5*A*a^4*b^9)*tan(d*x + c))/((a^14*b^3 + 4*a^12*b^5 +
6*a^10*b^7 + 4*a^8*b^9 + a^6*b^11)*d*tan(d*x + c)^5 + 3*(a^15*b^2 + 4*a^13*b^4 + 6*a^11*b^6 + 4*a^9*b^8 + a^7*
b^10)*d*tan(d*x + c)^4 + 3*(a^16*b + 4*a^14*b^3 + 6*a^12*b^5 + 4*a^10*b^7 + a^8*b^9)*d*tan(d*x + c)^3 + (a^17
+ 4*a^15*b^2 + 6*a^13*b^4 + 4*a^11*b^6 + a^9*b^8)*d*tan(d*x + c)^2)

Sympy [F(-2)]

Exception generated. \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\text {Exception raised: AttributeError} \]

[In]

integrate(cot(d*x+c)**3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))**4,x)

[Out]

Exception raised: AttributeError >> 'NoneType' object has no attribute 'primitive'

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 815, normalized size of antiderivative = 1.71 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=-\frac {\frac {6 \, {\left (B a^{4} - 4 \, A a^{3} b - 6 \, B a^{2} b^{2} + 4 \, A a b^{3} + B 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 (20 \, B a^{7} b^{3} - 35 \, A a^{6} b^{4} + 24 \, B a^{5} b^{5} - 56 \, A a^{4} b^{6} + 16 \, B a^{3} b^{7} - 39 \, A a^{2} b^{8} + 4 \, B a b^{9} - 10 \, A b^{10}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{14} + 4 \, a^{12} b^{2} + 6 \, a^{10} b^{4} + 4 \, a^{8} b^{6} + a^{6} b^{8}} - \frac {3 \, {\left (A a^{4} + 4 \, B a^{3} b - 6 \, A a^{2} b^{2} - 4 \, B a b^{3} + A 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 {3 \, A a^{10} + 9 \, A a^{8} b^{2} + 9 \, A a^{6} b^{4} + 3 \, A a^{4} b^{6} + 6 \, {\left (B a^{7} b^{3} - 4 \, A a^{6} b^{4} + 13 \, B a^{5} b^{5} - 27 \, A a^{4} b^{6} + 12 \, B a^{3} b^{7} - 29 \, A a^{2} b^{8} + 4 \, B a b^{9} - 10 \, A b^{10}\right )} \tan \left (d x + c\right )^{4} + 3 \, {\left (6 \, B a^{8} b^{2} - 23 \, A a^{7} b^{3} + 62 \, B a^{6} b^{4} - 134 \, A a^{5} b^{5} + 60 \, B a^{4} b^{6} - 145 \, A a^{3} b^{7} + 20 \, B a^{2} b^{8} - 50 \, A a b^{9}\right )} \tan \left (d x + c\right )^{3} + {\left (18 \, B a^{9} b - 63 \, A a^{8} b^{2} + 128 \, B a^{7} b^{3} - 296 \, A a^{6} b^{4} + 130 \, B a^{5} b^{5} - 319 \, A a^{4} b^{6} + 44 \, B a^{3} b^{7} - 110 \, A a^{2} b^{8}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (2 \, B a^{10} - 5 \, A a^{9} b + 6 \, B a^{8} b^{2} - 15 \, A a^{7} b^{3} + 6 \, B a^{6} b^{4} - 15 \, A a^{5} b^{5} + 2 \, B a^{4} b^{6} - 5 \, A a^{3} b^{7}\right )} \tan \left (d x + c\right )}{{\left (a^{11} b^{3} + 3 \, a^{9} b^{5} + 3 \, a^{7} b^{7} + a^{5} b^{9}\right )} \tan \left (d x + c\right )^{5} + 3 \, {\left (a^{12} b^{2} + 3 \, a^{10} b^{4} + 3 \, a^{8} b^{6} + a^{6} b^{8}\right )} \tan \left (d x + c\right )^{4} + 3 \, {\left (a^{13} b + 3 \, a^{11} b^{3} + 3 \, a^{9} b^{5} + a^{7} b^{7}\right )} \tan \left (d x + c\right )^{3} + {\left (a^{14} + 3 \, a^{12} b^{2} + 3 \, a^{10} b^{4} + a^{8} b^{6}\right )} \tan \left (d x + c\right )^{2}} + \frac {6 \, {\left (A a^{2} + 4 \, B a b - 10 \, A b^{2}\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{6}}}{6 \, d} \]

[In]

integrate(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

-1/6*(6*(B*a^4 - 4*A*a^3*b - 6*B*a^2*b^2 + 4*A*a*b^3 + B*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*(20*B*a^7*b^3 - 35*A*a^6*b^4 + 24*B*a^5*b^5 - 56*A*a^4*b^6 + 16*B*a^3*b^7 - 39*A*a^2*b^8 + 4*B*a
*b^9 - 10*A*b^10)*log(b*tan(d*x + c) + a)/(a^14 + 4*a^12*b^2 + 6*a^10*b^4 + 4*a^8*b^6 + a^6*b^8) - 3*(A*a^4 +
4*B*a^3*b - 6*A*a^2*b^2 - 4*B*a*b^3 + A*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) + (3*A*a^10 + 9*A*a^8*b^2 + 9*A*a^6*b^4 + 3*A*a^4*b^6 + 6*(B*a^7*b^3 - 4*A*a^6*b^4 + 13*B*a^5*b^5 - 27*
A*a^4*b^6 + 12*B*a^3*b^7 - 29*A*a^2*b^8 + 4*B*a*b^9 - 10*A*b^10)*tan(d*x + c)^4 + 3*(6*B*a^8*b^2 - 23*A*a^7*b^
3 + 62*B*a^6*b^4 - 134*A*a^5*b^5 + 60*B*a^4*b^6 - 145*A*a^3*b^7 + 20*B*a^2*b^8 - 50*A*a*b^9)*tan(d*x + c)^3 +
(18*B*a^9*b - 63*A*a^8*b^2 + 128*B*a^7*b^3 - 296*A*a^6*b^4 + 130*B*a^5*b^5 - 319*A*a^4*b^6 + 44*B*a^3*b^7 - 11
0*A*a^2*b^8)*tan(d*x + c)^2 + 3*(2*B*a^10 - 5*A*a^9*b + 6*B*a^8*b^2 - 15*A*a^7*b^3 + 6*B*a^6*b^4 - 15*A*a^5*b^
5 + 2*B*a^4*b^6 - 5*A*a^3*b^7)*tan(d*x + c))/((a^11*b^3 + 3*a^9*b^5 + 3*a^7*b^7 + a^5*b^9)*tan(d*x + c)^5 + 3*
(a^12*b^2 + 3*a^10*b^4 + 3*a^8*b^6 + a^6*b^8)*tan(d*x + c)^4 + 3*(a^13*b + 3*a^11*b^3 + 3*a^9*b^5 + a^7*b^7)*t
an(d*x + c)^3 + (a^14 + 3*a^12*b^2 + 3*a^10*b^4 + a^8*b^6)*tan(d*x + c)^2) + 6*(A*a^2 + 4*B*a*b - 10*A*b^2)*lo
g(tan(d*x + c))/a^6)/d

Giac [A] (verification not implemented)

none

Time = 1.11 (sec) , antiderivative size = 903, normalized size of antiderivative = 1.89 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=-\frac {\frac {6 \, {\left (B a^{4} - 4 \, A a^{3} b - 6 \, B a^{2} b^{2} + 4 \, A a b^{3} + B 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 (A a^{4} + 4 \, B a^{3} b - 6 \, A a^{2} b^{2} - 4 \, B a b^{3} + A 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 (20 \, B a^{7} b^{4} - 35 \, A a^{6} b^{5} + 24 \, B a^{5} b^{6} - 56 \, A a^{4} b^{7} + 16 \, B a^{3} b^{8} - 39 \, A a^{2} b^{9} + 4 \, B a b^{10} - 10 \, A b^{11}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{14} b + 4 \, a^{12} b^{3} + 6 \, a^{10} b^{5} + 4 \, a^{8} b^{7} + a^{6} b^{9}} + \frac {220 \, B a^{7} b^{6} \tan \left (d x + c\right )^{3} - 385 \, A a^{6} b^{7} \tan \left (d x + c\right )^{3} + 264 \, B a^{5} b^{8} \tan \left (d x + c\right )^{3} - 616 \, A a^{4} b^{9} \tan \left (d x + c\right )^{3} + 176 \, B a^{3} b^{10} \tan \left (d x + c\right )^{3} - 429 \, A a^{2} b^{11} \tan \left (d x + c\right )^{3} + 44 \, B a b^{12} \tan \left (d x + c\right )^{3} - 110 \, A b^{13} \tan \left (d x + c\right )^{3} + 720 \, B a^{8} b^{5} \tan \left (d x + c\right )^{2} - 1245 \, A a^{7} b^{6} \tan \left (d x + c\right )^{2} + 906 \, B a^{6} b^{7} \tan \left (d x + c\right )^{2} - 2040 \, A a^{5} b^{8} \tan \left (d x + c\right )^{2} + 600 \, B a^{4} b^{9} \tan \left (d x + c\right )^{2} - 1425 \, A a^{3} b^{10} \tan \left (d x + c\right )^{2} + 150 \, B a^{2} b^{11} \tan \left (d x + c\right )^{2} - 366 \, A a b^{12} \tan \left (d x + c\right )^{2} + 792 \, B a^{9} b^{4} \tan \left (d x + c\right ) - 1350 \, A a^{8} b^{5} \tan \left (d x + c\right ) + 1050 \, B a^{7} b^{6} \tan \left (d x + c\right ) - 2271 \, A a^{6} b^{7} \tan \left (d x + c\right ) + 696 \, B a^{5} b^{8} \tan \left (d x + c\right ) - 1596 \, A a^{4} b^{9} \tan \left (d x + c\right ) + 174 \, B a^{3} b^{10} \tan \left (d x + c\right ) - 411 \, A a^{2} b^{11} \tan \left (d x + c\right ) + 294 \, B a^{10} b^{3} - 492 \, A a^{9} b^{4} + 414 \, B a^{8} b^{5} - 853 \, A a^{7} b^{6} + 278 \, B a^{6} b^{7} - 606 \, A a^{5} b^{8} + 70 \, B a^{4} b^{9} - 157 \, A a^{3} b^{10}}{{\left (a^{14} + 4 \, a^{12} b^{2} + 6 \, a^{10} b^{4} + 4 \, a^{8} b^{6} + a^{6} b^{8}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3}} + \frac {6 \, {\left (A a^{2} + 4 \, B a b - 10 \, A b^{2}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{6}} - \frac {3 \, {\left (3 \, A a^{2} \tan \left (d x + c\right )^{2} + 12 \, B a b \tan \left (d x + c\right )^{2} - 30 \, A b^{2} \tan \left (d x + c\right )^{2} - 2 \, B a^{2} \tan \left (d x + c\right ) + 8 \, A a b \tan \left (d x + c\right ) - A a^{2}\right )}}{a^{6} \tan \left (d x + c\right )^{2}}}{6 \, d} \]

[In]

integrate(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x, algorithm="giac")

[Out]

-1/6*(6*(B*a^4 - 4*A*a^3*b - 6*B*a^2*b^2 + 4*A*a*b^3 + B*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*(A*a^4 + 4*B*a^3*b - 6*A*a^2*b^2 - 4*B*a*b^3 + A*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*(20*B*a^7*b^4 - 35*A*a^6*b^5 + 24*B*a^5*b^6 - 56*A*a^4*b^7 + 16*B*a^3*b^8 -
39*A*a^2*b^9 + 4*B*a*b^10 - 10*A*b^11)*log(abs(b*tan(d*x + c) + a))/(a^14*b + 4*a^12*b^3 + 6*a^10*b^5 + 4*a^8*
b^7 + a^6*b^9) + (220*B*a^7*b^6*tan(d*x + c)^3 - 385*A*a^6*b^7*tan(d*x + c)^3 + 264*B*a^5*b^8*tan(d*x + c)^3 -
616*A*a^4*b^9*tan(d*x + c)^3 + 176*B*a^3*b^10*tan(d*x + c)^3 - 429*A*a^2*b^11*tan(d*x + c)^3 + 44*B*a*b^12*ta
n(d*x + c)^3 - 110*A*b^13*tan(d*x + c)^3 + 720*B*a^8*b^5*tan(d*x + c)^2 - 1245*A*a^7*b^6*tan(d*x + c)^2 + 906*
B*a^6*b^7*tan(d*x + c)^2 - 2040*A*a^5*b^8*tan(d*x + c)^2 + 600*B*a^4*b^9*tan(d*x + c)^2 - 1425*A*a^3*b^10*tan(
d*x + c)^2 + 150*B*a^2*b^11*tan(d*x + c)^2 - 366*A*a*b^12*tan(d*x + c)^2 + 792*B*a^9*b^4*tan(d*x + c) - 1350*A
*a^8*b^5*tan(d*x + c) + 1050*B*a^7*b^6*tan(d*x + c) - 2271*A*a^6*b^7*tan(d*x + c) + 696*B*a^5*b^8*tan(d*x + c)
- 1596*A*a^4*b^9*tan(d*x + c) + 174*B*a^3*b^10*tan(d*x + c) - 411*A*a^2*b^11*tan(d*x + c) + 294*B*a^10*b^3 -
492*A*a^9*b^4 + 414*B*a^8*b^5 - 853*A*a^7*b^6 + 278*B*a^6*b^7 - 606*A*a^5*b^8 + 70*B*a^4*b^9 - 157*A*a^3*b^10)
/((a^14 + 4*a^12*b^2 + 6*a^10*b^4 + 4*a^8*b^6 + a^6*b^8)*(b*tan(d*x + c) + a)^3) + 6*(A*a^2 + 4*B*a*b - 10*A*b
^2)*log(abs(tan(d*x + c)))/a^6 - 3*(3*A*a^2*tan(d*x + c)^2 + 12*B*a*b*tan(d*x + c)^2 - 30*A*b^2*tan(d*x + c)^2
- 2*B*a^2*tan(d*x + c) + 8*A*a*b*tan(d*x + c) - A*a^2)/(a^6*tan(d*x + c)^2))/d

Mupad [B] (verification not implemented)

Time = 14.99 (sec) , antiderivative size = 664, normalized size of antiderivative = 1.39 \[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (5\,A\,b-2\,B\,a\right )}{2\,a^2}-\frac {A}{2\,a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (-B\,a^7\,b^3+4\,A\,a^6\,b^4-13\,B\,a^5\,b^5+27\,A\,a^4\,b^6-12\,B\,a^3\,b^7+29\,A\,a^2\,b^8-4\,B\,a\,b^9+10\,A\,b^{10}\right )}{a^5\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (-6\,B\,a^7\,b^2+23\,A\,a^6\,b^3-62\,B\,a^5\,b^4+134\,A\,a^4\,b^5-60\,B\,a^3\,b^6+145\,A\,a^2\,b^7-20\,B\,a\,b^8+50\,A\,b^9\right )}{2\,a^4\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-18\,B\,a^7\,b+63\,A\,a^6\,b^2-128\,B\,a^5\,b^3+296\,A\,a^4\,b^4-130\,B\,a^3\,b^5+319\,A\,a^2\,b^6-44\,B\,a\,b^7+110\,A\,b^8\right )}{6\,a^3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}}{d\,\left (a^3\,{\mathrm {tan}\left (c+d\,x\right )}^2+3\,a^2\,b\,{\mathrm {tan}\left (c+d\,x\right )}^3+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^4+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^5\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (A\,a^2+4\,B\,a\,b-10\,A\,b^2\right )}{a^6\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )}{2\,d\,\left (a^4\,1{}\mathrm {i}+4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}-4\,a\,b^3+b^4\,1{}\mathrm {i}\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (A+B\,1{}\mathrm {i}\right )}{2\,d\,\left (a^4+a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2-a\,b^3\,4{}\mathrm {i}+b^4\right )}-\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-20\,B\,a^7\,b^3+35\,A\,a^6\,b^4-24\,B\,a^5\,b^5+56\,A\,a^4\,b^6-16\,B\,a^3\,b^7+39\,A\,a^2\,b^8-4\,B\,a\,b^9+10\,A\,b^{10}\right )}{d\,\left (a^{14}+4\,a^{12}\,b^2+6\,a^{10}\,b^4+4\,a^8\,b^6+a^6\,b^8\right )} \]

[In]

int((cot(c + d*x)^3*(A + B*tan(c + d*x)))/(a + b*tan(c + d*x))^4,x)

[Out]

((tan(c + d*x)*(5*A*b - 2*B*a))/(2*a^2) - A/(2*a) + (tan(c + d*x)^4*(10*A*b^10 + 29*A*a^2*b^8 + 27*A*a^4*b^6 +
4*A*a^6*b^4 - 12*B*a^3*b^7 - 13*B*a^5*b^5 - B*a^7*b^3 - 4*B*a*b^9))/(a^5*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))
+ (tan(c + d*x)^3*(50*A*b^9 + 145*A*a^2*b^7 + 134*A*a^4*b^5 + 23*A*a^6*b^3 - 60*B*a^3*b^6 - 62*B*a^5*b^4 - 6*
B*a^7*b^2 - 20*B*a*b^8))/(2*a^4*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (tan(c + d*x)^2*(110*A*b^8 + 319*A*a^2*
b^6 + 296*A*a^4*b^4 + 63*A*a^6*b^2 - 130*B*a^3*b^5 - 128*B*a^5*b^3 - 44*B*a*b^7 - 18*B*a^7*b))/(6*a^3*(a^6 + b
^6 + 3*a^2*b^4 + 3*a^4*b^2)))/(d*(a^3*tan(c + d*x)^2 + b^3*tan(c + d*x)^5 + 3*a^2*b*tan(c + d*x)^3 + 3*a*b^2*t
an(c + d*x)^4)) - (log(tan(c + d*x))*(A*a^2 - 10*A*b^2 + 4*B*a*b))/(a^6*d) + (log(tan(c + d*x) + 1i)*(A*1i + B
))/(2*d*(4*a^3*b - 4*a*b^3 + a^4*1i + b^4*1i - a^2*b^2*6i)) + (log(tan(c + d*x) - 1i)*(A + B*1i))/(2*d*(a^3*b*
4i - a*b^3*4i + a^4 + b^4 - 6*a^2*b^2)) - (log(a + b*tan(c + d*x))*(10*A*b^10 + 39*A*a^2*b^8 + 56*A*a^4*b^6 +
35*A*a^6*b^4 - 16*B*a^3*b^7 - 24*B*a^5*b^5 - 20*B*a^7*b^3 - 4*B*a*b^9))/(d*(a^14 + a^6*b^8 + 4*a^8*b^6 + 6*a^1
0*b^4 + 4*a^12*b^2))