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

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

[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]))

________________________________________________________________________________________

Rubi [A]  time = 1.73836, antiderivative size = 477, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{b \left (27 a^4 A b^3+29 a^2 A b^5+4 a^6 A b-13 a^5 b^2 B-12 a^3 b^4 B+a^7 (-B)-4 a b^6 B+10 A b^7\right )}{a^5 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{b \left (19 a^2 A b^3+7 a^4 A b-8 a^3 b^2 B-2 a^5 B-4 a b^4 B+10 A b^5\right )}{2 a^4 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac{b \left (9 a^2 A b-3 a^3 B-4 a b^2 B+10 A b^3\right )}{3 a^3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}-\frac{\left (a^2 A+4 a b B-10 A b^2\right ) \log (\sin (c+d x))}{a^6 d}-\frac{b^3 \left (56 a^4 A b^3+39 a^2 A b^5+35 a^6 A b-24 a^5 b^2 B-16 a^3 b^4 B-20 a^7 B-4 a b^6 B+10 A b^7\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^6 d \left (a^2+b^2\right )^4}+\frac{x \left (4 a^3 A b+6 a^2 b^2 B+a^4 (-B)-4 a A b^3-b^4 B\right )}{\left (a^2+b^2\right )^4}+\frac{(5 A b-2 a B) \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))^3}-\frac{A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^3} \]

Antiderivative was successfully verified.

[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 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]

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 3475

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

Rubi steps

\begin{align*} \int \frac{\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx &=-\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]  time = 6.62844, size = 417, normalized size = 0.87 \[ \frac{b^3 \left (17 a^2 A b^3+15 a^4 A b-9 a^3 b^2 B-10 a^5 B-3 a b^4 B+6 A b^5\right )}{a^5 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{b^3 \left (5 a^2 A b-4 a^3 B-2 a b^2 B+3 A b^3\right )}{2 a^4 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac{b^3 (A b-a B)}{3 a^3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}-\frac{b^3 \left (56 a^4 A b^3+39 a^2 A b^5+35 a^6 A b-24 a^5 b^2 B-16 a^3 b^4 B-20 a^7 B-4 a b^6 B+10 A b^7\right ) \log (a+b \tan (c+d x))}{a^6 d \left (a^2+b^2\right )^4}-\frac{\left (a^2 A+4 a b B-10 A b^2\right ) \log (\tan (c+d x))}{a^6 d}+\frac{(4 A b-a B) \cot (c+d x)}{a^5 d}-\frac{A \cot ^2(c+d x)}{2 a^4 d}+\frac{(A+i B) \log (-\tan (c+d x)+i)}{2 d (a+i b)^4}+\frac{(A-i B) \log (\tan (c+d x)+i)}{2 d (a-i b)^4} \]

Antiderivative was successfully verified.

[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]))

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Maple [B]  time = 0.173, size = 1030, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/d/a^4*A/tan(d*x+c)^2-1/d/a^4/tan(d*x+c)*B-35/d/(a^2+b^2)^4*ln(a+b*tan(d*x+c))*A*b^4-10/d/(a^2+b^2)^3/(a+b
*tan(d*x+c))*B*b^3+1/2/d/(a^2+b^2)^4*ln(1+tan(d*x+c)^2)*A*a^4+1/2/d/(a^2+b^2)^4*ln(1+tan(d*x+c)^2)*A*b^4-1/d/(
a^2+b^2)^4*B*arctan(tan(d*x+c))*a^4-1/d/(a^2+b^2)^4*B*arctan(tan(d*x+c))*b^4-3/d/(a^2+b^2)^4*ln(1+tan(d*x+c)^2
)*A*a^2*b^2+2/d/(a^2+b^2)^4*ln(1+tan(d*x+c)^2)*B*a^3*b-10/d*b^10/a^6/(a^2+b^2)^4*ln(a+b*tan(d*x+c))*A-9/d*b^5/
a^2/(a^2+b^2)^3/(a+b*tan(d*x+c))*B-3/d*b^7/a^4/(a^2+b^2)^3/(a+b*tan(d*x+c))*B+24/d*b^5/a/(a^2+b^2)^4*ln(a+b*ta
n(d*x+c))*B-2/d/(a^2+b^2)^4*ln(1+tan(d*x+c)^2)*B*a*b^3+4/d/(a^2+b^2)^4*A*arctan(tan(d*x+c))*a^3*b+20/d/(a^2+b^
2)^4*ln(a+b*tan(d*x+c))*B*a*b^3+6/d/(a^2+b^2)^4*B*arctan(tan(d*x+c))*a^2*b^2-4/d/(a^2+b^2)^4*A*arctan(tan(d*x+
c))*a*b^3+6/d*b^8/a^5/(a^2+b^2)^3/(a+b*tan(d*x+c))*A+1/3/d*b^4/a^3/(a^2+b^2)/(a+b*tan(d*x+c))^3*A-1/3/d*b^3/a^
2/(a^2+b^2)/(a+b*tan(d*x+c))^3*B+3/2/d*b^6/a^4/(a^2+b^2)^2/(a+b*tan(d*x+c))^2*A-2/d*b^3/a/(a^2+b^2)^2/(a+b*tan
(d*x+c))^2*B-1/d*b^5/a^3/(a^2+b^2)^2/(a+b*tan(d*x+c))^2*B+16/d*b^7/a^3/(a^2+b^2)^4*ln(a+b*tan(d*x+c))*B+4/d*b^
9/a^5/(a^2+b^2)^4*ln(a+b*tan(d*x+c))*B-56/d*b^6/a^2/(a^2+b^2)^4*ln(a+b*tan(d*x+c))*A-39/d*b^8/a^4/(a^2+b^2)^4*
ln(a+b*tan(d*x+c))*A+5/2/d*b^4/a^2/(a^2+b^2)^2/(a+b*tan(d*x+c))^2*A+17/d*b^6/a^3/(a^2+b^2)^3/(a+b*tan(d*x+c))*
A+15/d*b^4/a/(a^2+b^2)^3/(a+b*tan(d*x+c))*A+10/d/a^6*ln(tan(d*x+c))*A*b^2-4/d/a^5*ln(tan(d*x+c))*B*b+4/d/a^5/t
an(d*x+c)*A*b-1/d/a^4*A*ln(tan(d*x+c))

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Maxima [A]  time = 1.6436, size = 1100, normalized size = 2.31 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 4.68492, size = 3954, normalized size = 8.29 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

Timed out

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Giac [A]  time = 1.3566, size = 1219, normalized size = 2.56 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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