\(\int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx\) [583]

Optimal result

Integrand size = 33, antiderivative size = 421 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {\left (3 a^2 b (A-B)-b^3 (A-B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \sqrt {\cot (c+d x)}}{d}+\frac {2 a \left (7 a^2 A-18 A b^2-21 a b B\right ) \cot ^{\frac {3}{2}}(c+d x)}{21 d}-\frac {2 a^2 (11 A b+7 a B) \cot ^{\frac {5}{2}}(c+d x)}{35 d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))^2}{7 d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}+\frac {\left (a^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d} \]

[Out]

2/21*a*(7*A*a^2-18*A*b^2-21*B*a*b)*cot(d*x+c)^(3/2)/d-2/35*a^2*(11*A*b+7*B*a)*cot(d*x+c)^(5/2)/d-2/7*a*A*cot(d
*x+c)^(3/2)*(b+a*cot(d*x+c))^2/d-1/2*(3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*arctan(-1+2^(1/2)*cot(d
*x+c)^(1/2))/d*2^(1/2)-1/2*(3*a^2*b*(A-B)-b^3*(A-B)+a^3*(A+B)-3*a*b^2*(A+B))*arctan(1+2^(1/2)*cot(d*x+c)^(1/2)
)/d*2^(1/2)-1/4*(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*ln(1+cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2))/d*
2^(1/2)+1/4*(a^3*(A-B)-3*a*b^2*(A-B)-3*a^2*b*(A+B)+b^3*(A+B))*ln(1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/2))/d*2^(1
/2)+2*(3*A*a^2*b-A*b^3+B*a^3-3*B*a*b^2)*cot(d*x+c)^(1/2)/d

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3662, 3688, 3718, 3711, 3609, 3615, 1182, 1176, 631, 210, 1179, 642} \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {2 a \left (7 a^2 A-21 a b B-18 A b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{21 d}-\frac {2 a^2 (7 a B+11 A b) \cot ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {\left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d}+\frac {2 \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \sqrt {\cot (c+d x)}}{d}-\frac {\left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {\left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)^2}{7 d} \]

[In]

Int[Cot[c + d*x]^(9/2)*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]

[Out]

((3*a^2*b*(A - B) - b^3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]])/(Sqrt
[2]*d) - ((3*a^2*b*(A - B) - b^3*(A - B) + a^3*(A + B) - 3*a*b^2*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]
]])/(Sqrt[2]*d) + (2*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Sqrt[Cot[c + d*x]])/d + (2*a*(7*a^2*A - 18*A*b^2
- 21*a*b*B)*Cot[c + d*x]^(3/2))/(21*d) - (2*a^2*(11*A*b + 7*a*B)*Cot[c + d*x]^(5/2))/(35*d) - (2*a*A*Cot[c + d
*x]^(3/2)*(b + a*Cot[c + d*x])^2)/(7*d) - ((a^3*(A - B) - 3*a*b^2*(A - B) - 3*a^2*b*(A + B) + b^3*(A + B))*Log
[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d) + ((a^3*(A - B) - 3*a*b^2*(A - B) - 3*a^2*b*(A
+ B) + b^3*(A + B))*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])/(2*Sqrt[2]*d)

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1182

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(-a)*c]

Rule 3609

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*
((a + b*Tan[e + f*x])^m/(f*m)), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[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] && GtQ[m, 0]

Rule 3615

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2/f, Subst[I
nt[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2,
0] && NeQ[c^2 + d^2, 0]

Rule 3662

Int[(cot[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*tan[(e_.)
+ (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[g^(m + n), Int[(g*Cot[e + f*x])^(p - m - n)*(b + a*Cot[e + f*x])^m*(d
+ c*Cot[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&  !IntegerQ[p] && IntegerQ[m] && IntegerQ
[n]

Rule 3688

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*B*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x])^(n + 1)/(d*f
*(m + n))), x] + Dist[1/(d*(m + n)), Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^n*Simp[a^2*A*d*(m +
n) - b*B*(b*c*(m - 1) + a*d*(n + 1)) + d*(m + n)*(2*a*A*b + B*(a^2 - b^2))*Tan[e + f*x] - (b*B*(b*c - a*d)*(m
- 1) - b*(A*b + a*B)*d*(m + n))*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] && GtQ[m, 1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&
!(IGtQ[n, 1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3711

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])
^m*Simp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0]
&&  !LeQ[m, -1]

Rule 3718

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*tan[(e
_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[b*C*Tan[e + f*x]*((c + d*Tan[e + f*x])
^(n + 1)/(d*f*(n + 2))), x] - Dist[1/(d*(n + 2)), Int[(c + d*Tan[e + f*x])^n*Simp[b*c*C - a*A*d*(n + 2) - (A*b
+ a*B - b*C)*d*(n + 2)*Tan[e + f*x] - (a*C*d*(n + 2) - b*(c*C - B*d*(n + 2)))*Tan[e + f*x]^2, x], x], x] /; F
reeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] &&  !LtQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = \int \sqrt {\cot (c+d x)} (b+a \cot (c+d x))^3 (B+A \cot (c+d x)) \, dx \\ & = -\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))^2}{7 d}-\frac {2}{7} \int \sqrt {\cot (c+d x)} (b+a \cot (c+d x)) \left (\frac {1}{2} b (3 a A-7 b B)+\frac {7}{2} \left (a^2 A-A b^2-2 a b B\right ) \cot (c+d x)-\frac {1}{2} a (11 A b+7 a B) \cot ^2(c+d x)\right ) \, dx \\ & = -\frac {2 a^2 (11 A b+7 a B) \cot ^{\frac {5}{2}}(c+d x)}{35 d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))^2}{7 d}-\frac {4}{35} \int \sqrt {\cot (c+d x)} \left (\frac {5}{4} b^2 (3 a A-7 b B)+\frac {35}{4} \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \cot (c+d x)+\frac {5}{4} a \left (7 a^2 A-18 A b^2-21 a b B\right ) \cot ^2(c+d x)\right ) \, dx \\ & = \frac {2 a \left (7 a^2 A-18 A b^2-21 a b B\right ) \cot ^{\frac {3}{2}}(c+d x)}{21 d}-\frac {2 a^2 (11 A b+7 a B) \cot ^{\frac {5}{2}}(c+d x)}{35 d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))^2}{7 d}-\frac {4}{35} \int \sqrt {\cot (c+d x)} \left (-\frac {35}{4} \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )+\frac {35}{4} \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \cot (c+d x)\right ) \, dx \\ & = \frac {2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \sqrt {\cot (c+d x)}}{d}+\frac {2 a \left (7 a^2 A-18 A b^2-21 a b B\right ) \cot ^{\frac {3}{2}}(c+d x)}{21 d}-\frac {2 a^2 (11 A b+7 a B) \cot ^{\frac {5}{2}}(c+d x)}{35 d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))^2}{7 d}-\frac {4}{35} \int \frac {-\frac {35}{4} \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )-\frac {35}{4} \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx \\ & = \frac {2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \sqrt {\cot (c+d x)}}{d}+\frac {2 a \left (7 a^2 A-18 A b^2-21 a b B\right ) \cot ^{\frac {3}{2}}(c+d x)}{21 d}-\frac {2 a^2 (11 A b+7 a B) \cot ^{\frac {5}{2}}(c+d x)}{35 d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))^2}{7 d}-\frac {8 \text {Subst}\left (\int \frac {\frac {35}{4} \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )+\frac {35}{4} \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{35 d} \\ & = \frac {2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \sqrt {\cot (c+d x)}}{d}+\frac {2 a \left (7 a^2 A-18 A b^2-21 a b B\right ) \cot ^{\frac {3}{2}}(c+d x)}{21 d}-\frac {2 a^2 (11 A b+7 a B) \cot ^{\frac {5}{2}}(c+d x)}{35 d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))^2}{7 d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d}+\frac {\left (a^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (A+B)\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d} \\ & = \frac {2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \sqrt {\cot (c+d x)}}{d}+\frac {2 a \left (7 a^2 A-18 A b^2-21 a b B\right ) \cot ^{\frac {3}{2}}(c+d x)}{21 d}-\frac {2 a^2 (11 A b+7 a B) \cot ^{\frac {5}{2}}(c+d x)}{35 d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))^2}{7 d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (A+B)\right ) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \sqrt {2} d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (A+B)\right ) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \sqrt {2} d} \\ & = \frac {2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \sqrt {\cot (c+d x)}}{d}+\frac {2 a \left (7 a^2 A-18 A b^2-21 a b B\right ) \cot ^{\frac {3}{2}}(c+d x)}{21 d}-\frac {2 a^2 (11 A b+7 a B) \cot ^{\frac {5}{2}}(c+d x)}{35 d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))^2}{7 d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}+\frac {\left (a^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d} \\ & = \frac {\left (3 a^2 b (A-B)-b^3 (A-B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \sqrt {\cot (c+d x)}}{d}+\frac {2 a \left (7 a^2 A-18 A b^2-21 a b B\right ) \cot ^{\frac {3}{2}}(c+d x)}{21 d}-\frac {2 a^2 (11 A b+7 a B) \cot ^{\frac {5}{2}}(c+d x)}{35 d}-\frac {2 a A \cot ^{\frac {3}{2}}(c+d x) (b+a \cot (c+d x))^2}{7 d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}+\frac {\left (a^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.91 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.77 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {2 \sqrt {\cot (c+d x)} \left (-\frac {\left (3 a^2 b (A-B)+b^3 (-A+B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \left (\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )\right )}{2 \sqrt {2}}-\frac {\left (a^3 (A-B)+3 a b^2 (-A+B)-3 a^2 b (A+B)+b^3 (A+B)\right ) \left (\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )\right )}{4 \sqrt {2}}-\frac {a^3 A}{7 \tan ^{\frac {7}{2}}(c+d x)}-\frac {a^2 (3 A b+a B)}{5 \tan ^{\frac {5}{2}}(c+d x)}+\frac {a \left (a^2 A-3 A b^2-3 a b B\right )}{3 \tan ^{\frac {3}{2}}(c+d x)}+\frac {3 a^2 A b-A b^3+a^3 B-3 a b^2 B}{\sqrt {\tan (c+d x)}}\right ) \sqrt {\tan (c+d x)}}{d} \]

[In]

Integrate[Cot[c + d*x]^(9/2)*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]

[Out]

(2*Sqrt[Cot[c + d*x]]*(-1/2*((3*a^2*b*(A - B) + b^3*(-A + B) + a^3*(A + B) - 3*a*b^2*(A + B))*(ArcTan[1 - Sqrt
[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]))/Sqrt[2] - ((a^3*(A - B) + 3*a*b^2*(-A + B)
- 3*a^2*b*(A + B) + b^3*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Ta
n[c + d*x]] + Tan[c + d*x]]))/(4*Sqrt[2]) - (a^3*A)/(7*Tan[c + d*x]^(7/2)) - (a^2*(3*A*b + a*B))/(5*Tan[c + d*
x]^(5/2)) + (a*(a^2*A - 3*A*b^2 - 3*a*b*B))/(3*Tan[c + d*x]^(3/2)) + (3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)/S
qrt[Tan[c + d*x]])*Sqrt[Tan[c + d*x]])/d

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1191\) vs. \(2(379)=758\).

Time = 1.18 (sec) , antiderivative size = 1192, normalized size of antiderivative = 2.83

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1192\)
default	\(\text {Expression too large to display}\)	\(1192\)

[In]

int(cot(d*x+c)^(9/2)*(a+b*tan(d*x+c))^3*(A+B*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/420/d*(1/tan(d*x+c))^(9/2)*tan(d*x+c)*(-840*A*tan(d*x+c)^3*b^3-168*B*tan(d*x+c)*a^3-630*B*arctan(1+2^(1/2)*t
an(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(7/2)*a*b^2-630*B*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(
7/2)*a^2*b-630*B*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(7/2)*a*b^2-315*B*ln(-(2^(1/2)*tan(d*x
+c)^(1/2)-tan(d*x+c)-1)/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*2^(1/2)*tan(d*x+c)^(7/2)*a*b^2-315*A*ln(-(1+2
^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*2^(1/2)*tan(d*x+c)^(7/2)*a*b^2+63
0*A*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(7/2)*a^2*b-630*A*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))
*2^(1/2)*tan(d*x+c)^(7/2)*a*b^2+630*A*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(7/2)*a^2*b-630*A
*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(7/2)*a*b^2+315*A*ln(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*
x+c)-1)/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*2^(1/2)*tan(d*x+c)^(7/2)*a^2*b-315*B*ln(-(1+2^(1/2)*tan(d*x+c
)^(1/2)+tan(d*x+c))/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*2^(1/2)*tan(d*x+c)^(7/2)*a^2*b-630*B*arctan(1+2^(
1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(7/2)*a^2*b+2520*A*tan(d*x+c)^3*a^2*b-120*A*a^3-2520*B*tan(d*x+c)^3*
a*b^2-840*A*tan(d*x+c)^2*a*b^2-840*B*tan(d*x+c)^2*a^2*b-504*A*tan(d*x+c)*a^2*b+840*B*tan(d*x+c)^3*a^3+280*A*ta
n(d*x+c)^2*a^3+210*A*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(7/2)*a^3-210*A*arctan(-1+2^(1/2)*
tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(7/2)*b^3-105*A*ln(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(1+2^(1/2)*ta
n(d*x+c)^(1/2)+tan(d*x+c)))*2^(1/2)*tan(d*x+c)^(7/2)*b^3+210*B*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(
d*x+c)^(7/2)*a^3+210*B*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(7/2)*b^3+210*B*arctan(-1+2^(1/2)
*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(7/2)*a^3+105*B*ln(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(1+2^(1/2)*t
an(d*x+c)^(1/2)+tan(d*x+c)))*2^(1/2)*tan(d*x+c)^(7/2)*a^3+105*B*ln(-(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(2
^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*2^(1/2)*tan(d*x+c)^(7/2)*b^3+105*A*ln(-(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(
d*x+c))/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*2^(1/2)*tan(d*x+c)^(7/2)*a^3+210*A*arctan(1+2^(1/2)*tan(d*x+c
)^(1/2))*2^(1/2)*tan(d*x+c)^(7/2)*a^3-210*A*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(7/2)*b^3+21
0*B*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(7/2)*b^3)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6214 vs. \(2 (379) = 758\).

Time = 6.61 (sec) , antiderivative size = 6214, normalized size of antiderivative = 14.76 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

integrate(cot(d*x+c)^(9/2)*(a+b*tan(d*x+c))^3*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(cot(d*x+c)**(9/2)*(a+b*tan(d*x+c))**3*(A+B*tan(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.45 (sec) , antiderivative size = 366, normalized size of antiderivative = 0.87 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=-\frac {210 \, \sqrt {2} {\left ({\left (A + B\right )} a^{3} + 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} - {\left (A - B\right )} b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 210 \, \sqrt {2} {\left ({\left (A + B\right )} a^{3} + 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} - {\left (A - B\right )} b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - 105 \, \sqrt {2} {\left ({\left (A - B\right )} a^{3} - 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} + {\left (A + B\right )} b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + 105 \, \sqrt {2} {\left ({\left (A - B\right )} a^{3} - 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} + {\left (A + B\right )} b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \frac {120 \, A a^{3}}{\tan \left (d x + c\right )^{\frac {7}{2}}} - \frac {840 \, {\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )}}{\sqrt {\tan \left (d x + c\right )}} - \frac {280 \, {\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2}\right )}}{\tan \left (d x + c\right )^{\frac {3}{2}}} + \frac {168 \, {\left (B a^{3} + 3 \, A a^{2} b\right )}}{\tan \left (d x + c\right )^{\frac {5}{2}}}}{420 \, d} \]

[In]

integrate(cot(d*x+c)^(9/2)*(a+b*tan(d*x+c))^3*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

-1/420*(210*sqrt(2)*((A + B)*a^3 + 3*(A - B)*a^2*b - 3*(A + B)*a*b^2 - (A - B)*b^3)*arctan(1/2*sqrt(2)*(sqrt(2
) + 2/sqrt(tan(d*x + c)))) + 210*sqrt(2)*((A + B)*a^3 + 3*(A - B)*a^2*b - 3*(A + B)*a*b^2 - (A - B)*b^3)*arcta
n(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) - 105*sqrt(2)*((A - B)*a^3 - 3*(A + B)*a^2*b - 3*(A - B)*a*b^
2 + (A + B)*b^3)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) + 105*sqrt(2)*((A - B)*a^3 - 3*(A + B)*a
^2*b - 3*(A - B)*a*b^2 + (A + B)*b^3)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) + 120*A*a^3/tan(d*
x + c)^(7/2) - 840*(B*a^3 + 3*A*a^2*b - 3*B*a*b^2 - A*b^3)/sqrt(tan(d*x + c)) - 280*(A*a^3 - 3*B*a^2*b - 3*A*a
*b^2)/tan(d*x + c)^(3/2) + 168*(B*a^3 + 3*A*a^2*b)/tan(d*x + c)^(5/2))/d

Giac [F]

\[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3} \cot \left (d x + c\right )^{\frac {9}{2}} \,d x } \]

[In]

integrate(cot(d*x+c)^(9/2)*(a+b*tan(d*x+c))^3*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

integrate((B*tan(d*x + c) + A)*(b*tan(d*x + c) + a)^3*cot(d*x + c)^(9/2), x)

Mupad [F(-1)]

Timed out. \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{9/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^3 \,d x \]

[In]

int(cot(c + d*x)^(9/2)*(A + B*tan(c + d*x))*(a + b*tan(c + d*x))^3,x)

[Out]

int(cot(c + d*x)^(9/2)*(A + B*tan(c + d*x))*(a + b*tan(c + d*x))^3, x)