\(\int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx\) [385]

Optimal result

Integrand size = 33, antiderivative size = 394 \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {\left (a^2 (A-B)-b^2 (A-B)-2 a b (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^2 (A-B)-b^2 (A-B)-2 a b (A+B)\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (2 a b (A-B)+a^2 (A+B)-b^2 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}+\frac {\left (2 a b (A-B)+a^2 (A+B)-b^2 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 \left (a^2 A-A b^2-2 a b B\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 b (9 A b+11 a B) \tan ^{\frac {7}{2}}(c+d x)}{63 d}+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d} \]

[Out]

-1/2*(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/d*2^(1/2)-1/2*(a^2*(A-B)-b^2*(A-B)-
2*a*b*(A+B))*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))/d*2^(1/2)-1/4*(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*ln(1-2^(1/2)*t
an(d*x+c)^(1/2)+tan(d*x+c))/d*2^(1/2)+1/4*(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(
d*x+c))/d*2^(1/2)-2*(2*A*a*b+B*a^2-B*b^2)*tan(d*x+c)^(1/2)/d+2/3*(A*a^2-A*b^2-2*B*a*b)*tan(d*x+c)^(3/2)/d+2/5*
(2*A*a*b+B*a^2-B*b^2)*tan(d*x+c)^(5/2)/d+2/63*b*(9*A*b+11*B*a)*tan(d*x+c)^(7/2)/d+2/9*b*B*tan(d*x+c)^(7/2)*(a+
b*tan(d*x+c))/d

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {3688, 3711, 3609, 3615, 1182, 1176, 631, 210, 1179, 642} \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {\left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d}+\frac {2 \left (a^2 B+2 a A b-b^2 B\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 \left (a^2 A-2 a b B-A b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 \left (a^2 B+2 a A b-b^2 B\right ) \sqrt {\tan (c+d x)}}{d}-\frac {\left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {\left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {2 b (11 a B+9 A b) \tan ^{\frac {7}{2}}(c+d x)}{63 d}+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d} \]

[In]

Int[Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]

[Out]

((a^2*(A - B) - b^2*(A - B) - 2*a*b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a^2*(A -
B) - b^2*(A - B) - 2*a*b*(A + B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((2*a*b*(A - B) + a^2*
(A + B) - b^2*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) + ((2*a*b*(A - B) + a
^2*(A + B) - b^2*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d) - (2*(2*a*A*b + a^
2*B - b^2*B)*Sqrt[Tan[c + d*x]])/d + (2*(a^2*A - A*b^2 - 2*a*b*B)*Tan[c + d*x]^(3/2))/(3*d) + (2*(2*a*A*b + a^
2*B - b^2*B)*Tan[c + d*x]^(5/2))/(5*d) + (2*b*(9*A*b + 11*a*B)*Tan[c + d*x]^(7/2))/(63*d) + (2*b*B*Tan[c + d*x
]^(7/2)*(a + b*Tan[c + d*x]))/(9*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 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]

Rubi steps \begin{align*} \text {integral}& = \frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}+\frac {2}{9} \int \tan ^{\frac {5}{2}}(c+d x) \left (\frac {1}{2} a (9 a A-7 b B)+\frac {9}{2} \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)+\frac {1}{2} b (9 A b+11 a B) \tan ^2(c+d x)\right ) \, dx \\ & = \frac {2 b (9 A b+11 a B) \tan ^{\frac {7}{2}}(c+d x)}{63 d}+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}+\frac {2}{9} \int \tan ^{\frac {5}{2}}(c+d x) \left (\frac {9}{2} \left (a^2 A-A b^2-2 a b B\right )+\frac {9}{2} \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)\right ) \, dx \\ & = \frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 b (9 A b+11 a B) \tan ^{\frac {7}{2}}(c+d x)}{63 d}+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}+\frac {2}{9} \int \tan ^{\frac {3}{2}}(c+d x) \left (-\frac {9}{2} \left (2 a A b+a^2 B-b^2 B\right )+\frac {9}{2} \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)\right ) \, dx \\ & = \frac {2 \left (a^2 A-A b^2-2 a b B\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 b (9 A b+11 a B) \tan ^{\frac {7}{2}}(c+d x)}{63 d}+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}+\frac {2}{9} \int \sqrt {\tan (c+d x)} \left (-\frac {9}{2} \left (a^2 A-A b^2-2 a b B\right )-\frac {9}{2} \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)\right ) \, dx \\ & = -\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 \left (a^2 A-A b^2-2 a b B\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 b (9 A b+11 a B) \tan ^{\frac {7}{2}}(c+d x)}{63 d}+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}+\frac {2}{9} \int \frac {\frac {9}{2} \left (2 a A b+a^2 B-b^2 B\right )-\frac {9}{2} \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx \\ & = -\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 \left (a^2 A-A b^2-2 a b B\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 b (9 A b+11 a B) \tan ^{\frac {7}{2}}(c+d x)}{63 d}+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}+\frac {4 \text {Subst}\left (\int \frac {\frac {9}{2} \left (2 a A b+a^2 B-b^2 B\right )-\frac {9}{2} \left (a^2 A-A b^2-2 a b B\right ) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{9 d} \\ & = -\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 \left (a^2 A-A b^2-2 a b B\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 b (9 A b+11 a B) \tan ^{\frac {7}{2}}(c+d x)}{63 d}+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}-\frac {\left (a^2 (A-B)-b^2 (A-B)-2 a b (A+B)\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}+\frac {\left (2 a b (A-B)+a^2 (A+B)-b^2 (A+B)\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = -\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 \left (a^2 A-A b^2-2 a b B\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 b (9 A b+11 a B) \tan ^{\frac {7}{2}}(c+d x)}{63 d}+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}-\frac {\left (a^2 (A-B)-b^2 (A-B)-2 a b (A+B)\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 d}-\frac {\left (a^2 (A-B)-b^2 (A-B)-2 a b (A+B)\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 d}-\frac {\left (2 a b (A-B)+a^2 (A+B)-b^2 (A+B)\right ) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} d}-\frac {\left (2 a b (A-B)+a^2 (A+B)-b^2 (A+B)\right ) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} d} \\ & = -\frac {\left (2 a b (A-B)+a^2 (A+B)-b^2 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}+\frac {\left (2 a b (A-B)+a^2 (A+B)-b^2 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 \left (a^2 A-A b^2-2 a b B\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 b (9 A b+11 a B) \tan ^{\frac {7}{2}}(c+d x)}{63 d}+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}-\frac {\left (a^2 (A-B)-b^2 (A-B)-2 a b (A+B)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (a^2 (A-B)-b^2 (A-B)-2 a b (A+B)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d} \\ & = \frac {\left (a^2 (A-B)-b^2 (A-B)-2 a b (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^2 (A-B)-b^2 (A-B)-2 a b (A+B)\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (2 a b (A-B)+a^2 (A+B)-b^2 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}+\frac {\left (2 a b (A-B)+a^2 (A+B)-b^2 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 \left (a^2 A-A b^2-2 a b B\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 b (9 A b+11 a B) \tan ^{\frac {7}{2}}(c+d x)}{63 d}+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.15 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.77 \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}+\frac {2}{9} \left (\frac {b (9 A b+11 a B) \tan ^{\frac {7}{2}}(c+d x)}{7 d}+\frac {i \left (\frac {9}{2} \left (a^2 A-A b^2-2 a b B\right )-\frac {9}{2} i \left (2 a A b+a^2 B-b^2 B\right )\right ) \left (-2 \sqrt [4]{-1} \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )-2 \sqrt {\tan (c+d x)}-\frac {2}{3} i \tan ^{\frac {3}{2}}(c+d x)+\frac {2}{5} \tan ^{\frac {5}{2}}(c+d x)\right )}{2 d}-\frac {i \left (\frac {9}{2} \left (a^2 A-A b^2-2 a b B\right )+\frac {9}{2} i \left (2 a A b+a^2 B-b^2 B\right )\right ) \left (-2 \sqrt [4]{-1} \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )-2 \sqrt {\tan (c+d x)}+\frac {2}{3} i \tan ^{\frac {3}{2}}(c+d x)+\frac {2}{5} \tan ^{\frac {5}{2}}(c+d x)\right )}{2 d}\right ) \]

[In]

Integrate[Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]

[Out]

(2*b*B*Tan[c + d*x]^(7/2)*(a + b*Tan[c + d*x]))/(9*d) + (2*((b*(9*A*b + 11*a*B)*Tan[c + d*x]^(7/2))/(7*d) + ((
I/2)*((9*(a^2*A - A*b^2 - 2*a*b*B))/2 - ((9*I)/2)*(2*a*A*b + a^2*B - b^2*B))*(-2*(-1)^(1/4)*ArcTan[(-1)^(3/4)*
Sqrt[Tan[c + d*x]]] - 2*Sqrt[Tan[c + d*x]] - ((2*I)/3)*Tan[c + d*x]^(3/2) + (2*Tan[c + d*x]^(5/2))/5))/d - ((I
/2)*((9*(a^2*A - A*b^2 - 2*a*b*B))/2 + ((9*I)/2)*(2*a*A*b + a^2*B - b^2*B))*(-2*(-1)^(1/4)*ArcTanh[(-1)^(3/4)*
Sqrt[Tan[c + d*x]]] - 2*Sqrt[Tan[c + d*x]] + ((2*I)/3)*Tan[c + d*x]^(3/2) + (2*Tan[c + d*x]^(5/2))/5))/d))/9

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.95

method	result	size
derivativedivides	\(\frac {\frac {2 B \,b^{2} \left (\tan ^{\frac {9}{2}}\left (d x +c \right )\right )}{9}+\frac {2 A \,b^{2} \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right )}{7}+\frac {4 B a b \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right )}{7}+\frac {4 A a b \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}+\frac {2 B \,a^{2} \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}-\frac {2 B \,b^{2} \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}+\frac {2 A \,a^{2} \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-\frac {2 A \,b^{2} \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-\frac {4 B a b \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-4 A a b \left (\sqrt {\tan }\left (d x +c \right )\right )-2 B \,a^{2} \left (\sqrt {\tan }\left (d x +c \right )\right )+2 \left (\sqrt {\tan }\left (d x +c \right )\right ) B \,b^{2}+\frac {\left (2 A a b +B \,a^{2}-B \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-A \,a^{2}+A \,b^{2}+2 B a b \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{d}\)	\(374\)
default	\(\frac {\frac {2 B \,b^{2} \left (\tan ^{\frac {9}{2}}\left (d x +c \right )\right )}{9}+\frac {2 A \,b^{2} \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right )}{7}+\frac {4 B a b \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right )}{7}+\frac {4 A a b \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}+\frac {2 B \,a^{2} \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}-\frac {2 B \,b^{2} \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}+\frac {2 A \,a^{2} \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-\frac {2 A \,b^{2} \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-\frac {4 B a b \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-4 A a b \left (\sqrt {\tan }\left (d x +c \right )\right )-2 B \,a^{2} \left (\sqrt {\tan }\left (d x +c \right )\right )+2 \left (\sqrt {\tan }\left (d x +c \right )\right ) B \,b^{2}+\frac {\left (2 A a b +B \,a^{2}-B \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-A \,a^{2}+A \,b^{2}+2 B a b \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{d}\)	\(374\)
parts	\(\frac {\left (A \,b^{2}+2 B a b \right ) \left (\frac {2 \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right )}{7}-\frac {2 \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+\frac {\sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}+\frac {\left (2 A a b +B \,a^{2}\right ) \left (\frac {2 \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}-2 \left (\sqrt {\tan }\left (d x +c \right )\right )+\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}+\frac {A \,a^{2} \left (\frac {2 \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-\frac {\sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}+\frac {B \,b^{2} \left (\frac {2 \left (\tan ^{\frac {9}{2}}\left (d x +c \right )\right )}{9}-\frac {2 \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}+2 \left (\sqrt {\tan }\left (d x +c \right )\right )-\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}\)	\(472\)

[In]

int(tan(d*x+c)^(5/2)*(a+b*tan(d*x+c))^2*(A+B*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(2/9*B*b^2*tan(d*x+c)^(9/2)+2/7*A*b^2*tan(d*x+c)^(7/2)+4/7*B*a*b*tan(d*x+c)^(7/2)+4/5*A*a*b*tan(d*x+c)^(5/
2)+2/5*B*a^2*tan(d*x+c)^(5/2)-2/5*B*b^2*tan(d*x+c)^(5/2)+2/3*A*a^2*tan(d*x+c)^(3/2)-2/3*A*b^2*tan(d*x+c)^(3/2)
-4/3*B*a*b*tan(d*x+c)^(3/2)-4*A*a*b*tan(d*x+c)^(1/2)-2*B*a^2*tan(d*x+c)^(1/2)+2*tan(d*x+c)^(1/2)*B*b^2+1/4*(2*
A*a*b+B*a^2-B*b^2)*2^(1/2)*(ln((1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))
)+2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2)))+1/4*(-A*a^2+A*b^2+2*B*a*b)*2^(1/
2)*(ln((1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))+2*arctan(1+2^(1/2)*tan
(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4337 vs. \(2 (348) = 696\).

Time = 0.75 (sec) , antiderivative size = 4337, normalized size of antiderivative = 11.01 \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

integrate(tan(d*x+c)^(5/2)*(a+b*tan(d*x+c))^2*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/630*(315*d*sqrt((2*A*B*a^4 - 12*A*B*a^2*b^2 + 2*A*B*b^4 + 4*(A^2 - B^2)*a^3*b - 4*(A^2 - B^2)*a*b^3 + d^2*sq
rt(-((A^4 - 2*A^2*B^2 + B^4)*a^8 - 16*(A^3*B - A*B^3)*a^7*b - 4*(3*A^4 - 22*A^2*B^2 + 3*B^4)*a^6*b^2 + 112*(A^
3*B - A*B^3)*a^5*b^3 + 2*(19*A^4 - 102*A^2*B^2 + 19*B^4)*a^4*b^4 - 112*(A^3*B - A*B^3)*a^3*b^5 - 4*(3*A^4 - 22
*A^2*B^2 + 3*B^4)*a^2*b^6 + 16*(A^3*B - A*B^3)*a*b^7 + (A^4 - 2*A^2*B^2 + B^4)*b^8)/d^4))/d^2)*log(((A*a^2 - 2
*B*a*b - A*b^2)*d^3*sqrt(-((A^4 - 2*A^2*B^2 + B^4)*a^8 - 16*(A^3*B - A*B^3)*a^7*b - 4*(3*A^4 - 22*A^2*B^2 + 3*
B^4)*a^6*b^2 + 112*(A^3*B - A*B^3)*a^5*b^3 + 2*(19*A^4 - 102*A^2*B^2 + 19*B^4)*a^4*b^4 - 112*(A^3*B - A*B^3)*a
^3*b^5 - 4*(3*A^4 - 22*A^2*B^2 + 3*B^4)*a^2*b^6 + 16*(A^3*B - A*B^3)*a*b^7 + (A^4 - 2*A^2*B^2 + B^4)*b^8)/d^4)
- ((A^2*B - B^3)*a^6 + 2*(A^3 - 5*A*B^2)*a^5*b - (23*A^2*B - 7*B^3)*a^4*b^2 - 4*(3*A^3 - 7*A*B^2)*a^3*b^3 + (
23*A^2*B - 7*B^3)*a^2*b^4 + 2*(A^3 - 5*A*B^2)*a*b^5 - (A^2*B - B^3)*b^6)*d)*sqrt((2*A*B*a^4 - 12*A*B*a^2*b^2 +
2*A*B*b^4 + 4*(A^2 - B^2)*a^3*b - 4*(A^2 - B^2)*a*b^3 + d^2*sqrt(-((A^4 - 2*A^2*B^2 + B^4)*a^8 - 16*(A^3*B -
A*B^3)*a^7*b - 4*(3*A^4 - 22*A^2*B^2 + 3*B^4)*a^6*b^2 + 112*(A^3*B - A*B^3)*a^5*b^3 + 2*(19*A^4 - 102*A^2*B^2
+ 19*B^4)*a^4*b^4 - 112*(A^3*B - A*B^3)*a^3*b^5 - 4*(3*A^4 - 22*A^2*B^2 + 3*B^4)*a^2*b^6 + 16*(A^3*B - A*B^3)*
a*b^7 + (A^4 - 2*A^2*B^2 + B^4)*b^8)/d^4))/d^2) - ((A^4 - B^4)*a^8 - 8*(A^3*B + A*B^3)*a^7*b - 4*(A^4 - B^4)*a
^6*b^2 - 8*(A^3*B + A*B^3)*a^5*b^3 - 10*(A^4 - B^4)*a^4*b^4 + 8*(A^3*B + A*B^3)*a^3*b^5 - 4*(A^4 - B^4)*a^2*b^
6 + 8*(A^3*B + A*B^3)*a*b^7 + (A^4 - B^4)*b^8)*sqrt(tan(d*x + c))) - 315*d*sqrt((2*A*B*a^4 - 12*A*B*a^2*b^2 +
2*A*B*b^4 + 4*(A^2 - B^2)*a^3*b - 4*(A^2 - B^2)*a*b^3 + d^2*sqrt(-((A^4 - 2*A^2*B^2 + B^4)*a^8 - 16*(A^3*B - A
*B^3)*a^7*b - 4*(3*A^4 - 22*A^2*B^2 + 3*B^4)*a^6*b^2 + 112*(A^3*B - A*B^3)*a^5*b^3 + 2*(19*A^4 - 102*A^2*B^2 +
19*B^4)*a^4*b^4 - 112*(A^3*B - A*B^3)*a^3*b^5 - 4*(3*A^4 - 22*A^2*B^2 + 3*B^4)*a^2*b^6 + 16*(A^3*B - A*B^3)*a
*b^7 + (A^4 - 2*A^2*B^2 + B^4)*b^8)/d^4))/d^2)*log(-((A*a^2 - 2*B*a*b - A*b^2)*d^3*sqrt(-((A^4 - 2*A^2*B^2 + B
^4)*a^8 - 16*(A^3*B - A*B^3)*a^7*b - 4*(3*A^4 - 22*A^2*B^2 + 3*B^4)*a^6*b^2 + 112*(A^3*B - A*B^3)*a^5*b^3 + 2*
(19*A^4 - 102*A^2*B^2 + 19*B^4)*a^4*b^4 - 112*(A^3*B - A*B^3)*a^3*b^5 - 4*(3*A^4 - 22*A^2*B^2 + 3*B^4)*a^2*b^6
+ 16*(A^3*B - A*B^3)*a*b^7 + (A^4 - 2*A^2*B^2 + B^4)*b^8)/d^4) - ((A^2*B - B^3)*a^6 + 2*(A^3 - 5*A*B^2)*a^5*b
- (23*A^2*B - 7*B^3)*a^4*b^2 - 4*(3*A^3 - 7*A*B^2)*a^3*b^3 + (23*A^2*B - 7*B^3)*a^2*b^4 + 2*(A^3 - 5*A*B^2)*a
*b^5 - (A^2*B - B^3)*b^6)*d)*sqrt((2*A*B*a^4 - 12*A*B*a^2*b^2 + 2*A*B*b^4 + 4*(A^2 - B^2)*a^3*b - 4*(A^2 - B^2
)*a*b^3 + d^2*sqrt(-((A^4 - 2*A^2*B^2 + B^4)*a^8 - 16*(A^3*B - A*B^3)*a^7*b - 4*(3*A^4 - 22*A^2*B^2 + 3*B^4)*a
^6*b^2 + 112*(A^3*B - A*B^3)*a^5*b^3 + 2*(19*A^4 - 102*A^2*B^2 + 19*B^4)*a^4*b^4 - 112*(A^3*B - A*B^3)*a^3*b^5
- 4*(3*A^4 - 22*A^2*B^2 + 3*B^4)*a^2*b^6 + 16*(A^3*B - A*B^3)*a*b^7 + (A^4 - 2*A^2*B^2 + B^4)*b^8)/d^4))/d^2)
- ((A^4 - B^4)*a^8 - 8*(A^3*B + A*B^3)*a^7*b - 4*(A^4 - B^4)*a^6*b^2 - 8*(A^3*B + A*B^3)*a^5*b^3 - 10*(A^4 -
B^4)*a^4*b^4 + 8*(A^3*B + A*B^3)*a^3*b^5 - 4*(A^4 - B^4)*a^2*b^6 + 8*(A^3*B + A*B^3)*a*b^7 + (A^4 - B^4)*b^8)*
sqrt(tan(d*x + c))) - 315*d*sqrt((2*A*B*a^4 - 12*A*B*a^2*b^2 + 2*A*B*b^4 + 4*(A^2 - B^2)*a^3*b - 4*(A^2 - B^2)
*a*b^3 - d^2*sqrt(-((A^4 - 2*A^2*B^2 + B^4)*a^8 - 16*(A^3*B - A*B^3)*a^7*b - 4*(3*A^4 - 22*A^2*B^2 + 3*B^4)*a^
6*b^2 + 112*(A^3*B - A*B^3)*a^5*b^3 + 2*(19*A^4 - 102*A^2*B^2 + 19*B^4)*a^4*b^4 - 112*(A^3*B - A*B^3)*a^3*b^5
- 4*(3*A^4 - 22*A^2*B^2 + 3*B^4)*a^2*b^6 + 16*(A^3*B - A*B^3)*a*b^7 + (A^4 - 2*A^2*B^2 + B^4)*b^8)/d^4))/d^2)*
log(((A*a^2 - 2*B*a*b - A*b^2)*d^3*sqrt(-((A^4 - 2*A^2*B^2 + B^4)*a^8 - 16*(A^3*B - A*B^3)*a^7*b - 4*(3*A^4 -
22*A^2*B^2 + 3*B^4)*a^6*b^2 + 112*(A^3*B - A*B^3)*a^5*b^3 + 2*(19*A^4 - 102*A^2*B^2 + 19*B^4)*a^4*b^4 - 112*(A
^3*B - A*B^3)*a^3*b^5 - 4*(3*A^4 - 22*A^2*B^2 + 3*B^4)*a^2*b^6 + 16*(A^3*B - A*B^3)*a*b^7 + (A^4 - 2*A^2*B^2 +
B^4)*b^8)/d^4) + ((A^2*B - B^3)*a^6 + 2*(A^3 - 5*A*B^2)*a^5*b - (23*A^2*B - 7*B^3)*a^4*b^2 - 4*(3*A^3 - 7*A*B
^2)*a^3*b^3 + (23*A^2*B - 7*B^3)*a^2*b^4 + 2*(A^3 - 5*A*B^2)*a*b^5 - (A^2*B - B^3)*b^6)*d)*sqrt((2*A*B*a^4 - 1
2*A*B*a^2*b^2 + 2*A*B*b^4 + 4*(A^2 - B^2)*a^3*b - 4*(A^2 - B^2)*a*b^3 - d^2*sqrt(-((A^4 - 2*A^2*B^2 + B^4)*a^8
- 16*(A^3*B - A*B^3)*a^7*b - 4*(3*A^4 - 22*A^2*B^2 + 3*B^4)*a^6*b^2 + 112*(A^3*B - A*B^3)*a^5*b^3 + 2*(19*A^4
- 102*A^2*B^2 + 19*B^4)*a^4*b^4 - 112*(A^3*B - A*B^3)*a^3*b^5 - 4*(3*A^4 - 22*A^2*B^2 + 3*B^4)*a^2*b^6 + 16*(
A^3*B - A*B^3)*a*b^7 + (A^4 - 2*A^2*B^2 + B^4)*b^8)/d^4))/d^2) - ((A^4 - B^4)*a^8 - 8*(A^3*B + A*B^3)*a^7*b -
4*(A^4 - B^4)*a^6*b^2 - 8*(A^3*B + A*B^3)*a^5*b^3 - 10*(A^4 - B^4)*a^4*b^4 + 8*(A^3*B + A*B^3)*a^3*b^5 - 4*(A^
4 - B^4)*a^2*b^6 + 8*(A^3*B + A*B^3)*a*b^7 + (A^4 - B^4)*b^8)*sqrt(tan(d*x + c))) + 315*d*sqrt((2*A*B*a^4 - 12
*A*B*a^2*b^2 + 2*A*B*b^4 + 4*(A^2 - B^2)*a^3*b - 4*(A^2 - B^2)*a*b^3 - d^2*sqrt(-((A^4 - 2*A^2*B^2 + B^4)*a^8
- 16*(A^3*B - A*B^3)*a^7*b - 4*(3*A^4 - 22*A^2*B^2 + 3*B^4)*a^6*b^2 + 112*(A^3*B - A*B^3)*a^5*b^3 + 2*(19*A^4
- 102*A^2*B^2 + 19*B^4)*a^4*b^4 - 112*(A^3*B - A*B^3)*a^3*b^5 - 4*(3*A^4 - 22*A^2*B^2 + 3*B^4)*a^2*b^6 + 16*(A
^3*B - A*B^3)*a*b^7 + (A^4 - 2*A^2*B^2 + B^4)*b^8)/d^4))/d^2)*log(-((A*a^2 - 2*B*a*b - A*b^2)*d^3*sqrt(-((A^4
- 2*A^2*B^2 + B^4)*a^8 - 16*(A^3*B - A*B^3)*a^7*b - 4*(3*A^4 - 22*A^2*B^2 + 3*B^4)*a^6*b^2 + 112*(A^3*B - A*B^
3)*a^5*b^3 + 2*(19*A^4 - 102*A^2*B^2 + 19*B^4)*a^4*b^4 - 112*(A^3*B - A*B^3)*a^3*b^5 - 4*(3*A^4 - 22*A^2*B^2 +
3*B^4)*a^2*b^6 + 16*(A^3*B - A*B^3)*a*b^7 + (A^4 - 2*A^2*B^2 + B^4)*b^8)/d^4) + ((A^2*B - B^3)*a^6 + 2*(A^3 -
5*A*B^2)*a^5*b - (23*A^2*B - 7*B^3)*a^4*b^2 - 4*(3*A^3 - 7*A*B^2)*a^3*b^3 + (23*A^2*B - 7*B^3)*a^2*b^4 + 2*(A
^3 - 5*A*B^2)*a*b^5 - (A^2*B - B^3)*b^6)*d)*sqrt((2*A*B*a^4 - 12*A*B*a^2*b^2 + 2*A*B*b^4 + 4*(A^2 - B^2)*a^3*b
- 4*(A^2 - B^2)*a*b^3 - d^2*sqrt(-((A^4 - 2*A^2*B^2 + B^4)*a^8 - 16*(A^3*B - A*B^3)*a^7*b - 4*(3*A^4 - 22*A^2
*B^2 + 3*B^4)*a^6*b^2 + 112*(A^3*B - A*B^3)*a^5*b^3 + 2*(19*A^4 - 102*A^2*B^2 + 19*B^4)*a^4*b^4 - 112*(A^3*B -
A*B^3)*a^3*b^5 - 4*(3*A^4 - 22*A^2*B^2 + 3*B^4)*a^2*b^6 + 16*(A^3*B - A*B^3)*a*b^7 + (A^4 - 2*A^2*B^2 + B^4)*
b^8)/d^4))/d^2) - ((A^4 - B^4)*a^8 - 8*(A^3*B + A*B^3)*a^7*b - 4*(A^4 - B^4)*a^6*b^2 - 8*(A^3*B + A*B^3)*a^5*b
^3 - 10*(A^4 - B^4)*a^4*b^4 + 8*(A^3*B + A*B^3)*a^3*b^5 - 4*(A^4 - B^4)*a^2*b^6 + 8*(A^3*B + A*B^3)*a*b^7 + (A
^4 - B^4)*b^8)*sqrt(tan(d*x + c))) + 4*(35*B*b^2*tan(d*x + c)^4 + 45*(2*B*a*b + A*b^2)*tan(d*x + c)^3 - 315*B*
a^2 - 630*A*a*b + 315*B*b^2 + 63*(B*a^2 + 2*A*a*b - B*b^2)*tan(d*x + c)^2 + 105*(A*a^2 - 2*B*a*b - A*b^2)*tan(
d*x + c))*sqrt(tan(d*x + c)))/d

Sympy [F]

\[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\int \left (A + B \tan {\left (c + d x \right )}\right ) \left (a + b \tan {\left (c + d x \right )}\right )^{2} \tan ^{\frac {5}{2}}{\left (c + d x \right )}\, dx \]

[In]

integrate(tan(d*x+c)**(5/2)*(a+b*tan(d*x+c))**2*(A+B*tan(d*x+c)),x)

[Out]

Integral((A + B*tan(c + d*x))*(a + b*tan(c + d*x))**2*tan(c + d*x)**(5/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.47 (sec) , antiderivative size = 329, normalized size of antiderivative = 0.84 \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {280 \, B b^{2} \tan \left (d x + c\right )^{\frac {9}{2}} + 360 \, {\left (2 \, B a b + A b^{2}\right )} \tan \left (d x + c\right )^{\frac {7}{2}} + 504 \, {\left (B a^{2} + 2 \, A a b - B b^{2}\right )} \tan \left (d x + c\right )^{\frac {5}{2}} - 630 \, \sqrt {2} {\left ({\left (A - B\right )} a^{2} - 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - 630 \, \sqrt {2} {\left ({\left (A - B\right )} a^{2} - 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 315 \, \sqrt {2} {\left ({\left (A + B\right )} a^{2} + 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - 315 \, \sqrt {2} {\left ({\left (A + B\right )} a^{2} + 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + 840 \, {\left (A a^{2} - 2 \, B a b - A b^{2}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} - 2520 \, {\left (B a^{2} + 2 \, A a b - B b^{2}\right )} \sqrt {\tan \left (d x + c\right )}}{1260 \, d} \]

[In]

integrate(tan(d*x+c)^(5/2)*(a+b*tan(d*x+c))^2*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/1260*(280*B*b^2*tan(d*x + c)^(9/2) + 360*(2*B*a*b + A*b^2)*tan(d*x + c)^(7/2) + 504*(B*a^2 + 2*A*a*b - B*b^2
)*tan(d*x + c)^(5/2) - 630*sqrt(2)*((A - B)*a^2 - 2*(A + B)*a*b - (A - B)*b^2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2
*sqrt(tan(d*x + c)))) - 630*sqrt(2)*((A - B)*a^2 - 2*(A + B)*a*b - (A - B)*b^2)*arctan(-1/2*sqrt(2)*(sqrt(2) -
2*sqrt(tan(d*x + c)))) + 315*sqrt(2)*((A + B)*a^2 + 2*(A - B)*a*b - (A + B)*b^2)*log(sqrt(2)*sqrt(tan(d*x + c
)) + tan(d*x + c) + 1) - 315*sqrt(2)*((A + B)*a^2 + 2*(A - B)*a*b - (A + B)*b^2)*log(-sqrt(2)*sqrt(tan(d*x + c
)) + tan(d*x + c) + 1) + 840*(A*a^2 - 2*B*a*b - A*b^2)*tan(d*x + c)^(3/2) - 2520*(B*a^2 + 2*A*a*b - B*b^2)*sqr
t(tan(d*x + c)))/d

Giac [F(-1)]

Timed out. \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(tan(d*x+c)^(5/2)*(a+b*tan(d*x+c))^2*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 29.53 (sec) , antiderivative size = 3914, normalized size of antiderivative = 9.93 \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

int(tan(c + d*x)^(5/2)*(A + B*tan(c + d*x))*(a + b*tan(c + d*x))^2,x)

[Out]

atan((B^2*a^4*tan(c + d*x)^(1/2)*((B^2*a*b^3)/d^2 - (12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a
^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2)/(4*d^4) - (B^2*a^3*b)/d^2)^(1/2)*32i)/((16*B*a^2*(12*B^4*a^2*b^6*d^4 -
B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2))/d^3 - (192*B^3*a^3*b^3)/d - (16*B*
b^2*(12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2))/d^3 + (3
2*B^3*a*b^5)/d + (32*B^3*a^5*b)/d) + (B^2*b^4*tan(c + d*x)^(1/2)*((B^2*a*b^3)/d^2 - (12*B^4*a^2*b^6*d^4 - B^4*
b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2)/(4*d^4) - (B^2*a^3*b)/d^2)^(1/2)*32i)/(
(16*B*a^2*(12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2))/d^
3 - (192*B^3*a^3*b^3)/d - (16*B*b^2*(12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*
B^4*a^6*b^2*d^4)^(1/2))/d^3 + (32*B^3*a*b^5)/d + (32*B^3*a^5*b)/d) - (B^2*a^2*b^2*tan(c + d*x)^(1/2)*((B^2*a*b
^3)/d^2 - (12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2)/(4*
d^4) - (B^2*a^3*b)/d^2)^(1/2)*192i)/((16*B*a^2*(12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^
4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2))/d^3 - (192*B^3*a^3*b^3)/d - (16*B*b^2*(12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B
^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2))/d^3 + (32*B^3*a*b^5)/d + (32*B^3*a^5*b)/d))*((B^2
*a*b^3)/d^2 - (12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2)
/(4*d^4) - (B^2*a^3*b)/d^2)^(1/2)*2i + atan((B^2*a^4*tan(c + d*x)^(1/2)*((12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B
^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2)/(4*d^4) + (B^2*a*b^3)/d^2 - (B^2*a^3*b)/d^2)^(1/2)
*32i)/((16*B*b^2*(12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1
/2))/d^3 - (16*B*a^2*(12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4
)^(1/2))/d^3 - (192*B^3*a^3*b^3)/d + (32*B^3*a*b^5)/d + (32*B^3*a^5*b)/d) + (B^2*b^4*tan(c + d*x)^(1/2)*((12*B
^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2)/(4*d^4) + (B^2*a*b
^3)/d^2 - (B^2*a^3*b)/d^2)^(1/2)*32i)/((16*B*b^2*(12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*
b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2))/d^3 - (16*B*a^2*(12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*
a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2))/d^3 - (192*B^3*a^3*b^3)/d + (32*B^3*a*b^5)/d + (32*B^3*a^5*b)/d) - (B
^2*a^2*b^2*tan(c + d*x)^(1/2)*((12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a
^6*b^2*d^4)^(1/2)/(4*d^4) + (B^2*a*b^3)/d^2 - (B^2*a^3*b)/d^2)^(1/2)*192i)/((16*B*b^2*(12*B^4*a^2*b^6*d^4 - B^
4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2))/d^3 - (16*B*a^2*(12*B^4*a^2*b^6*d^4
- B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2))/d^3 - (192*B^3*a^3*b^3)/d + (32*
B^3*a*b^5)/d + (32*B^3*a^5*b)/d))*((12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B
^4*a^6*b^2*d^4)^(1/2)/(4*d^4) + (B^2*a*b^3)/d^2 - (B^2*a^3*b)/d^2)^(1/2)*2i - atan((A^2*a^4*tan(c + d*x)^(1/2)
*((A^2*a^3*b)/d^2 - (A^2*a*b^3)/d^2 - (12*A^4*a^2*b^6*d^4 - A^4*b^8*d^4 - A^4*a^8*d^4 - 38*A^4*a^4*b^4*d^4 + 1
2*A^4*a^6*b^2*d^4)^(1/2)/(4*d^4))^(1/2)*32i)/((16*A^3*a^6)/d - (16*A^3*b^6)/d + (112*A^3*a^2*b^4)/d - (112*A^3
*a^4*b^2)/d + (32*A*a*b*(12*A^4*a^2*b^6*d^4 - A^4*b^8*d^4 - A^4*a^8*d^4 - 38*A^4*a^4*b^4*d^4 + 12*A^4*a^6*b^2*
d^4)^(1/2))/d^3) + (A^2*b^4*tan(c + d*x)^(1/2)*((A^2*a^3*b)/d^2 - (A^2*a*b^3)/d^2 - (12*A^4*a^2*b^6*d^4 - A^4*
b^8*d^4 - A^4*a^8*d^4 - 38*A^4*a^4*b^4*d^4 + 12*A^4*a^6*b^2*d^4)^(1/2)/(4*d^4))^(1/2)*32i)/((16*A^3*a^6)/d - (
16*A^3*b^6)/d + (112*A^3*a^2*b^4)/d - (112*A^3*a^4*b^2)/d + (32*A*a*b*(12*A^4*a^2*b^6*d^4 - A^4*b^8*d^4 - A^4*
a^8*d^4 - 38*A^4*a^4*b^4*d^4 + 12*A^4*a^6*b^2*d^4)^(1/2))/d^3) - (A^2*a^2*b^2*tan(c + d*x)^(1/2)*((A^2*a^3*b)/
d^2 - (A^2*a*b^3)/d^2 - (12*A^4*a^2*b^6*d^4 - A^4*b^8*d^4 - A^4*a^8*d^4 - 38*A^4*a^4*b^4*d^4 + 12*A^4*a^6*b^2*
d^4)^(1/2)/(4*d^4))^(1/2)*192i)/((16*A^3*a^6)/d - (16*A^3*b^6)/d + (112*A^3*a^2*b^4)/d - (112*A^3*a^4*b^2)/d +
(32*A*a*b*(12*A^4*a^2*b^6*d^4 - A^4*b^8*d^4 - A^4*a^8*d^4 - 38*A^4*a^4*b^4*d^4 + 12*A^4*a^6*b^2*d^4)^(1/2))/d
^3))*((A^2*a^3*b)/d^2 - (A^2*a*b^3)/d^2 - (12*A^4*a^2*b^6*d^4 - A^4*b^8*d^4 - A^4*a^8*d^4 - 38*A^4*a^4*b^4*d^4
+ 12*A^4*a^6*b^2*d^4)^(1/2)/(4*d^4))^(1/2)*2i + atan((A^2*a^4*tan(c + d*x)^(1/2)*((12*A^4*a^2*b^6*d^4 - A^4*b
^8*d^4 - A^4*a^8*d^4 - 38*A^4*a^4*b^4*d^4 + 12*A^4*a^6*b^2*d^4)^(1/2)/(4*d^4) - (A^2*a*b^3)/d^2 + (A^2*a^3*b)/
d^2)^(1/2)*32i)/((16*A^3*b^6)/d - (16*A^3*a^6)/d - (112*A^3*a^2*b^4)/d + (112*A^3*a^4*b^2)/d + (32*A*a*b*(12*A
^4*a^2*b^6*d^4 - A^4*b^8*d^4 - A^4*a^8*d^4 - 38*A^4*a^4*b^4*d^4 + 12*A^4*a^6*b^2*d^4)^(1/2))/d^3) + (A^2*b^4*t
an(c + d*x)^(1/2)*((12*A^4*a^2*b^6*d^4 - A^4*b^8*d^4 - A^4*a^8*d^4 - 38*A^4*a^4*b^4*d^4 + 12*A^4*a^6*b^2*d^4)^
(1/2)/(4*d^4) - (A^2*a*b^3)/d^2 + (A^2*a^3*b)/d^2)^(1/2)*32i)/((16*A^3*b^6)/d - (16*A^3*a^6)/d - (112*A^3*a^2*
b^4)/d + (112*A^3*a^4*b^2)/d + (32*A*a*b*(12*A^4*a^2*b^6*d^4 - A^4*b^8*d^4 - A^4*a^8*d^4 - 38*A^4*a^4*b^4*d^4
+ 12*A^4*a^6*b^2*d^4)^(1/2))/d^3) - (A^2*a^2*b^2*tan(c + d*x)^(1/2)*((12*A^4*a^2*b^6*d^4 - A^4*b^8*d^4 - A^4*a
^8*d^4 - 38*A^4*a^4*b^4*d^4 + 12*A^4*a^6*b^2*d^4)^(1/2)/(4*d^4) - (A^2*a*b^3)/d^2 + (A^2*a^3*b)/d^2)^(1/2)*192
i)/((16*A^3*b^6)/d - (16*A^3*a^6)/d - (112*A^3*a^2*b^4)/d + (112*A^3*a^4*b^2)/d + (32*A*a*b*(12*A^4*a^2*b^6*d^
4 - A^4*b^8*d^4 - A^4*a^8*d^4 - 38*A^4*a^4*b^4*d^4 + 12*A^4*a^6*b^2*d^4)^(1/2))/d^3))*((12*A^4*a^2*b^6*d^4 - A
^4*b^8*d^4 - A^4*a^8*d^4 - 38*A^4*a^4*b^4*d^4 + 12*A^4*a^6*b^2*d^4)^(1/2)/(4*d^4) - (A^2*a*b^3)/d^2 + (A^2*a^3
*b)/d^2)^(1/2)*2i + tan(c + d*x)^(3/2)*((2*A*a^2)/(3*d) - (2*A*b^2)/(3*d)) - tan(c + d*x)^(1/2)*((2*B*a^2)/d -
(2*B*b^2)/d) + tan(c + d*x)^(5/2)*((2*B*a^2)/(5*d) - (2*B*b^2)/(5*d)) + (2*A*b^2*tan(c + d*x)^(7/2))/(7*d) +
(2*B*b^2*tan(c + d*x)^(9/2))/(9*d) - (4*A*a*b*tan(c + d*x)^(1/2))/d + (4*A*a*b*tan(c + d*x)^(5/2))/(5*d) - (4*
B*a*b*tan(c + d*x)^(3/2))/(3*d) + (4*B*a*b*tan(c + d*x)^(7/2))/(7*d)