\(\int \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx\) [387]

Optimal result

Integrand size = 33, antiderivative size = 326 \[ \int \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\frac {\left (a^2 (A-B)-b^2 (A-B)-2 a b (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (a^2 (A-B)-b^2 (A-B)-2 a b (A+B)\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (2 a b (A-B)+a^2 (A+B)-b^2 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {\left (2 a b (A-B)+a^2 (A+B)-b^2 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}+\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 b (5 A b+7 a B) \tan ^{\frac {3}{2}}(c+d x)}{15 d}+\frac {2 b B \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}{5 d} \]

[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)*ta
n(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+2/15*b*(5*A*b+7*B*a)*tan(d*x+c)^(3/2)/d+2/5*b*B*ta
n(d*x+c)^(3/2)*(a+b*tan(d*x+c))/d

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps used = 13, 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} \[ \int \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\frac {\left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d}+\frac {2 \left (a^2 B+2 a A b-b^2 B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {\left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {\left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {2 b (7 a B+5 A b) \tan ^{\frac {3}{2}}(c+d x)}{15 d}+\frac {2 b B \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}{5 d} \]

[In]

Int[Sqrt[Tan[c + d*x]]*(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*b*(5*A*b + 7*a*B)*Tan[c + d*x]^(3/2))/(15*d) + (2*b*B*Tan[c + d*x]^
(3/2)*(a + b*Tan[c + d*x]))/(5*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 {3}{2}}(c+d x) (a+b \tan (c+d x))}{5 d}+\frac {2}{5} \int \sqrt {\tan (c+d x)} \left (\frac {1}{2} a (5 a A-3 b B)+\frac {5}{2} \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)+\frac {1}{2} b (5 A b+7 a B) \tan ^2(c+d x)\right ) \, dx \\ & = \frac {2 b (5 A b+7 a B) \tan ^{\frac {3}{2}}(c+d x)}{15 d}+\frac {2 b B \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}{5 d}+\frac {2}{5} \int \sqrt {\tan (c+d x)} \left (\frac {5}{2} \left (a^2 A-A b^2-2 a b B\right )+\frac {5}{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 b (5 A b+7 a B) \tan ^{\frac {3}{2}}(c+d x)}{15 d}+\frac {2 b B \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}{5 d}+\frac {2}{5} \int \frac {-\frac {5}{2} \left (2 a A b+a^2 B-b^2 B\right )+\frac {5}{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 b (5 A b+7 a B) \tan ^{\frac {3}{2}}(c+d x)}{15 d}+\frac {2 b B \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}{5 d}+\frac {4 \text {Subst}\left (\int \frac {-\frac {5}{2} \left (2 a A b+a^2 B-b^2 B\right )+\frac {5}{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 )}{5 d} \\ & = \frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 b (5 A b+7 a B) \tan ^{\frac {3}{2}}(c+d x)}{15 d}+\frac {2 b B \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}{5 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 b (5 A b+7 a B) \tan ^{\frac {3}{2}}(c+d x)}{15 d}+\frac {2 b B \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}{5 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 b (5 A b+7 a B) \tan ^{\frac {3}{2}}(c+d x)}{15 d}+\frac {2 b B \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}{5 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 b (5 A b+7 a B) \tan ^{\frac {3}{2}}(c+d x)}{15 d}+\frac {2 b B \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}{5 d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.31 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.46 \[ \int \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {15 \sqrt [4]{-1} (a-i b)^2 (i A+B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )-15 (-1)^{3/4} (a+i b)^2 (A+i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+2 \sqrt {\tan (c+d x)} \left (15 \left (2 a A b+a^2 B-b^2 B\right )+5 b (A b+2 a B) \tan (c+d x)+3 b^2 B \tan ^2(c+d x)\right )}{15 d} \]

[In]

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

[Out]

(15*(-1)^(1/4)*(a - I*b)^2*(I*A + B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] - 15*(-1)^(3/4)*(a + I*b)^2*(A + I*
B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*Sqrt[Tan[c + d*x]]*(15*(2*a*A*b + a^2*B - b^2*B) + 5*b*(A*b + 2*
a*B)*Tan[c + d*x] + 3*b^2*B*Tan[c + d*x]^2))/(15*d)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.90

method	result	size
derivativedivides	\(\frac {\frac {2 B \,b^{2} \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}+\frac {2 A \,b^{2} \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+\frac {4 B a b \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+4 A a b \left (\sqrt {\tan }\left (d x +c \right )\right )+2 B \,a^{2} \left (\sqrt {\tan }\left (d x +c \right )\right )-2 \left (\sqrt {\tan }\left (d x +c \right )\right ) B \,b^{2}+\frac {\left (-2 A a b -B \,a^{2}+B \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (A \,a^{2}-A \,b^{2}-2 B a b \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{d}\)	\(292\)
default	\(\frac {\frac {2 B \,b^{2} \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}+\frac {2 A \,b^{2} \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+\frac {4 B a b \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+4 A a b \left (\sqrt {\tan }\left (d x +c \right )\right )+2 B \,a^{2} \left (\sqrt {\tan }\left (d x +c \right )\right )-2 \left (\sqrt {\tan }\left (d x +c \right )\right ) B \,b^{2}+\frac {\left (-2 A a b -B \,a^{2}+B \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (A \,a^{2}-A \,b^{2}-2 B a b \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{d}\)	\(292\)
parts	\(\frac {\left (A \,b^{2}+2 B a b \right ) \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 {\left (2 A a b +B \,a^{2}\right ) \left (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} \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}+\frac {B \,b^{2} \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}\)	\(430\)

[In]

int(tan(d*x+c)^(1/2)*(a+b*tan(d*x+c))^2*(A+B*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(2/5*B*b^2*tan(d*x+c)^(5/2)+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. 4283 vs. \(2 (288) = 576\).

Time = 0.73 (sec) , antiderivative size = 4283, normalized size of antiderivative = 13.14 \[ \int \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

integrate(tan(d*x+c)^(1/2)*(a+b*tan(d*x+c))^2*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

-1/30*(15*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*sqr
t(-((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 + (2
3*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))) - 15*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 + 1
9*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*(1
9*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)*sq
rt(tan(d*x + c))) - 15*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))) + 15*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 - 10
2*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*(3*B*b^2*tan(d*x + c)^2 + 15*B*a^2 + 30*A*a*b - 15*B*b^2 + 5*(2*B*a*b + A*b
^2)*tan(d*x + c))*sqrt(tan(d*x + c)))/d

Sympy [F]

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

[In]

integrate(tan(d*x+c)**(1/2)*(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*sqrt(tan(c + d*x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.84 \[ \int \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {24 \, B b^{2} \tan \left (d x + c\right )^{\frac {5}{2}} + 30 \, \sqrt {2} {\left ({\left (A - B\right )} a^{2} - 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 30 \, \sqrt {2} {\left ({\left (A - B\right )} a^{2} - 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - 15 \, \sqrt {2} {\left ({\left (A + B\right )} a^{2} + 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + 15 \, \sqrt {2} {\left ({\left (A + B\right )} a^{2} + 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + 40 \, {\left (2 \, B a b + A b^{2}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} + 120 \, {\left (B a^{2} + 2 \, A a b - B b^{2}\right )} \sqrt {\tan \left (d x + c\right )}}{60 \, d} \]

[In]

integrate(tan(d*x+c)^(1/2)*(a+b*tan(d*x+c))^2*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/60*(24*B*b^2*tan(d*x + c)^(5/2) + 30*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)))) + 30*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)))) - 15*sqrt(2)*((A + B)*a^2 + 2*(A - B)*a*b - (A + B)*b^2)*log(sqrt(2)*sqrt(t
an(d*x + c)) + tan(d*x + c) + 1) + 15*sqrt(2)*((A + B)*a^2 + 2*(A - B)*a*b - (A + B)*b^2)*log(-sqrt(2)*sqrt(ta
n(d*x + c)) + tan(d*x + c) + 1) + 40*(2*B*a*b + A*b^2)*tan(d*x + c)^(3/2) + 120*(B*a^2 + 2*A*a*b - B*b^2)*sqrt
(tan(d*x + c)))/d

Giac [F(-1)]

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

[In]

integrate(tan(d*x+c)^(1/2)*(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 = 15.10 (sec) , antiderivative size = 3825, normalized size of antiderivative = 11.73 \[ \int \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

int(tan(c + d*x)^(1/2)*(A + B*tan(c + d*x))*(a + b*tan(c + d*x))^2,x)

[Out]

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 + 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*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((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 - 3
8*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((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 + (32*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((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)^(1/2)*((2*B*a^2)/d - (2*B*b^2)/d) + (2*A*b^2*tan(c + d*x)^(3/2))/(3*d) + (2*B
*b^2*tan(c + d*x)^(5/2))/(5*d) + (4*A*a*b*tan(c + d*x)^(1/2))/d + (4*B*a*b*tan(c + d*x)^(3/2))/(3*d)