3.381 \(\int \frac{(a+b \tan (c+d x)) (A+B \tan (c+d x))}{\sqrt{\tan (c+d x)}} \, dx\)

Optimal. Leaf size=205 \[ -\frac{(a (A+B)+b (A-B)) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d}+\frac{(a (A+B)+b (A-B)) \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{\sqrt{2} d}-\frac{(a (A-B)-b (A+B)) \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d}+\frac{(a (A-B)-b (A+B)) \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d}+\frac{2 b B \sqrt{\tan (c+d x)}}{d} \]

[Out]

-(((b*(A - B) + a*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + ((b*(A - B) + a*(A + B))*Arc
Tan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a*(A - B) - b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]
+ Tan[c + d*x]])/(2*Sqrt[2]*d) + ((a*(A - B) - b*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])
/(2*Sqrt[2]*d) + (2*b*B*Sqrt[Tan[c + d*x]])/d

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Rubi [A]  time = 0.198887, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.258, Rules used = {3592, 3534, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{(a (A+B)+b (A-B)) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d}+\frac{(a (A+B)+b (A-B)) \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{\sqrt{2} d}-\frac{(a (A-B)-b (A+B)) \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d}+\frac{(a (A-B)-b (A+B)) \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d}+\frac{2 b B \sqrt{\tan (c+d x)}}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((b*(A - B) + a*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d)) + ((b*(A - B) + a*(A + B))*Arc
Tan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d) - ((a*(A - B) - b*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]
+ Tan[c + d*x]])/(2*Sqrt[2]*d) + ((a*(A - B) - b*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])
/(2*Sqrt[2]*d) + (2*b*B*Sqrt[Tan[c + d*x]])/d

Rule 3592

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

Rule 3534

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 1168

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 1162

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 617

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 204

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

Rule 1165

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 628

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]

Rubi steps

\begin{align*} \int \frac{(a+b \tan (c+d x)) (A+B \tan (c+d x))}{\sqrt{\tan (c+d x)}} \, dx &=\frac{2 b B \sqrt{\tan (c+d x)}}{d}+\int \frac{a A-b B+(A b+a B) \tan (c+d x)}{\sqrt{\tan (c+d x)}} \, dx\\ &=\frac{2 b B \sqrt{\tan (c+d x)}}{d}+\frac{2 \operatorname{Subst}\left (\int \frac{a A-b B+(A b+a B) x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d}\\ &=\frac{2 b B \sqrt{\tan (c+d x)}}{d}+\frac{(b (A-B)+a (A+B)) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d}+\frac{(a (A-B)-b (A+B)) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d}\\ &=\frac{2 b B \sqrt{\tan (c+d x)}}{d}+\frac{(b (A-B)+a (A+B)) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 d}+\frac{(b (A-B)+a (A+B)) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 d}-\frac{(a (A-B)-b (A+B)) \operatorname{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{(a (A-B)-b (A+B)) \operatorname{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{(a (A-B)-b (A+B)) \log \left (1-\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} d}+\frac{(a (A-B)-b (A+B)) \log \left (1+\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} d}+\frac{2 b B \sqrt{\tan (c+d x)}}{d}+\frac{(b (A-B)+a (A+B)) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d}-\frac{(b (A-B)+a (A+B)) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d}\\ &=-\frac{(b (A-B)+a (A+B)) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d}+\frac{(b (A-B)+a (A+B)) \tan ^{-1}\left (1+\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d}-\frac{(a (A-B)-b (A+B)) \log \left (1-\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} d}+\frac{(a (A-B)-b (A+B)) \log \left (1+\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} d}+\frac{2 b B \sqrt{\tan (c+d x)}}{d}\\ \end{align*}

Mathematica [C]  time = 0.167372, size = 94, normalized size = 0.46 \[ -\frac{\sqrt [4]{-1} (a-i b) (A-i B) \tan ^{-1}\left ((-1)^{3/4} \sqrt{\tan (c+d x)}\right )+\sqrt [4]{-1} (a+i b) (A+i B) \tanh ^{-1}\left ((-1)^{3/4} \sqrt{\tan (c+d x)}\right )-2 b B \sqrt{\tan (c+d x)}}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.022, size = 437, normalized size = 2.1 \begin{align*} 2\,{\frac{Bb\sqrt{\tan \left ( dx+c \right ) }}{d}}+{\frac{Aa\sqrt{2}}{2\,d}\arctan \left ( 1+\sqrt{2}\sqrt{\tan \left ( dx+c \right ) } \right ) }+{\frac{Aa\sqrt{2}}{2\,d}\arctan \left ( -1+\sqrt{2}\sqrt{\tan \left ( dx+c \right ) } \right ) }+{\frac{Aa\sqrt{2}}{4\,d}\ln \left ({ \left ( 1+\sqrt{2}\sqrt{\tan \left ( dx+c \right ) }+\tan \left ( dx+c \right ) \right ) \left ( 1-\sqrt{2}\sqrt{\tan \left ( dx+c \right ) }+\tan \left ( dx+c \right ) \right ) ^{-1}} \right ) }-{\frac{B\sqrt{2}b}{2\,d}\arctan \left ( 1+\sqrt{2}\sqrt{\tan \left ( dx+c \right ) } \right ) }-{\frac{B\sqrt{2}b}{2\,d}\arctan \left ( -1+\sqrt{2}\sqrt{\tan \left ( dx+c \right ) } \right ) }-{\frac{B\sqrt{2}b}{4\,d}\ln \left ({ \left ( 1+\sqrt{2}\sqrt{\tan \left ( dx+c \right ) }+\tan \left ( dx+c \right ) \right ) \left ( 1-\sqrt{2}\sqrt{\tan \left ( dx+c \right ) }+\tan \left ( dx+c \right ) \right ) ^{-1}} \right ) }+{\frac{A\sqrt{2}b}{4\,d}\ln \left ({ \left ( 1-\sqrt{2}\sqrt{\tan \left ( dx+c \right ) }+\tan \left ( dx+c \right ) \right ) \left ( 1+\sqrt{2}\sqrt{\tan \left ( dx+c \right ) }+\tan \left ( dx+c \right ) \right ) ^{-1}} \right ) }+{\frac{A\sqrt{2}b}{2\,d}\arctan \left ( 1+\sqrt{2}\sqrt{\tan \left ( dx+c \right ) } \right ) }+{\frac{A\sqrt{2}b}{2\,d}\arctan \left ( -1+\sqrt{2}\sqrt{\tan \left ( dx+c \right ) } \right ) }+{\frac{aB\sqrt{2}}{4\,d}\ln \left ({ \left ( 1-\sqrt{2}\sqrt{\tan \left ( dx+c \right ) }+\tan \left ( dx+c \right ) \right ) \left ( 1+\sqrt{2}\sqrt{\tan \left ( dx+c \right ) }+\tan \left ( dx+c \right ) \right ) ^{-1}} \right ) }+{\frac{aB\sqrt{2}}{2\,d}\arctan \left ( 1+\sqrt{2}\sqrt{\tan \left ( dx+c \right ) } \right ) }+{\frac{aB\sqrt{2}}{2\,d}\arctan \left ( -1+\sqrt{2}\sqrt{\tan \left ( dx+c \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b*B*tan(d*x+c)^(1/2)/d+1/2/d*a*A*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)+1/2/d*a*A*arctan(-1+2^(1/2)*tan(
d*x+c)^(1/2))*2^(1/2)+1/4/d*a*A*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)))-1/2/d*B*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*b-1/2/d*B*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c
)^(1/2))*b-1/4/d*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)))
*b+1/4/d*A*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)))*b+1/2/d
*A*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*b+1/2/d*A*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*b+1/4/d*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)))+1/2/d*a*B*arctan
(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)+1/2/d*a*B*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)

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Maxima [A]  time = 1.70534, size = 235, normalized size = 1.15 \begin{align*} \frac{2 \, \sqrt{2}{\left ({\left (A + B\right )} a +{\left (A - B\right )} b\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt{2}{\left ({\left (A + B\right )} a +{\left (A - B\right )} b\right )} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{\tan \left (d x + c\right )}\right )}\right ) + \sqrt{2}{\left ({\left (A - B\right )} a -{\left (A + B\right )} b\right )} \log \left (\sqrt{2} \sqrt{\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt{2}{\left ({\left (A - B\right )} a -{\left (A + B\right )} b\right )} \log \left (-\sqrt{2} \sqrt{\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + 8 \, B b \sqrt{\tan \left (d x + c\right )}}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*sqrt(2)*((A + B)*a + (A - B)*b)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*((A +
B)*a + (A - B)*b)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c)))) + sqrt(2)*((A - B)*a - (A + B)*b)*log(
sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) - sqrt(2)*((A - B)*a - (A + B)*b)*log(-sqrt(2)*sqrt(tan(d*x + c
)) + tan(d*x + c) + 1) + 8*B*b*sqrt(tan(d*x + c)))/d

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Fricas [B]  time = 90.482, size = 27232, normalized size = 132.84 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*sqrt(2)*d^5*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 +
B^4)*b^4 + 2*(A*B*a^2 - A*B*b^2 + (A^2 - B^2)*a*b)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2
+ B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4))/((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2
*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4))*(((A^4 + 2*A^2*B^2
+ B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)^(3/4)*sqrt(((A^4 - 2*A^2*B
^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4
- 2*A^2*B^2 + B^4)*b^4)/d^4)*arctan(-(((A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*a^8 - 4*(A^7*B + 3*A^5*B^3 + 3*A^3*
B^5 + A*B^7)*a^7*b + 2*(A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*a^6*b^2 - 12*(A^7*B + 3*A^5*B^3 + 3*A^3*B^5 + A*B^7
)*a^5*b^3 - 12*(A^7*B + 3*A^5*B^3 + 3*A^3*B^5 + A*B^7)*a^3*b^5 - 2*(A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*a^2*b^6
- 4*(A^7*B + 3*A^5*B^3 + 3*A^3*B^5 + A*B^7)*a*b^7 - (A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*b^8)*d^4*sqrt(((A^4 +
2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)*sqrt(((A^4 - 2*A
^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (
A^4 - 2*A^2*B^2 + B^4)*b^4)/d^4) + sqrt(2)*((A*a - B*b)*d^7*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2
*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)*sqrt(((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*
a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4)/d^4) - ((A
^2*B + B^3)*a^3 + (A^3 + A*B^2)*a^2*b + (A^2*B + B^3)*a*b^2 + (A^3 + A*B^2)*b^3)*d^5*sqrt(((A^4 - 2*A^2*B^2 +
B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A
^2*B^2 + B^4)*b^4)/d^4))*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*
B^2 + B^4)*b^4 + 2*(A*B*a^2 - A*B*b^2 + (A^2 - B^2)*a*b)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^
2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4))/((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3
*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4))*sqrt((((A^6
- A^4*B^2 - A^2*B^4 + B^6)*a^6 - 8*(A^5*B - A*B^5)*a^5*b - (A^6 - 17*A^4*B^2 - 17*A^2*B^4 + B^6)*a^4*b^2 - (A^
6 - 17*A^4*B^2 - 17*A^2*B^4 + B^6)*a^2*b^4 + 8*(A^5*B - A*B^5)*a*b^5 + (A^6 - A^4*B^2 - A^2*B^4 + B^6)*b^6)*d^
2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)*co
s(d*x + c) + sqrt(2)*(((A^4*B - 2*A^2*B^3 + B^5)*a^5 + (A^5 - 10*A^3*B^2 + 9*A*B^4)*a^4*b - 2*(5*A^4*B - 14*A^
2*B^3 + B^5)*a^3*b^2 - 2*(A^5 - 14*A^3*B^2 + 5*A*B^4)*a^2*b^3 + (9*A^4*B - 10*A^2*B^3 + B^5)*a*b^4 + (A^5 - 2*
A^3*B^2 + A*B^4)*b^5)*d^3*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2
*B^2 + B^4)*b^4)/d^4)*cos(d*x + c) - ((A^7 - A^5*B^2 - A^3*B^4 + A*B^6)*a^7 - (9*A^6*B - A^4*B^3 - 9*A^2*B^5 +
B^7)*a^6*b - (A^7 - 25*A^5*B^2 - 17*A^3*B^4 + 9*A*B^6)*a^5*b^2 + (A^6*B - 17*A^4*B^3 - 17*A^2*B^5 + B^7)*a^4*
b^3 - (A^7 - 17*A^5*B^2 - 17*A^3*B^4 + A*B^6)*a^3*b^4 + (9*A^6*B - 17*A^4*B^3 - 25*A^2*B^5 + B^7)*a^2*b^5 + (A
^7 - 9*A^5*B^2 - A^3*B^4 + 9*A*B^6)*a*b^6 - (A^6*B - A^4*B^3 - A^2*B^5 + B^7)*b^7)*d*cos(d*x + c))*sqrt(((A^4
+ 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4 + 2*(A*B*a^2 - A*B*b^
2 + (A^2 - B^2)*a*b)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*
B^2 + B^4)*b^4)/d^4))/((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*
b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(((A^4 + 2*A^2*B
^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)^(1/4) + ((A^8 - 2*A^4*B^
4 + B^8)*a^8 - 8*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a^7*b + 16*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^6*b^2 - 8*(A
^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a^5*b^3 - 2*(A^8 - 16*A^6*B^2 - 34*A^4*B^4 - 16*A^2*B^6 + B^8)*a^4*b^4 + 8*(
A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a^3*b^5 + 16*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^2*b^6 + 8*(A^7*B + A^5*B^3 -
A^3*B^5 - A*B^7)*a*b^7 + (A^8 - 2*A^4*B^4 + B^8)*b^8)*sin(d*x + c))/cos(d*x + c))*(((A^4 + 2*A^2*B^2 + B^4)*a
^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)^(3/4) - sqrt(2)*(((A^5 - A*B^4)*a^5
- (5*A^4*B + 4*A^2*B^3 - B^5)*a^4*b + 4*(A^3*B^2 + A*B^4)*a^3*b^2 - 4*(A^4*B + A^2*B^3)*a^2*b^3 - (A^5 - 4*A^
3*B^2 - 5*A*B^4)*a*b^4 + (A^4*B - B^5)*b^5)*d^7*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*
a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)*sqrt(((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A
^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4)/d^4) - ((A^6*B + A^4*B
^3 - A^2*B^5 - B^7)*a^7 + (A^7 - 3*A^5*B^2 - 9*A^3*B^4 - 5*A*B^6)*a^6*b - (3*A^6*B + 7*A^4*B^3 + 5*A^2*B^5 + B
^7)*a^5*b^2 + (A^7 - 7*A^5*B^2 - 17*A^3*B^4 - 9*A*B^6)*a^4*b^3 - (9*A^6*B + 17*A^4*B^3 + 7*A^2*B^5 - B^7)*a^3*
b^4 - (A^7 + 5*A^5*B^2 + 7*A^3*B^4 + 3*A*B^6)*a^2*b^5 - (5*A^6*B + 9*A^4*B^3 + 3*A^2*B^5 - B^7)*a*b^6 - (A^7 +
A^5*B^2 - A^3*B^4 - A*B^6)*b^7)*d^5*sqrt(((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10
*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4)/d^4))*sqrt(((A^4 + 2*A^2*B^2
+ B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4 + 2*(A*B*a^2 - A*B*b^2 + (A^2 - B
^2)*a*b)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b
^4)/d^4))/((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3
*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(((A^4 + 2*A^2*B^2 + B^4)*a^
4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)^(3/4))/((A^12 + 2*A^10*B^2 - A^8*B^4
- 4*A^6*B^6 - A^4*B^8 + 2*A^2*B^10 + B^12)*a^12 - 8*(A^11*B + 3*A^9*B^3 + 2*A^7*B^5 - 2*A^5*B^7 - 3*A^3*B^9 -
A*B^11)*a^11*b + 2*(A^12 + 10*A^10*B^2 + 31*A^8*B^4 + 44*A^6*B^6 + 31*A^4*B^8 + 10*A^2*B^10 + B^12)*a^10*b^2
- 24*(A^11*B + 3*A^9*B^3 + 2*A^7*B^5 - 2*A^5*B^7 - 3*A^3*B^9 - A*B^11)*a^9*b^3 - (A^12 - 62*A^10*B^2 - 257*A^8
*B^4 - 388*A^6*B^6 - 257*A^4*B^8 - 62*A^2*B^10 + B^12)*a^8*b^4 - 16*(A^11*B + 3*A^9*B^3 + 2*A^7*B^5 - 2*A^5*B^
7 - 3*A^3*B^9 - A*B^11)*a^7*b^5 - 4*(A^12 - 22*A^10*B^2 - 97*A^8*B^4 - 148*A^6*B^6 - 97*A^4*B^8 - 22*A^2*B^10
+ B^12)*a^6*b^6 + 16*(A^11*B + 3*A^9*B^3 + 2*A^7*B^5 - 2*A^5*B^7 - 3*A^3*B^9 - A*B^11)*a^5*b^7 - (A^12 - 62*A^
10*B^2 - 257*A^8*B^4 - 388*A^6*B^6 - 257*A^4*B^8 - 62*A^2*B^10 + B^12)*a^4*b^8 + 24*(A^11*B + 3*A^9*B^3 + 2*A^
7*B^5 - 2*A^5*B^7 - 3*A^3*B^9 - A*B^11)*a^3*b^9 + 2*(A^12 + 10*A^10*B^2 + 31*A^8*B^4 + 44*A^6*B^6 + 31*A^4*B^8
+ 10*A^2*B^10 + B^12)*a^2*b^10 + 8*(A^11*B + 3*A^9*B^3 + 2*A^7*B^5 - 2*A^5*B^7 - 3*A^3*B^9 - A*B^11)*a*b^11 +
(A^12 + 2*A^10*B^2 - A^8*B^4 - 4*A^6*B^6 - A^4*B^8 + 2*A^2*B^10 + B^12)*b^12)) + 4*sqrt(2)*d^5*sqrt(((A^4 + 2
*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4 + 2*(A*B*a^2 - A*B*b^2 +
(A^2 - B^2)*a*b)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2
+ B^4)*b^4)/d^4))/((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2
+ 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4))*(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2
+ B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)^(3/4)*sqrt(((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)
*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4)/d^4)*arct
an((((A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*a^8 - 4*(A^7*B + 3*A^5*B^3 + 3*A^3*B^5 + A*B^7)*a^7*b + 2*(A^8 + 2*A^
6*B^2 - 2*A^2*B^6 - B^8)*a^6*b^2 - 12*(A^7*B + 3*A^5*B^3 + 3*A^3*B^5 + A*B^7)*a^5*b^3 - 12*(A^7*B + 3*A^5*B^3
+ 3*A^3*B^5 + A*B^7)*a^3*b^5 - 2*(A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*a^2*b^6 - 4*(A^7*B + 3*A^5*B^3 + 3*A^3*B^
5 + A*B^7)*a*b^7 - (A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*b^8)*d^4*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2
*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)*sqrt(((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B
^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4)/d^4) -
sqrt(2)*((A*a - B*b)*d^7*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2
*B^2 + B^4)*b^4)/d^4)*sqrt(((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)
*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4)/d^4) - ((A^2*B + B^3)*a^3 + (A^3 + A*B^2)*a^
2*b + (A^2*B + B^3)*a*b^2 + (A^3 + A*B^2)*b^3)*d^5*sqrt(((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b
- 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4)/d^4))*sqrt(((A^
4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4 + 2*(A*B*a^2 - A*B*
b^2 + (A^2 - B^2)*a*b)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^
2*B^2 + B^4)*b^4)/d^4))/((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^
2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4))*sqrt((((A^6 - A^4*B^2 - A^2*B^4 + B^6)*a^6 - 8
*(A^5*B - A*B^5)*a^5*b - (A^6 - 17*A^4*B^2 - 17*A^2*B^4 + B^6)*a^4*b^2 - (A^6 - 17*A^4*B^2 - 17*A^2*B^4 + B^6)
*a^2*b^4 + 8*(A^5*B - A*B^5)*a*b^5 + (A^6 - A^4*B^2 - A^2*B^4 + B^6)*b^6)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^
4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)*cos(d*x + c) - sqrt(2)*(((A^4*B - 2*
A^2*B^3 + B^5)*a^5 + (A^5 - 10*A^3*B^2 + 9*A*B^4)*a^4*b - 2*(5*A^4*B - 14*A^2*B^3 + B^5)*a^3*b^2 - 2*(A^5 - 14
*A^3*B^2 + 5*A*B^4)*a^2*b^3 + (9*A^4*B - 10*A^2*B^3 + B^5)*a*b^4 + (A^5 - 2*A^3*B^2 + A*B^4)*b^5)*d^3*sqrt(((A
^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)*cos(d*x + c)
- ((A^7 - A^5*B^2 - A^3*B^4 + A*B^6)*a^7 - (9*A^6*B - A^4*B^3 - 9*A^2*B^5 + B^7)*a^6*b - (A^7 - 25*A^5*B^2 -
17*A^3*B^4 + 9*A*B^6)*a^5*b^2 + (A^6*B - 17*A^4*B^3 - 17*A^2*B^5 + B^7)*a^4*b^3 - (A^7 - 17*A^5*B^2 - 17*A^3*B
^4 + A*B^6)*a^3*b^4 + (9*A^6*B - 17*A^4*B^3 - 25*A^2*B^5 + B^7)*a^2*b^5 + (A^7 - 9*A^5*B^2 - A^3*B^4 + 9*A*B^6
)*a*b^6 - (A^6*B - A^4*B^3 - A^2*B^5 + B^7)*b^7)*d*cos(d*x + c))*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 +
2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4 + 2*(A*B*a^2 - A*B*b^2 + (A^2 - B^2)*a*b)*d^2*sqrt(((A^
4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4))/((A^4 - 2*A^
2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A
^4 - 2*A^2*B^2 + B^4)*b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2
+ B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)^(1/4) + ((A^8 - 2*A^4*B^4 + B^8)*a^8 - 8*(A^7*B + A^5*B^3
- A^3*B^5 - A*B^7)*a^7*b + 16*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^6*b^2 - 8*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*
a^5*b^3 - 2*(A^8 - 16*A^6*B^2 - 34*A^4*B^4 - 16*A^2*B^6 + B^8)*a^4*b^4 + 8*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)
*a^3*b^5 + 16*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^2*b^6 + 8*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a*b^7 + (A^8 - 2
*A^4*B^4 + B^8)*b^8)*sin(d*x + c))/cos(d*x + c))*(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2
*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)^(3/4) + sqrt(2)*(((A^5 - A*B^4)*a^5 - (5*A^4*B + 4*A^2*B^3 - B^5)*a^4
*b + 4*(A^3*B^2 + A*B^4)*a^3*b^2 - 4*(A^4*B + A^2*B^3)*a^2*b^3 - (A^5 - 4*A^3*B^2 - 5*A*B^4)*a*b^4 + (A^4*B -
B^5)*b^5)*d^7*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*
b^4)/d^4)*sqrt(((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8
*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4)/d^4) - ((A^6*B + A^4*B^3 - A^2*B^5 - B^7)*a^7 + (A^7 - 3
*A^5*B^2 - 9*A^3*B^4 - 5*A*B^6)*a^6*b - (3*A^6*B + 7*A^4*B^3 + 5*A^2*B^5 + B^7)*a^5*b^2 + (A^7 - 7*A^5*B^2 - 1
7*A^3*B^4 - 9*A*B^6)*a^4*b^3 - (9*A^6*B + 17*A^4*B^3 + 7*A^2*B^5 - B^7)*a^3*b^4 - (A^7 + 5*A^5*B^2 + 7*A^3*B^4
+ 3*A*B^6)*a^2*b^5 - (5*A^6*B + 9*A^4*B^3 + 3*A^2*B^5 - B^7)*a*b^6 - (A^7 + A^5*B^2 - A^3*B^4 - A*B^6)*b^7)*d
^5*sqrt(((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B
- A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4)/d^4))*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 +
B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4 + 2*(A*B*a^2 - A*B*b^2 + (A^2 - B^2)*a*b)*d^2*sqrt(((A^4 + 2*A^2*B^
2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4))/((A^4 - 2*A^2*B^2 + B^4)
*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B
^2 + B^4)*b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*
b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)^(3/4))/((A^12 + 2*A^10*B^2 - A^8*B^4 - 4*A^6*B^6 - A^4*B^8 + 2*A^2*B^1
0 + B^12)*a^12 - 8*(A^11*B + 3*A^9*B^3 + 2*A^7*B^5 - 2*A^5*B^7 - 3*A^3*B^9 - A*B^11)*a^11*b + 2*(A^12 + 10*A^1
0*B^2 + 31*A^8*B^4 + 44*A^6*B^6 + 31*A^4*B^8 + 10*A^2*B^10 + B^12)*a^10*b^2 - 24*(A^11*B + 3*A^9*B^3 + 2*A^7*B
^5 - 2*A^5*B^7 - 3*A^3*B^9 - A*B^11)*a^9*b^3 - (A^12 - 62*A^10*B^2 - 257*A^8*B^4 - 388*A^6*B^6 - 257*A^4*B^8 -
62*A^2*B^10 + B^12)*a^8*b^4 - 16*(A^11*B + 3*A^9*B^3 + 2*A^7*B^5 - 2*A^5*B^7 - 3*A^3*B^9 - A*B^11)*a^7*b^5 -
4*(A^12 - 22*A^10*B^2 - 97*A^8*B^4 - 148*A^6*B^6 - 97*A^4*B^8 - 22*A^2*B^10 + B^12)*a^6*b^6 + 16*(A^11*B + 3*A
^9*B^3 + 2*A^7*B^5 - 2*A^5*B^7 - 3*A^3*B^9 - A*B^11)*a^5*b^7 - (A^12 - 62*A^10*B^2 - 257*A^8*B^4 - 388*A^6*B^6
- 257*A^4*B^8 - 62*A^2*B^10 + B^12)*a^4*b^8 + 24*(A^11*B + 3*A^9*B^3 + 2*A^7*B^5 - 2*A^5*B^7 - 3*A^3*B^9 - A*
B^11)*a^3*b^9 + 2*(A^12 + 10*A^10*B^2 + 31*A^8*B^4 + 44*A^6*B^6 + 31*A^4*B^8 + 10*A^2*B^10 + B^12)*a^2*b^10 +
8*(A^11*B + 3*A^9*B^3 + 2*A^7*B^5 - 2*A^5*B^7 - 3*A^3*B^9 - A*B^11)*a*b^11 + (A^12 + 2*A^10*B^2 - A^8*B^4 - 4*
A^6*B^6 - A^4*B^8 + 2*A^2*B^10 + B^12)*b^12)) + sqrt(2)*(2*(A*B*a^2 - A*B*b^2 + (A^2 - B^2)*a*b)*d^3*sqrt(((A^
4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4) - ((A^4 + 2*A
^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)*d)*sqrt(((A^4 + 2*A^2*B^2
+ B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4 + 2*(A*B*a^2 - A*B*b^2 + (A^2 -
B^2)*a*b)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*
b^4)/d^4))/((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^
3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4))*(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a
^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)^(1/4)*log((((A^6 - A^4*B^2 - A^2*B^4 + B^6)*a^6 - 8*(A^5*B - A*B^5)
*a^5*b - (A^6 - 17*A^4*B^2 - 17*A^2*B^4 + B^6)*a^4*b^2 - (A^6 - 17*A^4*B^2 - 17*A^2*B^4 + B^6)*a^2*b^4 + 8*(A^
5*B - A*B^5)*a*b^5 + (A^6 - A^4*B^2 - A^2*B^4 + B^6)*b^6)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A
^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)*cos(d*x + c) + sqrt(2)*(((A^4*B - 2*A^2*B^3 + B^5)*a
^5 + (A^5 - 10*A^3*B^2 + 9*A*B^4)*a^4*b - 2*(5*A^4*B - 14*A^2*B^3 + B^5)*a^3*b^2 - 2*(A^5 - 14*A^3*B^2 + 5*A*B
^4)*a^2*b^3 + (9*A^4*B - 10*A^2*B^3 + B^5)*a*b^4 + (A^5 - 2*A^3*B^2 + A*B^4)*b^5)*d^3*sqrt(((A^4 + 2*A^2*B^2 +
B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)*cos(d*x + c) - ((A^7 - A^5*B
^2 - A^3*B^4 + A*B^6)*a^7 - (9*A^6*B - A^4*B^3 - 9*A^2*B^5 + B^7)*a^6*b - (A^7 - 25*A^5*B^2 - 17*A^3*B^4 + 9*A
*B^6)*a^5*b^2 + (A^6*B - 17*A^4*B^3 - 17*A^2*B^5 + B^7)*a^4*b^3 - (A^7 - 17*A^5*B^2 - 17*A^3*B^4 + A*B^6)*a^3*
b^4 + (9*A^6*B - 17*A^4*B^3 - 25*A^2*B^5 + B^7)*a^2*b^5 + (A^7 - 9*A^5*B^2 - A^3*B^4 + 9*A*B^6)*a*b^6 - (A^6*B
- A^4*B^3 - A^2*B^5 + B^7)*b^7)*d*cos(d*x + c))*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)
*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4 + 2*(A*B*a^2 - A*B*b^2 + (A^2 - B^2)*a*b)*d^2*sqrt(((A^4 + 2*A^2*B^2 +
B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4))/((A^4 - 2*A^2*B^2 + B^4)*a^4
- 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 +
B^4)*b^4))*sqrt(sin(d*x + c)/cos(d*x + c))*(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2
+ (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)^(1/4) + ((A^8 - 2*A^4*B^4 + B^8)*a^8 - 8*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^
7)*a^7*b + 16*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^6*b^2 - 8*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a^5*b^3 - 2*(A^8
- 16*A^6*B^2 - 34*A^4*B^4 - 16*A^2*B^6 + B^8)*a^4*b^4 + 8*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a^3*b^5 + 16*(A
^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^2*b^6 + 8*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a*b^7 + (A^8 - 2*A^4*B^4 + B^8)*
b^8)*sin(d*x + c))/cos(d*x + c)) - sqrt(2)*(2*(A*B*a^2 - A*B*b^2 + (A^2 - B^2)*a*b)*d^3*sqrt(((A^4 + 2*A^2*B^2
+ B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4) - ((A^4 + 2*A^2*B^2 + B^4)
*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)*d)*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 +
2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4 + 2*(A*B*a^2 - A*B*b^2 + (A^2 - B^2)*a*b)*d^2
*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4))/((
A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B - A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*
a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4))*(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4
+ 2*A^2*B^2 + B^4)*b^4)/d^4)^(1/4)*log((((A^6 - A^4*B^2 - A^2*B^4 + B^6)*a^6 - 8*(A^5*B - A*B^5)*a^5*b - (A^6
- 17*A^4*B^2 - 17*A^2*B^4 + B^6)*a^4*b^2 - (A^6 - 17*A^4*B^2 - 17*A^2*B^4 + B^6)*a^2*b^4 + 8*(A^5*B - A*B^5)*
a*b^5 + (A^6 - A^4*B^2 - A^2*B^4 + B^6)*b^6)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)
*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)*cos(d*x + c) - sqrt(2)*(((A^4*B - 2*A^2*B^3 + B^5)*a^5 + (A^5 - 1
0*A^3*B^2 + 9*A*B^4)*a^4*b - 2*(5*A^4*B - 14*A^2*B^3 + B^5)*a^3*b^2 - 2*(A^5 - 14*A^3*B^2 + 5*A*B^4)*a^2*b^3 +
(9*A^4*B - 10*A^2*B^3 + B^5)*a*b^4 + (A^5 - 2*A^3*B^2 + A*B^4)*b^5)*d^3*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2
*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4)*cos(d*x + c) - ((A^7 - A^5*B^2 - A^3*B^4
+ A*B^6)*a^7 - (9*A^6*B - A^4*B^3 - 9*A^2*B^5 + B^7)*a^6*b - (A^7 - 25*A^5*B^2 - 17*A^3*B^4 + 9*A*B^6)*a^5*b^2
+ (A^6*B - 17*A^4*B^3 - 17*A^2*B^5 + B^7)*a^4*b^3 - (A^7 - 17*A^5*B^2 - 17*A^3*B^4 + A*B^6)*a^3*b^4 + (9*A^6*
B - 17*A^4*B^3 - 25*A^2*B^5 + B^7)*a^2*b^5 + (A^7 - 9*A^5*B^2 - A^3*B^4 + 9*A*B^6)*a*b^6 - (A^6*B - A^4*B^3 -
A^2*B^5 + B^7)*b^7)*d*cos(d*x + c))*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A
^4 + 2*A^2*B^2 + B^4)*b^4 + 2*(A*B*a^2 - A*B*b^2 + (A^2 - B^2)*a*b)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*
(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)/d^4))/((A^4 - 2*A^2*B^2 + B^4)*a^4 - 8*(A^3*B -
A*B^3)*a^3*b - 2*(A^4 - 10*A^2*B^2 + B^4)*a^2*b^2 + 8*(A^3*B - A*B^3)*a*b^3 + (A^4 - 2*A^2*B^2 + B^4)*b^4))*s
qrt(sin(d*x + c)/cos(d*x + c))*(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a^2*b^2 + (A^4 + 2*A^
2*B^2 + B^4)*b^4)/d^4)^(1/4) + ((A^8 - 2*A^4*B^4 + B^8)*a^8 - 8*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a^7*b + 16
*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^6*b^2 - 8*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a^5*b^3 - 2*(A^8 - 16*A^6*B^2
- 34*A^4*B^4 - 16*A^2*B^6 + B^8)*a^4*b^4 + 8*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a^3*b^5 + 16*(A^6*B^2 + 2*A^
4*B^4 + A^2*B^6)*a^2*b^6 + 8*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a*b^7 + (A^8 - 2*A^4*B^4 + B^8)*b^8)*sin(d*x
+ c))/cos(d*x + c)) + 8*((A^4*B + 2*A^2*B^3 + B^5)*a^4*b + 2*(A^4*B + 2*A^2*B^3 + B^5)*a^2*b^3 + (A^4*B + 2*A^
2*B^3 + B^5)*b^5)*sqrt(sin(d*x + c)/cos(d*x + c)))/(((A^4 + 2*A^2*B^2 + B^4)*a^4 + 2*(A^4 + 2*A^2*B^2 + B^4)*a
^2*b^2 + (A^4 + 2*A^2*B^2 + B^4)*b^4)*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + B \tan{\left (c + d x \right )}\right ) \left (a + b \tan{\left (c + d x \right )}\right )}{\sqrt{\tan{\left (c + d x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 2.12289, size = 306, normalized size = 1.49 \begin{align*} \frac{2 \, B b \sqrt{\tan \left (d x + c\right )}}{d} + \frac{{\left (\sqrt{2} A a + \sqrt{2} B a + \sqrt{2} A b - \sqrt{2} B b\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{\tan \left (d x + c\right )}\right )}\right )}{2 \, d} + \frac{{\left (\sqrt{2} A a + \sqrt{2} B a + \sqrt{2} A b - \sqrt{2} B b\right )} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{\tan \left (d x + c\right )}\right )}\right )}{2 \, d} + \frac{{\left (\sqrt{2} A a - \sqrt{2} B a - \sqrt{2} A b - \sqrt{2} B b\right )} \log \left (\sqrt{2} \sqrt{\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{4 \, d} - \frac{{\left (\sqrt{2} A a - \sqrt{2} B a - \sqrt{2} A b - \sqrt{2} B b\right )} \log \left (-\sqrt{2} \sqrt{\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*B*b*sqrt(tan(d*x + c))/d + 1/2*(sqrt(2)*A*a + sqrt(2)*B*a + sqrt(2)*A*b - sqrt(2)*B*b)*arctan(1/2*sqrt(2)*(s
qrt(2) + 2*sqrt(tan(d*x + c))))/d + 1/2*(sqrt(2)*A*a + sqrt(2)*B*a + sqrt(2)*A*b - sqrt(2)*B*b)*arctan(-1/2*sq
rt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c))))/d + 1/4*(sqrt(2)*A*a - sqrt(2)*B*a - sqrt(2)*A*b - sqrt(2)*B*b)*log(sq
rt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1)/d - 1/4*(sqrt(2)*A*a - sqrt(2)*B*a - sqrt(2)*A*b - sqrt(2)*B*b)*l
og(-sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1)/d