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

Optimal. Leaf size=278 \[ -\frac {(a (A-B)+b (A+B)) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )}+\frac {(a (A-B)+b (A+B)) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )}-\frac {2 \sqrt {a} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {b} d \left (a^2+b^2\right )}-\frac {(b (A-B)-a (A+B)) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}+\frac {(b (A-B)-a (A+B)) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.37, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3612, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ -\frac {(a (A-B)+b (A+B)) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )}+\frac {(a (A-B)+b (A+B)) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )}-\frac {2 \sqrt {a} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {b} d \left (a^2+b^2\right )}-\frac {(b (A-B)-a (A+B)) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}+\frac {(b (A-B)-a (A+B)) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 205

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

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 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]

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 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 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 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 3612

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

Rule 3634

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*
tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x]
/; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]

Rubi steps

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

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Mathematica [A]  time = 0.42, size = 195, normalized size = 0.70 \[ -\frac {2 \sqrt {2} (a (A-B)+b (A+B)) \left (\tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )\right )+\frac {8 \sqrt {a} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {b}}-\sqrt {2} (a (A+B)+b (B-A)) \left (\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )-\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )\right )}{4 d \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(2*Sqrt[2]*(a*(A - B) + b*(A + B))*(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - ArcTan[1 + Sqrt[2]*Sqrt[Tan[
c + d*x]]]) + (8*Sqrt[a]*(A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/Sqrt[b] - Sqrt[2]*(b*(-A +
B) + a*(A + B))*(Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan
[c + d*x]]))/((a^2 + b^2)*d)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [B]  time = 0.44, size = 607, normalized size = 2.18 \[ -\frac {2 a \arctan \left (\frac {\left (\sqrt {\tan }\left (d x +c \right )\right ) b}{\sqrt {a b}}\right ) A b}{d \left (a^{2}+b^{2}\right ) \sqrt {a b}}+\frac {2 a^{2} \arctan \left (\frac {\left (\sqrt {\tan }\left (d x +c \right )\right ) b}{\sqrt {a b}}\right ) B}{d \left (a^{2}+b^{2}\right ) \sqrt {a b}}+\frac {A \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) b}{2 d \left (a^{2}+b^{2}\right )}+\frac {A \sqrt {2}\, \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 ) b}{4 d \left (a^{2}+b^{2}\right )}+\frac {A \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) b}{2 d \left (a^{2}+b^{2}\right )}-\frac {B \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) a}{2 d \left (a^{2}+b^{2}\right )}-\frac {B \sqrt {2}\, \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 ) a}{4 d \left (a^{2}+b^{2}\right )}-\frac {B \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) a}{2 d \left (a^{2}+b^{2}\right )}+\frac {A \sqrt {2}\, \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 ) a}{4 d \left (a^{2}+b^{2}\right )}+\frac {A \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) a}{2 d \left (a^{2}+b^{2}\right )}+\frac {A \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) a}{2 d \left (a^{2}+b^{2}\right )}+\frac {B \sqrt {2}\, \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 ) b}{4 d \left (a^{2}+b^{2}\right )}+\frac {B \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) b}{2 d \left (a^{2}+b^{2}\right )}+\frac {B \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) b}{2 d \left (a^{2}+b^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.73, size = 216, normalized size = 0.78 \[ \frac {\frac {8 \, {\left (B a^{2} - A a b\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{2} + b^{2}\right )} \sqrt {a b}} + \frac {2 \, \sqrt {2} {\left ({\left (A - B\right )} a + {\left (A + B\right )} b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left ({\left (A - B\right )} a + {\left (A + B\right )} b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - \sqrt {2} {\left ({\left (A + B\right )} a - {\left (A - B\right )} b\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} {\left ({\left (A + B\right )} a - {\left (A - B\right )} b\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{2} + b^{2}}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 11.12, size = 15090, normalized size = 54.28 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(((((32*(13*A^3*a^2*b^4*d^2 + A^3*a^4*b^2*d^2))/d^5 + (((32*(12*A*a*b^7*d^4 + 24*A*a^3*b^5*d^4 + 12*A*a^5*
b^3*d^4))/d^5 - (32*tan(c + d*x)^(1/2)*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))
^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*
b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2)
- 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(20*A^2*a^3*b^4*d^2
+ 2*A^2*a^5*b^2*d^2 - 14*A^2*a*b^6*d^2))/d^4)*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^
2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(((64*A^4*a^2*b^2*d^4 - A^4*(1
6*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/
2) - (32*tan(c + d*x)^(1/2)*(A^4*b^5 - 2*A^4*a^2*b^3))/d^4)*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d
^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i - (((32*(13*A^
3*a^2*b^4*d^2 + A^3*a^4*b^2*d^2))/d^5 + (((32*(12*A*a*b^7*d^4 + 24*A*a^3*b^5*d^4 + 12*A*a^5*b^3*d^4))/d^5 + (3
2*tan(c + d*x)^(1/2)*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b
*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b
^3*d^4))/d^4)*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(
16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(20*A^2*a^3*b^4*d^2 + 2*A^2*a^5*b^2*d^
2 - 14*A^2*a*b^6*d^2))/d^4)*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*
A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4
*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d
*x)^(1/2)*(A^4*b^5 - 2*A^4*a^2*b^3))/d^4)*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^
4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i)/((((32*(13*A^3*a^2*b^4*d^2 + A^
3*a^4*b^2*d^2))/d^5 + (((32*(12*A*a*b^7*d^4 + 24*A*a^3*b^5*d^4 + 12*A*a^5*b^3*d^4))/d^5 - (32*tan(c + d*x)^(1/
2)*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4
+ b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(((6
4*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*
d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(20*A^2*a^3*b^4*d^2 + 2*A^2*a^5*b^2*d^2 - 14*A^2*a*b^6*d
^2))/d^4)*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(
a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*
d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(A^4*b^5
- 2*A^4*a^2*b^3))/d^4)*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*
a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (((32*(13*A^3*a^2*b^4*d^2 + A^3*a^4*b^2*d^2))/d^5 +
(((32*(12*A*a*b^7*d^4 + 24*A*a^3*b^5*d^4 + 12*A*a^5*b^3*d^4))/d^5 + (32*tan(c + d*x)^(1/2)*(((64*A^4*a^2*b^2*
d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^
2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(((64*A^4*a^2*b^2*d^4 - A
^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4))
)^(1/2) - (32*tan(c + d*x)^(1/2)*(20*A^2*a^3*b^4*d^2 + 2*A^2*a^5*b^2*d^2 - 14*A^2*a*b^6*d^2))/d^4)*(((64*A^4*a
^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2
*a^2*b^2*d^4)))^(1/2))*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a
*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(A^4*b^5 - 2*A^4*a^2*b^3))/d^
4)*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4
+ b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (64*A^5*a*b^3)/d^5))*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^
4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*2i - atan(((((((32
*(4*B*a^2*b^6*d^4 + 8*B*a^4*b^4*d^4 + 4*B*a^6*b^2*d^4))/d^5 - (32*tan(c + d*x)^(1/2)*(((64*B^4*a^2*b^2*d^4 - B
^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4))
)^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*
a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)
+ (32*tan(c + d*x)^(1/2)*(14*B^2*a^5*b^2*d^2 - 4*B^2*a^3*b^4*d^2 + 14*B^2*a*b^6*d^2))/d^4)*(((64*B^4*a^2*b^2*
d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^
2*d^4)))^(1/2) + (32*(B^3*a*b^5*d^2 - 15*B^3*a^3*b^3*d^2 + 4*B^3*a^5*b*d^2))/d^5)*(((64*B^4*a^2*b^2*d^4 - B^4*
(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(
1/2) - (32*tan(c + d*x)^(1/2)*(B^4*b^5 + 2*B^4*a^4*b))/d^4)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d
^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i - (((((32*(4*B
*a^2*b^6*d^4 + 8*B*a^4*b^4*d^4 + 4*B*a^6*b^2*d^4))/d^5 + (32*tan(c + d*x)^(1/2)*(((64*B^4*a^2*b^2*d^4 - B^4*(1
6*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/
2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d
^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (3
2*tan(c + d*x)^(1/2)*(14*B^2*a^5*b^2*d^2 - 4*B^2*a^3*b^4*d^2 + 14*B^2*a*b^6*d^2))/d^4)*(((64*B^4*a^2*b^2*d^4 -
B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4
)))^(1/2) + (32*(B^3*a*b^5*d^2 - 15*B^3*a^3*b^3*d^2 + 4*B^3*a^5*b*d^2))/d^5)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a
^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)
+ (32*tan(c + d*x)^(1/2)*(B^4*b^5 + 2*B^4*a^4*b))/d^4)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 +
32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i)/((((((32*(4*B*a^2*
b^6*d^4 + 8*B*a^4*b^4*d^4 + 4*B*a^6*b^2*d^4))/d^5 - (32*tan(c + d*x)^(1/2)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4
*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(1
6*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 +
16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan
(c + d*x)^(1/2)*(14*B^2*a^5*b^2*d^2 - 4*B^2*a^3*b^4*d^2 + 14*B^2*a*b^6*d^2))/d^4)*(((64*B^4*a^2*b^2*d^4 - B^4*
(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(
1/2) + (32*(B^3*a*b^5*d^2 - 15*B^3*a^3*b^3*d^2 + 4*B^3*a^5*b*d^2))/d^5)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^
4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32
*tan(c + d*x)^(1/2)*(B^4*b^5 + 2*B^4*a^4*b))/d^4)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^
2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (((((32*(4*B*a^2*b^6*d^4
+ 8*B*a^4*b^4*d^4 + 4*B*a^6*b^2*d^4))/d^5 + (32*tan(c + d*x)^(1/2)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 1
6*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^
4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d
^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x
)^(1/2)*(14*B^2*a^5*b^2*d^2 - 4*B^2*a^3*b^4*d^2 + 14*B^2*a*b^6*d^2))/d^4)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*
d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (
32*(B^3*a*b^5*d^2 - 15*B^3*a^3*b^3*d^2 + 4*B^3*a^5*b*d^2))/d^5)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b
^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c +
d*x)^(1/2)*(B^4*b^5 + 2*B^4*a^4*b))/d^4)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^
4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (64*B^5*a^2*b^2)/d^5))*(((64*B^4*
a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 +
2*a^2*b^2*d^4)))^(1/2)*2i - atan(((((((32*(4*B*a^2*b^6*d^4 + 8*B*a^4*b^4*d^4 + 4*B*a^6*b^2*d^4))/d^5 - (32*tan
(c + d*x)^(1/2)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2
)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d
^4))/d^4)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*
(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(14*B^2*a^5*b^2*d^2 - 4*B^2*a^3*b^4*d^2 +
14*B^2*a*b^6*d^2))/d^4)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^
2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*(B^3*a*b^5*d^2 - 15*B^3*a^3*b^3*d^2 + 4*B^3*a
^5*b*d^2))/d^5)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2
)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(B^4*b^5 + 2*B^4*a^4*b))/d^4)*(-((6
4*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*
d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i - (((((32*(4*B*a^2*b^6*d^4 + 8*B*a^4*b^4*d^4 + 4*B*a^6*b^2*d^4))/d^5 + (32*tan
(c + d*x)^(1/2)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2
)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d
^4))/d^4)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*
(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(14*B^2*a^5*b^2*d^2 - 4*B^2*a^3*b^4*d^2 +
14*B^2*a*b^6*d^2))/d^4)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^
2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*(B^3*a*b^5*d^2 - 15*B^3*a^3*b^3*d^2 + 4*B^3*a
^5*b*d^2))/d^5)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2
)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(B^4*b^5 + 2*B^4*a^4*b))/d^4)*(-((6
4*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*
d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i)/((((((32*(4*B*a^2*b^6*d^4 + 8*B*a^4*b^4*d^4 + 4*B*a^6*b^2*d^4))/d^5 - (32*tan
(c + d*x)^(1/2)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2
)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d
^4))/d^4)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*
(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(14*B^2*a^5*b^2*d^2 - 4*B^2*a^3*b^4*d^2 +
14*B^2*a*b^6*d^2))/d^4)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^
2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*(B^3*a*b^5*d^2 - 15*B^3*a^3*b^3*d^2 + 4*B^3*a
^5*b*d^2))/d^5)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2
)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(B^4*b^5 + 2*B^4*a^4*b))/d^4)*(-((6
4*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*
d^4 + 2*a^2*b^2*d^4)))^(1/2) + (((((32*(4*B*a^2*b^6*d^4 + 8*B*a^4*b^4*d^4 + 4*B*a^6*b^2*d^4))/d^5 + (32*tan(c
+ d*x)^(1/2)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(
16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4)
)/d^4)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^
4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(14*B^2*a^5*b^2*d^2 - 4*B^2*a^3*b^4*d^2 + 14
*B^2*a*b^6*d^2))/d^4)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a
*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*(B^3*a*b^5*d^2 - 15*B^3*a^3*b^3*d^2 + 4*B^3*a^5*
b*d^2))/d^5)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(
16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(B^4*b^5 + 2*B^4*a^4*b))/d^4)*(-((64*B
^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4
+ 2*a^2*b^2*d^4)))^(1/2) + (64*B^5*a^2*b^2)/d^5))*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*
a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*2i + atan(((((32*(13*A^3*
a^2*b^4*d^2 + A^3*a^4*b^2*d^2))/d^5 + (((32*(12*A*a*b^7*d^4 + 24*A*a^3*b^5*d^4 + 12*A*a^5*b^3*d^4))/d^5 - (32*
tan(c + d*x)^(1/2)*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*
d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^
3*d^4))/d^4)*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(
16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(20*A^2*a^3*b^4*d^2 + 2*A^2*a^5*b^2*d^
2 - 14*A^2*a*b^6*d^2))/d^4)*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8
*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b
^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c +
d*x)^(1/2)*(A^4*b^5 - 2*A^4*a^2*b^3))/d^4)*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2
*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i - (((32*(13*A^3*a^2*b^4*d^2 +
A^3*a^4*b^2*d^2))/d^5 + (((32*(12*A*a*b^7*d^4 + 24*A*a^3*b^5*d^4 + 12*A*a^5*b^3*d^4))/d^5 + (32*tan(c + d*x)^
(1/2)*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4
*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*
(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 +
b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(20*A^2*a^3*b^4*d^2 + 2*A^2*a^5*b^2*d^2 - 14*A^2*a*
b^6*d^2))/d^4)*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)
/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a
^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(
A^4*b^5 - 2*A^4*a^2*b^3))/d^4)*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2)
+ 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i)/((((32*(13*A^3*a^2*b^4*d^2 + A^3*a^4*b^2*
d^2))/d^5 + (((32*(12*A*a*b^7*d^4 + 24*A*a^3*b^5*d^4 + 12*A*a^5*b^3*d^4))/d^5 - (32*tan(c + d*x)^(1/2)*(-((64*
A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^
4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(-((64*A^4*a^
2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*
a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(20*A^2*a^3*b^4*d^2 + 2*A^2*a^5*b^2*d^2 - 14*A^2*a*b^6*d^2))/d^4
)*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4
+ b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^
(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(A^4*b^5 - 2*A
^4*a^2*b^3))/d^4)*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d
^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (((32*(13*A^3*a^2*b^4*d^2 + A^3*a^4*b^2*d^2))/d^5 + (((3
2*(12*A*a*b^7*d^4 + 24*A*a^3*b^5*d^4 + 12*A*a^5*b^3*d^4))/d^5 + (32*tan(c + d*x)^(1/2)*(-((64*A^4*a^2*b^2*d^4
- A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^
4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(-((64*A^4*a^2*b^2*d^4 - A^4*
(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(
1/2) - (32*tan(c + d*x)^(1/2)*(20*A^2*a^3*b^4*d^2 + 2*A^2*a^5*b^2*d^2 - 14*A^2*a*b^6*d^2))/d^4)*(-((64*A^4*a^2
*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a
^2*b^2*d^4)))^(1/2))*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*
b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(A^4*b^5 - 2*A^4*a^2*b^3))/d^4
)*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4
+ b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (64*A^5*a*b^3)/d^5))*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d
^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*2i + (B*a^3*atan(
((B*a^3*((32*tan(c + d*x)^(1/2)*(B^4*b^5 + 2*B^4*a^4*b))/d^4 - (B*a^3*((32*(B^3*a*b^5*d^2 - 15*B^3*a^3*b^3*d^2
+ 4*B^3*a^5*b*d^2))/d^5 + (B*a^3*((32*tan(c + d*x)^(1/2)*(14*B^2*a^5*b^2*d^2 - 4*B^2*a^3*b^4*d^2 + 14*B^2*a*b
^6*d^2))/d^4 + (B*a^3*((32*(4*B*a^2*b^6*d^4 + 8*B*a^4*b^4*d^4 + 4*B*a^6*b^2*d^4))/d^5 - (32*B*a^3*tan(c + d*x)
^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/(d^4*(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^
5*b^3*d^2)^(1/2))))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2)))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3
*d^2)^(1/2)))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2))*1i)/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^
2)^(1/2) + (B*a^3*((32*tan(c + d*x)^(1/2)*(B^4*b^5 + 2*B^4*a^4*b))/d^4 + (B*a^3*((32*(B^3*a*b^5*d^2 - 15*B^3*a
^3*b^3*d^2 + 4*B^3*a^5*b*d^2))/d^5 - (B*a^3*((32*tan(c + d*x)^(1/2)*(14*B^2*a^5*b^2*d^2 - 4*B^2*a^3*b^4*d^2 +
14*B^2*a*b^6*d^2))/d^4 - (B*a^3*((32*(4*B*a^2*b^6*d^4 + 8*B*a^4*b^4*d^4 + 4*B*a^6*b^2*d^4))/d^5 + (32*B*a^3*ta
n(c + d*x)^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/(d^4*(- a^7*b*d^2 - a^3*b^5*
d^2 - 2*a^5*b^3*d^2)^(1/2))))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2)))/(- a^7*b*d^2 - a^3*b^5*d^2 -
2*a^5*b^3*d^2)^(1/2)))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2))*1i)/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*
a^5*b^3*d^2)^(1/2))/((64*B^5*a^2*b^2)/d^5 - (B*a^3*((32*tan(c + d*x)^(1/2)*(B^4*b^5 + 2*B^4*a^4*b))/d^4 - (B*a
^3*((32*(B^3*a*b^5*d^2 - 15*B^3*a^3*b^3*d^2 + 4*B^3*a^5*b*d^2))/d^5 + (B*a^3*((32*tan(c + d*x)^(1/2)*(14*B^2*a
^5*b^2*d^2 - 4*B^2*a^3*b^4*d^2 + 14*B^2*a*b^6*d^2))/d^4 + (B*a^3*((32*(4*B*a^2*b^6*d^4 + 8*B*a^4*b^4*d^4 + 4*B
*a^6*b^2*d^4))/d^5 - (32*B*a^3*tan(c + d*x)^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d
^4))/(d^4*(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2))))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/
2)))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2)))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2)))/(
- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2) + (B*a^3*((32*tan(c + d*x)^(1/2)*(B^4*b^5 + 2*B^4*a^4*b))/d^4
+ (B*a^3*((32*(B^3*a*b^5*d^2 - 15*B^3*a^3*b^3*d^2 + 4*B^3*a^5*b*d^2))/d^5 - (B*a^3*((32*tan(c + d*x)^(1/2)*(1
4*B^2*a^5*b^2*d^2 - 4*B^2*a^3*b^4*d^2 + 14*B^2*a*b^6*d^2))/d^4 - (B*a^3*((32*(4*B*a^2*b^6*d^4 + 8*B*a^4*b^4*d^
4 + 4*B*a^6*b^2*d^4))/d^5 + (32*B*a^3*tan(c + d*x)^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^
6*b^3*d^4))/(d^4*(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2))))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d
^2)^(1/2)))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2)))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1
/2)))/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2)))*2i)/(- a^7*b*d^2 - a^3*b^5*d^2 - 2*a^5*b^3*d^2)^(1/2
) - (A*a*b*atan(((A*a*b*((32*tan(c + d*x)^(1/2)*(A^4*b^5 - 2*A^4*a^2*b^3))/d^4 - (A*a*b*((32*(13*A^3*a^2*b^4*d
^2 + A^3*a^4*b^2*d^2))/d^5 + (A*a*b*((32*tan(c + d*x)^(1/2)*(20*A^2*a^3*b^4*d^2 + 2*A^2*a^5*b^2*d^2 - 14*A^2*a
*b^6*d^2))/d^4 + (A*a*b*((32*(12*A*a*b^7*d^4 + 24*A*a^3*b^5*d^4 + 12*A*a^5*b^3*d^4))/d^5 - (32*A*a*b*tan(c + d
*x)^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/(d^4*(- a*b^5*d^2 - a^5*b*d^2 - 2*a
^3*b^3*d^2)^(1/2))))/(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2)))/(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^
2)^(1/2)))/(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2))*1i)/(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2
) + (A*a*b*((32*tan(c + d*x)^(1/2)*(A^4*b^5 - 2*A^4*a^2*b^3))/d^4 + (A*a*b*((32*(13*A^3*a^2*b^4*d^2 + A^3*a^4*
b^2*d^2))/d^5 - (A*a*b*((32*tan(c + d*x)^(1/2)*(20*A^2*a^3*b^4*d^2 + 2*A^2*a^5*b^2*d^2 - 14*A^2*a*b^6*d^2))/d^
4 - (A*a*b*((32*(12*A*a*b^7*d^4 + 24*A*a^3*b^5*d^4 + 12*A*a^5*b^3*d^4))/d^5 + (32*A*a*b*tan(c + d*x)^(1/2)*(16
*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/(d^4*(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(
1/2))))/(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2)))/(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2)))/(-
a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2))*1i)/(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2))/((64*A^5*a
*b^3)/d^5 - (A*a*b*((32*tan(c + d*x)^(1/2)*(A^4*b^5 - 2*A^4*a^2*b^3))/d^4 - (A*a*b*((32*(13*A^3*a^2*b^4*d^2 +
A^3*a^4*b^2*d^2))/d^5 + (A*a*b*((32*tan(c + d*x)^(1/2)*(20*A^2*a^3*b^4*d^2 + 2*A^2*a^5*b^2*d^2 - 14*A^2*a*b^6*
d^2))/d^4 + (A*a*b*((32*(12*A*a*b^7*d^4 + 24*A*a^3*b^5*d^4 + 12*A*a^5*b^3*d^4))/d^5 - (32*A*a*b*tan(c + d*x)^(
1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/(d^4*(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^
3*d^2)^(1/2))))/(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2)))/(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1
/2)))/(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2)))/(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2) + (A*a
*b*((32*tan(c + d*x)^(1/2)*(A^4*b^5 - 2*A^4*a^2*b^3))/d^4 + (A*a*b*((32*(13*A^3*a^2*b^4*d^2 + A^3*a^4*b^2*d^2)
)/d^5 - (A*a*b*((32*tan(c + d*x)^(1/2)*(20*A^2*a^3*b^4*d^2 + 2*A^2*a^5*b^2*d^2 - 14*A^2*a*b^6*d^2))/d^4 - (A*a
*b*((32*(12*A*a*b^7*d^4 + 24*A*a^3*b^5*d^4 + 12*A*a^5*b^3*d^4))/d^5 + (32*A*a*b*tan(c + d*x)^(1/2)*(16*b^9*d^4
+ 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/(d^4*(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2))))/
(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2)))/(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2)))/(- a*b^5*d
^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2)))/(- a*b^5*d^2 - a^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2)))*2i)/(- a*b^5*d^2 - a
^5*b*d^2 - 2*a^3*b^3*d^2)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \sqrt {\tan {\left (c + d x \right )}}}{a + b \tan {\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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