\(\int \frac {\sqrt {\tan (c+d x)}}{(a+b \tan (c+d x))^2} \, dx\) [595]

Optimal result

Integrand size = 23, antiderivative size = 316 \[ \int \frac {\sqrt {\tan (c+d x)}}{(a+b \tan (c+d x))^2} \, dx=-\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\sqrt {b} \left (3 a^2-b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2-2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2-2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {b \sqrt {\tan (c+d x)}}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))} \]

[Out]

1/2*(a^2+2*a*b-b^2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)^2/d*2^(1/2)+1/2*(a^2+2*a*b-b^2)*arctan(1+2^(
1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)^2/d*2^(1/2)+1/4*(a^2-2*a*b-b^2)*ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^
2+b^2)^2/d*2^(1/2)-1/4*(a^2-2*a*b-b^2)*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)^2/d*2^(1/2)-(3*a^2-
b^2)*arctan(b^(1/2)*tan(d*x+c)^(1/2)/a^(1/2))*b^(1/2)/(a^2+b^2)^2/d/a^(1/2)-b*tan(d*x+c)^(1/2)/(a^2+b^2)/d/(a+
b*tan(d*x+c))

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {3649, 3734, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \[ \int \frac {\sqrt {\tan (c+d x)}}{(a+b \tan (c+d x))^2} \, dx=-\frac {\sqrt {b} \left (3 a^2-b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d \left (a^2+b^2\right )^2}-\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}+\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^2}-\frac {b \sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {\left (a^2-2 a b-b^2\right ) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2}-\frac {\left (a^2-2 a b-b^2\right ) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )^2} \]

[In]

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

[Out]

-(((a^2 + 2*a*b - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d)) + ((a^2 + 2*a*b - b^
2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^2*d) - (Sqrt[b]*(3*a^2 - b^2)*ArcTan[(Sqrt[b]*
Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*(a^2 + b^2)^2*d) + ((a^2 - 2*a*b - b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*
x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - ((a^2 - 2*a*b - b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + T
an[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^2*d) - (b*Sqrt[Tan[c + d*x]])/((a^2 + b^2)*d*(a + b*Tan[c + d*x]))

Rule 65

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 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 211

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

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 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 3649

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[b*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^n/(f*(m + 1)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(a^2
+ b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)*Simp[a*c*(m + 1) - b*d*n - (b*c - a*d)*
(m + 1)*Tan[e + f*x] - b*d*(m + n + 1)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c -
a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[2*m]

Rule 3715

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]

Rule 3734

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

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.77 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.58 \[ \int \frac {\sqrt {\tan (c+d x)}}{(a+b \tan (c+d x))^2} \, dx=\frac {-\frac {\sqrt {a} \sqrt {b} \left (3 a^2-b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^2+b^2}+\frac {(-1)^{3/4} a \left ((a+i b)^2 \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )-(a-i b)^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )\right )}{a^2+b^2}-b \sqrt {\tan (c+d x)}+\frac {b^2 \tan ^{\frac {3}{2}}(c+d x)}{a+b \tan (c+d x)}}{a \left (a^2+b^2\right ) d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(\frac {\frac {\frac {a b \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{2}+\frac {\left (a^{2}-b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{2}}-\frac {2 b \left (\frac {\left (\frac {a^{2}}{2}+\frac {b^{2}}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{a +b \tan \left (d x +c \right )}+\frac {\left (3 a^{2}-b^{2}\right ) \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(278\)
default	\(\frac {\frac {\frac {a b \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{2}+\frac {\left (a^{2}-b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{2}}-\frac {2 b \left (\frac {\left (\frac {a^{2}}{2}+\frac {b^{2}}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{a +b \tan \left (d x +c \right )}+\frac {\left (3 a^{2}-b^{2}\right ) \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(278\)

[In]

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

[Out]

1/d*(2/(a^2+b^2)^2*(1/4*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)+ta
n(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)))+1/8*(a^2-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))))-2*b/(a^2+b^2)^2*((1/2*a^2+1/2*b^2)*tan(d*x+c)^(1/2)/(a+b*ta
n(d*x+c))+1/2*(3*a^2-b^2)/(a*b)^(1/2)*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2642 vs. \(2 (276) = 552\).

Time = 0.40 (sec) , antiderivative size = 5309, normalized size of antiderivative = 16.80 \[ \int \frac {\sqrt {\tan (c+d x)}}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(tan(c + d*x))/(a + b*tan(c + d*x))**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {\tan (c+d x)}}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {4 \, {\left (3 \, a^{2} b - b^{3}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a b}} + \frac {4 \, b \sqrt {\tan \left (d x + c\right )}}{a^{3} + a b^{2} + {\left (a^{2} b + b^{3}\right )} \tan \left (d x + c\right )} - \frac {2 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}}}{4 \, d} \]

[In]

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

[Out]

-1/4*(4*(3*a^2*b - b^3)*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^4 + 2*a^2*b^2 + b^4)*sqrt(a*b)) + 4*b*sqrt(
tan(d*x + c))/(a^3 + a*b^2 + (a^2*b + b^3)*tan(d*x + c)) - (2*sqrt(2)*(a^2 + 2*a*b - b^2)*arctan(1/2*sqrt(2)*(
sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*(a^2 + 2*a*b - b^2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x
+ c)))) - sqrt(2)*(a^2 - 2*a*b - b^2)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) + sqrt(2)*(a^2 - 2*a
*b - b^2)*log(-sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1))/(a^4 + 2*a^2*b^2 + b^4))/d

Giac [F]

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

[In]

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

[Out]

integrate(sqrt(tan(d*x + c))/(b*tan(d*x + c) + a)^2, x)

Mupad [B] (verification not implemented)

Time = 8.08 (sec) , antiderivative size = 10520, normalized size of antiderivative = 33.29 \[ \int \frac {\sqrt {\tan (c+d x)}}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

atan(((((((8*(52*a*b^10*d^2 - 128*a^3*b^8*d^2 - 24*a^5*b^6*d^2 + 160*a^7*b^4*d^2 + 4*a^9*b^2*d^2))/(a^8*d^5 +
b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) + (((((8*(320*a^6*b^9*d^4 - 96*a^2*b^13*d^4 - 32*b^15
*d^4 + 480*a^8*b^7*d^4 + 288*a^10*b^5*d^4 + 64*a^12*b^3*d^4))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d
^5 + 4*a^6*b^2*d^5) - (8*tan(c + d*x)^(1/2)*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*
d^2*6i))^(1/2)*(32*b^17*d^4 + 160*a^2*b^15*d^4 + 288*a^4*b^13*d^4 + 160*a^6*b^11*d^4 - 160*a^8*b^9*d^4 - 288*a
^10*b^7*d^4 - 160*a^12*b^5*d^4 - 32*a^14*b^3*d^4))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*
b^2*d^4))*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2))/2 + (16*tan(c + d*
x)^(1/2)*(68*a*b^12*d^2 + 20*a^3*b^10*d^2 - 88*a^5*b^8*d^2 + 40*a^7*b^6*d^2 + 84*a^9*b^4*d^2 + 4*a^11*b^2*d^2)
)/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d
^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2))/2)*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^
2*d^2*6i))^(1/2))/2 - (16*tan(c + d*x)^(1/2)*(3*b^9 - 3*a^2*b^7 + 17*a^4*b^5 - 9*a^6*b^3))/(a^8*d^4 + b^8*d^4
+ 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^
2*b^2*d^2*6i))^(1/2)*1i)/2 - (((((8*(52*a*b^10*d^2 - 128*a^3*b^8*d^2 - 24*a^5*b^6*d^2 + 160*a^7*b^4*d^2 + 4*a^
9*b^2*d^2))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) + (((((8*(320*a^6*b^9*d^4 - 96
*a^2*b^13*d^4 - 32*b^15*d^4 + 480*a^8*b^7*d^4 + 288*a^10*b^5*d^4 + 64*a^12*b^3*d^4))/(a^8*d^5 + b^8*d^5 + 4*a^
2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) + (8*tan(c + d*x)^(1/2)*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 -
4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2)*(32*b^17*d^4 + 160*a^2*b^15*d^4 + 288*a^4*b^13*d^4 + 160*a^6*b^11*d^4 -
160*a^8*b^9*d^4 - 288*a^10*b^7*d^4 - 160*a^12*b^5*d^4 - 32*a^14*b^3*d^4))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 +
6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1
/2))/2 - (16*tan(c + d*x)^(1/2)*(68*a*b^12*d^2 + 20*a^3*b^10*d^2 - 88*a^5*b^8*d^2 + 40*a^7*b^6*d^2 + 84*a^9*b^
4*d^2 + 4*a^11*b^2*d^2))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))*(1/(a^4*d^2*1i +
b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2))/2)*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2
- 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2))/2 + (16*tan(c + d*x)^(1/2)*(3*b^9 - 3*a^2*b^7 + 17*a^4*b^5 - 9*a^6*b^
3))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3
*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2)*1i)/2)/((((((8*(52*a*b^10*d^2 - 128*a^3*b^8*d^2 - 24*a^5*b^6*d^2 +
160*a^7*b^4*d^2 + 4*a^9*b^2*d^2))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) + (((((
8*(320*a^6*b^9*d^4 - 96*a^2*b^13*d^4 - 32*b^15*d^4 + 480*a^8*b^7*d^4 + 288*a^10*b^5*d^4 + 64*a^12*b^3*d^4))/(a
^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) - (8*tan(c + d*x)^(1/2)*(1/(a^4*d^2*1i + b^4
*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2)*(32*b^17*d^4 + 160*a^2*b^15*d^4 + 288*a^4*b^13*d^
4 + 160*a^6*b^11*d^4 - 160*a^8*b^9*d^4 - 288*a^10*b^7*d^4 - 160*a^12*b^5*d^4 - 32*a^14*b^3*d^4))/(a^8*d^4 + b^
8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^
2 - a^2*b^2*d^2*6i))^(1/2))/2 + (16*tan(c + d*x)^(1/2)*(68*a*b^12*d^2 + 20*a^3*b^10*d^2 - 88*a^5*b^8*d^2 + 40*
a^7*b^6*d^2 + 84*a^9*b^4*d^2 + 4*a^11*b^2*d^2))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2
*d^4))*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2))/2)*(1/(a^4*d^2*1i + b
^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2))/2 - (16*tan(c + d*x)^(1/2)*(3*b^9 - 3*a^2*b^7
+ 17*a^4*b^5 - 9*a^6*b^3))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))*(1/(a^4*d^2*1i
+ b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2))/2 + (((((8*(52*a*b^10*d^2 - 128*a^3*b^8*d^
2 - 24*a^5*b^6*d^2 + 160*a^7*b^4*d^2 + 4*a^9*b^2*d^2))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*
a^6*b^2*d^5) + (((((8*(320*a^6*b^9*d^4 - 96*a^2*b^13*d^4 - 32*b^15*d^4 + 480*a^8*b^7*d^4 + 288*a^10*b^5*d^4 +
64*a^12*b^3*d^4))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) + (8*tan(c + d*x)^(1/2)*
(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2)*(32*b^17*d^4 + 160*a^2*b^15*d
^4 + 288*a^4*b^13*d^4 + 160*a^6*b^11*d^4 - 160*a^8*b^9*d^4 - 288*a^10*b^7*d^4 - 160*a^12*b^5*d^4 - 32*a^14*b^3
*d^4))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*
b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2))/2 - (16*tan(c + d*x)^(1/2)*(68*a*b^12*d^2 + 20*a^3*b^10*d^2 -
88*a^5*b^8*d^2 + 40*a^7*b^6*d^2 + 84*a^9*b^4*d^2 + 4*a^11*b^2*d^2))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4
*b^4*d^4 + 4*a^6*b^2*d^4))*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2))/2
)*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2))/2 + (16*tan(c + d*x)^(1/2)
*(3*b^9 - 3*a^2*b^7 + 17*a^4*b^5 - 9*a^6*b^3))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*
d^4))*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2))/2 - (16*(b^7 - 9*a^4*b
^3))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5)))*(1/(a^4*d^2*1i + b^4*d^2*1i + 4*a*b
^3*d^2 - 4*a^3*b*d^2 - a^2*b^2*d^2*6i))^(1/2)*1i + atan(((1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*
4i - 6*a^2*b^2*d^2)))^(1/2)*((1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*
((8*(52*a*b^10*d^2 - 128*a^3*b^8*d^2 - 24*a^5*b^6*d^2 + 160*a^7*b^4*d^2 + 4*a^9*b^2*d^2))/(a^8*d^5 + b^8*d^5 +
4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) + (1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*
a^2*b^2*d^2)))^(1/2)*((1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*((8*(32
0*a^6*b^9*d^4 - 96*a^2*b^13*d^4 - 32*b^15*d^4 + 480*a^8*b^7*d^4 + 288*a^10*b^5*d^4 + 64*a^12*b^3*d^4))/(a^8*d^
5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) - (16*tan(c + d*x)^(1/2)*(1i/(4*(a^4*d^2 + b^4*d^
2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*(32*b^17*d^4 + 160*a^2*b^15*d^4 + 288*a^4*b^13*d^4 +
160*a^6*b^11*d^4 - 160*a^8*b^9*d^4 - 288*a^10*b^7*d^4 - 160*a^12*b^5*d^4 - 32*a^14*b^3*d^4))/(a^8*d^4 + b^8*d^
4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4)) + (16*tan(c + d*x)^(1/2)*(68*a*b^12*d^2 + 20*a^3*b^10*d^2
- 88*a^5*b^8*d^2 + 40*a^7*b^6*d^2 + 84*a^9*b^4*d^2 + 4*a^11*b^2*d^2))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a
^4*b^4*d^4 + 4*a^6*b^2*d^4))) - (16*tan(c + d*x)^(1/2)*(3*b^9 - 3*a^2*b^7 + 17*a^4*b^5 - 9*a^6*b^3))/(a^8*d^4
+ b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))*1i - (1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^
3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*((1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2))
)^(1/2)*((8*(52*a*b^10*d^2 - 128*a^3*b^8*d^2 - 24*a^5*b^6*d^2 + 160*a^7*b^4*d^2 + 4*a^9*b^2*d^2))/(a^8*d^5 + b
^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) + (1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2
*4i - 6*a^2*b^2*d^2)))^(1/2)*((1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)
*((8*(320*a^6*b^9*d^4 - 96*a^2*b^13*d^4 - 32*b^15*d^4 + 480*a^8*b^7*d^4 + 288*a^10*b^5*d^4 + 64*a^12*b^3*d^4))
/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) + (16*tan(c + d*x)^(1/2)*(1i/(4*(a^4*d^2
+ b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*(32*b^17*d^4 + 160*a^2*b^15*d^4 + 288*a^4*b^1
3*d^4 + 160*a^6*b^11*d^4 - 160*a^8*b^9*d^4 - 288*a^10*b^7*d^4 - 160*a^12*b^5*d^4 - 32*a^14*b^3*d^4))/(a^8*d^4
+ b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4)) - (16*tan(c + d*x)^(1/2)*(68*a*b^12*d^2 + 20*a^3*b
^10*d^2 - 88*a^5*b^8*d^2 + 40*a^7*b^6*d^2 + 84*a^9*b^4*d^2 + 4*a^11*b^2*d^2))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d
^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))) + (16*tan(c + d*x)^(1/2)*(3*b^9 - 3*a^2*b^7 + 17*a^4*b^5 - 9*a^6*b^3))/(
a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))*1i)/((1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2
*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*((1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b
^2*d^2)))^(1/2)*((8*(52*a*b^10*d^2 - 128*a^3*b^8*d^2 - 24*a^5*b^6*d^2 + 160*a^7*b^4*d^2 + 4*a^9*b^2*d^2))/(a^8
*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) + (1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a
^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*((1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)
))^(1/2)*((8*(320*a^6*b^9*d^4 - 96*a^2*b^13*d^4 - 32*b^15*d^4 + 480*a^8*b^7*d^4 + 288*a^10*b^5*d^4 + 64*a^12*b
^3*d^4))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) - (16*tan(c + d*x)^(1/2)*(1i/(4*(
a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*(32*b^17*d^4 + 160*a^2*b^15*d^4 + 288
*a^4*b^13*d^4 + 160*a^6*b^11*d^4 - 160*a^8*b^9*d^4 - 288*a^10*b^7*d^4 - 160*a^12*b^5*d^4 - 32*a^14*b^3*d^4))/(
a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4)) + (16*tan(c + d*x)^(1/2)*(68*a*b^12*d^2 +
20*a^3*b^10*d^2 - 88*a^5*b^8*d^2 + 40*a^7*b^6*d^2 + 84*a^9*b^4*d^2 + 4*a^11*b^2*d^2))/(a^8*d^4 + b^8*d^4 + 4*a
^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))) - (16*tan(c + d*x)^(1/2)*(3*b^9 - 3*a^2*b^7 + 17*a^4*b^5 - 9*a^6
*b^3))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4)) + (1i/(4*(a^4*d^2 + b^4*d^2 + a*b^
3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*((1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*
a^2*b^2*d^2)))^(1/2)*((8*(52*a*b^10*d^2 - 128*a^3*b^8*d^2 - 24*a^5*b^6*d^2 + 160*a^7*b^4*d^2 + 4*a^9*b^2*d^2))
/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) + (1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4
i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*((1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2
*d^2)))^(1/2)*((8*(320*a^6*b^9*d^4 - 96*a^2*b^13*d^4 - 32*b^15*d^4 + 480*a^8*b^7*d^4 + 288*a^10*b^5*d^4 + 64*a
^12*b^3*d^4))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) + (16*tan(c + d*x)^(1/2)*(1i
/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*(32*b^17*d^4 + 160*a^2*b^15*d^4
+ 288*a^4*b^13*d^4 + 160*a^6*b^11*d^4 - 160*a^8*b^9*d^4 - 288*a^10*b^7*d^4 - 160*a^12*b^5*d^4 - 32*a^14*b^3*d^
4))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4)) - (16*tan(c + d*x)^(1/2)*(68*a*b^12*d
^2 + 20*a^3*b^10*d^2 - 88*a^5*b^8*d^2 + 40*a^7*b^6*d^2 + 84*a^9*b^4*d^2 + 4*a^11*b^2*d^2))/(a^8*d^4 + b^8*d^4
+ 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))) + (16*tan(c + d*x)^(1/2)*(3*b^9 - 3*a^2*b^7 + 17*a^4*b^5 -
9*a^6*b^3))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4)) - (16*(b^7 - 9*a^4*b^3))/(a^8
*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5)))*(1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a
^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*2i + (atan((((-a*b)^(1/2)*(3*a^2 - b^2)*((16*tan(c + d*x)^(1/2)*(3*b^9 -
3*a^2*b^7 + 17*a^4*b^5 - 9*a^6*b^3))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4) - ((-
a*b)^(1/2)*(3*a^2 - b^2)*((8*(52*a*b^10*d^2 - 128*a^3*b^8*d^2 - 24*a^5*b^6*d^2 + 160*a^7*b^4*d^2 + 4*a^9*b^2*d
^2))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) + ((-a*b)^(1/2)*((16*tan(c + d*x)^(1/
2)*(68*a*b^12*d^2 + 20*a^3*b^10*d^2 - 88*a^5*b^8*d^2 + 40*a^7*b^6*d^2 + 84*a^9*b^4*d^2 + 4*a^11*b^2*d^2))/(a^8
*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4) + ((-a*b)^(1/2)*(3*a^2 - b^2)*((8*(320*a^6*b^9
*d^4 - 96*a^2*b^13*d^4 - 32*b^15*d^4 + 480*a^8*b^7*d^4 + 288*a^10*b^5*d^4 + 64*a^12*b^3*d^4))/(a^8*d^5 + b^8*d
^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) - (8*tan(c + d*x)^(1/2)*(-a*b)^(1/2)*(3*a^2 - b^2)*(32*b^1
7*d^4 + 160*a^2*b^15*d^4 + 288*a^4*b^13*d^4 + 160*a^6*b^11*d^4 - 160*a^8*b^9*d^4 - 288*a^10*b^7*d^4 - 160*a^12
*b^5*d^4 - 32*a^14*b^3*d^4))/((a^5*d + 2*a^3*b^2*d + a*b^4*d)*(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d
^4 + 4*a^6*b^2*d^4))))/(2*(a^5*d + 2*a^3*b^2*d + a*b^4*d)))*(3*a^2 - b^2))/(2*(a^5*d + 2*a^3*b^2*d + a*b^4*d))
))/(2*(a^5*d + 2*a^3*b^2*d + a*b^4*d)))*1i)/(2*(a^5*d + 2*a^3*b^2*d + a*b^4*d)) + ((-a*b)^(1/2)*(3*a^2 - b^2)*
((16*tan(c + d*x)^(1/2)*(3*b^9 - 3*a^2*b^7 + 17*a^4*b^5 - 9*a^6*b^3))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a
^4*b^4*d^4 + 4*a^6*b^2*d^4) + ((-a*b)^(1/2)*(3*a^2 - b^2)*((8*(52*a*b^10*d^2 - 128*a^3*b^8*d^2 - 24*a^5*b^6*d^
2 + 160*a^7*b^4*d^2 + 4*a^9*b^2*d^2))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) - ((
-a*b)^(1/2)*((16*tan(c + d*x)^(1/2)*(68*a*b^12*d^2 + 20*a^3*b^10*d^2 - 88*a^5*b^8*d^2 + 40*a^7*b^6*d^2 + 84*a^
9*b^4*d^2 + 4*a^11*b^2*d^2))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4) - ((-a*b)^(1/
2)*(3*a^2 - b^2)*((8*(320*a^6*b^9*d^4 - 96*a^2*b^13*d^4 - 32*b^15*d^4 + 480*a^8*b^7*d^4 + 288*a^10*b^5*d^4 + 6
4*a^12*b^3*d^4))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) + (8*tan(c + d*x)^(1/2)*(
-a*b)^(1/2)*(3*a^2 - b^2)*(32*b^17*d^4 + 160*a^2*b^15*d^4 + 288*a^4*b^13*d^4 + 160*a^6*b^11*d^4 - 160*a^8*b^9*
d^4 - 288*a^10*b^7*d^4 - 160*a^12*b^5*d^4 - 32*a^14*b^3*d^4))/((a^5*d + 2*a^3*b^2*d + a*b^4*d)*(a^8*d^4 + b^8*
d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))))/(2*(a^5*d + 2*a^3*b^2*d + a*b^4*d)))*(3*a^2 - b^2))/(2
*(a^5*d + 2*a^3*b^2*d + a*b^4*d))))/(2*(a^5*d + 2*a^3*b^2*d + a*b^4*d)))*1i)/(2*(a^5*d + 2*a^3*b^2*d + a*b^4*d
)))/((16*(b^7 - 9*a^4*b^3))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) + ((-a*b)^(1/2
)*(3*a^2 - b^2)*((16*tan(c + d*x)^(1/2)*(3*b^9 - 3*a^2*b^7 + 17*a^4*b^5 - 9*a^6*b^3))/(a^8*d^4 + b^8*d^4 + 4*a
^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4) - ((-a*b)^(1/2)*(3*a^2 - b^2)*((8*(52*a*b^10*d^2 - 128*a^3*b^8*d^2
- 24*a^5*b^6*d^2 + 160*a^7*b^4*d^2 + 4*a^9*b^2*d^2))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a
^6*b^2*d^5) + ((-a*b)^(1/2)*((16*tan(c + d*x)^(1/2)*(68*a*b^12*d^2 + 20*a^3*b^10*d^2 - 88*a^5*b^8*d^2 + 40*a^7
*b^6*d^2 + 84*a^9*b^4*d^2 + 4*a^11*b^2*d^2))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^
4) + ((-a*b)^(1/2)*(3*a^2 - b^2)*((8*(320*a^6*b^9*d^4 - 96*a^2*b^13*d^4 - 32*b^15*d^4 + 480*a^8*b^7*d^4 + 288*
a^10*b^5*d^4 + 64*a^12*b^3*d^4))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) - (8*tan(
c + d*x)^(1/2)*(-a*b)^(1/2)*(3*a^2 - b^2)*(32*b^17*d^4 + 160*a^2*b^15*d^4 + 288*a^4*b^13*d^4 + 160*a^6*b^11*d^
4 - 160*a^8*b^9*d^4 - 288*a^10*b^7*d^4 - 160*a^12*b^5*d^4 - 32*a^14*b^3*d^4))/((a^5*d + 2*a^3*b^2*d + a*b^4*d)
*(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))))/(2*(a^5*d + 2*a^3*b^2*d + a*b^4*d)))*(
3*a^2 - b^2))/(2*(a^5*d + 2*a^3*b^2*d + a*b^4*d))))/(2*(a^5*d + 2*a^3*b^2*d + a*b^4*d))))/(2*(a^5*d + 2*a^3*b^
2*d + a*b^4*d)) - ((-a*b)^(1/2)*(3*a^2 - b^2)*((16*tan(c + d*x)^(1/2)*(3*b^9 - 3*a^2*b^7 + 17*a^4*b^5 - 9*a^6*
b^3))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4) + ((-a*b)^(1/2)*(3*a^2 - b^2)*((8*(5
2*a*b^10*d^2 - 128*a^3*b^8*d^2 - 24*a^5*b^6*d^2 + 160*a^7*b^4*d^2 + 4*a^9*b^2*d^2))/(a^8*d^5 + b^8*d^5 + 4*a^2
*b^6*d^5 + 6*a^4*b^4*d^5 + 4*a^6*b^2*d^5) - ((-a*b)^(1/2)*((16*tan(c + d*x)^(1/2)*(68*a*b^12*d^2 + 20*a^3*b^10
*d^2 - 88*a^5*b^8*d^2 + 40*a^7*b^6*d^2 + 84*a^9*b^4*d^2 + 4*a^11*b^2*d^2))/(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4
+ 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4) - ((-a*b)^(1/2)*(3*a^2 - b^2)*((8*(320*a^6*b^9*d^4 - 96*a^2*b^13*d^4 - 32*b^1
5*d^4 + 480*a^8*b^7*d^4 + 288*a^10*b^5*d^4 + 64*a^12*b^3*d^4))/(a^8*d^5 + b^8*d^5 + 4*a^2*b^6*d^5 + 6*a^4*b^4*
d^5 + 4*a^6*b^2*d^5) + (8*tan(c + d*x)^(1/2)*(-a*b)^(1/2)*(3*a^2 - b^2)*(32*b^17*d^4 + 160*a^2*b^15*d^4 + 288*
a^4*b^13*d^4 + 160*a^6*b^11*d^4 - 160*a^8*b^9*d^4 - 288*a^10*b^7*d^4 - 160*a^12*b^5*d^4 - 32*a^14*b^3*d^4))/((
a^5*d + 2*a^3*b^2*d + a*b^4*d)*(a^8*d^4 + b^8*d^4 + 4*a^2*b^6*d^4 + 6*a^4*b^4*d^4 + 4*a^6*b^2*d^4))))/(2*(a^5*
d + 2*a^3*b^2*d + a*b^4*d)))*(3*a^2 - b^2))/(2*(a^5*d + 2*a^3*b^2*d + a*b^4*d))))/(2*(a^5*d + 2*a^3*b^2*d + a*
b^4*d))))/(2*(a^5*d + 2*a^3*b^2*d + a*b^4*d))))*(-a*b)^(1/2)*(3*a^2 - b^2)*1i)/(a^5*d + 2*a^3*b^2*d + a*b^4*d)
- (b*tan(c + d*x)^(1/2))/((a*d + b*d*tan(c + d*x))*(a^2 + b^2))