3.7.4 \(\int \frac {\sqrt {\tan (c+d x)}}{(a+b \tan (c+d x))^3} \, dx\) [604]

Optimal. Leaf size=389 \[ -\frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {ArcTan}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\sqrt {b} \left (15 a^4-18 a^2 b^2-b^4\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{3/2} \left (a^2+b^2\right )^3 d}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {b \sqrt {\tan (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {b \left (7 a^2-b^2\right ) \sqrt {\tan (c+d x)}}{4 a \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \]

[Out]

1/2*(a-b)*(a^2+4*a*b+b^2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)^3/d*2^(1/2)+1/2*(a-b)*(a^2+4*a*b+b^2)*
arctan(1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)^3/d*2^(1/2)+1/4*(a+b)*(a^2-4*a*b+b^2)*ln(1-2^(1/2)*tan(d*x+c)^(1/
2)+tan(d*x+c))/(a^2+b^2)^3/d*2^(1/2)-1/4*(a+b)*(a^2-4*a*b+b^2)*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+
b^2)^3/d*2^(1/2)-1/4*(15*a^4-18*a^2*b^2-b^4)*arctan(b^(1/2)*tan(d*x+c)^(1/2)/a^(1/2))*b^(1/2)/a^(3/2)/(a^2+b^2
)^3/d-1/2*b*tan(d*x+c)^(1/2)/(a^2+b^2)/d/(a+b*tan(d*x+c))^2-1/4*b*(7*a^2-b^2)*tan(d*x+c)^(1/2)/a/(a^2+b^2)^2/d
/(a+b*tan(d*x+c))

________________________________________________________________________________________

Rubi [A]
time = 0.53, antiderivative size = 389, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {3649, 3730, 3734, 3615, 1182, 1176, 631, 210, 1179, 642, 3715, 65, 211} \begin {gather*} -\frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {ArcTan}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )^3}-\frac {b \left (7 a^2-b^2\right ) \sqrt {\tan (c+d x)}}{4 a d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {b \sqrt {\tan (c+d x)}}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {(a+b) \left (a^2-4 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 )^3}-\frac {(a+b) \left (a^2-4 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 )^3}-\frac {\sqrt {b} \left (15 a^4-18 a^2 b^2-b^4\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{3/2} d \left (a^2+b^2\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[Tan[c + d*x]]/(a + b*Tan[c + d*x])^3,x]

[Out]

-(((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d)) + ((a - b)*(
a^2 + 4*a*b + b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)^3*d) - (Sqrt[b]*(15*a^4 - 18*a
^2*b^2 - b^4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(4*a^(3/2)*(a^2 + b^2)^3*d) + ((a + b)*(a^2 - 4*a*
b + b^2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - ((a + b)*(a^2 - 4*a
*b + b^2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)^3*d) - (b*Sqrt[Tan[c + d*
x]])/(2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (b*(7*a^2 - b^2)*Sqrt[Tan[c + d*x]])/(4*a*(a^2 + b^2)^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 3730

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Ta
n[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, 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] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

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*} \int \frac {\sqrt {\tan (c+d x)}}{(a+b \tan (c+d x))^3} \, dx &=-\frac {b \sqrt {\tan (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {\int \frac {-\frac {b}{2}-2 a \tan (c+d x)+\frac {3}{2} b \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2} \, dx}{2 \left (a^2+b^2\right )}\\ &=-\frac {b \sqrt {\tan (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {b \left (7 a^2-b^2\right ) \sqrt {\tan (c+d x)}}{4 a \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {\int \frac {-\frac {1}{4} b \left (9 a^2+b^2\right )-2 a \left (a^2-b^2\right ) \tan (c+d x)+\frac {1}{4} b \left (7 a^2-b^2\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{2 a \left (a^2+b^2\right )^2}\\ &=-\frac {b \sqrt {\tan (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {b \left (7 a^2-b^2\right ) \sqrt {\tan (c+d x)}}{4 a \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {\int \frac {-2 a b \left (3 a^2-b^2\right )-2 a^2 \left (a^2-3 b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{2 a \left (a^2+b^2\right )^3}-\frac {\left (b \left (15 a^4-18 a^2 b^2-b^4\right )\right ) \int \frac {1+\tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{8 a \left (a^2+b^2\right )^3}\\ &=-\frac {b \sqrt {\tan (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {b \left (7 a^2-b^2\right ) \sqrt {\tan (c+d x)}}{4 a \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {\text {Subst}\left (\int \frac {-2 a b \left (3 a^2-b^2\right )-2 a^2 \left (a^2-3 b^2\right ) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a \left (a^2+b^2\right )^3 d}-\frac {\left (b \left (15 a^4-18 a^2 b^2-b^4\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{8 a \left (a^2+b^2\right )^3 d}\\ &=-\frac {b \sqrt {\tan (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {b \left (7 a^2-b^2\right ) \sqrt {\tan (c+d x)}}{4 a \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\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 )^3 d}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\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 )^3 d}-\frac {\left (b \left (15 a^4-18 a^2 b^2-b^4\right )\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{4 a \left (a^2+b^2\right )^3 d}\\ &=-\frac {\sqrt {b} \left (15 a^4-18 a^2 b^2-b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{3/2} \left (a^2+b^2\right )^3 d}-\frac {b \sqrt {\tan (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {b \left (7 a^2-b^2\right ) \sqrt {\tan (c+d x)}}{4 a \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\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 )^3 d}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\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 )^3 d}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\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 )^3 d}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\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 )^3 d}\\ &=-\frac {\sqrt {b} \left (15 a^4-18 a^2 b^2-b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{3/2} \left (a^2+b^2\right )^3 d}+\frac {(a+b) \left (a^2-4 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 )^3 d}-\frac {(a+b) \left (a^2-4 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 )^3 d}-\frac {b \sqrt {\tan (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {b \left (7 a^2-b^2\right ) \sqrt {\tan (c+d x)}}{4 a \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\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 )^3 d}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\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 )^3 d}\\ &=-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {\sqrt {b} \left (15 a^4-18 a^2 b^2-b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{3/2} \left (a^2+b^2\right )^3 d}+\frac {(a+b) \left (a^2-4 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 )^3 d}-\frac {(a+b) \left (a^2-4 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 )^3 d}-\frac {b \sqrt {\tan (c+d x)}}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {b \left (7 a^2-b^2\right ) \sqrt {\tan (c+d x)}}{4 a \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 5.52, size = 259, normalized size = 0.67 \begin {gather*} -\frac {-\frac {b^{3/2} \left (-15 a^4+18 a^2 b^2+b^4\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} \left (a^2+b^2\right )^2}+\frac {4 \sqrt [4]{-1} a b \left ((i a-b)^3 \text {ArcTan}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )-(i a+b)^3 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )\right )}{\left (a^2+b^2\right )^2}-\frac {2 b^3 \tan ^{\frac {3}{2}}(c+d x)}{(a+b \tan (c+d x))^2}+\frac {2 b^2 \sqrt {\tan (c+d x)}}{a+b \tan (c+d x)}+\frac {\left (7 a^2 b^2-b^4\right ) \sqrt {\tan (c+d x)}}{\left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 a b \left (a^2+b^2\right ) d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Tan[c + d*x]]/(a + b*Tan[c + d*x])^3,x]

[Out]

-1/4*(-((b^(3/2)*(-15*a^4 + 18*a^2*b^2 + b^4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*(a^2 + b^
2)^2)) + (4*(-1)^(1/4)*a*b*((I*a - b)^3*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] - (I*a + b)^3*ArcTanh[(-1)^(3/4)
*Sqrt[Tan[c + d*x]]]))/(a^2 + b^2)^2 - (2*b^3*Tan[c + d*x]^(3/2))/(a + b*Tan[c + d*x])^2 + (2*b^2*Sqrt[Tan[c +
d*x]])/(a + b*Tan[c + d*x]) + ((7*a^2*b^2 - b^4)*Sqrt[Tan[c + d*x]])/((a^2 + b^2)*(a + b*Tan[c + d*x])))/(a*b
*(a^2 + b^2)*d)

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Maple [A]
time = 0.17, size = 343, normalized size = 0.88

method	result	size
derivativedivides	\(\frac {-\frac {2 b \left (\frac {\frac {b \left (7 a^{4}+6 a^{2} b^{2}-b^{4}\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{8 a}+\left (\frac {9}{8} a^{4}+\frac {5}{4} a^{2} b^{2}+\frac {1}{8} b^{4}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\left (15 a^{4}-18 a^{2} b^{2}-b^{4}\right ) \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{8 a \sqrt {a b}}\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {\left (3 a^{2} b -b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (a^{3}-3 b^{2} a \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{3}}}{d}\)	\(343\)
default	\(\frac {-\frac {2 b \left (\frac {\frac {b \left (7 a^{4}+6 a^{2} b^{2}-b^{4}\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{8 a}+\left (\frac {9}{8} a^{4}+\frac {5}{4} a^{2} b^{2}+\frac {1}{8} b^{4}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\left (15 a^{4}-18 a^{2} b^{2}-b^{4}\right ) \arctan \left (\frac {b \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {a b}}\right )}{8 a \sqrt {a b}}\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {\left (3 a^{2} b -b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (a^{3}-3 b^{2} a \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{3}}}{d}\)	\(343\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-2*b/(a^2+b^2)^3*((1/8*b*(7*a^4+6*a^2*b^2-b^4)/a*tan(d*x+c)^(3/2)+(9/8*a^4+5/4*a^2*b^2+1/8*b^4)*tan(d*x+c
)^(1/2))/(a+b*tan(d*x+c))^2+1/8*(15*a^4-18*a^2*b^2-b^4)/a/(a*b)^(1/2)*arctan(b*tan(d*x+c)^(1/2)/(a*b)^(1/2)))+
2/(a^2+b^2)^3*(1/8*(3*a^2*b-b^3)*2^(1/2)*(ln((1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1-2^(1/2)*tan(d*x+c)^(1/
2)+tan(d*x+c)))+2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2)))+1/8*(a^3-3*a*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)))))

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Maxima [A]
time = 0.50, size = 411, normalized size = 1.06 \begin {gather*} -\frac {\frac {{\left (15 \, a^{4} b - 18 \, a^{2} b^{3} - b^{5}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \sqrt {a b}} - \frac {2 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\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^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\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^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (7 \, a^{2} b^{2} - b^{4}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} + {\left (9 \, a^{3} b + a b^{3}\right )} \sqrt {\tan \left (d x + c\right )}}{a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4} + {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} \tan \left (d x + c\right )}}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/4*((15*a^4*b - 18*a^2*b^3 - b^5)*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b
^6)*sqrt(a*b)) - (2*sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c))
)) + 2*sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c)))) - sqrt(2)
*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) + sqrt(2)*(a^3 - 3*a^2*b -
3*a*b^2 + b^3)*log(-sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + ((7
*a^2*b^2 - b^4)*tan(d*x + c)^(3/2) + (9*a^3*b + a*b^3)*sqrt(tan(d*x + c)))/(a^7 + 2*a^5*b^2 + a^3*b^4 + (a^5*b
^2 + 2*a^3*b^4 + a*b^6)*tan(d*x + c)^2 + 2*(a^6*b + 2*a^4*b^3 + a^2*b^5)*tan(d*x + c)))/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 11386 vs. \(2 (341) = 682\).
time = 14.06, size = 22777, normalized size = 58.55 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

[1/16*(16*sqrt(2)*((a^23 + 3*a^21*b^2 - 17*a^19*b^4 - 123*a^17*b^6 - 342*a^15*b^8 - 546*a^13*b^10 - 546*a^11*b
^12 - 342*a^9*b^14 - 123*a^7*b^16 - 17*a^5*b^18 + 3*a^3*b^20 + a*b^22)*d^5*cos(d*x + c)^4 + 2*(3*a^21*b^2 + 26
*a^19*b^4 + 99*a^17*b^6 + 216*a^15*b^8 + 294*a^13*b^10 + 252*a^11*b^12 + 126*a^9*b^14 + 24*a^7*b^16 - 9*a^5*b^
18 - 6*a^3*b^20 - a*b^22)*d^5*cos(d*x + c)^2 + (a^19*b^4 + 9*a^17*b^6 + 36*a^15*b^8 + 84*a^13*b^10 + 126*a^11*
b^12 + 126*a^9*b^14 + 84*a^7*b^16 + 36*a^5*b^18 + 9*a^3*b^20 + a*b^22)*d^5 + 4*((a^22*b + 8*a^20*b^3 + 27*a^18
*b^5 + 48*a^16*b^7 + 42*a^14*b^9 - 42*a^10*b^13 - 48*a^8*b^15 - 27*a^6*b^17 - 8*a^4*b^19 - a^2*b^21)*d^5*cos(d
*x + c)^3 + (a^20*b^3 + 9*a^18*b^5 + 36*a^16*b^7 + 84*a^14*b^9 + 126*a^12*b^11 + 126*a^10*b^13 + 84*a^8*b^15 +
36*a^6*b^17 + 9*a^4*b^19 + a^2*b^21)*d^5*cos(d*x + c))*sin(d*x + c))*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 2
0*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12 + 2*(3*a^17*b + 8*a^15*b^3 - 12*a^13*b^5 - 72*a^11*b^7 - 110*a^9*b^
9 - 72*a^7*b^11 - 12*a^5*b^13 + 8*a^3*b^15 + 3*a*b^17)*d^2*sqrt(1/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^
6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))/(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*
a^2*b^10 + b^12))*sqrt((a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/((a
^24 + 12*a^22*b^2 + 66*a^20*b^4 + 220*a^18*b^6 + 495*a^16*b^8 + 792*a^14*b^10 + 924*a^12*b^12 + 792*a^10*b^14
+ 495*a^8*b^16 + 220*a^6*b^18 + 66*a^4*b^20 + 12*a^2*b^22 + b^24)*d^4))*(1/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 +
20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))^(3/4)*arctan(((a^24 - 6*a^22*b^2 - 84*a^20*b^4 - 322*a^18*b
^6 - 603*a^16*b^8 - 540*a^14*b^10 + 540*a^10*b^14 + 603*a^8*b^16 + 322*a^6*b^18 + 84*a^4*b^20 + 6*a^2*b^22 - b
^24)*d^4*sqrt((a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/((a^24 + 12*
a^22*b^2 + 66*a^20*b^4 + 220*a^18*b^6 + 495*a^16*b^8 + 792*a^14*b^10 + 924*a^12*b^12 + 792*a^10*b^14 + 495*a^8
*b^16 + 220*a^6*b^18 + 66*a^4*b^20 + 12*a^2*b^22 + b^24)*d^4))*sqrt(1/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^
6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) - sqrt(2)*((3*a^26*b + 35*a^24*b^3 + 186*a^22*b^5 + 594*a^20*b^7
+ 1265*a^18*b^9 + 1881*a^16*b^11 + 1980*a^14*b^13 + 1452*a^12*b^15 + 693*a^10*b^17 + 165*a^8*b^19 - 22*a^6*b^
21 - 30*a^4*b^23 - 9*a^2*b^25 - b^27)*d^7*sqrt((a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 -
30*a^2*b^10 + b^12)/((a^24 + 12*a^22*b^2 + 66*a^20*b^4 + 220*a^18*b^6 + 495*a^16*b^8 + 792*a^14*b^10 + 924*a^
12*b^12 + 792*a^10*b^14 + 495*a^8*b^16 + 220*a^6*b^18 + 66*a^4*b^20 + 12*a^2*b^22 + b^24)*d^4))*sqrt(1/((a^12
+ 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) - (a^21 + 6*a^19*b^2 + 9*a^17*b
^4 - 24*a^15*b^6 - 126*a^13*b^8 - 252*a^11*b^10 - 294*a^9*b^12 - 216*a^7*b^14 - 99*a^5*b^16 - 26*a^3*b^18 - 3*
a*b^20)*d^5*sqrt((a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/((a^24 +
12*a^22*b^2 + 66*a^20*b^4 + 220*a^18*b^6 + 495*a^16*b^8 + 792*a^14*b^10 + 924*a^12*b^12 + 792*a^10*b^14 + 495*
a^8*b^16 + 220*a^6*b^18 + 66*a^4*b^20 + 12*a^2*b^22 + b^24)*d^4)))*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a
^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12 + 2*(3*a^17*b + 8*a^15*b^3 - 12*a^13*b^5 - 72*a^11*b^7 - 110*a^9*b^9 -
72*a^7*b^11 - 12*a^5*b^13 + 8*a^3*b^15 + 3*a*b^17)*d^2*sqrt(1/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 +
15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))/(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2
*b^10 + b^12))*sqrt(((a^18 - 27*a^16*b^2 + 168*a^14*b^4 + 224*a^12*b^6 - 366*a^10*b^8 - 366*a^8*b^10 + 224*a^6
*b^12 + 168*a^4*b^14 - 27*a^2*b^16 + b^18)*d^2*sqrt(1/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b
^8 + 6*a^2*b^10 + b^12)*d^4))*cos(d*x + c) + sqrt(2)*((a^21 - 30*a^19*b^2 + 249*a^17*b^4 - 280*a^15*b^6 - 1038
*a^13*b^8 + 732*a^11*b^10 + 1322*a^9*b^12 - 504*a^7*b^14 - 531*a^5*b^16 + 82*a^3*b^18 - 3*a*b^20)*d^3*sqrt(1/(
(a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*cos(d*x + c) - (3*a^14*b
- 91*a^12*b^3 + 795*a^10*b^5 - 1611*a^8*b^7 + 1217*a^6*b^9 - 345*a^4*b^11 + 33*a^2*b^13 - b^15)*d*cos(d*x + c)
)*sqrt((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12 + 2*(3*a^17*b + 8*a^15*b^
3 - 12*a^13*b^5 - 72*a^11*b^7 - 110*a^9*b^9 - 72*a^7*b^11 - 12*a^5*b^13 + 8*a^3*b^15 + 3*a*b^17)*d^2*sqrt(1/((
a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))/(a^12 - 30*a^10*b^2 + 255
*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12))*sqrt(sin(d*x + c)/cos(d*x + c))*(1/((a^12 + 6*a^10
*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))^(1/4) + (a^12 - 30*a^10*b^2 + 255*a^8*b
^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)*sin(d*x + c))/cos(d*x + c))*(1/((a^12 + 6*a^10*b^2 + 15*a
^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))^(3/4) - sqrt(2)*((3*a^32*b - 10*a^30*b^3 - 294*a^2
8*b^5 - 1674*a^26*b^7 - 4890*a^24*b^9 - 8370*a^...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\tan {\left (c + d x \right )}}}{\left (a + b \tan {\left (c + d x \right )}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**(1/2)/(a+b*tan(d*x+c))**3,x)

[Out]

Integral(sqrt(tan(c + d*x))/(a + b*tan(c + d*x))**3, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^3,x, algorithm="giac")

[Out]

integrate(sqrt(tan(d*x + c))/(b*tan(d*x + c) + a)^3, x)

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Mupad [B]
time = 16.56, size = 2500, normalized size = 6.43 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)^(1/2)/(a + b*tan(c + d*x))^3,x)

[Out]

atan(((-1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2
*d^2)))^(1/2)*((-1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 1
5*a^4*b^2*d^2)))^(1/2)*(((-1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d
^2*20i + 15*a^4*b^2*d^2)))^(1/2)*((64*a*b^23*d^4 + 1472*a^3*b^21*d^4 + 8832*a^5*b^19*d^4 + 25344*a^7*b^17*d^4
+ 40320*a^9*b^15*d^4 + 34944*a^11*b^13*d^4 + 10752*a^13*b^11*d^4 - 8448*a^15*b^9*d^4 - 10176*a^17*b^7*d^4 - 41
60*a^19*b^5*d^4 - 640*a^21*b^3*d^4)/(a^18*d^5 + a^2*b^16*d^5 + 8*a^4*b^14*d^5 + 28*a^6*b^12*d^5 + 56*a^8*b^10*
d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16*b^2*d^5) + (tan(c + d*x)^(1/2)*(-1i/(4*(b^6
*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*(512
*a^2*b^25*d^4 + 4608*a^4*b^23*d^4 + 17920*a^6*b^21*d^4 + 38400*a^8*b^19*d^4 + 46080*a^10*b^17*d^4 + 21504*a^12
*b^15*d^4 - 21504*a^14*b^13*d^4 - 46080*a^16*b^11*d^4 - 38400*a^18*b^9*d^4 - 17920*a^20*b^7*d^4 - 4608*a^22*b^
5*d^4 - 512*a^24*b^3*d^4))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*
a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4)) - (tan(c + d*x)^(1/2)*(8*a*b^20*d^2 - 1152
*a^3*b^18*d^2 + 2528*a^5*b^16*d^2 + 15296*a^7*b^14*d^2 + 14128*a^9*b^12*d^2 - 5056*a^11*b^10*d^2 - 9248*a^13*b
^8*d^2 + 64*a^15*b^6*d^2 + 1800*a^17*b^4*d^2 + 64*a^19*b^2*d^2))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 2
8*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4))*(-1i
/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1
/2) - (2*b^18*d^2 - 138*a^2*b^16*d^2 - 3046*a^4*b^14*d^2 + 4862*a^6*b^12*d^2 + 9222*a^8*b^10*d^2 - 5246*a^10*b
^8*d^2 - 4290*a^12*b^6*d^2 + 2442*a^14*b^4*d^2 + 32*a^16*b^2*d^2)/(a^18*d^5 + a^2*b^16*d^5 + 8*a^4*b^14*d^5 +
28*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16*b^2*d^5)) + (
tan(c + d*x)^(1/2)*(2*a^2*b^13 - b^15 + 49*a^4*b^11 + 2460*a^6*b^9 - 3631*a^8*b^7 + 1922*a^10*b^5 - 225*a^12*b
^3))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12
*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4))*1i - (-1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i -
15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*((-1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*
d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*(((-1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6
i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*((64*a*b^23*d^4 + 1472*a^3*b^21*
d^4 + 8832*a^5*b^19*d^4 + 25344*a^7*b^17*d^4 + 40320*a^9*b^15*d^4 + 34944*a^11*b^13*d^4 + 10752*a^13*b^11*d^4
- 8448*a^15*b^9*d^4 - 10176*a^17*b^7*d^4 - 4160*a^19*b^5*d^4 - 640*a^21*b^3*d^4)/(a^18*d^5 + a^2*b^16*d^5 + 8*
a^4*b^14*d^5 + 28*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^1
6*b^2*d^5) - (tan(c + d*x)^(1/2)*(-1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a
^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2)*(512*a^2*b^25*d^4 + 4608*a^4*b^23*d^4 + 17920*a^6*b^21*d^4 + 38400*a^
8*b^19*d^4 + 46080*a^10*b^17*d^4 + 21504*a^12*b^15*d^4 - 21504*a^14*b^13*d^4 - 46080*a^16*b^11*d^4 - 38400*a^1
8*b^9*d^4 - 17920*a^20*b^7*d^4 - 4608*a^22*b^5*d^4 - 512*a^24*b^3*d^4))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*
d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4
)) + (tan(c + d*x)^(1/2)*(8*a*b^20*d^2 - 1152*a^3*b^18*d^2 + 2528*a^5*b^16*d^2 + 15296*a^7*b^14*d^2 + 14128*a^
9*b^12*d^2 - 5056*a^11*b^10*d^2 - 9248*a^13*b^8*d^2 + 64*a^15*b^6*d^2 + 1800*a^17*b^4*d^2 + 64*a^19*b^2*d^2))/
(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4 + 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*
d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4))*(-1i/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4
*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(1/2) - (2*b^18*d^2 - 138*a^2*b^16*d^2 - 3046*a^4*b^14*d^2 + 4862*a
^6*b^12*d^2 + 9222*a^8*b^10*d^2 - 5246*a^10*b^8*d^2 - 4290*a^12*b^6*d^2 + 2442*a^14*b^4*d^2 + 32*a^16*b^2*d^2)
/(a^18*d^5 + a^2*b^16*d^5 + 8*a^4*b^14*d^5 + 28*a^6*b^12*d^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6
*d^5 + 28*a^14*b^4*d^5 + 8*a^16*b^2*d^5)) - (tan(c + d*x)^(1/2)*(2*a^2*b^13 - b^15 + 49*a^4*b^11 + 2460*a^6*b^
9 - 3631*a^8*b^7 + 1922*a^10*b^5 - 225*a^12*b^3))/(a^18*d^4 + a^2*b^16*d^4 + 8*a^4*b^14*d^4 + 28*a^6*b^12*d^4
+ 56*a^8*b^10*d^4 + 70*a^10*b^8*d^4 + 56*a^12*b^6*d^4 + 28*a^14*b^4*d^4 + 8*a^16*b^2*d^4))*1i)/((7*a*b^11 + 11
6*a^3*b^9 - 270*a^5*b^7 + 420*a^7*b^5 - 225*a^9*b^3)/(a^18*d^5 + a^2*b^16*d^5 + 8*a^4*b^14*d^5 + 28*a^6*b^12*d
^5 + 56*a^8*b^10*d^5 + 70*a^10*b^8*d^5 + 56*a^12*b^6*d^5 + 28*a^14*b^4*d^5 + 8*a^16*b^2*d^5) + (-1i/(4*(b^6*d^
2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*...

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