3.13.33 \(\int \frac {\sqrt {c+d \tan (e+f x)}}{(a+b \tan (e+f x))^2} \, dx\) [1233]

Optimal. Leaf size=231 \[ -\frac {i \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(a-i b)^2 f}+\frac {i \sqrt {c+i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(a+i b)^2 f}-\frac {\sqrt {b} \left (4 a b c-3 a^2 d+b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right )^2 \sqrt {b c-a d} f}-\frac {b \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))} \]

[Out]

-I*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))*(c-I*d)^(1/2)/(a-I*b)^2/f+I*arctanh((c+d*tan(f*x+e))^(1/2)/(c
+I*d)^(1/2))*(c+I*d)^(1/2)/(a+I*b)^2/f-(-3*a^2*d+4*a*b*c+b^2*d)*arctanh(b^(1/2)*(c+d*tan(f*x+e))^(1/2)/(-a*d+b
*c)^(1/2))*b^(1/2)/(a^2+b^2)^2/f/(-a*d+b*c)^(1/2)-b*(c+d*tan(f*x+e))^(1/2)/(a^2+b^2)/f/(a+b*tan(f*x+e))

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Rubi [A]
time = 0.54, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {3649, 3734, 3620, 3618, 65, 214, 3715} \begin {gather*} -\frac {b \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}-\frac {\sqrt {b} \left (-3 a^2 d+4 a b c+b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{f \left (a^2+b^2\right )^2 \sqrt {b c-a d}}-\frac {i \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (a-i b)^2}+\frac {i \sqrt {c+i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (a+i b)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((a - I*b)^2*f) + (I*Sqrt[c + I*d]*ArcTan
h[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((a + I*b)^2*f) - (Sqrt[b]*(4*a*b*c - 3*a^2*d + b^2*d)*ArcTanh[(Sqr
t[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/((a^2 + b^2)^2*Sqrt[b*c - a*d]*f) - (b*Sqrt[c + d*Tan[e + f*x
]])/((a^2 + b^2)*f*(a + b*Tan[e + f*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 214

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

Rule 3618

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c*(
d/f), Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &&
NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]

Rule 3620

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

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*} \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+b \tan (e+f x))^2} \, dx &=-\frac {b \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}-\frac {\int \frac {\frac {1}{2} (-2 a c-b d)+(b c-a d) \tan (e+f x)+\frac {1}{2} b d \tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{a^2+b^2}\\ &=-\frac {b \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}-\frac {\int \frac {-a^2 c+b^2 c-2 a b d+\left (2 a b c-a^2 d+b^2 d\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{\left (a^2+b^2\right )^2}+\frac {\left (b \left (4 a b c-3 a^2 d+b^2 d\right )\right ) \int \frac {1+\tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{2 \left (a^2+b^2\right )^2}\\ &=-\frac {b \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}+\frac {(c-i d) \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (a-i b)^2}+\frac {(c+i d) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (a+i b)^2}+\frac {\left (b \left (4 a b c-3 a^2 d+b^2 d\right )\right ) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 \left (a^2+b^2\right )^2 f}\\ &=-\frac {b \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}-\frac {(i c-d) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 (a+i b)^2 f}+\frac {(i c+d) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 (a-i b)^2 f}+\frac {\left (b \left (4 a b c-3 a^2 d+b^2 d\right )\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{\left (a^2+b^2\right )^2 d f}\\ &=-\frac {\sqrt {b} \left (4 a b c-3 a^2 d+b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right )^2 \sqrt {b c-a d} f}-\frac {b \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}-\frac {(c-i d) \text {Subst}\left (\int \frac {1}{-1-\frac {i c}{d}+\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{(a-i b)^2 d f}-\frac {(c+i d) \text {Subst}\left (\int \frac {1}{-1+\frac {i c}{d}-\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{(a+i b)^2 d f}\\ &=-\frac {i \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(a-i b)^2 f}+\frac {i \sqrt {c+i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(a+i b)^2 f}-\frac {\sqrt {b} \left (4 a b c-3 a^2 d+b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right )^2 \sqrt {b c-a d} f}-\frac {b \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}\\ \end {align*}

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Mathematica [A]
time = 2.08, size = 276, normalized size = 1.19 \begin {gather*} -\frac {\frac {i \left ((a+i b)^2 \sqrt {c-i d} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )+(a-i b)^2 \sqrt {c+i d} (-b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )\right )}{a^2+b^2}+\frac {\sqrt {b} \sqrt {b c-a d} \left (4 a b c-3 a^2 d+b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{a^2+b^2}-b d \sqrt {c+d \tan (e+f x)}+\frac {b^2 (c+d \tan (e+f x))^{3/2}}{a+b \tan (e+f x)}}{\left (a^2+b^2\right ) (b c-a d) f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((I*((a + I*b)^2*Sqrt[c - I*d]*(b*c - a*d)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] + (a - I*b)^2*Sqr
t[c + I*d]*(-(b*c) + a*d)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]]))/(a^2 + b^2) + (Sqrt[b]*Sqrt[b*c -
a*d]*(4*a*b*c - 3*a^2*d + b^2*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(a^2 + b^2) - b*
d*Sqrt[c + d*Tan[e + f*x]] + (b^2*(c + d*Tan[e + f*x])^(3/2))/(a + b*Tan[e + f*x]))/((a^2 + b^2)*(b*c - a*d)*f
))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(991\) vs. \(2(199)=398\).
time = 0.52, size = 992, normalized size = 4.29

method	result	size
derivativedivides	\(\frac {2 d^{3} \left (\frac {\frac {-\frac {\left (-\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2}+\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2} c +2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a b d -\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2} c \right ) \ln \left (\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}-d \tan \left (f x +e \right )-c -\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (-4 \sqrt {c^{2}+d^{2}}\, a b d +\frac {\left (-\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2}+\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2} c +2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a b d -\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2} c \right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}-2 \sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{4 d}+\frac {\frac {\left (-\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2}+\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2} c +2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a b d -\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2} c \right ) \ln \left (d \tan \left (f x +e \right )+c +\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (4 \sqrt {c^{2}+d^{2}}\, a b d -\frac {\left (-\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2}+\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2} c +2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a b d -\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2} c \right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{4 d}}{d^{3} \left (a^{2}+b^{2}\right )^{2}}-\frac {b \left (\frac {\left (\frac {1}{2} a^{2} d +\frac {1}{2} b^{2} d \right ) \sqrt {c +d \tan \left (f x +e \right )}}{\left (c +d \tan \left (f x +e \right )\right ) b +a d -b c}+\frac {\left (3 a^{2} d -4 a b c -b^{2} d \right ) \arctan \left (\frac {b \sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{d^{3} \left (a^{2}+b^{2}\right )^{2}}\right )}{f}\)	\(992\)
default	\(\frac {2 d^{3} \left (\frac {\frac {-\frac {\left (-\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2}+\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2} c +2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a b d -\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2} c \right ) \ln \left (\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}-d \tan \left (f x +e \right )-c -\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (-4 \sqrt {c^{2}+d^{2}}\, a b d +\frac {\left (-\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2}+\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2} c +2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a b d -\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2} c \right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}-2 \sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{4 d}+\frac {\frac {\left (-\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2}+\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2} c +2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a b d -\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2} c \right ) \ln \left (d \tan \left (f x +e \right )+c +\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (4 \sqrt {c^{2}+d^{2}}\, a b d -\frac {\left (-\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2}+\sqrt {c^{2}+d^{2}}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a^{2} c +2 \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a b d -\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b^{2} c \right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{4 d}}{d^{3} \left (a^{2}+b^{2}\right )^{2}}-\frac {b \left (\frac {\left (\frac {1}{2} a^{2} d +\frac {1}{2} b^{2} d \right ) \sqrt {c +d \tan \left (f x +e \right )}}{\left (c +d \tan \left (f x +e \right )\right ) b +a d -b c}+\frac {\left (3 a^{2} d -4 a b c -b^{2} d \right ) \arctan \left (\frac {b \sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{d^{3} \left (a^{2}+b^{2}\right )^{2}}\right )}{f}\)	\(992\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more detail

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 16347 vs. \(2 (196) = 392\).
time = 240.14, size = 32681, normalized size = 141.48 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(4*sqrt(2)*((a^14 + 5*a^12*b^2 + 9*a^10*b^4 + 5*a^8*b^6 - 5*a^6*b^8 - 9*a^4*b^10 - 5*a^2*b^12 - b^14)*f^5
*cos(f*x + e)^2 + 2*(a^13*b + 6*a^11*b^3 + 15*a^9*b^5 + 20*a^7*b^7 + 15*a^5*b^9 + 6*a^3*b^11 + a*b^13)*f^5*cos
(f*x + e)*sin(f*x + e) + (a^12*b^2 + 6*a^10*b^4 + 15*a^8*b^6 + 20*a^6*b^8 + 15*a^4*b^10 + 6*a^2*b^12 + b^14)*f
^5)*sqrt((((a^12 - 2*a^10*b^2 - 17*a^8*b^4 - 28*a^6*b^6 - 17*a^4*b^8 - 2*a^2*b^10 + b^12)*c + 4*(a^11*b + 3*a^
9*b^3 + 2*a^7*b^5 - 2*a^5*b^7 - 3*a^3*b^9 - a*b^11)*d)*f^2*sqrt((c^2 + d^2)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*
a^2*b^6 + b^8)*f^4)) + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*c^2 + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*
a^2*b^6 + b^8)*d^2)/(16*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*c^2 - 8*(a^7*b - 7*a^5*b^3 + 7*a^3*b^5 - a*b^7)*c*d +
(a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*d^2))*sqrt((16*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*c^2 - 8*(a^7
*b - 7*a^5*b^3 + 7*a^3*b^5 - a*b^7)*c*d + (a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*d^2)/((a^16 + 8*a
^14*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 + b^16)*f^4))*((c^2
+ d^2)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*f^4))^(3/4)*arctan(((4*(a^15*b + 5*a^13*b^3 + 9*a^11*b
^5 + 5*a^9*b^7 - 5*a^7*b^9 - 9*a^5*b^11 - 5*a^3*b^13 - a*b^15)*c^3 - (a^16 - 20*a^12*b^4 - 64*a^10*b^6 - 90*a^
8*b^8 - 64*a^6*b^10 - 20*a^4*b^12 + b^16)*c^2*d + 4*(a^15*b + 5*a^13*b^3 + 9*a^11*b^5 + 5*a^9*b^7 - 5*a^7*b^9
- 9*a^5*b^11 - 5*a^3*b^13 - a*b^15)*c*d^2 - (a^16 - 20*a^12*b^4 - 64*a^10*b^6 - 90*a^8*b^8 - 64*a^6*b^10 - 20*
a^4*b^12 + b^16)*d^3)*f^4*sqrt((16*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*c^2 - 8*(a^7*b - 7*a^5*b^3 + 7*a^3*b^5 - a*
b^7)*c*d + (a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*d^2)/((a^16 + 8*a^14*b^2 + 28*a^12*b^4 + 56*a^10
*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 + b^16)*f^4))*sqrt((c^2 + d^2)/((a^8 + 4*a^6*b^2 +
6*a^4*b^4 + 4*a^2*b^6 + b^8)*f^4)) + (4*(a^11*b + 3*a^9*b^3 + 2*a^7*b^5 - 2*a^5*b^7 - 3*a^3*b^9 - a*b^11)*c^4
- (a^12 - 2*a^10*b^2 - 17*a^8*b^4 - 28*a^6*b^6 - 17*a^4*b^8 - 2*a^2*b^10 + b^12)*c^3*d + 4*(a^11*b + 3*a^9*b^3
+ 2*a^7*b^5 - 2*a^5*b^7 - 3*a^3*b^9 - a*b^11)*c^2*d^2 - (a^12 - 2*a^10*b^2 - 17*a^8*b^4 - 28*a^6*b^6 - 17*a^4
*b^8 - 2*a^2*b^10 + b^12)*c*d^3)*f^2*sqrt((16*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*c^2 - 8*(a^7*b - 7*a^5*b^3 + 7*a
^3*b^5 - a*b^7)*c*d + (a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*d^2)/((a^16 + 8*a^14*b^2 + 28*a^12*b^
4 + 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 + b^16)*f^4)) + sqrt(2)*(2*(4*(a^20*b^2
+ 7*a^18*b^4 + 20*a^16*b^6 + 28*a^14*b^8 + 14*a^12*b^10 - 14*a^10*b^12 - 28*a^8*b^14 - 20*a^6*b^16 - 7*a^4*b^1
8 - a^2*b^20)*c - (a^21*b + 2*a^19*b^3 - 19*a^17*b^5 - 104*a^15*b^7 - 238*a^13*b^9 - 308*a^11*b^11 - 238*a^9*b
^13 - 104*a^7*b^15 - 19*a^5*b^17 + 2*a^3*b^19 + a*b^21)*d)*f^7*sqrt((16*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*c^2 -
8*(a^7*b - 7*a^5*b^3 + 7*a^3*b^5 - a*b^7)*c*d + (a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*d^2)/((a^16
+ 8*a^14*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 + b^16)*f^4))*
sqrt((c^2 + d^2)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*f^4)) + (8*(a^16*b^2 + 5*a^14*b^4 + 9*a^12*b
^6 + 5*a^10*b^8 - 5*a^8*b^10 - 9*a^6*b^12 - 5*a^4*b^14 - a^2*b^16)*c^2 - 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)*c*d + (a^18 - a^16*b^2 - 20
*a^14*b^4 - 44*a^12*b^6 - 26*a^10*b^8 + 26*a^8*b^10 + 44*a^6*b^12 + 20*a^4*b^14 + a^2*b^16 - b^18)*d^2)*f^5*sq
rt((16*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*c^2 - 8*(a^7*b - 7*a^5*b^3 + 7*a^3*b^5 - a*b^7)*c*d + (a^8 - 12*a^6*b^2
+ 38*a^4*b^4 - 12*a^2*b^6 + b^8)*d^2)/((a^16 + 8*a^14*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b
^10 + 28*a^4*b^12 + 8*a^2*b^14 + b^16)*f^4)))*sqrt((((a^12 - 2*a^10*b^2 - 17*a^8*b^4 - 28*a^6*b^6 - 17*a^4*b^8
- 2*a^2*b^10 + b^12)*c + 4*(a^11*b + 3*a^9*b^3 + 2*a^7*b^5 - 2*a^5*b^7 - 3*a^3*b^9 - a*b^11)*d)*f^2*sqrt((c^2
+ d^2)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*f^4)) + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^
8)*c^2 + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^2)/(16*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*c^2 - 8*(a^7
*b - 7*a^5*b^3 + 7*a^3*b^5 - a*b^7)*c*d + (a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*d^2))*sqrt((c*cos
(f*x + e) + d*sin(f*x + e))/cos(f*x + e))*((c^2 + d^2)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*f^4))^
(3/4) + sqrt(2)*(2*(a^17*b + 8*a^15*b^3 + 28*a^13*b^5 + 56*a^11*b^7 + 70*a^9*b^9 + 56*a^7*b^11 + 28*a^5*b^13 +
8*a^3*b^15 + a*b^17)*f^7*sqrt((16*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*c^2 - 8*(a^7*b - 7*a^5*b^3 + 7*a^3*b^5 - a*
b^7)*c*d + (a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*d^2)/((a^16 + 8*a^14*b^2 + 28*a^12*b^4 + 56*a^10
*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 + b^16)*f^4))*sqrt((c^2 + d^2)/((a^8 + 4*a^6*b^2 +
6*a^4*b^4 + 4*a^2*b^6 + b^8)*f^4)) + (2*(a^13*b + 6*a^11*b^3 + 15*a^9*b^5 + 20*a^7*b^7 + 15*a^5*b^9 + 6*a^3*b^
11 + a*b^13)*c - (a^14 + 5*a^12*b^2 + 9*a^10*b^...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c + d \tan {\left (e + f x \right )}}}{\left (a + b \tan {\left (e + f x \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done

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Mupad [B]
time = 11.43, size = 2500, normalized size = 10.82 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan((((-b*(a*d - b*c))^(1/2)*((16*(c + d*tan(e + f*x))^(1/2)*(3*b^9*d^12 - 3*a^2*b^7*d^12 + 17*a^4*b^5*d^12
- 9*a^6*b^3*d^12 + 3*b^9*c^2*d^10 + 2*b^9*c^4*d^8 - 8*a*b^8*c^3*d^9 - 56*a^3*b^6*c*d^11 + 60*a^5*b^4*c*d^11 +
63*a^2*b^7*c^2*d^10 - 12*a^2*b^7*c^4*d^8 + 96*a^3*b^6*c^3*d^9 - 123*a^4*b^5*c^2*d^10 + 18*a^4*b^5*c^4*d^8 - 24
*a^5*b^4*c^3*d^9 + 9*a^6*b^3*c^2*d^10 + 12*a*b^8*c*d^11))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 +
4*a^6*b^2*f^4) - (((8*(20*b^11*c*d^11*f^2 - 52*a*b^10*d^12*f^2 + 128*a^3*b^8*d^12*f^2 + 24*a^5*b^6*d^12*f^2 -
160*a^7*b^4*d^12*f^2 - 4*a^9*b^2*d^12*f^2 + 20*b^11*c^3*d^9*f^2 - 256*a^2*b^9*c^3*d^9*f^2 - 128*a^3*b^8*c^4*d
^8*f^2 + 72*a^4*b^7*c^3*d^9*f^2 - 168*a^5*b^6*c^2*d^10*f^2 - 192*a^5*b^6*c^4*d^8*f^2 + 352*a^6*b^5*c^3*d^9*f^2
- 160*a^7*b^4*c^2*d^10*f^2 + 4*a^8*b^3*c^3*d^9*f^2 - 4*a^9*b^2*c^2*d^10*f^2 + 12*a*b^10*c^2*d^10*f^2 + 64*a*b
^10*c^4*d^8*f^2 - 256*a^2*b^9*c*d^11*f^2 + 72*a^4*b^7*c*d^11*f^2 + 352*a^6*b^5*c*d^11*f^2 + 4*a^8*b^3*c*d^11*f
^2))/(a^8*f^5 + b^8*f^5 + 4*a^2*b^6*f^5 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5) + ((-b*(a*d - b*c))^(1/2)*((16*(c + d
*tan(e + f*x))^(1/2)*(68*a*b^12*d^11*f^2 - 8*b^13*c*d^10*f^2 + 20*a^3*b^10*d^11*f^2 - 88*a^5*b^8*d^11*f^2 + 40
*a^7*b^6*d^11*f^2 + 84*a^9*b^4*d^11*f^2 + 4*a^11*b^2*d^11*f^2 - 20*b^13*c^3*d^8*f^2 + 116*a^2*b^11*c^3*d^8*f^2
+ 204*a^3*b^10*c^2*d^9*f^2 + 216*a^4*b^9*c^3*d^8*f^2 + 168*a^5*b^8*c^2*d^9*f^2 + 8*a^6*b^7*c^3*d^8*f^2 + 184*
a^7*b^6*c^2*d^9*f^2 - 68*a^8*b^5*c^3*d^8*f^2 + 100*a^9*b^4*c^2*d^9*f^2 + 4*a^10*b^3*c^3*d^8*f^2 - 4*a^11*b^2*c
^2*d^9*f^2 + 116*a*b^12*c^2*d^9*f^2 + 104*a^2*b^11*c*d^10*f^2 + 48*a^4*b^9*c*d^10*f^2 - 304*a^6*b^7*c*d^10*f^2
- 296*a^8*b^5*c*d^10*f^2 - 56*a^10*b^3*c*d^10*f^2))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^
6*b^2*f^4) + ((-b*(a*d - b*c))^(1/2)*((8*(32*b^15*d^11*f^4 + 96*a^2*b^13*d^11*f^4 - 320*a^6*b^9*d^11*f^4 - 480
*a^8*b^7*d^11*f^4 - 288*a^10*b^5*d^11*f^4 - 64*a^12*b^3*d^11*f^4 + 32*b^15*c^2*d^9*f^4 + 96*a^2*b^13*c^2*d^9*f
^4 + 320*a^3*b^12*c^3*d^8*f^4 + 640*a^5*b^10*c^3*d^8*f^4 - 320*a^6*b^9*c^2*d^9*f^4 + 640*a^7*b^8*c^3*d^8*f^4 -
480*a^8*b^7*c^2*d^9*f^4 + 320*a^9*b^6*c^3*d^8*f^4 - 288*a^10*b^5*c^2*d^9*f^4 + 64*a^11*b^4*c^3*d^8*f^4 - 64*a
^12*b^3*c^2*d^9*f^4 + 64*a*b^14*c*d^10*f^4 + 64*a*b^14*c^3*d^8*f^4 + 320*a^3*b^12*c*d^10*f^4 + 640*a^5*b^10*c*
d^10*f^4 + 640*a^7*b^8*c*d^10*f^4 + 320*a^9*b^6*c*d^10*f^4 + 64*a^11*b^4*c*d^10*f^4))/(a^8*f^5 + b^8*f^5 + 4*a
^2*b^6*f^5 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5) - (8*(-b*(a*d - b*c))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(b^2*d - 3*
a^2*d + 4*a*b*c)*(32*b^17*d^10*f^4 + 160*a^2*b^15*d^10*f^4 + 288*a^4*b^13*d^10*f^4 + 160*a^6*b^11*d^10*f^4 - 1
60*a^8*b^9*d^10*f^4 - 288*a^10*b^7*d^10*f^4 - 160*a^12*b^5*d^10*f^4 - 32*a^14*b^3*d^10*f^4 + 48*b^17*c^2*d^8*f
^4 + 272*a^2*b^15*c^2*d^8*f^4 + 624*a^4*b^13*c^2*d^8*f^4 + 720*a^6*b^11*c^2*d^8*f^4 + 400*a^8*b^9*c^2*d^8*f^4
+ 48*a^10*b^7*c^2*d^8*f^4 - 48*a^12*b^5*c^2*d^8*f^4 - 16*a^14*b^3*c^2*d^8*f^4 + 16*a*b^16*c*d^9*f^4 + 112*a^3*
b^14*c*d^9*f^4 + 336*a^5*b^12*c*d^9*f^4 + 560*a^7*b^10*c*d^9*f^4 + 560*a^9*b^8*c*d^9*f^4 + 336*a^11*b^6*c*d^9*
f^4 + 112*a^13*b^4*c*d^9*f^4 + 16*a^15*b^2*c*d^9*f^4))/((a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4
*a^6*b^2*f^4)*(a^5*d*f - b^5*c*f - a^4*b*c*f + a*b^4*d*f - 2*a^2*b^3*c*f + 2*a^3*b^2*d*f)))*(b^2*d - 3*a^2*d +
4*a*b*c))/(2*(a^5*d*f - b^5*c*f - a^4*b*c*f + a*b^4*d*f - 2*a^2*b^3*c*f + 2*a^3*b^2*d*f)))*(b^2*d - 3*a^2*d +
4*a*b*c))/(2*(a^5*d*f - b^5*c*f - a^4*b*c*f + a*b^4*d*f - 2*a^2*b^3*c*f + 2*a^3*b^2*d*f)))*(-b*(a*d - b*c))^(
1/2)*(b^2*d - 3*a^2*d + 4*a*b*c))/(2*(a^5*d*f - b^5*c*f - a^4*b*c*f + a*b^4*d*f - 2*a^2*b^3*c*f + 2*a^3*b^2*d*
f)))*(b^2*d - 3*a^2*d + 4*a*b*c)*1i)/(2*(a^5*d*f - b^5*c*f - a^4*b*c*f + a*b^4*d*f - 2*a^2*b^3*c*f + 2*a^3*b^2
*d*f)) + ((-b*(a*d - b*c))^(1/2)*((16*(c + d*tan(e + f*x))^(1/2)*(3*b^9*d^12 - 3*a^2*b^7*d^12 + 17*a^4*b^5*d^1
2 - 9*a^6*b^3*d^12 + 3*b^9*c^2*d^10 + 2*b^9*c^4*d^8 - 8*a*b^8*c^3*d^9 - 56*a^3*b^6*c*d^11 + 60*a^5*b^4*c*d^11
+ 63*a^2*b^7*c^2*d^10 - 12*a^2*b^7*c^4*d^8 + 96*a^3*b^6*c^3*d^9 - 123*a^4*b^5*c^2*d^10 + 18*a^4*b^5*c^4*d^8 -
24*a^5*b^4*c^3*d^9 + 9*a^6*b^3*c^2*d^10 + 12*a*b^8*c*d^11))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4
+ 4*a^6*b^2*f^4) + (((8*(20*b^11*c*d^11*f^2 - 52*a*b^10*d^12*f^2 + 128*a^3*b^8*d^12*f^2 + 24*a^5*b^6*d^12*f^2
- 160*a^7*b^4*d^12*f^2 - 4*a^9*b^2*d^12*f^2 + 20*b^11*c^3*d^9*f^2 - 256*a^2*b^9*c^3*d^9*f^2 - 128*a^3*b^8*c^4
*d^8*f^2 + 72*a^4*b^7*c^3*d^9*f^2 - 168*a^5*b^6*c^2*d^10*f^2 - 192*a^5*b^6*c^4*d^8*f^2 + 352*a^6*b^5*c^3*d^9*f
^2 - 160*a^7*b^4*c^2*d^10*f^2 + 4*a^8*b^3*c^3*d^9*f^2 - 4*a^9*b^2*c^2*d^10*f^2 + 12*a*b^10*c^2*d^10*f^2 + 64*a
*b^10*c^4*d^8*f^2 - 256*a^2*b^9*c*d^11*f^2 + 72*a^4*b^7*c*d^11*f^2 + 352*a^6*b^5*c*d^11*f^2 + 4*a^8*b^3*c*d^11
*f^2))/(a^8*f^5 + b^8*f^5 + 4*a^2*b^6*f^5 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5) - ((-b*(a*d - b*c))^(1/2)*((16*(c +
d*tan(e + f*x))^(1/2)*(68*a*b^12*d^11*f^2 - 8*b^13*c*d^10*f^2 + 20*a^3*b^10*d^11*f^2 - 88*a^5*b^8*d^11*f^2 +
40*a^7*b^6*d^11*f^2 + 84*a^9*b^4*d^11*f^2 + 4*a...

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