\(\int \frac {a+b \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx\) [1256]

Optimal result

Integrand size = 25, antiderivative size = 138 \[ \int \frac {a+b \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx=-\frac {(i a+b) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{3/2} f}+\frac {(i a-b) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(c+i d)^{3/2} f}+\frac {2 (b c-a d)}{\left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3610, 3620, 3618, 65, 214} \[ \int \frac {a+b \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx=-\frac {(b+i a) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (c-i d)^{3/2}}+\frac {(-b+i a) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (c+i d)^{3/2}}+\frac {2 (b c-a d)}{f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}} \]

[In]

Int[(a + b*Tan[e + f*x])/(c + d*Tan[e + f*x])^(3/2),x]

[Out]

-(((I*a + b)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(3/2)*f)) + ((I*a - b)*ArcTanh[Sqrt[c
+ d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(3/2)*f) + (2*(b*c - a*d))/((c^2 + d^2)*f*Sqrt[c + d*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 3610

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

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]

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.21 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.82 \[ \int \frac {a+b \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx=\frac {i \left (\frac {(a-i b) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {c+d \tan (e+f x)}{c-i d}\right )}{c-i d}-\frac {(a+i b) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {c+d \tan (e+f x)}{c+i d}\right )}{c+i d}\right )}{f \sqrt {c+d \tan (e+f x)}} \]

[In]

Integrate[(a + b*Tan[e + f*x])/(c + d*Tan[e + f*x])^(3/2),x]

[Out]

(I*(((a - I*b)*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c - I*d)])/(c - I*d) - ((a + I*b)*Hyperge
ometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c + I*d)])/(c + I*d)))/(f*Sqrt[c + d*Tan[e + f*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3682\) vs. \(2(118)=236\).

Time = 0.91 (sec) , antiderivative size = 3683, normalized size of antiderivative = 26.69

method	result	size
parts	\(\text {Expression too large to display}\)	\(3683\)
derivativedivides	\(\text {Expression too large to display}\)	\(7951\)
default	\(\text {Expression too large to display}\)	\(7951\)

[In]

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

[Out]

a*(-2/f*d/(c^2+d^2)/(c+d*tan(f*x+e))^(1/2)+1/4/f/d/(c^2+d^2)^2*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*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3+1/4/f*d/(c^2+d^2)^2*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*(c^2+d^2)^(1/2)+2*c)^(1/2)*c-1/4/f
/d/(c^2+d^2)^(5/2)*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*
(c^2+d^2)^(1/2)+2*c)^(1/2)*c^4+1/4/f*d^3/(c^2+d^2)^(5/2)*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*(c^2+d^2)^(1/2)+2*c)^(1/2)-1/f/d/(c^2+d^2)^(3/2)/(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))*c^3-1/f*
d/(c^2+d^2)^(3/2)/(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))*c-1/f*d/(c^2+d^2)^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))*c^2+1/f/d/(c^2+d^2)^(5/2)/(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))*c
^5-1/f*d^3/(c^2+d^2)^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))+3/f*d^3/(c^2+d^2)^(5/2)/(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))*c+4/f*d/(c^2+d^2)^(5/2)/(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))*c^3-1/4/f/d/(c^2+d^2)^2*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*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3-1/4/f*d/(c^2+d^2)^2*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*(c^2+d^2)^(1/2)+2*c)^(1/2)*c+1/4/f/d/(c^2+d^2)^(5/2)*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*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^4-1/
4/f*d^3/(c^2+d^2)^(5/2)*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*(c^2+d^2)^(1/2)+2*c)^(1/2)+1/f/d/(c^2+d^2)^(3/2)/(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))*c^3+1/f*d/(c^2+d^2)^(3/2)/(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))*
c+1/f*d/(c^2+d^2)^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))*c^2-1/f/d/(c^2+d^2)^(5/2)/(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))*c^5+1/f*d^3/(c^2+d^2)^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))-3/f*d^3/(c^2+d^2)^(5/2)/(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))*c-4/f*d/(c^2+d^2)^(5/2)/(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))*c^3)+b*(2/f*c/(c^2+d^2)/(c+
d*tan(f*x+e))^(1/2)+1/4/f/(c^2+d^2)^2*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*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2+1/4/f*d^2/(c^2+d^2)^2*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*(c^2+d^2)^(1/2)+2*c)^(1/2)-1/2/f/(c^2+d^2)^(5/2)*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*(c^2+d^2)^(1/2)+2*c)^(1/2)*
c^3-1/2/f*d^2/(c^2+d^2)^(5/2)*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*(c^2+d^2)^(1/2)+2*c)^(1/2)*c-1/f/(c^2+d^2)^(3/2)/(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))*c^2+1/f/(c^2+d^2)^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))*
c^3-1/f*d^2/(c^2+d^2)^(3/2)/(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))+1/f*d^2/(c^2+d^2)^2/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*ta
n(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))*c+2/f*d^2/(c^2+d^2)^(5/2)/(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))*c^2+2/f*d^4/(c^2+d^2)^(5/2)/(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))-1/4/f/(c^2+d^2)^2*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*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2-1/4/f*d^2/(c^2+d^2)^2*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*(c^2+d^2)^(1/2)+2*c)^(1/2
)+1/2/f/(c^2+d^2)^(5/2)*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*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3+1/2/f*d^2/(c^2+d^2)^(5/2)*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*(c^2+d^2)^(1/2)+2*c)^(1/2)*c+1/f/(c^2+d^2)^(3/2)/(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))*c^2
-1/f/(c^2+d^2)^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))*c^3+1/f*d^2/(c^2+d^2)^(3/2)/(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/f*d^2/(c^2+d^2)^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))*
c-2/f*d^2/(c^2+d^2)^(5/2)/(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))*c^2-2/f*d^4/(c^2+d^2)^(5/2)/(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)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4318 vs. \(2 (113) = 226\).

Time = 0.73 (sec) , antiderivative size = 4318, normalized size of antiderivative = 31.29 \[ \int \frac {a+b \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*(((c^2*d + d^3)*f*tan(f*x + e) + (c^3 + c*d^2)*f)*sqrt(-(6*a*b*c^2*d - 2*a*b*d^3 + (a^2 - b^2)*c^3 - 3*(a^
2 - b^2)*c*d^2 + (c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6)*f^2*sqrt(-(4*a^2*b^2*c^6 - 12*(a^3*b - a*b^3)*c^5*d + 3*(
3*a^4 - 14*a^2*b^2 + 3*b^4)*c^4*d^2 + 40*(a^3*b - a*b^3)*c^3*d^3 - 6*(a^4 - 8*a^2*b^2 + b^4)*c^2*d^4 - 12*(a^3
*b - a*b^3)*c*d^5 + (a^4 - 2*a^2*b^2 + b^4)*d^6)/((c^12 + 6*c^10*d^2 + 15*c^8*d^4 + 20*c^6*d^6 + 15*c^4*d^8 +
6*c^2*d^10 + d^12)*f^4)))/((c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6)*f^2))*log(-(2*(a^3*b + a*b^3)*c^3 - 3*(a^4 - b^
4)*c^2*d - 6*(a^3*b + a*b^3)*c*d^2 + (a^4 - b^4)*d^3)*sqrt(d*tan(f*x + e) + c) + ((a*c^8 + 2*b*c^7*d + 2*a*c^6
*d^2 + 6*b*c^5*d^3 + 6*b*c^3*d^5 - 2*a*c^2*d^6 + 2*b*c*d^7 - a*d^8)*f^3*sqrt(-(4*a^2*b^2*c^6 - 12*(a^3*b - a*b
^3)*c^5*d + 3*(3*a^4 - 14*a^2*b^2 + 3*b^4)*c^4*d^2 + 40*(a^3*b - a*b^3)*c^3*d^3 - 6*(a^4 - 8*a^2*b^2 + b^4)*c^
2*d^4 - 12*(a^3*b - a*b^3)*c*d^5 + (a^4 - 2*a^2*b^2 + b^4)*d^6)/((c^12 + 6*c^10*d^2 + 15*c^8*d^4 + 20*c^6*d^6
+ 15*c^4*d^8 + 6*c^2*d^10 + d^12)*f^4)) + (2*a*b^2*c^5 - (7*a^2*b - 3*b^3)*c^4*d + 2*(3*a^3 - 7*a*b^2)*c^3*d^2
+ 4*(4*a^2*b - b^3)*c^2*d^3 - 2*(a^3 - 4*a*b^2)*c*d^4 - (a^2*b - b^3)*d^5)*f)*sqrt(-(6*a*b*c^2*d - 2*a*b*d^3
+ (a^2 - b^2)*c^3 - 3*(a^2 - b^2)*c*d^2 + (c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6)*f^2*sqrt(-(4*a^2*b^2*c^6 - 12*(a
^3*b - a*b^3)*c^5*d + 3*(3*a^4 - 14*a^2*b^2 + 3*b^4)*c^4*d^2 + 40*(a^3*b - a*b^3)*c^3*d^3 - 6*(a^4 - 8*a^2*b^2
+ b^4)*c^2*d^4 - 12*(a^3*b - a*b^3)*c*d^5 + (a^4 - 2*a^2*b^2 + b^4)*d^6)/((c^12 + 6*c^10*d^2 + 15*c^8*d^4 + 2
0*c^6*d^6 + 15*c^4*d^8 + 6*c^2*d^10 + d^12)*f^4)))/((c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6)*f^2))) - ((c^2*d + d^3
)*f*tan(f*x + e) + (c^3 + c*d^2)*f)*sqrt(-(6*a*b*c^2*d - 2*a*b*d^3 + (a^2 - b^2)*c^3 - 3*(a^2 - b^2)*c*d^2 + (
c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6)*f^2*sqrt(-(4*a^2*b^2*c^6 - 12*(a^3*b - a*b^3)*c^5*d + 3*(3*a^4 - 14*a^2*b^2
+ 3*b^4)*c^4*d^2 + 40*(a^3*b - a*b^3)*c^3*d^3 - 6*(a^4 - 8*a^2*b^2 + b^4)*c^2*d^4 - 12*(a^3*b - a*b^3)*c*d^5
+ (a^4 - 2*a^2*b^2 + b^4)*d^6)/((c^12 + 6*c^10*d^2 + 15*c^8*d^4 + 20*c^6*d^6 + 15*c^4*d^8 + 6*c^2*d^10 + d^12)
*f^4)))/((c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6)*f^2))*log(-(2*(a^3*b + a*b^3)*c^3 - 3*(a^4 - b^4)*c^2*d - 6*(a^3*
b + a*b^3)*c*d^2 + (a^4 - b^4)*d^3)*sqrt(d*tan(f*x + e) + c) - ((a*c^8 + 2*b*c^7*d + 2*a*c^6*d^2 + 6*b*c^5*d^3
+ 6*b*c^3*d^5 - 2*a*c^2*d^6 + 2*b*c*d^7 - a*d^8)*f^3*sqrt(-(4*a^2*b^2*c^6 - 12*(a^3*b - a*b^3)*c^5*d + 3*(3*a
^4 - 14*a^2*b^2 + 3*b^4)*c^4*d^2 + 40*(a^3*b - a*b^3)*c^3*d^3 - 6*(a^4 - 8*a^2*b^2 + b^4)*c^2*d^4 - 12*(a^3*b
- a*b^3)*c*d^5 + (a^4 - 2*a^2*b^2 + b^4)*d^6)/((c^12 + 6*c^10*d^2 + 15*c^8*d^4 + 20*c^6*d^6 + 15*c^4*d^8 + 6*c
^2*d^10 + d^12)*f^4)) + (2*a*b^2*c^5 - (7*a^2*b - 3*b^3)*c^4*d + 2*(3*a^3 - 7*a*b^2)*c^3*d^2 + 4*(4*a^2*b - b^
3)*c^2*d^3 - 2*(a^3 - 4*a*b^2)*c*d^4 - (a^2*b - b^3)*d^5)*f)*sqrt(-(6*a*b*c^2*d - 2*a*b*d^3 + (a^2 - b^2)*c^3
- 3*(a^2 - b^2)*c*d^2 + (c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6)*f^2*sqrt(-(4*a^2*b^2*c^6 - 12*(a^3*b - a*b^3)*c^5*
d + 3*(3*a^4 - 14*a^2*b^2 + 3*b^4)*c^4*d^2 + 40*(a^3*b - a*b^3)*c^3*d^3 - 6*(a^4 - 8*a^2*b^2 + b^4)*c^2*d^4 -
12*(a^3*b - a*b^3)*c*d^5 + (a^4 - 2*a^2*b^2 + b^4)*d^6)/((c^12 + 6*c^10*d^2 + 15*c^8*d^4 + 20*c^6*d^6 + 15*c^4
*d^8 + 6*c^2*d^10 + d^12)*f^4)))/((c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6)*f^2))) - ((c^2*d + d^3)*f*tan(f*x + e) +
(c^3 + c*d^2)*f)*sqrt(-(6*a*b*c^2*d - 2*a*b*d^3 + (a^2 - b^2)*c^3 - 3*(a^2 - b^2)*c*d^2 - (c^6 + 3*c^4*d^2 +
3*c^2*d^4 + d^6)*f^2*sqrt(-(4*a^2*b^2*c^6 - 12*(a^3*b - a*b^3)*c^5*d + 3*(3*a^4 - 14*a^2*b^2 + 3*b^4)*c^4*d^2
+ 40*(a^3*b - a*b^3)*c^3*d^3 - 6*(a^4 - 8*a^2*b^2 + b^4)*c^2*d^4 - 12*(a^3*b - a*b^3)*c*d^5 + (a^4 - 2*a^2*b^2
+ b^4)*d^6)/((c^12 + 6*c^10*d^2 + 15*c^8*d^4 + 20*c^6*d^6 + 15*c^4*d^8 + 6*c^2*d^10 + d^12)*f^4)))/((c^6 + 3*
c^4*d^2 + 3*c^2*d^4 + d^6)*f^2))*log(-(2*(a^3*b + a*b^3)*c^3 - 3*(a^4 - b^4)*c^2*d - 6*(a^3*b + a*b^3)*c*d^2 +
(a^4 - b^4)*d^3)*sqrt(d*tan(f*x + e) + c) + ((a*c^8 + 2*b*c^7*d + 2*a*c^6*d^2 + 6*b*c^5*d^3 + 6*b*c^3*d^5 - 2
*a*c^2*d^6 + 2*b*c*d^7 - a*d^8)*f^3*sqrt(-(4*a^2*b^2*c^6 - 12*(a^3*b - a*b^3)*c^5*d + 3*(3*a^4 - 14*a^2*b^2 +
3*b^4)*c^4*d^2 + 40*(a^3*b - a*b^3)*c^3*d^3 - 6*(a^4 - 8*a^2*b^2 + b^4)*c^2*d^4 - 12*(a^3*b - a*b^3)*c*d^5 + (
a^4 - 2*a^2*b^2 + b^4)*d^6)/((c^12 + 6*c^10*d^2 + 15*c^8*d^4 + 20*c^6*d^6 + 15*c^4*d^8 + 6*c^2*d^10 + d^12)*f^
4)) - (2*a*b^2*c^5 - (7*a^2*b - 3*b^3)*c^4*d + 2*(3*a^3 - 7*a*b^2)*c^3*d^2 + 4*(4*a^2*b - b^3)*c^2*d^3 - 2*(a^
3 - 4*a*b^2)*c*d^4 - (a^2*b - b^3)*d^5)*f)*sqrt(-(6*a*b*c^2*d - 2*a*b*d^3 + (a^2 - b^2)*c^3 - 3*(a^2 - b^2)*c*
d^2 - (c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6)*f^2*sqrt(-(4*a^2*b^2*c^6 - 12*(a^3*b - a*b^3)*c^5*d + 3*(3*a^4 - 14*
a^2*b^2 + 3*b^4)*c^4*d^2 + 40*(a^3*b - a*b^3)*c^3*d^3 - 6*(a^4 - 8*a^2*b^2 + b^4)*c^2*d^4 - 12*(a^3*b - a*b^3)
*c*d^5 + (a^4 - 2*a^2*b^2 + b^4)*d^6)/((c^12 + 6*c^10*d^2 + 15*c^8*d^4 + 20*c^6*d^6 + 15*c^4*d^8 + 6*c^2*d^10
+ d^12)*f^4)))/((c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6)*f^2))) + ((c^2*d + d^3)*f*tan(f*x + e) + (c^3 + c*d^2)*f)*
sqrt(-(6*a*b*c^2*d - 2*a*b*d^3 + (a^2 - b^2)*c^3 - 3*(a^2 - b^2)*c*d^2 - (c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6)*f
^2*sqrt(-(4*a^2*b^2*c^6 - 12*(a^3*b - a*b^3)*c^5*d + 3*(3*a^4 - 14*a^2*b^2 + 3*b^4)*c^4*d^2 + 40*(a^3*b - a*b^
3)*c^3*d^3 - 6*(a^4 - 8*a^2*b^2 + b^4)*c^2*d^4 - 12*(a^3*b - a*b^3)*c*d^5 + (a^4 - 2*a^2*b^2 + b^4)*d^6)/((c^1
2 + 6*c^10*d^2 + 15*c^8*d^4 + 20*c^6*d^6 + 15*c^4*d^8 + 6*c^2*d^10 + d^12)*f^4)))/((c^6 + 3*c^4*d^2 + 3*c^2*d^
4 + d^6)*f^2))*log(-(2*(a^3*b + a*b^3)*c^3 - 3*(a^4 - b^4)*c^2*d - 6*(a^3*b + a*b^3)*c*d^2 + (a^4 - b^4)*d^3)*
sqrt(d*tan(f*x + e) + c) - ((a*c^8 + 2*b*c^7*d + 2*a*c^6*d^2 + 6*b*c^5*d^3 + 6*b*c^3*d^5 - 2*a*c^2*d^6 + 2*b*c
*d^7 - a*d^8)*f^3*sqrt(-(4*a^2*b^2*c^6 - 12*(a^3*b - a*b^3)*c^5*d + 3*(3*a^4 - 14*a^2*b^2 + 3*b^4)*c^4*d^2 + 4
0*(a^3*b - a*b^3)*c^3*d^3 - 6*(a^4 - 8*a^2*b^2 + b^4)*c^2*d^4 - 12*(a^3*b - a*b^3)*c*d^5 + (a^4 - 2*a^2*b^2 +
b^4)*d^6)/((c^12 + 6*c^10*d^2 + 15*c^8*d^4 + 20*c^6*d^6 + 15*c^4*d^8 + 6*c^2*d^10 + d^12)*f^4)) - (2*a*b^2*c^5
- (7*a^2*b - 3*b^3)*c^4*d + 2*(3*a^3 - 7*a*b^2)*c^3*d^2 + 4*(4*a^2*b - b^3)*c^2*d^3 - 2*(a^3 - 4*a*b^2)*c*d^4
- (a^2*b - b^3)*d^5)*f)*sqrt(-(6*a*b*c^2*d - 2*a*b*d^3 + (a^2 - b^2)*c^3 - 3*(a^2 - b^2)*c*d^2 - (c^6 + 3*c^4
*d^2 + 3*c^2*d^4 + d^6)*f^2*sqrt(-(4*a^2*b^2*c^6 - 12*(a^3*b - a*b^3)*c^5*d + 3*(3*a^4 - 14*a^2*b^2 + 3*b^4)*c
^4*d^2 + 40*(a^3*b - a*b^3)*c^3*d^3 - 6*(a^4 - 8*a^2*b^2 + b^4)*c^2*d^4 - 12*(a^3*b - a*b^3)*c*d^5 + (a^4 - 2*
a^2*b^2 + b^4)*d^6)/((c^12 + 6*c^10*d^2 + 15*c^8*d^4 + 20*c^6*d^6 + 15*c^4*d^8 + 6*c^2*d^10 + d^12)*f^4)))/((c
^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6)*f^2))) + 4*(b*c - a*d)*sqrt(d*tan(f*x + e) + c))/((c^2*d + d^3)*f*tan(f*x +
e) + (c^3 + c*d^2)*f)

Sympy [F]

\[ \int \frac {a+b \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx=\int \frac {a + b \tan {\left (e + f x \right )}}{\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

Integral((a + b*tan(e + f*x))/(c + d*tan(e + f*x))**(3/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a+b*tan(f*x+e))/(c+d*tan(f*x+e))^(3/2),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(d-c>0)', see `assume?` for mor
e details)Is

Giac [F(-1)]

Timed out. \[ \int \frac {a+b \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 12.27 (sec) , antiderivative size = 5737, normalized size of antiderivative = 41.57 \[ \int \frac {a+b \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(log(- (((c + d*tan(e + f*x))^(1/2)*(16*b^2*d^10*f^3 + 32*b^2*c^2*d^8*f^3 - 32*b^2*c^6*d^4*f^3 - 16*b^2*c^8*d^
2*f^3) + ((((96*b^4*c^2*d^4*f^4 - 16*b^4*d^6*f^4 - 144*b^4*c^4*d^2*f^4)^(1/2) + 4*b^2*c^3*f^2 - 12*b^2*c*d^2*f
^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2)*(32*b*d^12*f^4 + ((((96*b^4*c^2*d^4*f^4 - 16*b^
4*d^6*f^4 - 144*b^4*c^4*d^2*f^4)^(1/2) + 4*b^2*c^3*f^2 - 12*b^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4
+ 3*c^4*d^2*f^4))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d^10*f^5 + 640*c^5*d^8*f^5 + 640*c
^7*d^6*f^5 + 320*c^9*d^4*f^5 + 64*c^11*d^2*f^5))/4 + 96*b*c^2*d^10*f^4 + 64*b*c^4*d^8*f^4 - 64*b*c^6*d^6*f^4 -
96*b*c^8*d^4*f^4 - 32*b*c^10*d^2*f^4))/4)*(((96*b^4*c^2*d^4*f^4 - 16*b^4*d^6*f^4 - 144*b^4*c^4*d^2*f^4)^(1/2)
+ 4*b^2*c^3*f^2 - 12*b^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2))/4 - 8*b^3*c*d
^8*f^2 - 24*b^3*c^3*d^6*f^2 - 24*b^3*c^5*d^4*f^2 - 8*b^3*c^7*d^2*f^2)*(((96*b^4*c^2*d^4*f^4 - 16*b^4*d^6*f^4 -
144*b^4*c^4*d^2*f^4)^(1/2) + 4*b^2*c^3*f^2 - 12*b^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2
*f^4))^(1/2))/4 + (log(- (((c + d*tan(e + f*x))^(1/2)*(16*b^2*d^10*f^3 + 32*b^2*c^2*d^8*f^3 - 32*b^2*c^6*d^4*f
^3 - 16*b^2*c^8*d^2*f^3) + ((-((96*b^4*c^2*d^4*f^4 - 16*b^4*d^6*f^4 - 144*b^4*c^4*d^2*f^4)^(1/2) - 4*b^2*c^3*f
^2 + 12*b^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2)*(32*b*d^12*f^4 + ((-((96*b^4
*c^2*d^4*f^4 - 16*b^4*d^6*f^4 - 144*b^4*c^4*d^2*f^4)^(1/2) - 4*b^2*c^3*f^2 + 12*b^2*c*d^2*f^2)/(c^6*f^4 + d^6*
f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d^10*f^5 + 640
*c^5*d^8*f^5 + 640*c^7*d^6*f^5 + 320*c^9*d^4*f^5 + 64*c^11*d^2*f^5))/4 + 96*b*c^2*d^10*f^4 + 64*b*c^4*d^8*f^4
- 64*b*c^6*d^6*f^4 - 96*b*c^8*d^4*f^4 - 32*b*c^10*d^2*f^4))/4)*(-((96*b^4*c^2*d^4*f^4 - 16*b^4*d^6*f^4 - 144*b
^4*c^4*d^2*f^4)^(1/2) - 4*b^2*c^3*f^2 + 12*b^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))
^(1/2))/4 - 8*b^3*c*d^8*f^2 - 24*b^3*c^3*d^6*f^2 - 24*b^3*c^5*d^4*f^2 - 8*b^3*c^7*d^2*f^2)*(-((96*b^4*c^2*d^4*
f^4 - 16*b^4*d^6*f^4 - 144*b^4*c^4*d^2*f^4)^(1/2) - 4*b^2*c^3*f^2 + 12*b^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c
^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2))/4 - log(((c + d*tan(e + f*x))^(1/2)*(16*b^2*d^10*f^3 + 32*b^2*c^2*d^8*f^3
- 32*b^2*c^6*d^4*f^3 - 16*b^2*c^8*d^2*f^3) + (((96*b^4*c^2*d^4*f^4 - 16*b^4*d^6*f^4 - 144*b^4*c^4*d^2*f^4)^(1/
2) + 4*b^2*c^3*f^2 - 12*b^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2)*((((
96*b^4*c^2*d^4*f^4 - 16*b^4*d^6*f^4 - 144*b^4*c^4*d^2*f^4)^(1/2) + 4*b^2*c^3*f^2 - 12*b^2*c*d^2*f^2)/(16*c^6*f
^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(64*c*d^12*f^5 + 320*c^3*
d^10*f^5 + 640*c^5*d^8*f^5 + 640*c^7*d^6*f^5 + 320*c^9*d^4*f^5 + 64*c^11*d^2*f^5) - 32*b*d^12*f^4 - 96*b*c^2*d
^10*f^4 - 64*b*c^4*d^8*f^4 + 64*b*c^6*d^6*f^4 + 96*b*c^8*d^4*f^4 + 32*b*c^10*d^2*f^4))*(((96*b^4*c^2*d^4*f^4 -
16*b^4*d^6*f^4 - 144*b^4*c^4*d^2*f^4)^(1/2) + 4*b^2*c^3*f^2 - 12*b^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48
*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2) - 8*b^3*c*d^8*f^2 - 24*b^3*c^3*d^6*f^2 - 24*b^3*c^5*d^4*f^2 - 8*b^3*c^7*
d^2*f^2)*(((96*b^4*c^2*d^4*f^4 - 16*b^4*d^6*f^4 - 144*b^4*c^4*d^2*f^4)^(1/2) + 4*b^2*c^3*f^2 - 12*b^2*c*d^2*f^
2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2) - log(((c + d*tan(e + f*x))^(1/2)*(16*b^
2*d^10*f^3 + 32*b^2*c^2*d^8*f^3 - 32*b^2*c^6*d^4*f^3 - 16*b^2*c^8*d^2*f^3) + (-((96*b^4*c^2*d^4*f^4 - 16*b^4*d
^6*f^4 - 144*b^4*c^4*d^2*f^4)^(1/2) - 4*b^2*c^3*f^2 + 12*b^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*
f^4 + 48*c^4*d^2*f^4))^(1/2)*((-((96*b^4*c^2*d^4*f^4 - 16*b^4*d^6*f^4 - 144*b^4*c^4*d^2*f^4)^(1/2) - 4*b^2*c^3
*f^2 + 12*b^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2)*(c + d*tan(e + f*x
))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d^10*f^5 + 640*c^5*d^8*f^5 + 640*c^7*d^6*f^5 + 320*c^9*d^4*f^5 + 64*c^11*d^2
*f^5) - 32*b*d^12*f^4 - 96*b*c^2*d^10*f^4 - 64*b*c^4*d^8*f^4 + 64*b*c^6*d^6*f^4 + 96*b*c^8*d^4*f^4 + 32*b*c^10
*d^2*f^4))*(-((96*b^4*c^2*d^4*f^4 - 16*b^4*d^6*f^4 - 144*b^4*c^4*d^2*f^4)^(1/2) - 4*b^2*c^3*f^2 + 12*b^2*c*d^2
*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2) - 8*b^3*c*d^8*f^2 - 24*b^3*c^3*d^6*f^
2 - 24*b^3*c^5*d^4*f^2 - 8*b^3*c^7*d^2*f^2)*(-((96*b^4*c^2*d^4*f^4 - 16*b^4*d^6*f^4 - 144*b^4*c^4*d^2*f^4)^(1/
2) - 4*b^2*c^3*f^2 + 12*b^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2) + (l
og(((((96*a^4*c^2*d^4*f^4 - 16*a^4*d^6*f^4 - 144*a^4*c^4*d^2*f^4)^(1/2) - 4*a^2*c^3*f^2 + 12*a^2*c*d^2*f^2)/(c
^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2)*((c + d*tan(e + f*x))^(1/2)*(16*a^2*d^10*f^3 + 32*a^2
*c^2*d^8*f^3 - 32*a^2*c^6*d^4*f^3 - 16*a^2*c^8*d^2*f^3) - ((((96*a^4*c^2*d^4*f^4 - 16*a^4*d^6*f^4 - 144*a^4*c^
4*d^2*f^4)^(1/2) - 4*a^2*c^3*f^2 + 12*a^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2
)*(((((96*a^4*c^2*d^4*f^4 - 16*a^4*d^6*f^4 - 144*a^4*c^4*d^2*f^4)^(1/2) - 4*a^2*c^3*f^2 + 12*a^2*c*d^2*f^2)/(c
^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d
^10*f^5 + 640*c^5*d^8*f^5 + 640*c^7*d^6*f^5 + 320*c^9*d^4*f^5 + 64*c^11*d^2*f^5))/4 + 256*a*c^3*d^9*f^4 + 384*
a*c^5*d^7*f^4 + 256*a*c^7*d^5*f^4 + 64*a*c^9*d^3*f^4 + 64*a*c*d^11*f^4))/4))/4 + 8*a^3*d^9*f^2 + 24*a^3*c^2*d^
7*f^2 + 24*a^3*c^4*d^5*f^2 + 8*a^3*c^6*d^3*f^2)*(((96*a^4*c^2*d^4*f^4 - 16*a^4*d^6*f^4 - 144*a^4*c^4*d^2*f^4)^
(1/2) - 4*a^2*c^3*f^2 + 12*a^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2))/4 + (log
(((-((96*a^4*c^2*d^4*f^4 - 16*a^4*d^6*f^4 - 144*a^4*c^4*d^2*f^4)^(1/2) + 4*a^2*c^3*f^2 - 12*a^2*c*d^2*f^2)/(c^
6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2)*((c + d*tan(e + f*x))^(1/2)*(16*a^2*d^10*f^3 + 32*a^2*
c^2*d^8*f^3 - 32*a^2*c^6*d^4*f^3 - 16*a^2*c^8*d^2*f^3) - ((-((96*a^4*c^2*d^4*f^4 - 16*a^4*d^6*f^4 - 144*a^4*c^
4*d^2*f^4)^(1/2) + 4*a^2*c^3*f^2 - 12*a^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2
)*(((-((96*a^4*c^2*d^4*f^4 - 16*a^4*d^6*f^4 - 144*a^4*c^4*d^2*f^4)^(1/2) + 4*a^2*c^3*f^2 - 12*a^2*c*d^2*f^2)/(
c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(64*c*d^12*f^5 + 320*c^3*
d^10*f^5 + 640*c^5*d^8*f^5 + 640*c^7*d^6*f^5 + 320*c^9*d^4*f^5 + 64*c^11*d^2*f^5))/4 + 256*a*c^3*d^9*f^4 + 384
*a*c^5*d^7*f^4 + 256*a*c^7*d^5*f^4 + 64*a*c^9*d^3*f^4 + 64*a*c*d^11*f^4))/4))/4 + 8*a^3*d^9*f^2 + 24*a^3*c^2*d
^7*f^2 + 24*a^3*c^4*d^5*f^2 + 8*a^3*c^6*d^3*f^2)*(-((96*a^4*c^2*d^4*f^4 - 16*a^4*d^6*f^4 - 144*a^4*c^4*d^2*f^4
)^(1/2) + 4*a^2*c^3*f^2 - 12*a^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2))/4 - lo
g(8*a^3*d^9*f^2 - (((96*a^4*c^2*d^4*f^4 - 16*a^4*d^6*f^4 - 144*a^4*c^4*d^2*f^4)^(1/2) - 4*a^2*c^3*f^2 + 12*a^2
*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2)*((c + d*tan(e + f*x))^(1/2)*(16
*a^2*d^10*f^3 + 32*a^2*c^2*d^8*f^3 - 32*a^2*c^6*d^4*f^3 - 16*a^2*c^8*d^2*f^3) + (((96*a^4*c^2*d^4*f^4 - 16*a^4
*d^6*f^4 - 144*a^4*c^4*d^2*f^4)^(1/2) - 4*a^2*c^3*f^2 + 12*a^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^
4*f^4 + 48*c^4*d^2*f^4))^(1/2)*(256*a*c^3*d^9*f^4 - (((96*a^4*c^2*d^4*f^4 - 16*a^4*d^6*f^4 - 144*a^4*c^4*d^2*f
^4)^(1/2) - 4*a^2*c^3*f^2 + 12*a^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/
2)*(c + d*tan(e + f*x))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d^10*f^5 + 640*c^5*d^8*f^5 + 640*c^7*d^6*f^5 + 320*c^9*
d^4*f^5 + 64*c^11*d^2*f^5) + 384*a*c^5*d^7*f^4 + 256*a*c^7*d^5*f^4 + 64*a*c^9*d^3*f^4 + 64*a*c*d^11*f^4)) + 24
*a^3*c^2*d^7*f^2 + 24*a^3*c^4*d^5*f^2 + 8*a^3*c^6*d^3*f^2)*(((96*a^4*c^2*d^4*f^4 - 16*a^4*d^6*f^4 - 144*a^4*c^
4*d^2*f^4)^(1/2) - 4*a^2*c^3*f^2 + 12*a^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^
4))^(1/2) - log(8*a^3*d^9*f^2 - (-((96*a^4*c^2*d^4*f^4 - 16*a^4*d^6*f^4 - 144*a^4*c^4*d^2*f^4)^(1/2) + 4*a^2*c
^3*f^2 - 12*a^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2)*((c + d*tan(e +
f*x))^(1/2)*(16*a^2*d^10*f^3 + 32*a^2*c^2*d^8*f^3 - 32*a^2*c^6*d^4*f^3 - 16*a^2*c^8*d^2*f^3) + (-((96*a^4*c^2*
d^4*f^4 - 16*a^4*d^6*f^4 - 144*a^4*c^4*d^2*f^4)^(1/2) + 4*a^2*c^3*f^2 - 12*a^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6
*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2)*(256*a*c^3*d^9*f^4 - (-((96*a^4*c^2*d^4*f^4 - 16*a^4*d^6*f^4 -
144*a^4*c^4*d^2*f^4)^(1/2) + 4*a^2*c^3*f^2 - 12*a^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*
c^4*d^2*f^4))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d^10*f^5 + 640*c^5*d^8*f^5 + 640*c^7*d
^6*f^5 + 320*c^9*d^4*f^5 + 64*c^11*d^2*f^5) + 384*a*c^5*d^7*f^4 + 256*a*c^7*d^5*f^4 + 64*a*c^9*d^3*f^4 + 64*a*
c*d^11*f^4)) + 24*a^3*c^2*d^7*f^2 + 24*a^3*c^4*d^5*f^2 + 8*a^3*c^6*d^3*f^2)*(-((96*a^4*c^2*d^4*f^4 - 16*a^4*d^
6*f^4 - 144*a^4*c^4*d^2*f^4)^(1/2) + 4*a^2*c^3*f^2 - 12*a^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f
^4 + 48*c^4*d^2*f^4))^(1/2) - (2*a*d)/(f*(c^2 + d^2)*(c + d*tan(e + f*x))^(1/2)) + (2*b*c)/(f*(c^2 + d^2)*(c +
d*tan(e + f*x))^(1/2))