\(\int \frac {(a+b \tan (e+f x))^3}{\sqrt {c+d \tan (e+f x)}} \, dx\) [1248]

Optimal result

Integrand size = 27, antiderivative size = 178 \[ \int \frac {(a+b \tan (e+f x))^3}{\sqrt {c+d \tan (e+f x)}} \, dx=\frac {(i a+b)^3 \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d} f}-\frac {(i a-b)^3 \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d} f}-\frac {4 b^2 (b c-4 a d) \sqrt {c+d \tan (e+f x)}}{3 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}{3 d f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3647, 3711, 3620, 3618, 65, 214} \[ \int \frac {(a+b \tan (e+f x))^3}{\sqrt {c+d \tan (e+f x)}} \, dx=-\frac {(-b+i a)^3 \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}+\frac {(b+i a)^3 \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}-\frac {4 b^2 (b c-4 a d) \sqrt {c+d \tan (e+f x)}}{3 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}{3 d f} \]

[In]

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

[Out]

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

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 3647

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[b^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Dist[1/(d*(m + n -
1)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(
1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2,
x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || IntegerQ[m]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0]
&& NeQ[a, 0])))

Rule 3711

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

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

Mathematica [A] (verified)

Time = 1.14 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \tan (e+f x))^3}{\sqrt {c+d \tan (e+f x)}} \, dx=\frac {2 \left (-\frac {3 i (a-i b)^3 d \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{2 \sqrt {c-i d}}+\frac {3 i (a+i b)^3 d \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{2 \sqrt {c+i d}}+\frac {2 b^2 (-b c+4 a d) \sqrt {c+d \tan (e+f x)}}{d}+b^2 (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}\right )}{3 d f} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3850\) vs. \(2(152)=304\).

Time = 1.02 (sec) , antiderivative size = 3851, normalized size of antiderivative = 21.63

method	result	size
parts	\(\text {Expression too large to display}\)	\(3851\)
derivativedivides	\(\text {Expression too large to display}\)	\(8262\)
default	\(\text {Expression too large to display}\)	\(8262\)

[In]

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

[Out]

a^3*(1/4/f/d/(c^2+d^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/(c^2+d^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)-1/4/f/d/(c^2+d^2)^(3/2)*ln(d*tan(f*x+e)+c+(c+d*ta
n(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)^(3/2)*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/d/(c^2+d^2)^(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))*c^2-1/f*d/(c^2+d^2)^(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))+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^4+3/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^2+2/f*d^3/(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))-
1/4/f/d/(c^2+d^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/(c^2+d^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)+1/4/f/d/(c^2+d^2)^(3/2)*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)^
(3/2)*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/d/(c^2+d^2)^(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))*c^2-1/f*d/(c^2+d^2)^(1/2)/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*a
rctan((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/(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^4+3/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^2+2/f*d^3/(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)))+2*b^
3/f/d^2*(1/3*(c+d*tan(f*x+e))^(3/2)-(c+d*tan(f*x+e))^(1/2)*c-d^2*(1/4/(c^2+d^2)^(1/2)*(-1/2*(2*(c^2+d^2)^(1/2)
+2*c)^(1/2)*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)-c)/(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/(c^2+d^2)^(1/2)*(1/2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*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)-c)/(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*a*b^2*(
2/d/f*(c+d*tan(f*x+e))^(1/2)-1/4/f/d/(c^2+d^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/(c^2+d^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)+1/4/f/d/(c^2+d^2)^(3/2)*l
n(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)^(3/2)*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/d/(c^2+d^2)^(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))*c^2+1/f*d/(c^2+d^2)^(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))-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^4-3/f*d/(c^2+d^2)^(3/2)/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arct
an((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^3/(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))+1/4/f/d/(c^2+d^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/(c^2+d^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)-1/4/f/d/(c^2+d^2)^(3/2)*ln(d*t
an(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)^(3/2)*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/d/(c^2+d^2)^(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))*c^2+1/f*d/(c^2+d^2)^(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))-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^4-3/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^2-2/f*d^3/(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)))+3*a^2*b/f*(1/2/(c^2+d^2)^(1/2)*(-1/2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*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)-c)/(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)))+1/2/(c^
2+d^2)^(1/2)*(1/2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*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)-c)/(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))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3928 vs. \(2 (146) = 292\).

Time = 0.51 (sec) , antiderivative size = 3928, normalized size of antiderivative = 22.07 \[ \int \frac {(a+b \tan (e+f x))^3}{\sqrt {c+d \tan (e+f x)}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/6*(3*d^2*f*sqrt(-((c^2 + d^2)*f^2*sqrt(-(4*(9*a^10*b^2 - 60*a^8*b^4 + 118*a^6*b^6 - 60*a^4*b^8 + 9*a^2*b^10
)*c^2 - 4*(3*a^11*b - 55*a^9*b^3 + 198*a^7*b^5 - 198*a^5*b^7 + 55*a^3*b^9 - 3*a*b^11)*c*d + (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)*d^2)/((c^4 + 2*c^2*d^2 + d^4)*f^4)) + (a^6 -
15*a^4*b^2 + 15*a^2*b^4 - b^6)*c + 2*(3*a^5*b - 10*a^3*b^3 + 3*a*b^5)*d)/((c^2 + d^2)*f^2))*log((2*(3*a^11*b
- a^9*b^3 - 18*a^7*b^5 - 18*a^5*b^7 - a^3*b^9 + 3*a*b^11)*c - (a^12 - 12*a^10*b^2 - 27*a^8*b^4 + 27*a^4*b^8 +
12*a^2*b^10 - b^12)*d)*sqrt(d*tan(f*x + e) + c) + (((a^3 - 3*a*b^2)*c^3 + (3*a^2*b - b^3)*c^2*d + (a^3 - 3*a*b
^2)*c*d^2 + (3*a^2*b - b^3)*d^3)*f^3*sqrt(-(4*(9*a^10*b^2 - 60*a^8*b^4 + 118*a^6*b^6 - 60*a^4*b^8 + 9*a^2*b^10
)*c^2 - 4*(3*a^11*b - 55*a^9*b^3 + 198*a^7*b^5 - 198*a^5*b^7 + 55*a^3*b^9 - 3*a*b^11)*c*d + (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)*d^2)/((c^4 + 2*c^2*d^2 + d^4)*f^4)) + (2*(9*
a^7*b^2 - 33*a^5*b^4 + 19*a^3*b^6 - 3*a*b^8)*c^2 - (9*a^8*b - 84*a^6*b^3 + 126*a^4*b^5 - 36*a^2*b^7 + b^9)*c*d
+ (a^9 - 18*a^7*b^2 + 60*a^5*b^4 - 46*a^3*b^6 + 3*a*b^8)*d^2)*f)*sqrt(-((c^2 + d^2)*f^2*sqrt(-(4*(9*a^10*b^2
- 60*a^8*b^4 + 118*a^6*b^6 - 60*a^4*b^8 + 9*a^2*b^10)*c^2 - 4*(3*a^11*b - 55*a^9*b^3 + 198*a^7*b^5 - 198*a^5*b
^7 + 55*a^3*b^9 - 3*a*b^11)*c*d + (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)*d^2)/((c^4 + 2*c^2*d^2 + d^4)*f^4)) + (a^6 - 15*a^4*b^2 + 15*a^2*b^4 - b^6)*c + 2*(3*a^5*b - 10*a^3*b^
3 + 3*a*b^5)*d)/((c^2 + d^2)*f^2))) - 3*d^2*f*sqrt(-((c^2 + d^2)*f^2*sqrt(-(4*(9*a^10*b^2 - 60*a^8*b^4 + 118*a
^6*b^6 - 60*a^4*b^8 + 9*a^2*b^10)*c^2 - 4*(3*a^11*b - 55*a^9*b^3 + 198*a^7*b^5 - 198*a^5*b^7 + 55*a^3*b^9 - 3*
a*b^11)*c*d + (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)*d^2)/((c^4 +
2*c^2*d^2 + d^4)*f^4)) + (a^6 - 15*a^4*b^2 + 15*a^2*b^4 - b^6)*c + 2*(3*a^5*b - 10*a^3*b^3 + 3*a*b^5)*d)/((c^
2 + d^2)*f^2))*log((2*(3*a^11*b - a^9*b^3 - 18*a^7*b^5 - 18*a^5*b^7 - a^3*b^9 + 3*a*b^11)*c - (a^12 - 12*a^10*
b^2 - 27*a^8*b^4 + 27*a^4*b^8 + 12*a^2*b^10 - b^12)*d)*sqrt(d*tan(f*x + e) + c) - (((a^3 - 3*a*b^2)*c^3 + (3*a
^2*b - b^3)*c^2*d + (a^3 - 3*a*b^2)*c*d^2 + (3*a^2*b - b^3)*d^3)*f^3*sqrt(-(4*(9*a^10*b^2 - 60*a^8*b^4 + 118*a
^6*b^6 - 60*a^4*b^8 + 9*a^2*b^10)*c^2 - 4*(3*a^11*b - 55*a^9*b^3 + 198*a^7*b^5 - 198*a^5*b^7 + 55*a^3*b^9 - 3*
a*b^11)*c*d + (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)*d^2)/((c^4 +
2*c^2*d^2 + d^4)*f^4)) + (2*(9*a^7*b^2 - 33*a^5*b^4 + 19*a^3*b^6 - 3*a*b^8)*c^2 - (9*a^8*b - 84*a^6*b^3 + 126
*a^4*b^5 - 36*a^2*b^7 + b^9)*c*d + (a^9 - 18*a^7*b^2 + 60*a^5*b^4 - 46*a^3*b^6 + 3*a*b^8)*d^2)*f)*sqrt(-((c^2
+ d^2)*f^2*sqrt(-(4*(9*a^10*b^2 - 60*a^8*b^4 + 118*a^6*b^6 - 60*a^4*b^8 + 9*a^2*b^10)*c^2 - 4*(3*a^11*b - 55*a
^9*b^3 + 198*a^7*b^5 - 198*a^5*b^7 + 55*a^3*b^9 - 3*a*b^11)*c*d + (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)*d^2)/((c^4 + 2*c^2*d^2 + d^4)*f^4)) + (a^6 - 15*a^4*b^2 + 15*a^2*b^4 -
b^6)*c + 2*(3*a^5*b - 10*a^3*b^3 + 3*a*b^5)*d)/((c^2 + d^2)*f^2))) - 3*d^2*f*sqrt(((c^2 + d^2)*f^2*sqrt(-(4*(
9*a^10*b^2 - 60*a^8*b^4 + 118*a^6*b^6 - 60*a^4*b^8 + 9*a^2*b^10)*c^2 - 4*(3*a^11*b - 55*a^9*b^3 + 198*a^7*b^5
- 198*a^5*b^7 + 55*a^3*b^9 - 3*a*b^11)*c*d + (a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 3
0*a^2*b^10 + b^12)*d^2)/((c^4 + 2*c^2*d^2 + d^4)*f^4)) - (a^6 - 15*a^4*b^2 + 15*a^2*b^4 - b^6)*c - 2*(3*a^5*b
- 10*a^3*b^3 + 3*a*b^5)*d)/((c^2 + d^2)*f^2))*log((2*(3*a^11*b - a^9*b^3 - 18*a^7*b^5 - 18*a^5*b^7 - a^3*b^9 +
3*a*b^11)*c - (a^12 - 12*a^10*b^2 - 27*a^8*b^4 + 27*a^4*b^8 + 12*a^2*b^10 - b^12)*d)*sqrt(d*tan(f*x + e) + c)
+ (((a^3 - 3*a*b^2)*c^3 + (3*a^2*b - b^3)*c^2*d + (a^3 - 3*a*b^2)*c*d^2 + (3*a^2*b - b^3)*d^3)*f^3*sqrt(-(4*(
9*a^10*b^2 - 60*a^8*b^4 + 118*a^6*b^6 - 60*a^4*b^8 + 9*a^2*b^10)*c^2 - 4*(3*a^11*b - 55*a^9*b^3 + 198*a^7*b^5
- 198*a^5*b^7 + 55*a^3*b^9 - 3*a*b^11)*c*d + (a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 3
0*a^2*b^10 + b^12)*d^2)/((c^4 + 2*c^2*d^2 + d^4)*f^4)) - (2*(9*a^7*b^2 - 33*a^5*b^4 + 19*a^3*b^6 - 3*a*b^8)*c^
2 - (9*a^8*b - 84*a^6*b^3 + 126*a^4*b^5 - 36*a^2*b^7 + b^9)*c*d + (a^9 - 18*a^7*b^2 + 60*a^5*b^4 - 46*a^3*b^6
+ 3*a*b^8)*d^2)*f)*sqrt(((c^2 + d^2)*f^2*sqrt(-(4*(9*a^10*b^2 - 60*a^8*b^4 + 118*a^6*b^6 - 60*a^4*b^8 + 9*a^2*
b^10)*c^2 - 4*(3*a^11*b - 55*a^9*b^3 + 198*a^7*b^5 - 198*a^5*b^7 + 55*a^3*b^9 - 3*a*b^11)*c*d + (a^12 - 30*a^1
0*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)*d^2)/((c^4 + 2*c^2*d^2 + d^4)*f^4)) - (a
^6 - 15*a^4*b^2 + 15*a^2*b^4 - b^6)*c - 2*(3*a^5*b - 10*a^3*b^3 + 3*a*b^5)*d)/((c^2 + d^2)*f^2))) + 3*d^2*f*sq
rt(((c^2 + d^2)*f^2*sqrt(-(4*(9*a^10*b^2 - 60*a^8*b^4 + 118*a^6*b^6 - 60*a^4*b^8 + 9*a^2*b^10)*c^2 - 4*(3*a^11
*b - 55*a^9*b^3 + 198*a^7*b^5 - 198*a^5*b^7 + 55*a^3*b^9 - 3*a*b^11)*c*d + (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)*d^2)/((c^4 + 2*c^2*d^2 + d^4)*f^4)) - (a^6 - 15*a^4*b^2 + 15*
a^2*b^4 - b^6)*c - 2*(3*a^5*b - 10*a^3*b^3 + 3*a*b^5)*d)/((c^2 + d^2)*f^2))*log((2*(3*a^11*b - a^9*b^3 - 18*a^
7*b^5 - 18*a^5*b^7 - a^3*b^9 + 3*a*b^11)*c - (a^12 - 12*a^10*b^2 - 27*a^8*b^4 + 27*a^4*b^8 + 12*a^2*b^10 - b^1
2)*d)*sqrt(d*tan(f*x + e) + c) - (((a^3 - 3*a*b^2)*c^3 + (3*a^2*b - b^3)*c^2*d + (a^3 - 3*a*b^2)*c*d^2 + (3*a^
2*b - b^3)*d^3)*f^3*sqrt(-(4*(9*a^10*b^2 - 60*a^8*b^4 + 118*a^6*b^6 - 60*a^4*b^8 + 9*a^2*b^10)*c^2 - 4*(3*a^11
*b - 55*a^9*b^3 + 198*a^7*b^5 - 198*a^5*b^7 + 55*a^3*b^9 - 3*a*b^11)*c*d + (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)*d^2)/((c^4 + 2*c^2*d^2 + d^4)*f^4)) - (2*(9*a^7*b^2 - 33*a^5*
b^4 + 19*a^3*b^6 - 3*a*b^8)*c^2 - (9*a^8*b - 84*a^6*b^3 + 126*a^4*b^5 - 36*a^2*b^7 + b^9)*c*d + (a^9 - 18*a^7*
b^2 + 60*a^5*b^4 - 46*a^3*b^6 + 3*a*b^8)*d^2)*f)*sqrt(((c^2 + d^2)*f^2*sqrt(-(4*(9*a^10*b^2 - 60*a^8*b^4 + 118
*a^6*b^6 - 60*a^4*b^8 + 9*a^2*b^10)*c^2 - 4*(3*a^11*b - 55*a^9*b^3 + 198*a^7*b^5 - 198*a^5*b^7 + 55*a^3*b^9 -
3*a*b^11)*c*d + (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)*d^2)/((c^4
+ 2*c^2*d^2 + d^4)*f^4)) - (a^6 - 15*a^4*b^2 + 15*a^2*b^4 - b^6)*c - 2*(3*a^5*b - 10*a^3*b^3 + 3*a*b^5)*d)/((
c^2 + d^2)*f^2))) - 4*(b^3*d*tan(f*x + e) - 2*b^3*c + 9*a*b^2*d)*sqrt(d*tan(f*x + e) + c))/(d^2*f)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 12.03 (sec) , antiderivative size = 3017, normalized size of antiderivative = 16.95 \[ \int \frac {(a+b \tan (e+f x))^3}{\sqrt {c+d \tan (e+f x)}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

atan(((((8*(4*a^3*d^3*f^2 - 12*a*b^2*d^3*f^2 + 4*b^3*c*d^2*f^2 - 12*a^2*b*c*d^2*f^2))/f^3 - 64*c*d^2*(c + d*ta
n(e + f*x))^(1/2)*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*(c*f^2*1i
- d*f^2)))^(1/2))*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*(c*f^2*1
i - d*f^2)))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(a^6*d^2 - b^6*d^2 + 15*a^2*b^4*d^2 - 15*a^4*b^2*d^2))/f^2
)*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*(c*f^2*1i - d*f^2)))^(1/2
)*1i - (((8*(4*a^3*d^3*f^2 - 12*a*b^2*d^3*f^2 + 4*b^3*c*d^2*f^2 - 12*a^2*b*c*d^2*f^2))/f^3 + 64*c*d^2*(c + d*t
an(e + f*x))^(1/2)*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*(c*f^2*1
i - d*f^2)))^(1/2))*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*(c*f^2*
1i - d*f^2)))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(a^6*d^2 - b^6*d^2 + 15*a^2*b^4*d^2 - 15*a^4*b^2*d^2))/f^
2)*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*(c*f^2*1i - d*f^2)))^(1/
2)*1i)/((((8*(4*a^3*d^3*f^2 - 12*a*b^2*d^3*f^2 + 4*b^3*c*d^2*f^2 - 12*a^2*b*c*d^2*f^2))/f^3 - 64*c*d^2*(c + d*
tan(e + f*x))^(1/2)*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*(c*f^2*
1i - d*f^2)))^(1/2))*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*(c*f^2
*1i - d*f^2)))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(a^6*d^2 - b^6*d^2 + 15*a^2*b^4*d^2 - 15*a^4*b^2*d^2))/f
^2)*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*(c*f^2*1i - d*f^2)))^(1
/2) + (((8*(4*a^3*d^3*f^2 - 12*a*b^2*d^3*f^2 + 4*b^3*c*d^2*f^2 - 12*a^2*b*c*d^2*f^2))/f^3 + 64*c*d^2*(c + d*ta
n(e + f*x))^(1/2)*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*(c*f^2*1i
- d*f^2)))^(1/2))*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*(c*f^2*1
i - d*f^2)))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(a^6*d^2 - b^6*d^2 + 15*a^2*b^4*d^2 - 15*a^4*b^2*d^2))/f^2
)*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*(c*f^2*1i - d*f^2)))^(1/2
) + (16*(3*a^8*b*d^2 - b^9*d^2 + 6*a^4*b^5*d^2 + 8*a^6*b^3*d^2))/f^3))*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i -
a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*(c*f^2*1i - d*f^2)))^(1/2)*2i + atan(((((8*(4*a^3*d^3*f^2 - 12*a*b
^2*d^3*f^2 + 4*b^3*c*d^2*f^2 - 12*a^2*b*c*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((a*b^5*6i + a^5
*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2))*((a*b^5*6i + a^5*b*6
i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2) - (16*(c + d*tan(e + f*x)
)^(1/2)*(a^6*d^2 - b^6*d^2 + 15*a^2*b^4*d^2 - 15*a^4*b^2*d^2))/f^2)*((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2
*b^4 - a^3*b^3*20i + 15*a^4*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2)*1i - (((8*(4*a^3*d^3*f^2 - 12*a*b^2*d^3*f^2 + 4
*b^3*c*d^2*f^2 - 12*a^2*b*c*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((a*b^5*6i + a^5*b*6i - a^6 +
b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2))*((a*b^5*6i + a^5*b*6i - a^6 + b^6
- 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(a^6*d
^2 - b^6*d^2 + 15*a^2*b^4*d^2 - 15*a^4*b^2*d^2))/f^2)*((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3
*20i + 15*a^4*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2)*1i)/((((8*(4*a^3*d^3*f^2 - 12*a*b^2*d^3*f^2 + 4*b^3*c*d^2*f^2
- 12*a^2*b*c*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b
^4 - a^3*b^3*20i + 15*a^4*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2))*((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 -
a^3*b^3*20i + 15*a^4*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(a^6*d^2 - b^6*d^2 +
15*a^2*b^4*d^2 - 15*a^4*b^2*d^2))/f^2)*((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*
b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2) + (((8*(4*a^3*d^3*f^2 - 12*a*b^2*d^3*f^2 + 4*b^3*c*d^2*f^2 - 12*a^2*b*c*d^2
*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i
+ 15*a^4*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2))*((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15
*a^4*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(a^6*d^2 - b^6*d^2 + 15*a^2*b^4*d^2 -
15*a^4*b^2*d^2))/f^2)*((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)/(4*(c*f^2 -
d*f^2*1i)))^(1/2) + (16*(3*a^8*b*d^2 - b^9*d^2 + 6*a^4*b^5*d^2 + 8*a^6*b^3*d^2))/f^3))*((a*b^5*6i + a^5*b*6i -
a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)/(4*(c*f^2 - d*f^2*1i)))^(1/2)*2i - ((6*b^3*c - 6*a*b^2*d)/
(d^2*f) - (4*b^3*c)/(d^2*f))*(c + d*tan(e + f*x))^(1/2) + (2*b^3*(c + d*tan(e + f*x))^(3/2))/(3*d^2*f)