\(\int (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)} \, dx\) [1229]
Optimal result
Integrand size = 27, antiderivative size = 209 \[
\int (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)} \, dx=\frac {(i a+b)^3 \sqrt {c-i d} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}-\frac {(i a-b)^3 \sqrt {c+i d} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 b \left (3 a^2-b^2\right ) \sqrt {c+d \tan (e+f x)}}{f}-\frac {4 b^2 (b c-6 a d) (c+d \tan (e+f x))^{3/2}}{15 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}{5 d f}
\]
[Out]
(I*a+b)^3*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))*(c-I*d)^(1/2)/f-(I*a-b)^3*arctanh((c+d*tan(f*x+e))^(1/
2)/(c+I*d)^(1/2))*(c+I*d)^(1/2)/f+2*b*(3*a^2-b^2)*(c+d*tan(f*x+e))^(1/2)/f-4/15*b^2*(-6*a*d+b*c)*(c+d*tan(f*x+
e))^(3/2)/d^2/f+2/5*b^2*(a+b*tan(f*x+e))*(c+d*tan(f*x+e))^(3/2)/d/f
Rubi [A] (verified)
Time = 0.82 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of
steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {3647, 3711, 3609, 3620, 3618,
65, 214} \[
\int (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)} \, dx=\frac {2 b \left (3 a^2-b^2\right ) \sqrt {c+d \tan (e+f x)}}{f}-\frac {(-b+i a)^3 \sqrt {c+i d} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {(b+i a)^3 \sqrt {c-i d} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}-\frac {4 b^2 (b c-6 a d) (c+d \tan (e+f x))^{3/2}}{15 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}{5 d f}
\]
[In]
Int[(a + b*Tan[e + f*x])^3*Sqrt[c + d*Tan[e + f*x]],x]
[Out]
((I*a + b)^3*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f - ((I*a - b)^3*Sqrt[c + I*d]*Arc
Tanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (2*b*(3*a^2 - b^2)*Sqrt[c + d*Tan[e + f*x]])/f - (4*b^2*(b*c
- 6*a*d)*(c + d*Tan[e + f*x])^(3/2))/(15*d^2*f) + (2*b^2*(a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2))/(5*
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 3609
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*
((a + b*Tan[e + f*x])^m/(f*m)), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]
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)) (c+d \tan (e+f x))^{3/2}}{5 d f}+\frac {2 \int \sqrt {c+d \tan (e+f x)} \left (\frac {1}{2} \left (5 a^3 d-2 b^2 \left (b c+\frac {3 a d}{2}\right )\right )+\frac {5}{2} b \left (3 a^2-b^2\right ) d \tan (e+f x)-b^2 (b c-6 a d) \tan ^2(e+f x)\right ) \, dx}{5 d} \\ & = -\frac {4 b^2 (b c-6 a d) (c+d \tan (e+f x))^{3/2}}{15 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}{5 d f}+\frac {2 \int \sqrt {c+d \tan (e+f x)} \left (\frac {5}{2} a \left (a^2-3 b^2\right ) d+\frac {5}{2} b \left (3 a^2-b^2\right ) d \tan (e+f x)\right ) \, dx}{5 d} \\ & = \frac {2 b \left (3 a^2-b^2\right ) \sqrt {c+d \tan (e+f x)}}{f}-\frac {4 b^2 (b c-6 a d) (c+d \tan (e+f x))^{3/2}}{15 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}{5 d f}+\frac {2 \int \frac {\frac {5}{2} d \left (a^3 c-3 a b^2 c-3 a^2 b d+b^3 d\right )+\frac {5}{2} d \left (3 a^2 b c-b^3 c+a^3 d-3 a b^2 d\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{5 d} \\ & = \frac {2 b \left (3 a^2-b^2\right ) \sqrt {c+d \tan (e+f x)}}{f}-\frac {4 b^2 (b c-6 a d) (c+d \tan (e+f x))^{3/2}}{15 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}{5 d f}+\frac {1}{2} \left ((a-i b)^3 (c-i d)\right ) \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx+\frac {1}{2} \left ((a+i b)^3 (c+i d)\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx \\ & = \frac {2 b \left (3 a^2-b^2\right ) \sqrt {c+d \tan (e+f x)}}{f}-\frac {4 b^2 (b c-6 a d) (c+d \tan (e+f x))^{3/2}}{15 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}{5 d f}-\frac {\left (i (a+i b)^3 (c+i d)\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 f}+\frac {\left ((a-i b)^3 (i c+d)\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 f} \\ & = \frac {2 b \left (3 a^2-b^2\right ) \sqrt {c+d \tan (e+f x)}}{f}-\frac {4 b^2 (b c-6 a d) (c+d \tan (e+f x))^{3/2}}{15 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}{5 d f}-\frac {\left ((a+i b)^3 (c+i d)\right ) \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 {\left ((i a+b)^3 (i c+d)\right ) \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 \sqrt {c-i d} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}-\frac {(i a-b)^3 \sqrt {c+i d} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 b \left (3 a^2-b^2\right ) \sqrt {c+d \tan (e+f x)}}{f}-\frac {4 b^2 (b c-6 a d) (c+d \tan (e+f x))^{3/2}}{15 d^2 f}+\frac {2 b^2 (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}}{5 d f} \\
\end{align*}
Mathematica [A] (verified)
Time = 2.37 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.93
\[
\int (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)} \, dx=\frac {-15 i (a-i b)^3 \sqrt {c-i d} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )+15 i (a+i b)^3 \sqrt {c+i d} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )+\frac {2 b \sqrt {c+d \tan (e+f x)} \left (15 a b c d+45 a^2 d^2-b^2 \left (2 c^2+15 d^2\right )+b d (b c+15 a d) \tan (e+f x)+3 b^2 d^2 \tan ^2(e+f x)\right )}{d^2}}{15 f}
\]
[In]
Integrate[(a + b*Tan[e + f*x])^3*Sqrt[c + d*Tan[e + f*x]],x]
[Out]
((-15*I)*(a - I*b)^3*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] + (15*I)*(a + I*b)^3*Sqrt[c
+ I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]] + (2*b*Sqrt[c + d*Tan[e + f*x]]*(15*a*b*c*d + 45*a^2*d
^2 - b^2*(2*c^2 + 15*d^2) + b*d*(b*c + 15*a*d)*Tan[e + f*x] + 3*b^2*d^2*Tan[e + f*x]^2))/d^2)/(15*f)
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(1704\) vs. \(2(181)=362\).
Time = 0.99 (sec) , antiderivative size = 1705, normalized size of antiderivative =
8.16
| | |
method | result | size |
| | |
parts |
\(\text {Expression too large to display}\) |
\(1705\) |
derivativedivides |
\(\text {Expression too large to display}\) |
\(2073\) |
default |
\(\text {Expression too large to display}\) |
\(2073\) |
| | |
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|
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[In]
int((c+d*tan(f*x+e))^(1/2)*(a+b*tan(f*x+e))^3,x,method=_RETURNVERBOSE)
[Out]
-1/4/f/d*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+d^2)^(1/2)*a^3+1/f*d/(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))*a^3+1/4/f/d*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)*a^3*c+1/4/f/d*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+d^2)^
(1/2)*a^3-1/f*d/(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))*a^3-1/4/f/d*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)*a^3*c+2*b^3/f/d^2*(1/5*(c+d*tan(f*x+e))^(5/2)-1/3*(c+d*tan(
f*x+e))^(3/2)*c-d^2*(c+d*tan(f*x+e))^(1/2)-d^2*(-1/8*(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))+1/2*(-(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))+1/8*(2*(c
^2+d^2)^(1/2)+2*c)^(1/2)*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
))+1/2*((c^2+d^2)^(1/2)-c)/(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*a*b^2*(2/3/d/f*(c+d*tan(f*x+e))^(3/2)+1/4/f/d*(2*(c^2+d^2)^(1/2)
+2*c)^(1/2)*(c^2+d^2)^(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))-1/f*d/(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*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*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))-1/4/f/d*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*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))+1/f*d/(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/f/d*(2*(
c^2+d^2)^(1/2)+2*c)^(1/2)*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)))+3*a^2*b/f*(2*(c+d*tan(f*x+e))^(1/2)-1/4*(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))+(-(c^2+d^2)^(1/2)+c)/(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))+1/4*(2*(c^2+d^2)^(1
/2)+2*c)^(1/2)*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))+((c^2+d
^2)^(1/2)-c)/(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. 3264 vs. \(2 (175) = 350\).
Time = 0.48 (sec) , antiderivative size = 3264, normalized size of antiderivative = 15.62
\[
\int (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)} \, dx=\text {Too large to display}
\]
[In]
integrate((c+d*tan(f*x+e))^(1/2)*(a+b*tan(f*x+e))^3,x, algorithm="fricas")
[Out]
1/30*(15*d^2*f*sqrt(-(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)/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)/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)*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^1
1*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)/f^4) + (2*(9*a^7*b^2 - 33*a^5*b^4 + 19*a^3*b^6 - 3*a*b^
8)*c + (3*a^8*b - 46*a^6*b^3 + 60*a^4*b^5 - 18*a^2*b^7 + b^9)*d)*f)*sqrt(-(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)/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)/f^2)) - 15*d^2*f*sqr
t(-(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)/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)/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)*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 + 25
5*a^4*b^8 - 30*a^2*b^10 + b^12)*d^2)/f^4) + (2*(9*a^7*b^2 - 33*a^5*b^4 + 19*a^3*b^6 - 3*a*b^8)*c + (3*a^8*b -
46*a^6*b^3 + 60*a^4*b^5 - 18*a^2*b^7 + b^9)*d)*f)*sqrt(-(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)/f^4) + (a^6 - 1
5*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)/f^2)) - 15*d^2*f*sqrt((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)/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)/
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)*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)/f^4) - (2*(9*a^7*b^2 - 33*a^5*b^4 + 19*a^3*b^6 - 3*a*b^8)*c + (3*a^8*b - 46*a^6*b^3 + 60*a^4
*b^5 - 18*a^2*b^7 + b^9)*d)*f)*sqrt((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)/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)/f^2)) + 15*d^2*f*sqrt((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)/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)/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)*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)/f^
4) - (2*(9*a^7*b^2 - 33*a^5*b^4 + 19*a^3*b^6 - 3*a*b^8)*c + (3*a^8*b - 46*a^6*b^3 + 60*a^4*b^5 - 18*a^2*b^7 +
b^9)*d)*f)*sqrt((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^1
1*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)/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)/f^2)) + 4*(3*b^3*d^2*tan(f*x + e)^2 - 2*b^3*c^2 + 15*a*b^2*c*d + 15*(3*a^2*b -
b^3)*d^2 + (b^3*c*d + 15*a*b^2*d^2)*tan(f*x + e))*sqrt(d*tan(f*x + e) + c))/(d^2*f)
Sympy [F]
\[
\int (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)} \, dx=\int \left (a + b \tan {\left (e + f x \right )}\right )^{3} \sqrt {c + d \tan {\left (e + f x \right )}}\, dx
\]
[In]
integrate((c+d*tan(f*x+e))**(1/2)*(a+b*tan(f*x+e))**3,x)
[Out]
Integral((a + b*tan(e + f*x))**3*sqrt(c + d*tan(e + f*x)), x)
Maxima [F]
\[
\int (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)} \, dx=\int { {\left (b \tan \left (f x + e\right ) + a\right )}^{3} \sqrt {d \tan \left (f x + e\right ) + c} \,d x }
\]
[In]
integrate((c+d*tan(f*x+e))^(1/2)*(a+b*tan(f*x+e))^3,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. \[
\int (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)} \, dx=\text {Timed out}
\]
[In]
integrate((c+d*tan(f*x+e))^(1/2)*(a+b*tan(f*x+e))^3,x, algorithm="giac")
[Out]
Timed out
Mupad [B] (verification not implemented)
Time = 25.51 (sec) , antiderivative size = 10306, normalized size of antiderivative = 49.31
\[
\int (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)} \, dx=\text {Too large to display}
\]
[In]
int((a + b*tan(e + f*x))^3*(c + d*tan(e + f*x))^(1/2),x)
[Out]
atan(((((8*(4*b^3*d^4*f^2 - 12*a^2*b*d^4*f^2 + 4*b^3*c^2*d^2*f^2 - 12*a^2*b*c^2*d^2*f^2))/f^3 - 64*c*d^2*(c +
d*tan(e + f*x))^(1/2)*(-(((8*b^6*c*f^2 - 8*a^6*c*f^2 - 120*a^2*b^4*c*f^2 + 120*a^4*b^2*c*f^2 - 160*a^3*b^3*d*f
^2 + 48*a*b^5*d*f^2 + 48*a^5*b*d*f^2)^2/64 - f^4*(a^12*c^2 + a^12*d^2 + b^12*c^2 + b^12*d^2 + 6*a^2*b^10*c^2 +
15*a^4*b^8*c^2 + 20*a^6*b^6*c^2 + 15*a^8*b^4*c^2 + 6*a^10*b^2*c^2 + 6*a^2*b^10*d^2 + 15*a^4*b^8*d^2 + 20*a^6*
b^6*d^2 + 15*a^8*b^4*d^2 + 6*a^10*b^2*d^2))^(1/2) + a^6*c*f^2 - b^6*c*f^2 + 15*a^2*b^4*c*f^2 - 15*a^4*b^2*c*f^
2 + 20*a^3*b^3*d*f^2 - 6*a*b^5*d*f^2 - 6*a^5*b*d*f^2)/(4*f^4))^(1/2))*(-(((8*b^6*c*f^2 - 8*a^6*c*f^2 - 120*a^2
*b^4*c*f^2 + 120*a^4*b^2*c*f^2 - 160*a^3*b^3*d*f^2 + 48*a*b^5*d*f^2 + 48*a^5*b*d*f^2)^2/64 - f^4*(a^12*c^2 + a
^12*d^2 + b^12*c^2 + b^12*d^2 + 6*a^2*b^10*c^2 + 15*a^4*b^8*c^2 + 20*a^6*b^6*c^2 + 15*a^8*b^4*c^2 + 6*a^10*b^2
*c^2 + 6*a^2*b^10*d^2 + 15*a^4*b^8*d^2 + 20*a^6*b^6*d^2 + 15*a^8*b^4*d^2 + 6*a^10*b^2*d^2))^(1/2) + a^6*c*f^2
- b^6*c*f^2 + 15*a^2*b^4*c*f^2 - 15*a^4*b^2*c*f^2 + 20*a^3*b^3*d*f^2 - 6*a*b^5*d*f^2 - 6*a^5*b*d*f^2)/(4*f^4))
^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(a^6*d^4 - b^6*d^4 + 15*a^2*b^4*d^4 - 15*a^4*b^2*d^4 - a^6*c^2*d^2 + b
^6*c^2*d^2 - 40*a^3*b^3*c*d^3 - 15*a^2*b^4*c^2*d^2 + 15*a^4*b^2*c^2*d^2 + 12*a*b^5*c*d^3 + 12*a^5*b*c*d^3))/f^
2)*(-(((8*b^6*c*f^2 - 8*a^6*c*f^2 - 120*a^2*b^4*c*f^2 + 120*a^4*b^2*c*f^2 - 160*a^3*b^3*d*f^2 + 48*a*b^5*d*f^2
+ 48*a^5*b*d*f^2)^2/64 - f^4*(a^12*c^2 + a^12*d^2 + b^12*c^2 + b^12*d^2 + 6*a^2*b^10*c^2 + 15*a^4*b^8*c^2 + 2
0*a^6*b^6*c^2 + 15*a^8*b^4*c^2 + 6*a^10*b^2*c^2 + 6*a^2*b^10*d^2 + 15*a^4*b^8*d^2 + 20*a^6*b^6*d^2 + 15*a^8*b^
4*d^2 + 6*a^10*b^2*d^2))^(1/2) + a^6*c*f^2 - b^6*c*f^2 + 15*a^2*b^4*c*f^2 - 15*a^4*b^2*c*f^2 + 20*a^3*b^3*d*f^
2 - 6*a*b^5*d*f^2 - 6*a^5*b*d*f^2)/(4*f^4))^(1/2)*1i - (((8*(4*b^3*d^4*f^2 - 12*a^2*b*d^4*f^2 + 4*b^3*c^2*d^2*
f^2 - 12*a^2*b*c^2*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(((8*b^6*c*f^2 - 8*a^6*c*f^2 - 120*a^
2*b^4*c*f^2 + 120*a^4*b^2*c*f^2 - 160*a^3*b^3*d*f^2 + 48*a*b^5*d*f^2 + 48*a^5*b*d*f^2)^2/64 - f^4*(a^12*c^2 +
a^12*d^2 + b^12*c^2 + b^12*d^2 + 6*a^2*b^10*c^2 + 15*a^4*b^8*c^2 + 20*a^6*b^6*c^2 + 15*a^8*b^4*c^2 + 6*a^10*b^
2*c^2 + 6*a^2*b^10*d^2 + 15*a^4*b^8*d^2 + 20*a^6*b^6*d^2 + 15*a^8*b^4*d^2 + 6*a^10*b^2*d^2))^(1/2) + a^6*c*f^2
- b^6*c*f^2 + 15*a^2*b^4*c*f^2 - 15*a^4*b^2*c*f^2 + 20*a^3*b^3*d*f^2 - 6*a*b^5*d*f^2 - 6*a^5*b*d*f^2)/(4*f^4)
)^(1/2))*(-(((8*b^6*c*f^2 - 8*a^6*c*f^2 - 120*a^2*b^4*c*f^2 + 120*a^4*b^2*c*f^2 - 160*a^3*b^3*d*f^2 + 48*a*b^5
*d*f^2 + 48*a^5*b*d*f^2)^2/64 - f^4*(a^12*c^2 + a^12*d^2 + b^12*c^2 + b^12*d^2 + 6*a^2*b^10*c^2 + 15*a^4*b^8*c
^2 + 20*a^6*b^6*c^2 + 15*a^8*b^4*c^2 + 6*a^10*b^2*c^2 + 6*a^2*b^10*d^2 + 15*a^4*b^8*d^2 + 20*a^6*b^6*d^2 + 15*
a^8*b^4*d^2 + 6*a^10*b^2*d^2))^(1/2) + a^6*c*f^2 - b^6*c*f^2 + 15*a^2*b^4*c*f^2 - 15*a^4*b^2*c*f^2 + 20*a^3*b^
3*d*f^2 - 6*a*b^5*d*f^2 - 6*a^5*b*d*f^2)/(4*f^4))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(a^6*d^4 - b^6*d^4 +
15*a^2*b^4*d^4 - 15*a^4*b^2*d^4 - a^6*c^2*d^2 + b^6*c^2*d^2 - 40*a^3*b^3*c*d^3 - 15*a^2*b^4*c^2*d^2 + 15*a^4*b
^2*c^2*d^2 + 12*a*b^5*c*d^3 + 12*a^5*b*c*d^3))/f^2)*(-(((8*b^6*c*f^2 - 8*a^6*c*f^2 - 120*a^2*b^4*c*f^2 + 120*a
^4*b^2*c*f^2 - 160*a^3*b^3*d*f^2 + 48*a*b^5*d*f^2 + 48*a^5*b*d*f^2)^2/64 - f^4*(a^12*c^2 + a^12*d^2 + b^12*c^2
+ b^12*d^2 + 6*a^2*b^10*c^2 + 15*a^4*b^8*c^2 + 20*a^6*b^6*c^2 + 15*a^8*b^4*c^2 + 6*a^10*b^2*c^2 + 6*a^2*b^10*
d^2 + 15*a^4*b^8*d^2 + 20*a^6*b^6*d^2 + 15*a^8*b^4*d^2 + 6*a^10*b^2*d^2))^(1/2) + a^6*c*f^2 - b^6*c*f^2 + 15*a
^2*b^4*c*f^2 - 15*a^4*b^2*c*f^2 + 20*a^3*b^3*d*f^2 - 6*a*b^5*d*f^2 - 6*a^5*b*d*f^2)/(4*f^4))^(1/2)*1i)/((((8*(
4*b^3*d^4*f^2 - 12*a^2*b*d^4*f^2 + 4*b^3*c^2*d^2*f^2 - 12*a^2*b*c^2*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*
x))^(1/2)*(-(((8*b^6*c*f^2 - 8*a^6*c*f^2 - 120*a^2*b^4*c*f^2 + 120*a^4*b^2*c*f^2 - 160*a^3*b^3*d*f^2 + 48*a*b^
5*d*f^2 + 48*a^5*b*d*f^2)^2/64 - f^4*(a^12*c^2 + a^12*d^2 + b^12*c^2 + b^12*d^2 + 6*a^2*b^10*c^2 + 15*a^4*b^8*
c^2 + 20*a^6*b^6*c^2 + 15*a^8*b^4*c^2 + 6*a^10*b^2*c^2 + 6*a^2*b^10*d^2 + 15*a^4*b^8*d^2 + 20*a^6*b^6*d^2 + 15
*a^8*b^4*d^2 + 6*a^10*b^2*d^2))^(1/2) + a^6*c*f^2 - b^6*c*f^2 + 15*a^2*b^4*c*f^2 - 15*a^4*b^2*c*f^2 + 20*a^3*b
^3*d*f^2 - 6*a*b^5*d*f^2 - 6*a^5*b*d*f^2)/(4*f^4))^(1/2))*(-(((8*b^6*c*f^2 - 8*a^6*c*f^2 - 120*a^2*b^4*c*f^2 +
120*a^4*b^2*c*f^2 - 160*a^3*b^3*d*f^2 + 48*a*b^5*d*f^2 + 48*a^5*b*d*f^2)^2/64 - f^4*(a^12*c^2 + a^12*d^2 + b^
12*c^2 + b^12*d^2 + 6*a^2*b^10*c^2 + 15*a^4*b^8*c^2 + 20*a^6*b^6*c^2 + 15*a^8*b^4*c^2 + 6*a^10*b^2*c^2 + 6*a^2
*b^10*d^2 + 15*a^4*b^8*d^2 + 20*a^6*b^6*d^2 + 15*a^8*b^4*d^2 + 6*a^10*b^2*d^2))^(1/2) + a^6*c*f^2 - b^6*c*f^2
+ 15*a^2*b^4*c*f^2 - 15*a^4*b^2*c*f^2 + 20*a^3*b^3*d*f^2 - 6*a*b^5*d*f^2 - 6*a^5*b*d*f^2)/(4*f^4))^(1/2) + (16
*(c + d*tan(e + f*x))^(1/2)*(a^6*d^4 - b^6*d^4 + 15*a^2*b^4*d^4 - 15*a^4*b^2*d^4 - a^6*c^2*d^2 + b^6*c^2*d^2 -
40*a^3*b^3*c*d^3 - 15*a^2*b^4*c^2*d^2 + 15*a^4*b^2*c^2*d^2 + 12*a*b^5*c*d^3 + 12*a^5*b*c*d^3))/f^2)*(-(((8*b^
6*c*f^2 - 8*a^6*c*f^2 - 120*a^2*b^4*c*f^2 + 120*a^4*b^2*c*f^2 - 160*a^3*b^3*d*f^2 + 48*a*b^5*d*f^2 + 48*a^5*b*
d*f^2)^2/64 - f^4*(a^12*c^2 + a^12*d^2 + b^12*c^2 + b^12*d^2 + 6*a^2*b^10*c^2 + 15*a^4*b^8*c^2 + 20*a^6*b^6*c^
2 + 15*a^8*b^4*c^2 + 6*a^10*b^2*c^2 + 6*a^2*b^10*d^2 + 15*a^4*b^8*d^2 + 20*a^6*b^6*d^2 + 15*a^8*b^4*d^2 + 6*a^
10*b^2*d^2))^(1/2) + a^6*c*f^2 - b^6*c*f^2 + 15*a^2*b^4*c*f^2 - 15*a^4*b^2*c*f^2 + 20*a^3*b^3*d*f^2 - 6*a*b^5*
d*f^2 - 6*a^5*b*d*f^2)/(4*f^4))^(1/2) + (((8*(4*b^3*d^4*f^2 - 12*a^2*b*d^4*f^2 + 4*b^3*c^2*d^2*f^2 - 12*a^2*b*
c^2*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(((8*b^6*c*f^2 - 8*a^6*c*f^2 - 120*a^2*b^4*c*f^2 + 1
20*a^4*b^2*c*f^2 - 160*a^3*b^3*d*f^2 + 48*a*b^5*d*f^2 + 48*a^5*b*d*f^2)^2/64 - f^4*(a^12*c^2 + a^12*d^2 + b^12
*c^2 + b^12*d^2 + 6*a^2*b^10*c^2 + 15*a^4*b^8*c^2 + 20*a^6*b^6*c^2 + 15*a^8*b^4*c^2 + 6*a^10*b^2*c^2 + 6*a^2*b
^10*d^2 + 15*a^4*b^8*d^2 + 20*a^6*b^6*d^2 + 15*a^8*b^4*d^2 + 6*a^10*b^2*d^2))^(1/2) + a^6*c*f^2 - b^6*c*f^2 +
15*a^2*b^4*c*f^2 - 15*a^4*b^2*c*f^2 + 20*a^3*b^3*d*f^2 - 6*a*b^5*d*f^2 - 6*a^5*b*d*f^2)/(4*f^4))^(1/2))*(-(((8
*b^6*c*f^2 - 8*a^6*c*f^2 - 120*a^2*b^4*c*f^2 + 120*a^4*b^2*c*f^2 - 160*a^3*b^3*d*f^2 + 48*a*b^5*d*f^2 + 48*a^5
*b*d*f^2)^2/64 - f^4*(a^12*c^2 + a^12*d^2 + b^12*c^2 + b^12*d^2 + 6*a^2*b^10*c^2 + 15*a^4*b^8*c^2 + 20*a^6*b^6
*c^2 + 15*a^8*b^4*c^2 + 6*a^10*b^2*c^2 + 6*a^2*b^10*d^2 + 15*a^4*b^8*d^2 + 20*a^6*b^6*d^2 + 15*a^8*b^4*d^2 + 6
*a^10*b^2*d^2))^(1/2) + a^6*c*f^2 - b^6*c*f^2 + 15*a^2*b^4*c*f^2 - 15*a^4*b^2*c*f^2 + 20*a^3*b^3*d*f^2 - 6*a*b
^5*d*f^2 - 6*a^5*b*d*f^2)/(4*f^4))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(a^6*d^4 - b^6*d^4 + 15*a^2*b^4*d^4
- 15*a^4*b^2*d^4 - a^6*c^2*d^2 + b^6*c^2*d^2 - 40*a^3*b^3*c*d^3 - 15*a^2*b^4*c^2*d^2 + 15*a^4*b^2*c^2*d^2 + 12
*a*b^5*c*d^3 + 12*a^5*b*c*d^3))/f^2)*(-(((8*b^6*c*f^2 - 8*a^6*c*f^2 - 120*a^2*b^4*c*f^2 + 120*a^4*b^2*c*f^2 -
160*a^3*b^3*d*f^2 + 48*a*b^5*d*f^2 + 48*a^5*b*d*f^2)^2/64 - f^4*(a^12*c^2 + a^12*d^2 + b^12*c^2 + b^12*d^2 + 6
*a^2*b^10*c^2 + 15*a^4*b^8*c^2 + 20*a^6*b^6*c^2 + 15*a^8*b^4*c^2 + 6*a^10*b^2*c^2 + 6*a^2*b^10*d^2 + 15*a^4*b^
8*d^2 + 20*a^6*b^6*d^2 + 15*a^8*b^4*d^2 + 6*a^10*b^2*d^2))^(1/2) + a^6*c*f^2 - b^6*c*f^2 + 15*a^2*b^4*c*f^2 -
15*a^4*b^2*c*f^2 + 20*a^3*b^3*d*f^2 - 6*a*b^5*d*f^2 - 6*a^5*b*d*f^2)/(4*f^4))^(1/2) + (16*(a^9*d^5 - 3*a*b^8*d
^5 - b^9*c*d^4 - 8*a^3*b^6*d^5 - 6*a^5*b^4*d^5 + a^9*c^2*d^3 - b^9*c^3*d^2 - 3*a*b^8*c^2*d^3 + 6*a^4*b^5*c*d^4
+ 8*a^6*b^3*c*d^4 + 3*a^8*b*c^3*d^2 - 8*a^3*b^6*c^2*d^3 + 6*a^4*b^5*c^3*d^2 - 6*a^5*b^4*c^2*d^3 + 8*a^6*b^3*c
^3*d^2 + 3*a^8*b*c*d^4))/f^3))*(-(((8*b^6*c*f^2 - 8*a^6*c*f^2 - 120*a^2*b^4*c*f^2 + 120*a^4*b^2*c*f^2 - 160*a^
3*b^3*d*f^2 + 48*a*b^5*d*f^2 + 48*a^5*b*d*f^2)^2/64 - f^4*(a^12*c^2 + a^12*d^2 + b^12*c^2 + b^12*d^2 + 6*a^2*b
^10*c^2 + 15*a^4*b^8*c^2 + 20*a^6*b^6*c^2 + 15*a^8*b^4*c^2 + 6*a^10*b^2*c^2 + 6*a^2*b^10*d^2 + 15*a^4*b^8*d^2
+ 20*a^6*b^6*d^2 + 15*a^8*b^4*d^2 + 6*a^10*b^2*d^2))^(1/2) + a^6*c*f^2 - b^6*c*f^2 + 15*a^2*b^4*c*f^2 - 15*a^4
*b^2*c*f^2 + 20*a^3*b^3*d*f^2 - 6*a*b^5*d*f^2 - 6*a^5*b*d*f^2)/(4*f^4))^(1/2)*2i + atan(((((8*(4*b^3*d^4*f^2 -
12*a^2*b*d^4*f^2 + 4*b^3*c^2*d^2*f^2 - 12*a^2*b*c^2*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((((8
*b^6*c*f^2 - 8*a^6*c*f^2 - 120*a^2*b^4*c*f^2 + 120*a^4*b^2*c*f^2 - 160*a^3*b^3*d*f^2 + 48*a*b^5*d*f^2 + 48*a^5
*b*d*f^2)^2/64 - f^4*(a^12*c^2 + a^12*d^2 + b^12*c^2 + b^12*d^2 + 6*a^2*b^10*c^2 + 15*a^4*b^8*c^2 + 20*a^6*b^6
*c^2 + 15*a^8*b^4*c^2 + 6*a^10*b^2*c^2 + 6*a^2*b^10*d^2 + 15*a^4*b^8*d^2 + 20*a^6*b^6*d^2 + 15*a^8*b^4*d^2 + 6
*a^10*b^2*d^2))^(1/2) - a^6*c*f^2 + b^6*c*f^2 - 15*a^2*b^4*c*f^2 + 15*a^4*b^2*c*f^2 - 20*a^3*b^3*d*f^2 + 6*a*b
^5*d*f^2 + 6*a^5*b*d*f^2)/(4*f^4))^(1/2))*((((8*b^6*c*f^2 - 8*a^6*c*f^2 - 120*a^2*b^4*c*f^2 + 120*a^4*b^2*c*f^
2 - 160*a^3*b^3*d*f^2 + 48*a*b^5*d*f^2 + 48*a^5*b*d*f^2)^2/64 - f^4*(a^12*c^2 + a^12*d^2 + b^12*c^2 + b^12*d^2
+ 6*a^2*b^10*c^2 + 15*a^4*b^8*c^2 + 20*a^6*b^6*c^2 + 15*a^8*b^4*c^2 + 6*a^10*b^2*c^2 + 6*a^2*b^10*d^2 + 15*a^
4*b^8*d^2 + 20*a^6*b^6*d^2 + 15*a^8*b^4*d^2 + 6*a^10*b^2*d^2))^(1/2) - a^6*c*f^2 + b^6*c*f^2 - 15*a^2*b^4*c*f^
2 + 15*a^4*b^2*c*f^2 - 20*a^3*b^3*d*f^2 + 6*a*b^5*d*f^2 + 6*a^5*b*d*f^2)/(4*f^4))^(1/2) + (16*(c + d*tan(e + f
*x))^(1/2)*(a^6*d^4 - b^6*d^4 + 15*a^2*b^4*d^4 - 15*a^4*b^2*d^4 - a^6*c^2*d^2 + b^6*c^2*d^2 - 40*a^3*b^3*c*d^3
- 15*a^2*b^4*c^2*d^2 + 15*a^4*b^2*c^2*d^2 + 12*a*b^5*c*d^3 + 12*a^5*b*c*d^3))/f^2)*((((8*b^6*c*f^2 - 8*a^6*c*
f^2 - 120*a^2*b^4*c*f^2 + 120*a^4*b^2*c*f^2 - 160*a^3*b^3*d*f^2 + 48*a*b^5*d*f^2 + 48*a^5*b*d*f^2)^2/64 - f^4*
(a^12*c^2 + a^12*d^2 + b^12*c^2 + b^12*d^2 + 6*a^2*b^10*c^2 + 15*a^4*b^8*c^2 + 20*a^6*b^6*c^2 + 15*a^8*b^4*c^2
+ 6*a^10*b^2*c^2 + 6*a^2*b^10*d^2 + 15*a^4*b^8*d^2 + 20*a^6*b^6*d^2 + 15*a^8*b^4*d^2 + 6*a^10*b^2*d^2))^(1/2)
- a^6*c*f^2 + b^6*c*f^2 - 15*a^2*b^4*c*f^2 + 15*a^4*b^2*c*f^2 - 20*a^3*b^3*d*f^2 + 6*a*b^5*d*f^2 + 6*a^5*b*d*
f^2)/(4*f^4))^(1/2)*1i - (((8*(4*b^3*d^4*f^2 - 12*a^2*b*d^4*f^2 + 4*b^3*c^2*d^2*f^2 - 12*a^2*b*c^2*d^2*f^2))/f
^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((((8*b^6*c*f^2 - 8*a^6*c*f^2 - 120*a^2*b^4*c*f^2 + 120*a^4*b^2*c*f^2
- 160*a^3*b^3*d*f^2 + 48*a*b^5*d*f^2 + 48*a^5*b*d*f^2)^2/64 - f^4*(a^12*c^2 + a^12*d^2 + b^12*c^2 + b^12*d^2
+ 6*a^2*b^10*c^2 + 15*a^4*b^8*c^2 + 20*a^6*b^6*c^2 + 15*a^8*b^4*c^2 + 6*a^10*b^2*c^2 + 6*a^2*b^10*d^2 + 15*a^4
*b^8*d^2 + 20*a^6*b^6*d^2 + 15*a^8*b^4*d^2 + 6*a^10*b^2*d^2))^(1/2) - a^6*c*f^2 + b^6*c*f^2 - 15*a^2*b^4*c*f^2
+ 15*a^4*b^2*c*f^2 - 20*a^3*b^3*d*f^2 + 6*a*b^5*d*f^2 + 6*a^5*b*d*f^2)/(4*f^4))^(1/2))*((((8*b^6*c*f^2 - 8*a^
6*c*f^2 - 120*a^2*b^4*c*f^2 + 120*a^4*b^2*c*f^2 - 160*a^3*b^3*d*f^2 + 48*a*b^5*d*f^2 + 48*a^5*b*d*f^2)^2/64 -
f^4*(a^12*c^2 + a^12*d^2 + b^12*c^2 + b^12*d^2 + 6*a^2*b^10*c^2 + 15*a^4*b^8*c^2 + 20*a^6*b^6*c^2 + 15*a^8*b^4
*c^2 + 6*a^10*b^2*c^2 + 6*a^2*b^10*d^2 + 15*a^4*b^8*d^2 + 20*a^6*b^6*d^2 + 15*a^8*b^4*d^2 + 6*a^10*b^2*d^2))^(
1/2) - a^6*c*f^2 + b^6*c*f^2 - 15*a^2*b^4*c*f^2 + 15*a^4*b^2*c*f^2 - 20*a^3*b^3*d*f^2 + 6*a*b^5*d*f^2 + 6*a^5*
b*d*f^2)/(4*f^4))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(a^6*d^4 - b^6*d^4 + 15*a^2*b^4*d^4 - 15*a^4*b^2*d^4
- a^6*c^2*d^2 + b^6*c^2*d^2 - 40*a^3*b^3*c*d^3 - 15*a^2*b^4*c^2*d^2 + 15*a^4*b^2*c^2*d^2 + 12*a*b^5*c*d^3 + 12
*a^5*b*c*d^3))/f^2)*((((8*b^6*c*f^2 - 8*a^6*c*f^2 - 120*a^2*b^4*c*f^2 + 120*a^4*b^2*c*f^2 - 160*a^3*b^3*d*f^2
+ 48*a*b^5*d*f^2 + 48*a^5*b*d*f^2)^2/64 - f^4*(a^12*c^2 + a^12*d^2 + b^12*c^2 + b^12*d^2 + 6*a^2*b^10*c^2 + 15
*a^4*b^8*c^2 + 20*a^6*b^6*c^2 + 15*a^8*b^4*c^2 + 6*a^10*b^2*c^2 + 6*a^2*b^10*d^2 + 15*a^4*b^8*d^2 + 20*a^6*b^6
*d^2 + 15*a^8*b^4*d^2 + 6*a^10*b^2*d^2))^(1/2) - a^6*c*f^2 + b^6*c*f^2 - 15*a^2*b^4*c*f^2 + 15*a^4*b^2*c*f^2 -
20*a^3*b^3*d*f^2 + 6*a*b^5*d*f^2 + 6*a^5*b*d*f^2)/(4*f^4))^(1/2)*1i)/((((8*(4*b^3*d^4*f^2 - 12*a^2*b*d^4*f^2
+ 4*b^3*c^2*d^2*f^2 - 12*a^2*b*c^2*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((((8*b^6*c*f^2 - 8*a^6
*c*f^2 - 120*a^2*b^4*c*f^2 + 120*a^4*b^2*c*f^2 - 160*a^3*b^3*d*f^2 + 48*a*b^5*d*f^2 + 48*a^5*b*d*f^2)^2/64 - f
^4*(a^12*c^2 + a^12*d^2 + b^12*c^2 + b^12*d^2 + 6*a^2*b^10*c^2 + 15*a^4*b^8*c^2 + 20*a^6*b^6*c^2 + 15*a^8*b^4*
c^2 + 6*a^10*b^2*c^2 + 6*a^2*b^10*d^2 + 15*a^4*b^8*d^2 + 20*a^6*b^6*d^2 + 15*a^8*b^4*d^2 + 6*a^10*b^2*d^2))^(1
/2) - a^6*c*f^2 + b^6*c*f^2 - 15*a^2*b^4*c*f^2 + 15*a^4*b^2*c*f^2 - 20*a^3*b^3*d*f^2 + 6*a*b^5*d*f^2 + 6*a^5*b
*d*f^2)/(4*f^4))^(1/2))*((((8*b^6*c*f^2 - 8*a^6*c*f^2 - 120*a^2*b^4*c*f^2 + 120*a^4*b^2*c*f^2 - 160*a^3*b^3*d*
f^2 + 48*a*b^5*d*f^2 + 48*a^5*b*d*f^2)^2/64 - f^4*(a^12*c^2 + a^12*d^2 + b^12*c^2 + b^12*d^2 + 6*a^2*b^10*c^2
+ 15*a^4*b^8*c^2 + 20*a^6*b^6*c^2 + 15*a^8*b^4*c^2 + 6*a^10*b^2*c^2 + 6*a^2*b^10*d^2 + 15*a^4*b^8*d^2 + 20*a^6
*b^6*d^2 + 15*a^8*b^4*d^2 + 6*a^10*b^2*d^2))^(1/2) - a^6*c*f^2 + b^6*c*f^2 - 15*a^2*b^4*c*f^2 + 15*a^4*b^2*c*f
^2 - 20*a^3*b^3*d*f^2 + 6*a*b^5*d*f^2 + 6*a^5*b*d*f^2)/(4*f^4))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(a^6*d^
4 - b^6*d^4 + 15*a^2*b^4*d^4 - 15*a^4*b^2*d^4 - a^6*c^2*d^2 + b^6*c^2*d^2 - 40*a^3*b^3*c*d^3 - 15*a^2*b^4*c^2*
d^2 + 15*a^4*b^2*c^2*d^2 + 12*a*b^5*c*d^3 + 12*a^5*b*c*d^3))/f^2)*((((8*b^6*c*f^2 - 8*a^6*c*f^2 - 120*a^2*b^4*
c*f^2 + 120*a^4*b^2*c*f^2 - 160*a^3*b^3*d*f^2 + 48*a*b^5*d*f^2 + 48*a^5*b*d*f^2)^2/64 - f^4*(a^12*c^2 + a^12*d
^2 + b^12*c^2 + b^12*d^2 + 6*a^2*b^10*c^2 + 15*a^4*b^8*c^2 + 20*a^6*b^6*c^2 + 15*a^8*b^4*c^2 + 6*a^10*b^2*c^2
+ 6*a^2*b^10*d^2 + 15*a^4*b^8*d^2 + 20*a^6*b^6*d^2 + 15*a^8*b^4*d^2 + 6*a^10*b^2*d^2))^(1/2) - a^6*c*f^2 + b^6
*c*f^2 - 15*a^2*b^4*c*f^2 + 15*a^4*b^2*c*f^2 - 20*a^3*b^3*d*f^2 + 6*a*b^5*d*f^2 + 6*a^5*b*d*f^2)/(4*f^4))^(1/2
) + (((8*(4*b^3*d^4*f^2 - 12*a^2*b*d^4*f^2 + 4*b^3*c^2*d^2*f^2 - 12*a^2*b*c^2*d^2*f^2))/f^3 + 64*c*d^2*(c + d*
tan(e + f*x))^(1/2)*((((8*b^6*c*f^2 - 8*a^6*c*f^2 - 120*a^2*b^4*c*f^2 + 120*a^4*b^2*c*f^2 - 160*a^3*b^3*d*f^2
+ 48*a*b^5*d*f^2 + 48*a^5*b*d*f^2)^2/64 - f^4*(a^12*c^2 + a^12*d^2 + b^12*c^2 + b^12*d^2 + 6*a^2*b^10*c^2 + 15
*a^4*b^8*c^2 + 20*a^6*b^6*c^2 + 15*a^8*b^4*c^2 + 6*a^10*b^2*c^2 + 6*a^2*b^10*d^2 + 15*a^4*b^8*d^2 + 20*a^6*b^6
*d^2 + 15*a^8*b^4*d^2 + 6*a^10*b^2*d^2))^(1/2) - a^6*c*f^2 + b^6*c*f^2 - 15*a^2*b^4*c*f^2 + 15*a^4*b^2*c*f^2 -
20*a^3*b^3*d*f^2 + 6*a*b^5*d*f^2 + 6*a^5*b*d*f^2)/(4*f^4))^(1/2))*((((8*b^6*c*f^2 - 8*a^6*c*f^2 - 120*a^2*b^4
*c*f^2 + 120*a^4*b^2*c*f^2 - 160*a^3*b^3*d*f^2 + 48*a*b^5*d*f^2 + 48*a^5*b*d*f^2)^2/64 - f^4*(a^12*c^2 + a^12*
d^2 + b^12*c^2 + b^12*d^2 + 6*a^2*b^10*c^2 + 15*a^4*b^8*c^2 + 20*a^6*b^6*c^2 + 15*a^8*b^4*c^2 + 6*a^10*b^2*c^2
+ 6*a^2*b^10*d^2 + 15*a^4*b^8*d^2 + 20*a^6*b^6*d^2 + 15*a^8*b^4*d^2 + 6*a^10*b^2*d^2))^(1/2) - a^6*c*f^2 + b^
6*c*f^2 - 15*a^2*b^4*c*f^2 + 15*a^4*b^2*c*f^2 - 20*a^3*b^3*d*f^2 + 6*a*b^5*d*f^2 + 6*a^5*b*d*f^2)/(4*f^4))^(1/
2) - (16*(c + d*tan(e + f*x))^(1/2)*(a^6*d^4 - b^6*d^4 + 15*a^2*b^4*d^4 - 15*a^4*b^2*d^4 - a^6*c^2*d^2 + b^6*c
^2*d^2 - 40*a^3*b^3*c*d^3 - 15*a^2*b^4*c^2*d^2 + 15*a^4*b^2*c^2*d^2 + 12*a*b^5*c*d^3 + 12*a^5*b*c*d^3))/f^2)*(
(((8*b^6*c*f^2 - 8*a^6*c*f^2 - 120*a^2*b^4*c*f^2 + 120*a^4*b^2*c*f^2 - 160*a^3*b^3*d*f^2 + 48*a*b^5*d*f^2 + 48
*a^5*b*d*f^2)^2/64 - f^4*(a^12*c^2 + a^12*d^2 + b^12*c^2 + b^12*d^2 + 6*a^2*b^10*c^2 + 15*a^4*b^8*c^2 + 20*a^6
*b^6*c^2 + 15*a^8*b^4*c^2 + 6*a^10*b^2*c^2 + 6*a^2*b^10*d^2 + 15*a^4*b^8*d^2 + 20*a^6*b^6*d^2 + 15*a^8*b^4*d^2
+ 6*a^10*b^2*d^2))^(1/2) - a^6*c*f^2 + b^6*c*f^2 - 15*a^2*b^4*c*f^2 + 15*a^4*b^2*c*f^2 - 20*a^3*b^3*d*f^2 + 6
*a*b^5*d*f^2 + 6*a^5*b*d*f^2)/(4*f^4))^(1/2) + (16*(a^9*d^5 - 3*a*b^8*d^5 - b^9*c*d^4 - 8*a^3*b^6*d^5 - 6*a^5*
b^4*d^5 + a^9*c^2*d^3 - b^9*c^3*d^2 - 3*a*b^8*c^2*d^3 + 6*a^4*b^5*c*d^4 + 8*a^6*b^3*c*d^4 + 3*a^8*b*c^3*d^2 -
8*a^3*b^6*c^2*d^3 + 6*a^4*b^5*c^3*d^2 - 6*a^5*b^4*c^2*d^3 + 8*a^6*b^3*c^3*d^2 + 3*a^8*b*c*d^4))/f^3))*((((8*b^
6*c*f^2 - 8*a^6*c*f^2 - 120*a^2*b^4*c*f^2 + 120*a^4*b^2*c*f^2 - 160*a^3*b^3*d*f^2 + 48*a*b^5*d*f^2 + 48*a^5*b*
d*f^2)^2/64 - f^4*(a^12*c^2 + a^12*d^2 + b^12*c^2 + b^12*d^2 + 6*a^2*b^10*c^2 + 15*a^4*b^8*c^2 + 20*a^6*b^6*c^
2 + 15*a^8*b^4*c^2 + 6*a^10*b^2*c^2 + 6*a^2*b^10*d^2 + 15*a^4*b^8*d^2 + 20*a^6*b^6*d^2 + 15*a^8*b^4*d^2 + 6*a^
10*b^2*d^2))^(1/2) - a^6*c*f^2 + b^6*c*f^2 - 15*a^2*b^4*c*f^2 + 15*a^4*b^2*c*f^2 - 20*a^3*b^3*d*f^2 + 6*a*b^5*
d*f^2 + 6*a^5*b*d*f^2)/(4*f^4))^(1/2)*2i - ((6*b^3*c - 6*a*b^2*d)/(3*d^2*f) - (4*b^3*c)/(3*d^2*f))*(c + d*tan(
e + f*x))^(3/2) - (c + d*tan(e + f*x))^(1/2)*(2*c*((6*b^3*c - 6*a*b^2*d)/(d^2*f) - (4*b^3*c)/(d^2*f)) - (6*b*(
a*d - b*c)^2)/(d^2*f) + (2*b^3*(c^2 + d^2))/(d^2*f)) + (2*b^3*(c + d*tan(e + f*x))^(5/2))/(5*d^2*f)