3.13.59 \(\int \frac {(a+b \tan (e+f x))^4}{(c+d \tan (e+f x))^{5/2}} \, dx\) [1259]
Optimal. Leaf size=290 \[ -\frac {i (a-i b)^4 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{5/2} f}+\frac {i (a+i b)^4 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(c+i d)^{5/2} f}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^2}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {4 (b c-a d)^3 \left (2 b c^2+3 a c d+5 b d^2\right )}{3 d^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}-\frac {2 b^2 \left (a d (2 b c-a d)-b^2 \left (4 c^2+3 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 d^3 \left (c^2+d^2\right ) f} \]
[Out]
-I*(a-I*b)^4*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/(c-I*d)^(5/2)/f+I*(a+I*b)^4*arctanh((c+d*tan(f*x+e)
)^(1/2)/(c+I*d)^(1/2))/(c+I*d)^(5/2)/f+4/3*(-a*d+b*c)^3*(3*a*c*d+2*b*c^2+5*b*d^2)/d^3/(c^2+d^2)^2/f/(c+d*tan(f
*x+e))^(1/2)-2/3*b^2*(a*d*(-a*d+2*b*c)-b^2*(4*c^2+3*d^2))*(c+d*tan(f*x+e))^(1/2)/d^3/(c^2+d^2)/f-2/3*(-a*d+b*c
)^2*(a+b*tan(f*x+e))^2/d/(c^2+d^2)/f/(c+d*tan(f*x+e))^(3/2)
________________________________________________________________________________________
Rubi [A]
time = 0.72, antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {3646, 3716,
3711, 3620, 3618, 65, 214} \begin {gather*} -\frac {2 b^2 \left (a d (2 b c-a d)-b^2 \left (4 c^2+3 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 d^3 f \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^2}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}+\frac {4 (b c-a d)^3 \left (3 a c d+2 b c^2+5 b d^2\right )}{3 d^3 f \left (c^2+d^2\right )^2 \sqrt {c+d \tan (e+f x)}}-\frac {i (a-i b)^4 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (c-i d)^{5/2}}+\frac {i (a+i b)^4 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (c+i d)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + b*Tan[e + f*x])^4/(c + d*Tan[e + f*x])^(5/2),x]
[Out]
((-I)*(a - I*b)^4*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(5/2)*f) + (I*(a + I*b)^4*ArcTan
h[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(5/2)*f) - (2*(b*c - a*d)^2*(a + b*Tan[e + f*x])^2)/(3*d
*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) + (4*(b*c - a*d)^3*(2*b*c^2 + 3*a*c*d + 5*b*d^2))/(3*d^3*(c^2 + d^2
)^2*f*Sqrt[c + d*Tan[e + f*x]]) - (2*b^2*(a*d*(2*b*c - a*d) - b^2*(4*c^2 + 3*d^2))*Sqrt[c + d*Tan[e + f*x]])/(
3*d^3*(c^2 + d^2)*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 3646
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]
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]
Rule 3716
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e
_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(b*c - a*d))*(c^2*C - B*c*d + A*d^2)
*((c + d*Tan[e + f*x])^(n + 1)/(d^2*f*(n + 1)*(c^2 + d^2))), x] + Dist[1/(d*(c^2 + d^2)), Int[(c + d*Tan[e + f
*x])^(n + 1)*Simp[a*d*(A*c - c*C + B*d) + b*(c^2*C - B*c*d + A*d^2) + d*(A*b*c + a*B*c - b*c*C - a*A*d + b*B*d
+ a*C*d)*Tan[e + f*x] + b*C*(c^2 + d^2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] &
& NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] && LtQ[n, -1]
Rubi steps
\begin {align*} \int \frac {(a+b \tan (e+f x))^4}{(c+d \tan (e+f x))^{5/2}} \, dx &=-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^2}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 \int \frac {(a+b \tan (e+f x)) \left (\frac {1}{2} \left (4 b^3 c^2+3 a^3 c d-11 a b^2 c d+10 a^2 b d^2\right )+\frac {3}{2} d \left (3 a^2 b c-b^3 c-a^3 d+3 a b^2 d\right ) \tan (e+f x)-\frac {1}{2} b \left (a d (2 b c-a d)-b^2 \left (4 c^2+3 d^2\right )\right ) \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx}{3 d \left (c^2+d^2\right )}\\ &=-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^2}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {4 (b c-a d)^3 \left (2 b c^2+3 a c d+5 b d^2\right )}{3 d^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {2 \int \frac {\frac {1}{2} \left (24 a^3 b c d^3-a^2 b^2 d^2 \left (17 c^2-19 d^2\right )+3 a^4 d^2 \left (c^2-d^2\right )-2 a b^3 c d \left (c^2+13 d^2\right )+2 b^4 \left (2 c^4+5 c^2 d^2\right )\right )+3 d^2 \left (a^2 c-b^2 c+2 a b d\right ) \left (2 a b c-a^2 d+b^2 d\right ) \tan (e+f x)-\frac {1}{2} b^2 \left (c^2+d^2\right ) \left (a d (2 b c-a d)-b^2 \left (4 c^2+3 d^2\right )\right ) \tan ^2(e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{3 d^2 \left (c^2+d^2\right )^2}\\ &=-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^2}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {4 (b c-a d)^3 \left (2 b c^2+3 a c d+5 b d^2\right )}{3 d^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}-\frac {2 b^2 \left (a d (2 b c-a d)-b^2 \left (4 c^2+3 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 d^3 \left (c^2+d^2\right ) f}+\frac {2 \int \frac {\frac {3}{2} d^2 \left (8 a^3 b c d-8 a b^3 c d+a^4 \left (c^2-d^2\right )-6 a^2 b^2 \left (c^2-d^2\right )+b^4 \left (c^2-d^2\right )\right )+3 d^2 \left (a^2 c-b^2 c+2 a b d\right ) \left (2 a b c-a^2 d+b^2 d\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{3 d^2 \left (c^2+d^2\right )^2}\\ &=-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^2}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {4 (b c-a d)^3 \left (2 b c^2+3 a c d+5 b d^2\right )}{3 d^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}-\frac {2 b^2 \left (a d (2 b c-a d)-b^2 \left (4 c^2+3 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 d^3 \left (c^2+d^2\right ) f}+\frac {(a-i b)^4 \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (c-i d)^2}+\frac {(a+i b)^4 \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (c+i d)^2}\\ &=-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^2}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {4 (b c-a d)^3 \left (2 b c^2+3 a c d+5 b d^2\right )}{3 d^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}-\frac {2 b^2 \left (a d (2 b c-a d)-b^2 \left (4 c^2+3 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 d^3 \left (c^2+d^2\right ) f}+\frac {\left (i (a-i b)^4\right ) \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)^2 f}-\frac {\left (i (a+i b)^4\right ) \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)^2 f}\\ &=-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^2}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {4 (b c-a d)^3 \left (2 b c^2+3 a c d+5 b d^2\right )}{3 d^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}-\frac {2 b^2 \left (a d (2 b c-a d)-b^2 \left (4 c^2+3 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 d^3 \left (c^2+d^2\right ) f}-\frac {(a-i b)^4 \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)^2 d f}-\frac {(a+i b)^4 \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)^2 d f}\\ &=-\frac {i (a-i b)^4 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{5/2} f}+\frac {i (a+i b)^4 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(c+i d)^{5/2} f}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))^2}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {4 (b c-a d)^3 \left (2 b c^2+3 a c d+5 b d^2\right )}{3 d^3 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}-\frac {2 b^2 \left (a d (2 b c-a d)-b^2 \left (4 c^2+3 d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{3 d^3 \left (c^2+d^2\right ) f}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 3.66, size = 368, normalized size = 1.27 \begin {gather*} -\frac {-2 b^2 (c-i d) (c+i d) \left (-20 a b c d+9 a^2 d^2+b^2 \left (8 c^2+d^2\right )\right )+d^2 \left (-4 a^3 b c+4 a b^3 c+a^4 d-6 a^2 b^2 d+b^4 d\right ) \left ((-i c+d) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {c+d \tan (e+f x)}{c-i d}\right )+(i c+d) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {c+d \tan (e+f x)}{c+i d}\right )\right )-12 b^2 (c-i d) (c+i d) d (2 b c-3 a d) (a+b \tan (e+f x))-6 b^2 (c-i d) (c+i d) d^2 (a+b \tan (e+f x))^2-12 a b \left (a^2-b^2\right ) d^2 \left (i (c+i d) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c+d \tan (e+f x)}{c-i d}\right )-(i c+d) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c+d \tan (e+f x)}{c+i d}\right )\right ) (c+d \tan (e+f x))}{3 d^3 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Tan[e + f*x])^4/(c + d*Tan[e + f*x])^(5/2),x]
[Out]
-1/3*(-2*b^2*(c - I*d)*(c + I*d)*(-20*a*b*c*d + 9*a^2*d^2 + b^2*(8*c^2 + d^2)) + d^2*(-4*a^3*b*c + 4*a*b^3*c +
a^4*d - 6*a^2*b^2*d + b^4*d)*(((-I)*c + d)*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c - I*d)] +
(I*c + d)*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c + I*d)]) - 12*b^2*(c - I*d)*(c + I*d)*d*(2
*b*c - 3*a*d)*(a + b*Tan[e + f*x]) - 6*b^2*(c - I*d)*(c + I*d)*d^2*(a + b*Tan[e + f*x])^2 - 12*a*b*(a^2 - b^2)
*d^2*(I*(c + I*d)*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c - I*d)] - (I*c + d)*Hypergeometric2F
1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c + I*d)])*(c + d*Tan[e + f*x]))/(d^3*(c^2 + d^2)*f*(c + d*Tan[e + f*x])
^(3/2))
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(8857\) vs.
\(2(258)=516\).
time = 0.55, size = 8858, normalized size = 30.54
| | |
method |
result |
size |
| | |
derivativedivides |
\(\text {Expression too large to display}\) |
\(8858\) |
default |
\(\text {Expression too large to display}\) |
\(8858\) |
| | |
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*tan(f*x+e))^4/(c+d*tan(f*x+e))^(5/2),x,method=_RETURNVERBOSE)
[Out]
result too large to display
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*tan(f*x+e))^4/(c+d*tan(f*x+e))^(5/2),x, algorithm="maxima")
[Out]
Timed out
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*tan(f*x+e))^4/(c+d*tan(f*x+e))^(5/2),x, algorithm="fricas")
[Out]
Timed out
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{4}}{\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*tan(f*x+e))**4/(c+d*tan(f*x+e))**(5/2),x)
[Out]
Integral((a + b*tan(e + f*x))**4/(c + d*tan(e + f*x))**(5/2), x)
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*tan(f*x+e))^4/(c+d*tan(f*x+e))^(5/2),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done
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Mupad [B]
time = 35.67, size = 2500, normalized size = 8.62 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*tan(e + f*x))^4/(c + d*tan(e + f*x))^(5/2),x)
[Out]
(2*b^4*(c + d*tan(e + f*x))^(1/2))/(d^3*f) - atan((((-(((8*a^8*c^5*f^2 + 8*b^8*c^5*f^2 - 64*a*b^7*d^5*f^2 + 64
*a^7*b*d^5*f^2 + 40*a^8*c*d^4*f^2 + 40*b^8*c*d^4*f^2 - 224*a^2*b^6*c^5*f^2 + 560*a^4*b^4*c^5*f^2 - 224*a^6*b^2
*c^5*f^2 + 448*a^3*b^5*d^5*f^2 - 448*a^5*b^3*d^5*f^2 - 80*a^8*c^3*d^2*f^2 - 80*b^8*c^3*d^2*f^2 + 2240*a^2*b^6*
c^3*d^2*f^2 - 4480*a^3*b^5*c^2*d^3*f^2 - 5600*a^4*b^4*c^3*d^2*f^2 + 4480*a^5*b^3*c^2*d^3*f^2 + 2240*a^6*b^2*c^
3*d^2*f^2 - 320*a*b^7*c^4*d*f^2 + 320*a^7*b*c^4*d*f^2 + 640*a*b^7*c^2*d^3*f^2 - 1120*a^2*b^6*c*d^4*f^2 + 2240*
a^3*b^5*c^4*d*f^2 + 2800*a^4*b^4*c*d^4*f^2 - 2240*a^5*b^3*c^4*d*f^2 - 1120*a^6*b^2*c*d^4*f^2 - 640*a^7*b*c^2*d
^3*f^2)^2/4 - (16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4
)*(a^16 + b^16 + 8*a^2*b^14 + 28*a^4*b^12 + 56*a^6*b^10 + 70*a^8*b^8 + 56*a^10*b^6 + 28*a^12*b^4 + 8*a^14*b^2)
)^(1/2) + 4*a^8*c^5*f^2 + 4*b^8*c^5*f^2 - 32*a*b^7*d^5*f^2 + 32*a^7*b*d^5*f^2 + 20*a^8*c*d^4*f^2 + 20*b^8*c*d^
4*f^2 - 112*a^2*b^6*c^5*f^2 + 280*a^4*b^4*c^5*f^2 - 112*a^6*b^2*c^5*f^2 + 224*a^3*b^5*d^5*f^2 - 224*a^5*b^3*d^
5*f^2 - 40*a^8*c^3*d^2*f^2 - 40*b^8*c^3*d^2*f^2 + 1120*a^2*b^6*c^3*d^2*f^2 - 2240*a^3*b^5*c^2*d^3*f^2 - 2800*a
^4*b^4*c^3*d^2*f^2 + 2240*a^5*b^3*c^2*d^3*f^2 + 1120*a^6*b^2*c^3*d^2*f^2 - 160*a*b^7*c^4*d*f^2 + 160*a^7*b*c^4
*d*f^2 + 320*a*b^7*c^2*d^3*f^2 - 560*a^2*b^6*c*d^4*f^2 + 1120*a^3*b^5*c^4*d*f^2 + 1400*a^4*b^4*c*d^4*f^2 - 112
0*a^5*b^3*c^4*d*f^2 - 560*a^6*b^2*c*d^4*f^2 - 320*a^7*b*c^2*d^3*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4
+ 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4)))^(1/2)*((c + d*tan(e + f*x))^(1/2)*(-(((8*a^8*c^5*f^2 + 8*
b^8*c^5*f^2 - 64*a*b^7*d^5*f^2 + 64*a^7*b*d^5*f^2 + 40*a^8*c*d^4*f^2 + 40*b^8*c*d^4*f^2 - 224*a^2*b^6*c^5*f^2
+ 560*a^4*b^4*c^5*f^2 - 224*a^6*b^2*c^5*f^2 + 448*a^3*b^5*d^5*f^2 - 448*a^5*b^3*d^5*f^2 - 80*a^8*c^3*d^2*f^2 -
80*b^8*c^3*d^2*f^2 + 2240*a^2*b^6*c^3*d^2*f^2 - 4480*a^3*b^5*c^2*d^3*f^2 - 5600*a^4*b^4*c^3*d^2*f^2 + 4480*a^
5*b^3*c^2*d^3*f^2 + 2240*a^6*b^2*c^3*d^2*f^2 - 320*a*b^7*c^4*d*f^2 + 320*a^7*b*c^4*d*f^2 + 640*a*b^7*c^2*d^3*f
^2 - 1120*a^2*b^6*c*d^4*f^2 + 2240*a^3*b^5*c^4*d*f^2 + 2800*a^4*b^4*c*d^4*f^2 - 2240*a^5*b^3*c^4*d*f^2 - 1120*
a^6*b^2*c*d^4*f^2 - 640*a^7*b*c^2*d^3*f^2)^2/4 - (16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4
+ 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4)*(a^16 + b^16 + 8*a^2*b^14 + 28*a^4*b^12 + 56*a^6*b^10 + 70*a^8*b^8 + 56*a
^10*b^6 + 28*a^12*b^4 + 8*a^14*b^2))^(1/2) + 4*a^8*c^5*f^2 + 4*b^8*c^5*f^2 - 32*a*b^7*d^5*f^2 + 32*a^7*b*d^5*f
^2 + 20*a^8*c*d^4*f^2 + 20*b^8*c*d^4*f^2 - 112*a^2*b^6*c^5*f^2 + 280*a^4*b^4*c^5*f^2 - 112*a^6*b^2*c^5*f^2 + 2
24*a^3*b^5*d^5*f^2 - 224*a^5*b^3*d^5*f^2 - 40*a^8*c^3*d^2*f^2 - 40*b^8*c^3*d^2*f^2 + 1120*a^2*b^6*c^3*d^2*f^2
- 2240*a^3*b^5*c^2*d^3*f^2 - 2800*a^4*b^4*c^3*d^2*f^2 + 2240*a^5*b^3*c^2*d^3*f^2 + 1120*a^6*b^2*c^3*d^2*f^2 -
160*a*b^7*c^4*d*f^2 + 160*a^7*b*c^4*d*f^2 + 320*a*b^7*c^2*d^3*f^2 - 560*a^2*b^6*c*d^4*f^2 + 1120*a^3*b^5*c^4*d
*f^2 + 1400*a^4*b^4*c*d^4*f^2 - 1120*a^5*b^3*c^4*d*f^2 - 560*a^6*b^2*c*d^4*f^2 - 320*a^7*b*c^2*d^3*f^2)/(16*(c
^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4)))^(1/2)*(64*c*d^22*f^5 +
640*c^3*d^20*f^5 + 2880*c^5*d^18*f^5 + 7680*c^7*d^16*f^5 + 13440*c^9*d^14*f^5 + 16128*c^11*d^12*f^5 + 13440*c
^13*d^10*f^5 + 7680*c^15*d^8*f^5 + 2880*c^17*d^6*f^5 + 640*c^19*d^4*f^5 + 64*c^21*d^2*f^5) - 32*a^4*d^21*f^4 -
32*b^4*d^21*f^4 + 192*a^2*b^2*d^21*f^4 - 160*a^4*c^2*d^19*f^4 - 128*a^4*c^4*d^17*f^4 + 896*a^4*c^6*d^15*f^4 +
3136*a^4*c^8*d^13*f^4 + 4928*a^4*c^10*d^11*f^4 + 4480*a^4*c^12*d^9*f^4 + 2432*a^4*c^14*d^7*f^4 + 736*a^4*c^16
*d^5*f^4 + 96*a^4*c^18*d^3*f^4 - 160*b^4*c^2*d^19*f^4 - 128*b^4*c^4*d^17*f^4 + 896*b^4*c^6*d^15*f^4 + 3136*b^4
*c^8*d^13*f^4 + 4928*b^4*c^10*d^11*f^4 + 4480*b^4*c^12*d^9*f^4 + 2432*b^4*c^14*d^7*f^4 + 736*b^4*c^16*d^5*f^4
+ 96*b^4*c^18*d^3*f^4 + 960*a^2*b^2*c^2*d^19*f^4 + 768*a^2*b^2*c^4*d^17*f^4 - 5376*a^2*b^2*c^6*d^15*f^4 - 1881
6*a^2*b^2*c^8*d^13*f^4 - 29568*a^2*b^2*c^10*d^11*f^4 - 26880*a^2*b^2*c^12*d^9*f^4 - 14592*a^2*b^2*c^14*d^7*f^4
- 4416*a^2*b^2*c^16*d^5*f^4 - 576*a^2*b^2*c^18*d^3*f^4 - 384*a*b^3*c*d^20*f^4 + 384*a^3*b*c*d^20*f^4 - 2944*a
*b^3*c^3*d^18*f^4 - 9728*a*b^3*c^5*d^16*f^4 - 17920*a*b^3*c^7*d^14*f^4 - 19712*a*b^3*c^9*d^12*f^4 - 12544*a*b^
3*c^11*d^10*f^4 - 3584*a*b^3*c^13*d^8*f^4 + 512*a*b^3*c^15*d^6*f^4 + 640*a*b^3*c^17*d^4*f^4 + 128*a*b^3*c^19*d
^2*f^4 + 2944*a^3*b*c^3*d^18*f^4 + 9728*a^3*b*c^5*d^16*f^4 + 17920*a^3*b*c^7*d^14*f^4 + 19712*a^3*b*c^9*d^12*f
^4 + 12544*a^3*b*c^11*d^10*f^4 + 3584*a^3*b*c^13*d^8*f^4 - 512*a^3*b*c^15*d^6*f^4 - 640*a^3*b*c^17*d^4*f^4 - 1
28*a^3*b*c^19*d^2*f^4) - (c + d*tan(e + f*x))^(1/2)*(448*a^2*b^6*d^18*f^3 - 16*b^8*d^18*f^3 - 16*a^8*d^18*f^3
- 1120*a^4*b^4*d^18*f^3 + 448*a^6*b^2*d^18*f^3 + 320*a^8*c^4*d^14*f^3 + 1024*a^8*c^6*d^12*f^3 + 1440*a^8*c^8*d
^10*f^3 + 1024*a^8*c^10*d^8*f^3 + 320*a^8*c^12*d^6*f^3 - 16*a^8*c^16*d^2*f^3 + 320*b^8*c^4*d^14*f^3 + 1024*b^8
*c^6*d^12*f^3 + 1440*b^8*c^8*d^10*f^3 + 1024*b^...
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