∫ (c+d tan(e+fx))5∕2 ____________________________

3.1246 \(\int \frac {(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^3} \, dx\)

Optimal. Leaf size=355 \[ -\frac {(b c-a d) \left (a^2 d+8 a b c+9 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 b f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))}-\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {\sqrt {b c-a d} \left (a^4 d^2+8 a^3 b c d-6 a^2 b^2 \left (4 c^2-3 d^2\right )-56 a b^3 c d+b^4 \left (8 c^2-15 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{4 b^{3/2} f \left (a^2+b^2\right )^3}-\frac {(c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (b+i a)^3}+\frac {(c+i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (-b+i a)^3} \]

[Out]

-(c-I*d)^(5/2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/(I*a+b)^3/f+(c+I*d)^(5/2)*arctanh((c+d*tan(f*x+e)

)^(1/2)/(c+I*d)^(1/2))/(I*a-b)^3/f+1/4*(8*a^3*b*c*d-56*a*b^3*c*d+a^4*d^2+b^4*(8*c^2-15*d^2)-6*a^2*b^2*(4*c^2-3

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

*d+b*c)^2*(c+d*tan(f*x+e))^(1/2)/b/(a^2+b^2)/f/(a+b*tan(f*x+e))^2-1/4*(-a*d+b*c)*(a^2*d+8*a*b*c+9*b^2*d)*(c+d*

tan(f*x+e))^(1/2)/b/(a^2+b^2)^2/f/(a+b*tan(f*x+e))

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Rubi [A] time = 1.86, antiderivative size = 355, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {3565, 3649, 3653, 3539, 3537, 63, 208, 3634} \[ \frac {\sqrt {b c-a d} \left (-6 a^2 b^2 \left (4 c^2-3 d^2\right )+8 a^3 b c d+a^4 d^2-56 a b^3 c d+b^4 \left (8 c^2-15 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{4 b^{3/2} f \left (a^2+b^2\right )^3}-\frac {(b c-a d) \left (a^2 d+8 a b c+9 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 b f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))}-\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}-\frac {(c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (b+i a)^3}+\frac {(c+i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (-b+i a)^3} \]

Antiderivative was successfully veriﬁed.

[In]

Int[(c + d*Tan[e + f*x])^(5/2)/(a + b*Tan[e + f*x])^3,x]

[Out]

-(((c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((I*a + b)^3*f)) + ((c + I*d)^(5/2)*ArcTan

h[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((I*a - b)^3*f) + (Sqrt[b*c - a*d]*(8*a^3*b*c*d - 56*a*b^3*c*d + a^

4*d^2 + b^4*(8*c^2 - 15*d^2) - 6*a^2*b^2*(4*c^2 - 3*d^2))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c

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

e + f*x])^2) - ((b*c - a*d)*(8*a*b*c + a^2*d + 9*b^2*d)*Sqrt[c + d*Tan[e + f*x]])/(4*b*(a^2 + b^2)^2*f*(a + b*

Tan[e + f*x]))

Rule 63

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 208

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

b}, x] && NegQ[a/b]

Rule 3537

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c*

d)/f, Subst[Int[(a + (b*x)/d)^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 3539

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 3565

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 3634

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*

tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x]

/; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]

Rule 3649

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t

an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[((A*b^2 - a*(b*B - a*C))*(a + b*T

an[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2)), x] + Dist[1/((m + 1)*(

b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)

- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e

+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,

n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !I

ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3653

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (

f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*

x])^n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 +

b^2), Int[((c + d*Tan[e + f*x])^n*(1 + Tan[e + f*x]^2))/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e,

f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !GtQ[n, 0] && !LeQ[n, -

Rubi steps

\begin {align*} \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^3} \, dx &=-\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}+\frac {\int \frac {\frac {1}{2} \left (9 b^2 c^2 d+a^2 d^3+a b \left (4 c^3-6 c d^2\right )\right )+2 b \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right ) \tan (e+f x)+\frac {1}{2} d \left (\left (a^2+4 b^2\right ) d^2-3 b c (b c-2 a d)\right ) \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}} \, dx}{2 b \left (a^2+b^2\right )}\\ &=-\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {(b c-a d) \left (8 a b c+a^2 d+9 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 b \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\int \frac {-\frac {1}{4} (b c-a d) \left (8 a^2 b c^3-8 b^3 c^3+40 a b^2 c^2 d-17 a^2 b c d^2+15 b^3 c d^2+a^3 d^3-7 a b^2 d^3\right )+2 b (b c-a d) \left (2 a b c \left (c^2-3 d^2\right )+b^2 d \left (3 c^2-d^2\right )-a^2 \left (3 c^2 d-d^3\right )\right ) \tan (e+f x)+\frac {1}{4} d (b c-a d)^2 \left (8 a b c+a^2 d+9 b^2 d\right ) \tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{2 b \left (a^2+b^2\right )^2 (b c-a d)}\\ &=-\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {(b c-a d) \left (8 a b c+a^2 d+9 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 b \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\int \frac {-2 b (b c-a d) (a c+b d) \left (a^2 c^2-3 b^2 c^2+8 a b c d-3 a^2 d^2+b^2 d^2\right )+2 b (b c-a d)^2 \left (8 a b c d-b^2 \left (c^2-3 d^2\right )+a^2 \left (3 c^2-d^2\right )\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 b \left (a^2+b^2\right )^3 (b c-a d)}-\frac {\left ((b c-a d) \left (8 a^3 b c d-56 a b^3 c d+a^4 d^2+b^4 \left (8 c^2-15 d^2\right )-6 a^2 b^2 \left (4 c^2-3 d^2\right )\right )\right ) \int \frac {1+\tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{8 b \left (a^2+b^2\right )^3}\\ &=-\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {(b c-a d) \left (8 a b c+a^2 d+9 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 b \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}+\frac {(c-i d)^3 \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (a-i b)^3}+\frac {(c+i d)^3 \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (a+i b)^3}-\frac {\left ((b c-a d) \left (8 a^3 b c d-56 a b^3 c d+a^4 d^2+b^4 \left (8 c^2-15 d^2\right )-6 a^2 b^2 \left (4 c^2-3 d^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{8 b \left (a^2+b^2\right )^3 f}\\ &=-\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {(b c-a d) \left (8 a b c+a^2 d+9 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 b \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}+\frac {(c-i d)^3 \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 (i a+b)^3 f}-\frac {(c+i d)^3 \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 (i a-b)^3 f}-\frac {\left ((b c-a d) \left (8 a^3 b c d-56 a b^3 c d+a^4 d^2+b^4 \left (8 c^2-15 d^2\right )-6 a^2 b^2 \left (4 c^2-3 d^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{4 b \left (a^2+b^2\right )^3 d f}\\ &=\frac {\sqrt {b c-a d} \left (8 a^3 b c d-56 a b^3 c d+a^4 d^2+b^4 \left (8 c^2-15 d^2\right )-6 a^2 b^2 \left (4 c^2-3 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{4 b^{3/2} \left (a^2+b^2\right )^3 f}-\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {(b c-a d) \left (8 a b c+a^2 d+9 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 b \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {(c-i d)^3 \operatorname {Subst}\left (\int \frac {1}{-1-\frac {i c}{d}+\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{(a-i b)^3 d f}-\frac {(c+i d)^3 \operatorname {Subst}\left (\int \frac {1}{-1+\frac {i c}{d}-\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{(a+i b)^3 d f}\\ &=-\frac {(c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(i a+b)^3 f}+\frac {(c+i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(i a-b)^3 f}+\frac {\sqrt {b c-a d} \left (8 a^3 b c d-56 a b^3 c d+a^4 d^2+b^4 \left (8 c^2-15 d^2\right )-6 a^2 b^2 \left (4 c^2-3 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{4 b^{3/2} \left (a^2+b^2\right )^3 f}-\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {(b c-a d) \left (8 a b c+a^2 d+9 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 b \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}\\ \end {align*}

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Mathematica [B] time = 6.47, size = 2946, normalized size = 8.30 \[ \text {Result too large to show} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[(c + d*Tan[e + f*x])^(5/2)/(a + b*Tan[e + f*x])^3,x]

[Out]

-1/2*(b^2*(c + d*Tan[e + f*x])^(7/2))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Tan[e + f*x])^2) - (-((b*d*(c + d*Tan[

e + f*x])^(5/2))/(f*(a + b*Tan[e + f*x]))) + (2*((-3*b*d*(b*c - a*d)*(c + d*Tan[e + f*x])^(3/2))/(2*f*(a + b*T

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

rt[c - I*d]*(b*(b*c - a*d)*((3*a*b^3*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*b^3*d*(b*c -

a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16 + (3*b^3*(b*c - a*d)*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3

- 6*c*d^2)))/16) + a*((3*b^2*(b*c - a*d)*((b^2*d)/2 - a*(b*c - a*d))*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c

*d^2)))/16 + (-(b*c) + (a*d)/2)*((-3*b^4*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*a*b^2*d*(

b*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16) - (d*((3*b^4*(b*c - a*d)*(9*b^2*c^2*d + a^2*d^3 +

a*b*(4*c^3 - 6*c*d^2)))/16 - a*((-3*b^4*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*a*b^2*d*(b

*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16)))/2) - I*(a*(b*c - a*d)*((3*a*b^3*(b*c - a*d)*(b*c^

3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*b^3*d*(b*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16 +

(3*b^3*(b*c - a*d)*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16) - b*((3*b^2*(b*c - a*d)*((b^2*d)/2 -

a*(b*c - a*d))*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16 + (-(b*c) + (a*d)/2)*((-3*b^4*(b*c - a*d)*(

b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*a*b^2*d*(b*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2))

)/16) - (d*((3*b^4*(b*c - a*d)*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16 - a*((-3*b^4*(b*c - a*d)*(b

*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*a*b^2*d*(b*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))

/16)))/2)))*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((-c + I*d)*f) - (I*Sqrt[c + I*d]*(b*(b*c - a*d)*

((3*a*b^3*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*b^3*d*(b*c - a*d)*(6*a*b*c*d + a^2*d^2 -

b^2*(3*c^2 - 4*d^2)))/16 + (3*b^3*(b*c - a*d)*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16) + a*((3*b^

2*(b*c - a*d)*((b^2*d)/2 - a*(b*c - a*d))*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16 + (-(b*c) + (a*d

)/2)*((-3*b^4*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*a*b^2*d*(b*c - a*d)*(6*a*b*c*d + a^2

*d^2 - b^2*(3*c^2 - 4*d^2)))/16) - (d*((3*b^4*(b*c - a*d)*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16

- a*((-3*b^4*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*a*b^2*d*(b*c - a*d)*(6*a*b*c*d + a^2*

d^2 - b^2*(3*c^2 - 4*d^2)))/16)))/2) + I*(a*(b*c - a*d)*((3*a*b^3*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 +

a*d^3))/4 - (3*b^3*d*(b*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16 + (3*b^3*(b*c - a*d)*(9*b^2*

c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16) - b*((3*b^2*(b*c - a*d)*((b^2*d)/2 - a*(b*c - a*d))*(9*b^2*c^2*d

+ a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16 + (-(b*c) + (a*d)/2)*((-3*b^4*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d

^2 + a*d^3))/4 - (3*a*b^2*d*(b*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16) - (d*((3*b^4*(b*c - a

*d)*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16 - a*((-3*b^4*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^

2 + a*d^3))/4 - (3*a*b^2*d*(b*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16)))/2)))*ArcTanh[Sqrt[c

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

3*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*b^3*d*(b*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*

c^2 - 4*d^2)))/16 + (3*b^3*(b*c - a*d)*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16)) + (a^2*d*((3*b^4*

(b*c - a*d)*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16 - a*((-3*b^4*(b*c - a*d)*(b*c^3 - 3*a*c^2*d -

3*b*c*d^2 + a*d^3))/4 - (3*a*b^2*d*(b*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16)))/2 + b^2*((3*

b^2*(b*c - a*d)*((b^2*d)/2 - a*(b*c - a*d))*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16 + (-(b*c) + (a

*d)/2)*((-3*b^4*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*a*b^2*d*(b*c - a*d)*(6*a*b*c*d + a

^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16)))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(Sqrt[b]*(a^

2 + b^2)*(-(b*c) + a*d)*f))/((a^2 + b^2)*(b*c - a*d))) - (((3*b^4*(b*c - a*d)*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*

c^3 - 6*c*d^2)))/16 - a*((-3*b^4*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*a*b^2*d*(b*c - a*

d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16))*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*(b*c - a*d)*f*(a +

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

________________________________________________________________________________________

fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^(5/2)/(a+b*tan(f*x+e))^3,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^(5/2)/(a+b*tan(f*x+e))^3,x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [B] time = 0.49, size = 6771, normalized size = 19.07 \[ \text {output too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*tan(f*x+e))^(5/2)/(a+b*tan(f*x+e))^3,x)

[Out]

result too large to display

________________________________________________________________________________________

maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^(5/2)/(a+b*tan(f*x+e))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a

ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for

more details)Is a*d-b*c positive or negative?

________________________________________________________________________________________

mupad [B] time = 44.60, size = 116010, normalized size = 326.79 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*tan(e + f*x))^(5/2)/(a + b*tan(e + f*x))^3,x)

[Out]

atan(((((20*a^16*b*d^18*f^2 + 8*a^17*c*d^17*f^2 - 4796*a^2*b^15*d^18*f^2 + 10476*a^4*b^13*d^18*f^2 + 14772*a^6

*b^11*d^18*f^2 - 16644*a^8*b^9*d^18*f^2 - 10996*a^10*b^7*d^18*f^2 + 5892*a^12*b^5*d^18*f^2 + 764*a^14*b^3*d^18

*f^2 + 8*a^17*c^3*d^15*f^2 - 3708*b^17*c^2*d^16*f^2 + 6912*b^17*c^4*d^14*f^2 + 5820*b^17*c^6*d^12*f^2 - 4608*b

^17*c^8*d^10*f^2 + 192*b^17*c^10*d^8*f^2 + 125788*a^2*b^15*c^2*d^16*f^2 - 223956*a^2*b^15*c^4*d^14*f^2 - 20369

2*a^2*b^15*c^6*d^12*f^2 + 145600*a^2*b^15*c^8*d^10*f^2 - 5248*a^2*b^15*c^10*d^8*f^2 + 470816*a^3*b^14*c^3*d^15

*f^2 - 4848*a^3*b^14*c^5*d^13*f^2 - 482048*a^3*b^14*c^7*d^11*f^2 + 73728*a^3*b^14*c^9*d^9*f^2 - 274940*a^4*b^1

3*c^2*d^16*f^2 + 324004*a^4*b^13*c^4*d^14*f^2 + 420684*a^4*b^13*c^6*d^12*f^2 - 183040*a^4*b^13*c^8*d^10*f^2 +

5696*a^4*b^13*c^10*d^8*f^2 + 125696*a^5*b^12*c^3*d^15*f^2 - 188624*a^5*b^12*c^5*d^13*f^2 - 262656*a^5*b^12*c^7

*d^11*f^2 + 51968*a^5*b^12*c^9*d^9*f^2 - 474836*a^6*b^11*c^2*d^16*f^2 + 859132*a^6*b^11*c^4*d^14*f^2 + 822084*

a^6*b^11*c^6*d^12*f^2 - 508992*a^6*b^11*c^8*d^10*f^2 + 17664*a^6*b^11*c^10*d^8*f^2 - 1071584*a^7*b^10*c^3*d^15

*f^2 - 235184*a^7*b^10*c^5*d^13*f^2 + 891392*a^7*b^10*c^7*d^11*f^2 - 133120*a^7*b^10*c^9*d^9*f^2 + 325404*a^8*

b^9*c^2*d^16*f^2 - 108972*a^8*b^9*c^4*d^14*f^2 - 350476*a^8*b^9*c^6*d^12*f^2 + 96768*a^8*b^9*c^8*d^10*f^2 - 37

76*a^8*b^9*c^10*d^8*f^2 - 346352*a^9*b^8*c^3*d^15*f^2 + 31056*a^9*b^8*c^5*d^13*f^2 + 327936*a^9*b^8*c^7*d^11*f

^2 - 64256*a^9*b^8*c^9*d^9*f^2 + 323668*a^10*b^7*c^2*d^16*f^2 - 342716*a^10*b^7*c^4*d^14*f^2 - 434500*a^10*b^7

*c^6*d^12*f^2 + 232512*a^10*b^7*c^8*d^10*f^2 - 10368*a^10*b^7*c^10*d^8*f^2 + 292448*a^11*b^6*c^3*d^15*f^2 + 83

760*a^11*b^6*c^5*d^13*f^2 - 257792*a^11*b^6*c^7*d^11*f^2 + 34816*a^11*b^6*c^9*d^9*f^2 - 72724*a^12*b^5*c^2*d^1

6*f^2 + 65964*a^12*b^5*c^4*d^14*f^2 + 104452*a^12*b^5*c^6*d^12*f^2 - 40192*a^12*b^5*c^8*d^10*f^2 - 64*a^12*b^5

*c^10*d^8*f^2 - 24512*a^13*b^4*c^3*d^15*f^2 - 6896*a^13*b^4*c^5*d^13*f^2 + 19456*a^13*b^4*c^7*d^11*f^2 + 256*a

^13*b^4*c^9*d^9*f^2 - 476*a^14*b^3*c^2*d^16*f^2 - 4460*a^14*b^3*c^4*d^14*f^2 - 3412*a^14*b^3*c^6*d^12*f^2 - 19

2*a^14*b^3*c^8*d^10*f^2 + 96*a^15*b^2*c^3*d^15*f^2 - 400*a^15*b^2*c^5*d^13*f^2 + 8504*a*b^16*c*d^17*f^2 - 6306

4*a*b^16*c^3*d^15*f^2 + 7792*a*b^16*c^5*d^13*f^2 + 66816*a*b^16*c^7*d^11*f^2 - 12544*a*b^16*c^9*d^9*f^2 - 8011

2*a^3*b^14*c*d^17*f^2 - 304*a^5*b^12*c*d^17*f^2 + 188112*a^7*b^10*c*d^17*f^2 + 14784*a^9*b^8*c*d^17*f^2 - 8392

0*a^11*b^6*c*d^17*f^2 + 1584*a^13*b^4*c*d^17*f^2 + 496*a^15*b^2*c*d^17*f^2 + 112*a^16*b*c^2*d^16*f^2 + 92*a^16

*b*c^4*d^14*f^2)/(2*(b^17*f^5 + a^16*b*f^5 + 8*a^2*b^15*f^5 + 28*a^4*b^13*f^5 + 56*a^6*b^11*f^5 + 70*a^8*b^9*f

^5 + 56*a^10*b^7*f^5 + 28*a^12*b^5*f^5 + 8*a^14*b^3*f^5)) + (((1664*b^23*c*d^12*f^4 - 1664*a*b^22*d^13*f^4 - 1

1904*a^3*b^20*d^13*f^4 - 35328*a^5*b^18*d^13*f^4 - 53760*a^7*b^16*d^13*f^4 - 37632*a^9*b^14*d^13*f^4 + 5376*a^

11*b^12*d^13*f^4 + 32256*a^13*b^10*d^13*f^4 + 26112*a^15*b^8*d^13*f^4 + 9600*a^17*b^6*d^13*f^4 + 1408*a^19*b^4

*d^13*f^4 + 896*b^23*c^3*d^10*f^4 - 768*b^23*c^5*d^8*f^4 + 2432*a^2*b^21*c^3*d^10*f^4 - 4864*a^2*b^21*c^5*d^8*

f^4 + 29312*a^3*b^20*c^2*d^11*f^4 + 41216*a^3*b^20*c^4*d^9*f^4 - 12288*a^4*b^19*c^3*d^10*f^4 - 11264*a^4*b^19*

c^5*d^8*f^4 + 100864*a^5*b^18*c^2*d^11*f^4 + 136192*a^5*b^18*c^4*d^9*f^4 - 78336*a^6*b^17*c^3*d^10*f^4 - 7168*

a^6*b^17*c^5*d^8*f^4 + 197120*a^7*b^16*c^2*d^11*f^4 + 250880*a^7*b^16*c^4*d^9*f^4 - 188160*a^8*b^15*c^3*d^10*f

^4 + 17920*a^8*b^15*c^5*d^8*f^4 + 238336*a^9*b^14*c^2*d^11*f^4 + 275968*a^9*b^14*c^4*d^9*f^4 - 252672*a^10*b^1

3*c^3*d^10*f^4 + 46592*a^10*b^13*c^5*d^8*f^4 + 180992*a^11*b^12*c^2*d^11*f^4 + 175616*a^11*b^12*c^4*d^9*f^4 -

204288*a^12*b^11*c^3*d^10*f^4 + 50176*a^12*b^11*c^5*d^8*f^4 + 82432*a^13*b^10*c^2*d^11*f^4 + 50176*a^13*b^10*c

^4*d^9*f^4 - 96768*a^14*b^9*c^3*d^10*f^4 + 29696*a^14*b^9*c^5*d^8*f^4 + 18944*a^15*b^8*c^2*d^11*f^4 - 7168*a^1

5*b^8*c^4*d^9*f^4 - 22656*a^16*b^7*c^3*d^10*f^4 + 9472*a^16*b^7*c^5*d^8*f^4 + 640*a^17*b^6*c^2*d^11*f^4 - 8960

*a^17*b^6*c^4*d^9*f^4 - 640*a^18*b^5*c^3*d^10*f^4 + 1280*a^18*b^5*c^5*d^8*f^4 - 384*a^19*b^4*c^2*d^11*f^4 - 17

92*a^19*b^4*c^4*d^9*f^4 + 512*a^20*b^3*c^3*d^10*f^4 + 3712*a*b^22*c^2*d^11*f^4 + 5376*a*b^22*c^4*d^9*f^4 + 729

6*a^2*b^21*c*d^12*f^4 - 1024*a^4*b^19*c*d^12*f^4 - 71168*a^6*b^17*c*d^12*f^4 - 206080*a^8*b^15*c*d^12*f^4 - 29

9264*a^10*b^13*c*d^12*f^4 - 254464*a^12*b^11*c*d^12*f^4 - 126464*a^14*b^9*c*d^12*f^4 - 32128*a^16*b^7*c*d^12*f

^4 - 1920*a^18*b^5*c*d^12*f^4 + 512*a^20*b^3*c*d^12*f^4)/(2*(b^17*f^5 + a^16*b*f^5 + 8*a^2*b^15*f^5 + 28*a^4*b

^13*f^5 + 56*a^6*b^11*f^5 + 70*a^8*b^9*f^5 + 56*a^10*b^7*f^5 + 28*a^12*b^5*f^5 + 8*a^14*b^3*f^5)) + ((c + d*ta

n(e + f*x))^(1/2)*((((8*a^6*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5*d^5*f^2 + 48*a^5*b*d^5*f^2 + 40*a^6*c*d^4*f^2 -

40*b^6*c*d^4*f^2 + 120*a^2*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2 - 160*a^3*b^3*d^5*f^2 - 80*a^6*c^3*d^2*f^2 + 80*

b^6*c^3*d^2*f^2 - 1200*a^2*b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2*d^3*f^2 + 1200*a^4*b^2*c^3*d^2*f^2 + 240*a*b^5*c

^4*d*f^2 + 240*a^5*b*c^4*d*f^2 - 480*a*b^5*c^2*d^3*f^2 + 600*a^2*b^4*c*d^4*f^2 - 800*a^3*b^3*c^4*d*f^2 - 600*a

^4*b^2*c*d^4*f^2 - 480*a^5*b*c^2*d^3*f^2)^2/4 - (c^10 + d^10 + 5*c^2*d^8 + 10*c^4*d^6 + 10*c^6*d^4 + 5*c^8*d^2

)*(16*a^12*f^4 + 16*b^12*f^4 + 96*a^2*b^10*f^4 + 240*a^4*b^8*f^4 + 320*a^6*b^6*f^4 + 240*a^8*b^4*f^4 + 96*a^10

*b^2*f^4))^(1/2) - 4*a^6*c^5*f^2 + 4*b^6*c^5*f^2 - 24*a*b^5*d^5*f^2 - 24*a^5*b*d^5*f^2 - 20*a^6*c*d^4*f^2 + 20

*b^6*c*d^4*f^2 - 60*a^2*b^4*c^5*f^2 + 60*a^4*b^2*c^5*f^2 + 80*a^3*b^3*d^5*f^2 + 40*a^6*c^3*d^2*f^2 - 40*b^6*c^

3*d^2*f^2 + 600*a^2*b^4*c^3*d^2*f^2 - 800*a^3*b^3*c^2*d^3*f^2 - 600*a^4*b^2*c^3*d^2*f^2 - 120*a*b^5*c^4*d*f^2

- 120*a^5*b*c^4*d*f^2 + 240*a*b^5*c^2*d^3*f^2 - 300*a^2*b^4*c*d^4*f^2 + 400*a^3*b^3*c^4*d*f^2 + 300*a^4*b^2*c*

d^4*f^2 + 240*a^5*b*c^2*d^3*f^2)/(16*(a^12*f^4 + b^12*f^4 + 6*a^2*b^10*f^4 + 15*a^4*b^8*f^4 + 20*a^6*b^6*f^4 +

15*a^8*b^4*f^4 + 6*a^10*b^2*f^4)))^(1/2)*(512*b^26*d^10*f^4 + 4608*a^2*b^24*d^10*f^4 + 17920*a^4*b^22*d^10*f^

4 + 38400*a^6*b^20*d^10*f^4 + 46080*a^8*b^18*d^10*f^4 + 21504*a^10*b^16*d^10*f^4 - 21504*a^12*b^14*d^10*f^4 -

46080*a^14*b^12*d^10*f^4 - 38400*a^16*b^10*d^10*f^4 - 17920*a^18*b^8*d^10*f^4 - 4608*a^20*b^6*d^10*f^4 - 512*a

^22*b^4*d^10*f^4 + 768*b^26*c^2*d^8*f^4 + 7424*a^2*b^24*c^2*d^8*f^4 + 32000*a^4*b^22*c^2*d^8*f^4 + 80640*a^6*b

^20*c^2*d^8*f^4 + 130560*a^8*b^18*c^2*d^8*f^4 + 139776*a^10*b^16*c^2*d^8*f^4 + 96768*a^12*b^14*c^2*d^8*f^4 + 3

8400*a^14*b^12*c^2*d^8*f^4 + 3840*a^16*b^10*c^2*d^8*f^4 - 3840*a^18*b^8*c^2*d^8*f^4 - 1792*a^20*b^6*c^2*d^8*f^

4 - 256*a^22*b^4*c^2*d^8*f^4 + 256*a*b^25*c*d^9*f^4 + 2816*a^3*b^23*c*d^9*f^4 + 14080*a^5*b^21*c*d^9*f^4 + 422

40*a^7*b^19*c*d^9*f^4 + 84480*a^9*b^17*c*d^9*f^4 + 118272*a^11*b^15*c*d^9*f^4 + 118272*a^13*b^13*c*d^9*f^4 + 8

4480*a^15*b^11*c*d^9*f^4 + 42240*a^17*b^9*c*d^9*f^4 + 14080*a^19*b^7*c*d^9*f^4 + 2816*a^21*b^5*c*d^9*f^4 + 256

*a^23*b^3*c*d^9*f^4))/(b^17*f^4 + a^16*b*f^4 + 8*a^2*b^15*f^4 + 28*a^4*b^13*f^4 + 56*a^6*b^11*f^4 + 70*a^8*b^9

*f^4 + 56*a^10*b^7*f^4 + 28*a^12*b^5*f^4 + 8*a^14*b^3*f^4))*((((8*a^6*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5*d^5*f

^2 + 48*a^5*b*d^5*f^2 + 40*a^6*c*d^4*f^2 - 40*b^6*c*d^4*f^2 + 120*a^2*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2 - 160*

a^3*b^3*d^5*f^2 - 80*a^6*c^3*d^2*f^2 + 80*b^6*c^3*d^2*f^2 - 1200*a^2*b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2*d^3*f^

2 + 1200*a^4*b^2*c^3*d^2*f^2 + 240*a*b^5*c^4*d*f^2 + 240*a^5*b*c^4*d*f^2 - 480*a*b^5*c^2*d^3*f^2 + 600*a^2*b^4

*c*d^4*f^2 - 800*a^3*b^3*c^4*d*f^2 - 600*a^4*b^2*c*d^4*f^2 - 480*a^5*b*c^2*d^3*f^2)^2/4 - (c^10 + d^10 + 5*c^2

*d^8 + 10*c^4*d^6 + 10*c^6*d^4 + 5*c^8*d^2)*(16*a^12*f^4 + 16*b^12*f^4 + 96*a^2*b^10*f^4 + 240*a^4*b^8*f^4 + 3

20*a^6*b^6*f^4 + 240*a^8*b^4*f^4 + 96*a^10*b^2*f^4))^(1/2) - 4*a^6*c^5*f^2 + 4*b^6*c^5*f^2 - 24*a*b^5*d^5*f^2

- 24*a^5*b*d^5*f^2 - 20*a^6*c*d^4*f^2 + 20*b^6*c*d^4*f^2 - 60*a^2*b^4*c^5*f^2 + 60*a^4*b^2*c^5*f^2 + 80*a^3*b^

3*d^5*f^2 + 40*a^6*c^3*d^2*f^2 - 40*b^6*c^3*d^2*f^2 + 600*a^2*b^4*c^3*d^2*f^2 - 800*a^3*b^3*c^2*d^3*f^2 - 600*

a^4*b^2*c^3*d^2*f^2 - 120*a*b^5*c^4*d*f^2 - 120*a^5*b*c^4*d*f^2 + 240*a*b^5*c^2*d^3*f^2 - 300*a^2*b^4*c*d^4*f^

2 + 400*a^3*b^3*c^4*d*f^2 + 300*a^4*b^2*c*d^4*f^2 + 240*a^5*b*c^2*d^3*f^2)/(16*(a^12*f^4 + b^12*f^4 + 6*a^2*b^

10*f^4 + 15*a^4*b^8*f^4 + 20*a^6*b^6*f^4 + 15*a^8*b^4*f^4 + 6*a^10*b^2*f^4)))^(1/2) - ((c + d*tan(e + f*x))^(1

/2)*(8*a^19*b*d^15*f^2 - 1472*a*b^19*d^15*f^2 - 4*a^20*c*d^14*f^2 + 1216*b^20*c*d^14*f^2 + 776*a^3*b^17*d^15*f

^2 + 11328*a^5*b^15*d^15*f^2 + 10208*a^7*b^13*d^15*f^2 - 5056*a^9*b^11*d^15*f^2 - 5328*a^11*b^9*d^15*f^2 + 403

2*a^13*b^7*d^15*f^2 + 3552*a^15*b^5*d^15*f^2 + 384*a^17*b^3*d^15*f^2 + 580*b^20*c^3*d^12*f^2 - 4224*b^20*c^5*d

^10*f^2 + 576*b^20*c^7*d^8*f^2 + 12380*a^2*b^18*c^3*d^12*f^2 + 39808*a^2*b^18*c^5*d^10*f^2 - 3584*a^2*b^18*c^7

*d^8*f^2 - 18744*a^3*b^17*c^2*d^13*f^2 + 51200*a^3*b^17*c^4*d^11*f^2 - 11776*a^3*b^17*c^6*d^9*f^2 - 29360*a^4*

b^16*c^3*d^12*f^2 + 151680*a^4*b^16*c^5*d^10*f^2 - 13056*a^4*b^16*c^7*d^8*f^2 - 114112*a^5*b^15*c^2*d^13*f^2 +

2560*a^5*b^15*c^4*d^11*f^2 + 60160*a^5*b^15*c^6*d^9*f^2 - 219280*a^6*b^14*c^3*d^12*f^2 + 152960*a^6*b^14*c^5*

d^10*f^2 - 7680*a^6*b^14*c^7*d^8*f^2 - 41760*a^7*b^13*c^2*d^13*f^2 - 117760*a^7*b^13*c^4*d^11*f^2 + 109056*a^7

*b^13*c^6*d^9*f^2 - 345160*a^8*b^12*c^3*d^12*f^2 + 31104*a^8*b^12*c^5*d^10*f^2 + 8064*a^8*b^12*c^7*d^8*f^2 + 2

01280*a^9*b^11*c^2*d^13*f^2 - 186880*a^9*b^11*c^4*d^11*f^2 + 49280*a^9*b^11*c^6*d^9*f^2 - 175800*a^10*b^10*c^3

*d^12*f^2 - 1408*a^10*b^10*c^5*d^10*f^2 + 5632*a^10*b^10*c^7*d^8*f^2 + 230832*a^11*b^9*c^2*d^13*f^2 - 163840*a

^11*b^9*c^4*d^11*f^2 + 1536*a^11*b^9*c^6*d^9*f^2 + 34960*a^12*b^8*c^3*d^12*f^2 + 17792*a^12*b^8*c^5*d^10*f^2 -

3840*a^12*b^8*c^7*d^8*f^2 + 68032*a^13*b^7*c^2*d^13*f^2 - 84480*a^13*b^7*c^4*d^11*f^2 + 9984*a^13*b^7*c^6*d^9

*f^2 + 54960*a^14*b^6*c^3*d^12*f^2 - 2944*a^14*b^6*c^5*d^10*f^2 - 2560*a^14*b^6*c^7*d^8*f^2 - 5408*a^15*b^5*c^

2*d^13*f^2 - 15360*a^15*b^5*c^4*d^11*f^2 + 7680*a^15*b^5*c^6*d^9*f^2 + 11300*a^16*b^4*c^3*d^12*f^2 - 7936*a^16

*b^4*c^5*d^10*f^2 + 64*a^16*b^4*c^7*d^8*f^2 - 1664*a^17*b^3*c^2*d^13*f^2 + 2560*a^17*b^3*c^4*d^11*f^2 - 64*a^1

7*b^3*c^6*d^9*f^2 + 60*a^18*b^2*c^3*d^12*f^2 + 11328*a*b^19*c^2*d^13*f^2 + 20480*a*b^19*c^4*d^11*f^2 - 12864*a

*b^19*c^6*d^9*f^2 - 15244*a^2*b^18*c*d^14*f^2 - 32740*a^4*b^16*c*d^14*f^2 + 23824*a^6*b^14*c*d^14*f^2 + 73456*

a^8*b^12*c*d^14*f^2 + 2392*a^10*b^10*c*d^14*f^2 - 58424*a^12*b^8*c*d^14*f^2 - 29360*a^14*b^6*c*d^14*f^2 - 1936

*a^16*b^4*c*d^14*f^2 - 44*a^18*b^2*c*d^14*f^2 - 56*a^19*b*c^2*d^13*f^2))/(b^17*f^4 + a^16*b*f^4 + 8*a^2*b^15*f

^4 + 28*a^4*b^13*f^4 + 56*a^6*b^11*f^4 + 70*a^8*b^9*f^4 + 56*a^10*b^7*f^4 + 28*a^12*b^5*f^4 + 8*a^14*b^3*f^4))

*((((8*a^6*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5*d^5*f^2 + 48*a^5*b*d^5*f^2 + 40*a^6*c*d^4*f^2 - 40*b^6*c*d^4*f^2

+ 120*a^2*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2 - 160*a^3*b^3*d^5*f^2 - 80*a^6*c^3*d^2*f^2 + 80*b^6*c^3*d^2*f^2 -

1200*a^2*b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2*d^3*f^2 + 1200*a^4*b^2*c^3*d^2*f^2 + 240*a*b^5*c^4*d*f^2 + 240*a^

5*b*c^4*d*f^2 - 480*a*b^5*c^2*d^3*f^2 + 600*a^2*b^4*c*d^4*f^2 - 800*a^3*b^3*c^4*d*f^2 - 600*a^4*b^2*c*d^4*f^2

- 480*a^5*b*c^2*d^3*f^2)^2/4 - (c^10 + d^10 + 5*c^2*d^8 + 10*c^4*d^6 + 10*c^6*d^4 + 5*c^8*d^2)*(16*a^12*f^4 +

16*b^12*f^4 + 96*a^2*b^10*f^4 + 240*a^4*b^8*f^4 + 320*a^6*b^6*f^4 + 240*a^8*b^4*f^4 + 96*a^10*b^2*f^4))^(1/2)

- 4*a^6*c^5*f^2 + 4*b^6*c^5*f^2 - 24*a*b^5*d^5*f^2 - 24*a^5*b*d^5*f^2 - 20*a^6*c*d^4*f^2 + 20*b^6*c*d^4*f^2 -

60*a^2*b^4*c^5*f^2 + 60*a^4*b^2*c^5*f^2 + 80*a^3*b^3*d^5*f^2 + 40*a^6*c^3*d^2*f^2 - 40*b^6*c^3*d^2*f^2 + 600*a

^2*b^4*c^3*d^2*f^2 - 800*a^3*b^3*c^2*d^3*f^2 - 600*a^4*b^2*c^3*d^2*f^2 - 120*a*b^5*c^4*d*f^2 - 120*a^5*b*c^4*d

*f^2 + 240*a*b^5*c^2*d^3*f^2 - 300*a^2*b^4*c*d^4*f^2 + 400*a^3*b^3*c^4*d*f^2 + 300*a^4*b^2*c*d^4*f^2 + 240*a^5

*b*c^2*d^3*f^2)/(16*(a^12*f^4 + b^12*f^4 + 6*a^2*b^10*f^4 + 15*a^4*b^8*f^4 + 20*a^6*b^6*f^4 + 15*a^8*b^4*f^4 +

6*a^10*b^2*f^4)))^(1/2))*((((8*a^6*c^5*f^2 - 8*b^6*c^5*f^2 + 48*a*b^5*d^5*f^2 + 48*a^5*b*d^5*f^2 + 40*a^6*c*d

^4*f^2 - 40*b^6*c*d^4*f^2 + 120*a^2*b^4*c^5*f^2 - 120*a^4*b^2*c^5*f^2 - 160*a^3*b^3*d^5*f^2 - 80*a^6*c^3*d^2*f

^2 + 80*b^6*c^3*d^2*f^2 - 1200*a^2*b^4*c^3*d^2*f^2 + 1600*a^3*b^3*c^2*d^3*f^2 + 1200*a^4*b^2*c^3*d^2*f^2 + 240

*a*b^5*c^4*d*f^2 + 240*a^5*b*c^4*d*f^2 - 480*a*b^5*c^2*d^3*f^2 + 600*a^2*b^4*c*d^4*f^2 - 800*a^3*b^3*c^4*d*f^2

- 600*a^4*b^2*c*d^4*f^2 - 480*a^5*b*c^2*d^3*f^2)^2/4 - (c^10 + d^10 + 5*c^2*d^8 + 10*c^4*d^6 + 10*c^6*d^4 + 5

*c^8*d^2)*(16*a^12*f^4 + 16*b^12*f^4 + 96*a^2*b^10*f^4 + 240*a^4*b^8*f^4 + 320*a^6*b^6*f^4 + 240*a^8*b^4*f^4 +

96*a^10*b^2*f^4))^(1/2) - 4*a^6*c^5*f^2 + 4*b^6*c^5*f^2 - 24*a*b^5*d^5*f^2 - 24*a^5*b*d^5*f^2 - 20*a^6*c*d^4*

f^2 + 20*b^6*c*d^4*f^2 - 60*a^2*b^4*c^5*f^2 + 60*a^4*b^2*c^5*f^2 + 80*a^3*b^3*d^5*f^2 + 40*a^6*c^3*d^2*f^2 - 4

0*b^6*c^3*d^2*f^2 + 600*a^2*b^4*c^3*d^2*f^2 - 800*a^3*b^3*c^2*d^3*f^2 - 600*a^4*b^2*c^3*d^2*f^2 - 120*a*b^5*c^

4*d*f^2 - 120*a^5*b*c^4*d*f^2 + 240*a*b^5*c^2*d^3*f^2 - 300*a^2*b^4*c*d^4*f^2 + 400*a^3*b^3*c^4*d*f^2 + 300*a^

4*b^2*c*d^4*f^2 + 240*a^5*b*c^2*d^3*f^2)/(16*(a^12*f^4 + b^12*f^4 + 6*a^2*b^10*f^4 + 15*a^4*b^8*f^4 + 20*a^6*b

^6*f^4 + 15*a^8*b^4*f^4 + 6*a^10*b^2*f^4)))^(1/2) + ((c + d*tan(e + f*x))^(1/2)*(32*b^14*d^20 - a^14*d^20 - 97

*a^2*b^12*d^20 + 2082*a^4*b^10*d^20 - 3631*a^6*b^8*d^20 + 2300*a^8*b^6*d^20 - 79*a^10*b^4*d^20 - 30*a^12*b^2*d

^20 + 15*a^14*c^2*d^18 - 15*a^14*c^4*d^16 + a^14*c^6*d^14 - 33*b^14*c^2*d^18 + 4095*b^14*c^4*d^16 - 6399*b^14*

c^6*d^14 + 5265*b^14*c^8*d^12 - 1008*b^14*c^10*d^10 + 96*b^14*c^12*d^8 - 14310*a*b^13*c^3*d^17 + 63934*a*b^13*

c^5*d^15 - 74346*a*b^13*c^7*d^13 + 28400*a*b^13*c^9*d^11 - 2560*a*b^13*c^11*d^9 - 10860*a^3*b^11*c*d^19 + 3821

4*a^5*b^9*c*d^19 - 39944*a^7*b^7*c*d^19 + 9646*a^9*b^5*c*d^19 + 820*a^11*b^3*c*d^19 + 130*a^13*b*c^3*d^17 - 18

6*a^13*b*c^5*d^15 + 14*a^13*b*c^7*d^13 + 20433*a^2*b^12*c^2*d^18 - 179085*a^2*b^12*c^4*d^16 + 405679*a^2*b^12*

c^6*d^14 - 260450*a^2*b^12*c^8*d^12 + 44304*a^2*b^12*c^10*d^10 - 640*a^2*b^12*c^12*d^8 + 229060*a^3*b^11*c^3*d

^17 - 961988*a^3*b^11*c^5*d^15 + 1132092*a^3*b^11*c^7*d^13 - 355920*a^3*b^11*c^9*d^11 + 20480*a^3*b^11*c^11*d^

9 - 138253*a^4*b^10*c^2*d^18 + 1168655*a^4*b^10*c^4*d^16 - 2446851*a^4*b^10*c^6*d^14 + 1436095*a^4*b^10*c^8*d^

12 - 197600*a^4*b^10*c^10*d^10 + 3136*a^4*b^10*c^12*d^8 - 750450*a^5*b^9*c^3*d^17 + 2876874*a^5*b^9*c^5*d^15 -

2978014*a^5*b^9*c^7*d^13 + 845280*a^5*b^9*c^9*d^11 - 47104*a^5*b^9*c^11*d^9 + 251549*a^6*b^8*c^2*d^18 - 18852

85*a^6*b^8*c^4*d^16 + 3408435*a^6*b^8*c^6*d^14 - 1764060*a^6*b^8*c^8*d^12 + 235168*a^6*b^8*c^10*d^10 - 3712*a^

6*b^8*c^12*d^8 + 680600*a^7*b^7*c^3*d^17 - 2192216*a^7*b^7*c^5*d^15 + 1980392*a^7*b^7*c^7*d^13 - 507680*a^7*b^

7*c^9*d^11 + 26624*a^7*b^7*c^11*d^9 - 125875*a^8*b^6*c^2*d^18 + 770805*a^8*b^6*c^4*d^16 - 1218445*a^8*b^6*c^6*

d^14 + 557215*a^8*b^6*c^8*d^12 - 61232*a^8*b^6*c^10*d^10 + 608*a^8*b^6*c^12*d^8 - 130410*a^9*b^5*c^3*d^17 + 38

8978*a^9*b^5*c^5*d^15 - 317574*a^9*b^5*c^7*d^13 + 64560*a^9*b^5*c^9*d^11 - 1536*a^9*b^5*c^11*d^9 + 5315*a^10*b

^4*c^2*d^18 - 48895*a^10*b^4*c^4*d^16 + 84189*a^10*b^4*c^6*d^14 - 31970*a^10*b^4*c^8*d^12 + 1360*a^10*b^4*c^10

*d^10 - 2460*a^11*b^3*c^3*d^17 - 4260*a^11*b^3*c^5*d^15 + 5660*a^11*b^3*c^7*d^13 - 400*a^11*b^3*c^9*d^11 + 801

*a^12*b^2*c^2*d^18 - 1795*a^12*b^2*c^4*d^16 + 591*a^12*b^2*c^6*d^14 - 15*a^12*b^2*c^8*d^12 + 450*a*b^13*c*d^19

+ 10*a^13*b*c*d^19))/(b^17*f^4 + a^16*b*f^4 + 8*a^2*b^15*f^4 + 28*a^4*b^13*f^4 + 56*a^6*b^11*f^4 + 70*a^8*b^9