3.13.46 \(\int \frac {(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^3} \, dx\) [1246]

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

[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 = 1.29, 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 = {3646, 3730, 3734, 3620, 3618, 65, 214, 3715} \begin {gather*} -\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} \end {gather*}

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 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 3715

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 3730

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*Ta
n[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 3734

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, -
1]

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 ) \text {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 \text {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 \text {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 ) \text {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 \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 )}{(a-i b)^3 d f}-\frac {(c+i d)^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 )}{(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] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2946\) vs. \(2(355)=710\).
time = 6.47, size = 2946, normalized size = 8.30 \begin {gather*} \text {Result too large to show} \end {gather*}

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))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2484\) vs. \(2(317)=634\).
time = 0.55, size = 2485, normalized size = 7.00

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2485\)
default	\(\text {Expression too large to display}\)	\(2485\)

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,method=_RETURNVERBOSE)

[Out]

2/f*d^4*(1/d^4/(a^2+b^2)^3*(1/4/d*(1/2*(-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^3*c^2+(2*(c^2+d^2)^(1
/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^3*d^2-6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^2*b*c*d+3*(2*(c^2+d^2
)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*b^2*c^2-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*b^2*d^2+2*(2*(c
^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b^3*c*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^3-3*(2*(c^2+d^2)^(1/2)+2*
c)^(1/2)*a^3*c*d^2+9*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c^2*d-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*d^3-3*(2*
(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c^3+9*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c*d^2-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/
2)*b^3*c^2*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*d^3)*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*(-4*(c^2+d^2)^(1/2)*a^3*c*d^2+6*(c^2+d^2)^(1/2)*a^2*b*c^2*d-6*(c^2+d^2)^(1/2)*
a^2*b*d^3+12*(c^2+d^2)^(1/2)*a*b^2*c*d^2-2*(c^2+d^2)^(1/2)*b^3*c^2*d+2*(c^2+d^2)^(1/2)*b^3*d^3-1/2*(-(2*(c^2+d
^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^3*c^2+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^3*d^2-6*(2*(c^2+d
^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^2*b*c*d+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*b^2*c^2-3*(2*
(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*b^2*d^2+2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b^3*c*d+(
2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^3-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c*d^2+9*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)
*a^2*b*c^2*d-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*d^3-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c^3+9*(2*(c^2+d^2
)^(1/2)+2*c)^(1/2)*a*b^2*c*d^2-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c^2*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*d^3
)*(2*(c^2+d^2)^(1/2)+2*c)^(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)))+1/4/d*(-1/2*(-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a
^3*c^2+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^3*d^2-6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a
^2*b*c*d+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*b^2*c^2-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(
1/2)*a*b^2*d^2+2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b^3*c*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^3-3
*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c*d^2+9*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c^2*d-3*(2*(c^2+d^2)^(1/2)+2*c)
^(1/2)*a^2*b*d^3-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c^3+9*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c*d^2-3*(2*(c
^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c^2*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*d^3)*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*(4*(c^2+d^2)^(1/2)*a^3*c*d^2-6*(c^2+d^2)^(1/2)*a^2*b*c^2
*d+6*(c^2+d^2)^(1/2)*a^2*b*d^3-12*(c^2+d^2)^(1/2)*a*b^2*c*d^2+2*(c^2+d^2)^(1/2)*b^3*c^2*d-2*(c^2+d^2)^(1/2)*b^
3*d^3+1/2*(-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^3*c^2+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2
)*a^3*d^2-6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^2*b*c*d+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^
(1/2)*a*b^2*c^2-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*b^2*d^2+2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2
+d^2)^(1/2)*b^3*c*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^3-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c*d^2+9*(2*(c^2+
d^2)^(1/2)+2*c)^(1/2)*a^2*b*c^2*d-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*d^3-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*
b^2*c^3+9*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c*d^2-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c^2*d+(2*(c^2+d^2)^(1/
2)+2*c)^(1/2)*b^3*d^3)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan(((2*(c^2+d^2)^(1/2)
+2*c)^(1/2)-2*(c+d*tan(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))))+(a*d-b*c)/d^4/(a^2+b^2)^3*(((1/8*a^4*d^
2+a^3*b*c*d+5/4*a^2*b^2*d^2+a*b^3*c*d+9/8*b^4*d^2)*(c+d*tan(f*x+e))^(3/2)-1/8*d*(a^5*d^2-9*a^4*b*c*d+8*a^3*b^2
*c^2-6*a^3*b^2*d^2-2*a^2*b^3*c*d+8*a*b^4*c^2-7*a*b^4*d^2+7*b^5*c*d)/b*(c+d*tan(f*x+e))^(1/2))/((c+d*tan(f*x+e)
)*b+a*d-b*c)^2+1/8*(a^4*d^2+8*a^3*b*c*d-24*a^2*b^2*c^2+18*a^2*b^2*d^2-56*a*b^3*c*d+8*b^4*c^2-15*b^4*d^2)/b/((a
*d-b*c)*b)^(1/2)*arctan(b*(c+d*tan(f*x+e))^(1/2)/((a*d-b*c)*b)^(1/2))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 detail

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

[Out]

Timed out

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

[Out]

Timed out

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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((c+d*tan(f*x+e))^(5/2)/(a+b*tan(f*x+e))^3,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 = 44.60, size = 2500, normalized size = 7.04 \begin {gather*} \text {Too large to display} \end {gather*}

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*...

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