3.13.52 \(\int \frac {1}{(a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}} \, dx\) [1252]

Optimal. Leaf size=244 \[ -\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(a-i b)^2 \sqrt {c-i d} f}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(a+i b)^2 \sqrt {c+i d} f}-\frac {b^{3/2} \left (4 a b c-5 a^2 d-b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right )^2 (b c-a d)^{3/2} f}-\frac {b^2 \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))} \]

[Out]

-b^(3/2)*(-5*a^2*d+4*a*b*c-b^2*d)*arctanh(b^(1/2)*(c+d*tan(f*x+e))^(1/2)/(-a*d+b*c)^(1/2))/(a^2+b^2)^2/(-a*d+b
*c)^(3/2)/f-I*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/(a-I*b)^2/f/(c-I*d)^(1/2)+I*arctanh((c+d*tan(f*x+e
))^(1/2)/(c+I*d)^(1/2))/(a+I*b)^2/f/(c+I*d)^(1/2)-b^2*(c+d*tan(f*x+e))^(1/2)/(a^2+b^2)/(-a*d+b*c)/f/(a+b*tan(f
*x+e))

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Rubi [A]
time = 0.61, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {3650, 3734, 3620, 3618, 65, 214, 3715} \begin {gather*} -\frac {b^2 \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))}-\frac {b^{3/2} \left (-5 a^2 d+4 a b c-b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{f \left (a^2+b^2\right )^2 (b c-a d)^{3/2}}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (a-i b)^2 \sqrt {c-i d}}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (a+i b)^2 \sqrt {c+i d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((a - I*b)^2*Sqrt[c - I*d]*f) + (I*ArcTanh[Sqrt[c + d*T
an[e + f*x]]/Sqrt[c + I*d]])/((a + I*b)^2*Sqrt[c + I*d]*f) - (b^(3/2)*(4*a*b*c - 5*a^2*d - b^2*d)*ArcTanh[(Sqr
t[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/((a^2 + b^2)^2*(b*c - a*d)^(3/2)*f) - (b^2*Sqrt[c + d*Tan[e +
f*x]])/((a^2 + b^2)*(b*c - a*d)*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 3650

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 + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d))), x] + D
ist[1/((m + 1)*(a^2 + b^2)*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c -
a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*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] && I
ntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] &&
NeQ[a, 0])))

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 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 {1}{(a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}} \, dx &=-\frac {b^2 \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))}-\frac {\int \frac {\frac {1}{2} \left (-2 a b c+2 a^2 d+b^2 d\right )+b (b c-a d) \tan (e+f x)+\frac {1}{2} b^2 d \tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{\left (a^2+b^2\right ) (b c-a d)}\\ &=-\frac {b^2 \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))}-\frac {\int \frac {-\left (a^2-b^2\right ) (b c-a d)+2 a b (b c-a d) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{\left (a^2+b^2\right )^2 (b c-a d)}+\frac {\left (b^2 \left (4 a b c-5 a^2 d-b^2 d\right )\right ) \int \frac {1+\tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{2 \left (a^2+b^2\right )^2 (b c-a d)}\\ &=-\frac {b^2 \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))}+\frac {\int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (a-i b)^2}+\frac {\int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (a+i b)^2}+\frac {\left (b^2 \left (4 a b c-5 a^2 d-b^2 d\right )\right ) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 \left (a^2+b^2\right )^2 (b c-a d) f}\\ &=-\frac {b^2 \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))}+\frac {i \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 (a-i b)^2 f}-\frac {i \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 (a+i b)^2 f}+\frac {\left (b^2 \left (4 a b c-5 a^2 d-b^2 d\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 )}{\left (a^2+b^2\right )^2 d (b c-a d) f}\\ &=-\frac {b^{3/2} \left (4 a b c-5 a^2 d-b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right )^2 (b c-a d)^{3/2} f}-\frac {b^2 \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))}-\frac {\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)^2 d f}-\frac {\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)^2 d f}\\ &=-\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(a-i b)^2 \sqrt {c-i d} f}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(a+i b)^2 \sqrt {c+i d} f}-\frac {b^{3/2} \left (4 a b c-5 a^2 d-b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right )^2 (b c-a d)^{3/2} f}-\frac {b^2 \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))}\\ \end {align*}

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Mathematica [A]
time = 2.43, size = 258, normalized size = 1.06 \begin {gather*} \frac {-\frac {i \left (\frac {(a+i b)^2 (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d}}+\frac {(a-i b)^2 (-b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d}}\right )}{a^2+b^2}+\frac {b^{3/2} \left (-4 a b c+5 a^2 d+b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right ) \sqrt {b c-a d}}-\frac {b^2 \sqrt {c+d \tan (e+f x)}}{a+b \tan (e+f x)}}{\left (a^2+b^2\right ) (b c-a d) f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]]),x]

[Out]

(((-I)*(((a + I*b)^2*(b*c - a*d)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/Sqrt[c - I*d] + ((a - I*b)^2
*(-(b*c) + a*d)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/Sqrt[c + I*d]))/(a^2 + b^2) + (b^(3/2)*(-4*a*
b*c + 5*a^2*d + b^2*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/((a^2 + b^2)*Sqrt[b*c - a*
d]) - (b^2*Sqrt[c + d*Tan[e + f*x]])/(a + b*Tan[e + f*x]))/((a^2 + b^2)*(b*c - a*d)*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2143\) vs. \(2(212)=424\).
time = 0.56, size = 2144, normalized size = 8.79

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2144\)
default	\(\text {Expression too large to display}\)	\(2144\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/f*d^3*(1/(a^2+b^2)^2/d^3*(1/4/(c^2+d^2)^(3/2)/d^2*(1/2*(-2*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b
*c+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^2*d+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*d
^3+2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^3+2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b
*c*d^2-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^2*d-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b
^2*d^3-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^3*d-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c*d^3+2*(2*(c^2+d^2)^(1/2)+2*
c)^(1/2)*a*b*c^2*d^2+2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*d^4+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^3*d+(2*(c^2+d
^2)^(1/2)+2*c)^(1/2)*b^2*c*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*(2*a^2*c^2*d^3+2*a^2*d^5+4*a*b*c^3*d^2+4*a*b*c*d^4-2*b^2*c^2*d^3-2*b^2*d^5-1/2*(-2*(c^2+d^2)^(3/2)
*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^2*d+(c^2+d^2)^(1/2)*(
2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*d^3+2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^3+2*(c^2+d^2)^(1/2)
*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c*d^2-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^2*d-(c^2+d^2)^(1/
2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*d^3-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^3*d-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)
*a^2*c*d^3+2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^2*d^2+2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*d^4+(2*(c^2+d^2)^(1
/2)+2*c)^(1/2)*b^2*c^3*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c*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/(c^2+d^2)^(3/2)/d^2*(1/2*(2*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c-(c^2+d^2)^(1/2)*(2*(c^2
+d^2)^(1/2)+2*c)^(1/2)*a^2*c^2*d-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*d^3-2*(c^2+d^2)^(1/2)*(2*(c
^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^3-2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c*d^2+(c^2+d^2)^(1/2)*(2*
(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^2*d+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*d^3+(2*(c^2+d^2)^(1/2)+
2*c)^(1/2)*a^2*c^3*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c*d^3-2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^2*d^2-2*(2*
(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*d^4-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^3*d-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c
*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*(2*a^2*c^2*d^3
+2*a^2*d^5+4*a*b*c^3*d^2+4*a*b*c*d^4-2*b^2*c^2*d^3-2*b^2*d^5+1/2*(2*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1
/2)*a*b*c-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^2*d-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2
)*a^2*d^3-2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^3-2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1
/2)*a*b*c*d^2+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^2*d+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^
(1/2)*b^2*d^3+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^3*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c*d^3-2*(2*(c^2+d^2)^(
1/2)+2*c)^(1/2)*a*b*c^2*d^2-2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*d^4-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^3*d-(2
*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c*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))))+b^2/d^3/(a^2+b^2)^2*(1/
2*d*(a^2+b^2)/(a*d-b*c)*(c+d*tan(f*x+e))^(1/2)/((c+d*tan(f*x+e))*b+a*d-b*c)+1/2*(5*a^2*d-4*a*b*c+b^2*d)/(a*d-b
*c)/((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(1/(c+d*tan(f*x+e))^(1/2)/(a+b*tan(f*x+e))^2,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 [B] Leaf count of result is larger than twice the leaf count of optimal. 23867 vs. \(2 (209) = 418\).
time = 174.50, size = 47721, normalized size = 195.58 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(4*sqrt(2)*(((a^14*b + 5*a^12*b^3 + 9*a^10*b^5 + 5*a^8*b^7 - 5*a^6*b^9 - 9*a^4*b^11 - 5*a^2*b^13 - b^15)*
c^3 - (a^15 + 5*a^13*b^2 + 9*a^11*b^4 + 5*a^9*b^6 - 5*a^7*b^8 - 9*a^5*b^10 - 5*a^3*b^12 - a*b^14)*c^2*d + (a^1
4*b + 5*a^12*b^3 + 9*a^10*b^5 + 5*a^8*b^7 - 5*a^6*b^9 - 9*a^4*b^11 - 5*a^2*b^13 - b^15)*c*d^2 - (a^15 + 5*a^13
*b^2 + 9*a^11*b^4 + 5*a^9*b^6 - 5*a^7*b^8 - 9*a^5*b^10 - 5*a^3*b^12 - a*b^14)*d^3)*f^5*cos(f*x + e)^2 + 2*((a^
13*b^2 + 6*a^11*b^4 + 15*a^9*b^6 + 20*a^7*b^8 + 15*a^5*b^10 + 6*a^3*b^12 + a*b^14)*c^3 - (a^14*b + 6*a^12*b^3
+ 15*a^10*b^5 + 20*a^8*b^7 + 15*a^6*b^9 + 6*a^4*b^11 + a^2*b^13)*c^2*d + (a^13*b^2 + 6*a^11*b^4 + 15*a^9*b^6 +
20*a^7*b^8 + 15*a^5*b^10 + 6*a^3*b^12 + a*b^14)*c*d^2 - (a^14*b + 6*a^12*b^3 + 15*a^10*b^5 + 20*a^8*b^7 + 15*
a^6*b^9 + 6*a^4*b^11 + a^2*b^13)*d^3)*f^5*cos(f*x + e)*sin(f*x + e) + ((a^12*b^3 + 6*a^10*b^5 + 15*a^8*b^7 + 2
0*a^6*b^9 + 15*a^4*b^11 + 6*a^2*b^13 + b^15)*c^3 - (a^13*b^2 + 6*a^11*b^4 + 15*a^9*b^6 + 20*a^7*b^8 + 15*a^5*b
^10 + 6*a^3*b^12 + a*b^14)*c^2*d + (a^12*b^3 + 6*a^10*b^5 + 15*a^8*b^7 + 20*a^6*b^9 + 15*a^4*b^11 + 6*a^2*b^13
+ b^15)*c*d^2 - (a^13*b^2 + 6*a^11*b^4 + 15*a^9*b^6 + 20*a^7*b^8 + 15*a^5*b^10 + 6*a^3*b^12 + a*b^14)*d^3)*f^
5)*sqrt((((a^12 - 2*a^10*b^2 - 17*a^8*b^4 - 28*a^6*b^6 - 17*a^4*b^8 - 2*a^2*b^10 + b^12)*c^3 - 4*(a^11*b + 3*a
^9*b^3 + 2*a^7*b^5 - 2*a^5*b^7 - 3*a^3*b^9 - a*b^11)*c^2*d + (a^12 - 2*a^10*b^2 - 17*a^8*b^4 - 28*a^6*b^6 - 17
*a^4*b^8 - 2*a^2*b^10 + b^12)*c*d^2 - 4*(a^11*b + 3*a^9*b^3 + 2*a^7*b^5 - 2*a^5*b^7 - 3*a^3*b^9 - a*b^11)*d^3)
*f^2*sqrt(1/(((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*c^2 + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 +
b^8)*d^2)*f^4)) + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*c^2 + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*
b^6 + b^8)*d^2)/(16*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*c^2 + 8*(a^7*b - 7*a^5*b^3 + 7*a^3*b^5 - a*b^7)*c*d + (a^8
- 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*d^2))*sqrt((16*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*c^2 + 8*(a^7*b -
7*a^5*b^3 + 7*a^3*b^5 - a*b^7)*c*d + (a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*d^2)/(((a^16 + 8*a^14
*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 + b^16)*c^4 + 2*(a^16 +
8*a^14*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 + b^16)*c^2*d^2
+ (a^16 + 8*a^14*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 + b^16)
*d^4)*f^4))*(1/(((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*c^2 + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^
6 + b^8)*d^2)*f^4))^(3/4)*arctan(((4*(a^15*b + 5*a^13*b^3 + 9*a^11*b^5 + 5*a^9*b^7 - 5*a^7*b^9 - 9*a^5*b^11 -
5*a^3*b^13 - a*b^15)*c^5 + (a^16 - 20*a^12*b^4 - 64*a^10*b^6 - 90*a^8*b^8 - 64*a^6*b^10 - 20*a^4*b^12 + b^16)*
c^4*d + 8*(a^15*b + 5*a^13*b^3 + 9*a^11*b^5 + 5*a^9*b^7 - 5*a^7*b^9 - 9*a^5*b^11 - 5*a^3*b^13 - a*b^15)*c^3*d^
2 + 2*(a^16 - 20*a^12*b^4 - 64*a^10*b^6 - 90*a^8*b^8 - 64*a^6*b^10 - 20*a^4*b^12 + b^16)*c^2*d^3 + 4*(a^15*b +
5*a^13*b^3 + 9*a^11*b^5 + 5*a^9*b^7 - 5*a^7*b^9 - 9*a^5*b^11 - 5*a^3*b^13 - a*b^15)*c*d^4 + (a^16 - 20*a^12*b
^4 - 64*a^10*b^6 - 90*a^8*b^8 - 64*a^6*b^10 - 20*a^4*b^12 + b^16)*d^5)*f^4*sqrt((16*(a^6*b^2 - 2*a^4*b^4 + a^2
*b^6)*c^2 + 8*(a^7*b - 7*a^5*b^3 + 7*a^3*b^5 - a*b^7)*c*d + (a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)
*d^2)/(((a^16 + 8*a^14*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 +
b^16)*c^4 + 2*(a^16 + 8*a^14*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a^2
*b^14 + b^16)*c^2*d^2 + (a^16 + 8*a^14*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^1
2 + 8*a^2*b^14 + b^16)*d^4)*f^4))*sqrt(1/(((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*c^2 + (a^8 + 4*a^6*
b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^2)*f^4)) + (4*(a^11*b + 3*a^9*b^3 + 2*a^7*b^5 - 2*a^5*b^7 - 3*a^3*b^9 - a
*b^11)*c^4 + (a^12 - 2*a^10*b^2 - 17*a^8*b^4 - 28*a^6*b^6 - 17*a^4*b^8 - 2*a^2*b^10 + b^12)*c^3*d + 4*(a^11*b
+ 3*a^9*b^3 + 2*a^7*b^5 - 2*a^5*b^7 - 3*a^3*b^9 - a*b^11)*c^2*d^2 + (a^12 - 2*a^10*b^2 - 17*a^8*b^4 - 28*a^6*b
^6 - 17*a^4*b^8 - 2*a^2*b^10 + b^12)*c*d^3)*f^2*sqrt((16*(a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*c^2 + 8*(a^7*b - 7*a^
5*b^3 + 7*a^3*b^5 - a*b^7)*c*d + (a^8 - 12*a^6*b^2 + 38*a^4*b^4 - 12*a^2*b^6 + b^8)*d^2)/(((a^16 + 8*a^14*b^2
+ 28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 + b^16)*c^4 + 2*(a^16 + 8*a^
14*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 + b^16)*c^2*d^2 + (a^
16 + 8*a^14*b^2 + 28*a^12*b^4 + 56*a^10*b^6 + 70*a^8*b^8 + 56*a^6*b^10 + 28*a^4*b^12 + 8*a^2*b^14 + b^16)*d^4)
*f^4)) - sqrt(2)*((2*(a^17*b + 8*a^15*b^3 + 28*a^13*b^5 + 56*a^11*b^7 + 70*a^9*b^9 + 56*a^7*b^11 + 28*a^5*b^13
+ 8*a^3*b^15 + a*b^17)*c^5 + (a^18 + 7*a^16*b^2 + 20*a^14*b^4 + 28*a^12*b^6 + 14*a^10*b^8 - 14*a^8*b^10 - 28*
a^6*b^12 - 20*a^4*b^14 - 7*a^2*b^16 - b^18)*c^4*d + 4*(a^17*b + 8*a^15*b^3 + 28*a^13*b^5 + 56*a^11*b^7 + 70*a^
9*b^9 + 56*a^7*b^11 + 28*a^5*b^13 + 8*a^3*b^15 ...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b \tan {\left (e + f x \right )}\right )^{2} \sqrt {c + d \tan {\left (e + f x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((a + b*tan(e + f*x))**2*sqrt(c + d*tan(e + f*x))), 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(1/(c+d*tan(f*x+e))^(1/2)/(a+b*tan(f*x+e))^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 = 15.26, size = 2500, normalized size = 10.25 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan(((((((16*(2*b^13*d^11*f^2 - 24*a^2*b^11*d^11*f^2 - 196*a^4*b^9*d^11*f^2 - 120*a^6*b^7*d^11*f^2 + 50*a^8*
b^5*d^11*f^2 + 8*b^13*c^2*d^9*f^2 - 8*a^2*b^11*c^2*d^9*f^2 + 64*a^3*b^10*c^3*d^8*f^2 - 232*a^4*b^9*c^2*d^9*f^2
+ 96*a^5*b^8*c^3*d^8*f^2 - 216*a^6*b^7*c^2*d^9*f^2 - 32*a*b^12*c^3*d^8*f^2 + 208*a^3*b^10*c*d^10*f^2 + 288*a^
5*b^8*c*d^10*f^2 + 80*a^7*b^6*c*d^10*f^2))/(a^10*d^2*f^5 + b^10*c^2*f^5 + 4*a^2*b^8*c^2*f^5 + 6*a^4*b^6*c^2*f^
5 + 4*a^6*b^4*c^2*f^5 + a^8*b^2*c^2*f^5 + a^2*b^8*d^2*f^5 + 4*a^4*b^6*d^2*f^5 + 6*a^6*b^4*d^2*f^5 + 4*a^8*b^2*
d^2*f^5 - 2*a*b^9*c*d*f^5 - 2*a^9*b*c*d*f^5 - 8*a^3*b^7*c*d*f^5 - 12*a^5*b^5*c*d*f^5 - 8*a^7*b^3*c*d*f^5) + ((
(16*(c + d*tan(e + f*x))^(1/2)*(8*a*b^14*d^11*f^2 + 4*b^15*c*d^10*f^2 + 36*a^3*b^12*d^11*f^2 + 316*a^5*b^10*d^
11*f^2 + 552*a^7*b^8*d^11*f^2 + 256*a^9*b^6*d^11*f^2 - 12*a^11*b^4*d^11*f^2 - 4*a^13*b^2*d^11*f^2 - 20*b^15*c^
3*d^8*f^2 + 116*a^2*b^13*c^3*d^8*f^2 - 220*a^3*b^12*c^2*d^9*f^2 + 216*a^4*b^11*c^3*d^8*f^2 - 104*a^5*b^10*c^2*
d^9*f^2 + 8*a^6*b^9*c^3*d^8*f^2 + 232*a^7*b^8*c^2*d^9*f^2 - 68*a^8*b^7*c^3*d^8*f^2 + 156*a^9*b^6*c^2*d^9*f^2 +
4*a^10*b^5*c^3*d^8*f^2 - 12*a^11*b^4*c^2*d^9*f^2 - 52*a*b^14*c^2*d^9*f^2 + 80*a^2*b^13*c*d^10*f^2 - 156*a^4*b
^11*c*d^10*f^2 - 640*a^6*b^9*c*d^10*f^2 - 500*a^8*b^7*c*d^10*f^2 - 80*a^10*b^5*c*d^10*f^2 + 12*a^12*b^3*c*d^10
*f^2))/(a^10*d^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^8*b^2*c^2*
f^4 + a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2*f^4 + 6*a^6*b^4*d^2*f^4 + 4*a^8*b^2*d^2*f^4 - 2*a*b^9*c*d*f^4 - 2*a^9*b*
c*d*f^4 - 8*a^3*b^7*c*d*f^4 - 12*a^5*b^5*c*d*f^4 - 8*a^7*b^3*c*d*f^4) - (((16*(16*a*b^16*d^12*f^4 - 16*b^17*c*
d^11*f^4 + 136*a^3*b^14*d^12*f^4 + 432*a^5*b^12*d^12*f^4 + 680*a^7*b^10*d^12*f^4 + 560*a^9*b^8*d^12*f^4 + 216*
a^11*b^6*d^12*f^4 + 16*a^13*b^4*d^12*f^4 - 8*a^15*b^2*d^12*f^4 - 8*b^17*c^3*d^9*f^4 - 128*a^2*b^15*c^3*d^9*f^4
+ 352*a^3*b^14*c^2*d^10*f^4 + 160*a^3*b^14*c^4*d^8*f^4 - 520*a^4*b^13*c^3*d^9*f^4 + 920*a^5*b^12*c^2*d^10*f^4
+ 320*a^5*b^12*c^4*d^8*f^4 - 960*a^6*b^11*c^3*d^9*f^4 + 1280*a^7*b^10*c^2*d^10*f^4 + 320*a^7*b^10*c^4*d^8*f^4
- 920*a^8*b^9*c^3*d^9*f^4 + 1000*a^9*b^8*c^2*d^10*f^4 + 160*a^9*b^8*c^4*d^8*f^4 - 448*a^10*b^7*c^3*d^9*f^4 +
416*a^11*b^6*c^2*d^10*f^4 + 32*a^11*b^6*c^4*d^8*f^4 - 88*a^12*b^5*c^3*d^9*f^4 + 72*a^13*b^4*c^2*d^10*f^4 + 56*
a*b^16*c^2*d^10*f^4 + 32*a*b^16*c^4*d^8*f^4 - 184*a^2*b^15*c*d^11*f^4 - 688*a^4*b^13*c*d^11*f^4 - 1240*a^6*b^1
1*c*d^11*f^4 - 1200*a^8*b^9*c*d^11*f^4 - 616*a^10*b^7*c*d^11*f^4 - 144*a^12*b^5*c*d^11*f^4 - 8*a^14*b^3*c*d^11
*f^4))/(a^10*d^2*f^5 + b^10*c^2*f^5 + 4*a^2*b^8*c^2*f^5 + 6*a^4*b^6*c^2*f^5 + 4*a^6*b^4*c^2*f^5 + a^8*b^2*c^2*
f^5 + a^2*b^8*d^2*f^5 + 4*a^4*b^6*d^2*f^5 + 6*a^6*b^4*d^2*f^5 + 4*a^8*b^2*d^2*f^5 - 2*a*b^9*c*d*f^5 - 2*a^9*b*
c*d*f^5 - 8*a^3*b^7*c*d*f^5 - 12*a^5*b^5*c*d*f^5 - 8*a^7*b^3*c*d*f^5) + (16*(-(b^7*d^2 + 16*a^2*b^5*c^2 + 10*a
^2*b^5*d^2 + 25*a^4*b^3*d^2 - 8*a*b^6*c*d - 40*a^3*b^4*c*d)*(a^11*d^3*f^2 - b^11*c^3*f^2 - 4*a^2*b^9*c^3*f^2 -
6*a^4*b^7*c^3*f^2 - 4*a^6*b^5*c^3*f^2 - a^8*b^3*c^3*f^2 + a^3*b^8*d^3*f^2 + 4*a^5*b^6*d^3*f^2 + 6*a^7*b^4*d^3
*f^2 + 4*a^9*b^2*d^3*f^2 + 3*a*b^10*c^2*d*f^2 - 3*a^10*b*c*d^2*f^2 - 3*a^2*b^9*c*d^2*f^2 + 12*a^3*b^8*c^2*d*f^
2 - 12*a^4*b^7*c*d^2*f^2 + 18*a^5*b^6*c^2*d*f^2 - 18*a^6*b^5*c*d^2*f^2 + 12*a^7*b^4*c^2*d*f^2 - 12*a^8*b^3*c*d
^2*f^2 + 3*a^9*b^2*c^2*d*f^2))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(32*a^2*b^17*d^12*f^4 + 160*a^4*b^15*d^12*f^4
+ 288*a^6*b^13*d^12*f^4 + 160*a^8*b^11*d^12*f^4 - 160*a^10*b^9*d^12*f^4 - 288*a^12*b^7*d^12*f^4 - 160*a^14*b^5
*d^12*f^4 - 32*a^16*b^3*d^12*f^4 + 32*b^19*c^2*d^10*f^4 + 48*b^19*c^4*d^8*f^4 + 176*a^2*b^17*c^2*d^10*f^4 + 27
2*a^2*b^17*c^4*d^8*f^4 - 432*a^3*b^16*c^3*d^9*f^4 + 336*a^4*b^15*c^2*d^10*f^4 + 624*a^4*b^15*c^4*d^8*f^4 - 912
*a^5*b^14*c^3*d^9*f^4 + 112*a^6*b^13*c^2*d^10*f^4 + 720*a^6*b^13*c^4*d^8*f^4 - 880*a^7*b^12*c^3*d^9*f^4 - 560*
a^8*b^11*c^2*d^10*f^4 + 400*a^8*b^11*c^4*d^8*f^4 - 240*a^9*b^10*c^3*d^9*f^4 - 1008*a^10*b^9*c^2*d^10*f^4 + 48*
a^10*b^9*c^4*d^8*f^4 + 240*a^11*b^8*c^3*d^9*f^4 - 784*a^12*b^7*c^2*d^10*f^4 - 48*a^12*b^7*c^4*d^8*f^4 + 208*a^
13*b^6*c^3*d^9*f^4 - 304*a^14*b^5*c^2*d^10*f^4 - 16*a^14*b^5*c^4*d^8*f^4 + 48*a^15*b^4*c^3*d^9*f^4 - 48*a^16*b
^3*c^2*d^10*f^4 - 64*a*b^18*c*d^11*f^4 - 80*a*b^18*c^3*d^9*f^4 - 304*a^3*b^16*c*d^11*f^4 - 464*a^5*b^14*c*d^11
*f^4 + 16*a^7*b^12*c*d^11*f^4 + 880*a^9*b^10*c*d^11*f^4 + 1136*a^11*b^8*c*d^11*f^4 + 656*a^13*b^6*c*d^11*f^4 +
176*a^15*b^4*c*d^11*f^4 + 16*a^17*b^2*c*d^11*f^4))/((b^9*(8*a^2*c^3*f^2 + 6*a^2*c*d^2*f^2) + b^3*(2*a^8*c^3*f
^2 + 24*a^8*c*d^2*f^2) + b^7*(12*a^4*c^3*f^2 + 24*a^4*c*d^2*f^2) + b^5*(8*a^6*c^3*f^2 + 36*a^6*c*d^2*f^2) - b^
2*(8*a^9*d^3*f^2 + 6*a^9*c^2*d*f^2) - b^8*(2*a^3*d^3*f^2 + 24*a^3*c^2*d*f^2) - b^4*(12*a^7*d^3*f^2 + 24*a^7*c^
2*d*f^2) - b^6*(8*a^5*d^3*f^2 + 36*a^5*c^2*d*f^2) - 2*a^11*d^3*f^2 + 2*b^11*c^3*f^2 - 6*a*b^10*c^2*d*f^2 + 6*a
^10*b*c*d^2*f^2)*(a^10*d^2*f^4 + b^10*c^2*f^4 + 4*a^2*b^8*c^2*f^4 + 6*a^4*b^6*c^2*f^4 + 4*a^6*b^4*c^2*f^4 + a^
8*b^2*c^2*f^4 + a^2*b^8*d^2*f^4 + 4*a^4*b^6*d^2...

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