3.13.57 \(\int \frac {1}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}} \, dx\) [1257]

Optimal. Leaf size=211 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(i a+b) (c-i d)^{3/2} f}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(i a-b) (c+i d)^{3/2} f}-\frac {2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right ) (b c-a d)^{3/2} f}+\frac {2 d^2}{(b c-a d) \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}} \]

[Out]

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

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Rubi [A]
time = 0.65, antiderivative size = 211, 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 {2 b^{5/2} \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 ) (b c-a d)^{3/2}}+\frac {2 d^2}{f \left (c^2+d^2\right ) (b c-a d) \sqrt {c+d \tan (e+f x)}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (b+i a) (c-i d)^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (-b+i a) (c+i d)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2)),x]

[Out]

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

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Mathematica [A]
time = 2.13, size = 247, normalized size = 1.17 \begin {gather*} \frac {-\frac {i \left (\frac {(a+i b) (c+i d) (-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) (c-i d) (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 {2 b^{5/2} \left (c^2+d^2\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 {2 d^2}{\sqrt {c+d \tan (e+f x)}}}{(-b c+a d) \left (c^2+d^2\right ) f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2)),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2352\) vs. \(2(181)=362\).
time = 0.57, size = 2353, normalized size = 11.15

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2353\)
default	\(\text {Expression too large to display}\)	\(2353\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*tan(f*x+e))/(c+d*tan(f*x+e))^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/f*d^2*(-1/(a*d-b*c)/(c^2+d^2)/(c+d*tan(f*x+e))^(1/2)-1/(a*d-b*c)*b^3/d^2/(a^2+b^2)/((a*d-b*c)*b)^(1/2)*arcta
n(b*(c+d*tan(f*x+e))^(1/2)/((a*d-b*c)*b)^(1/2))+1/(c^2+d^2)/d^2/(a^2+b^2)*(1/4/d^2/(3*c^2-d^2)/(c^2+d^2)^(3/2)
*(-1/2*(-(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^4+(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d
^4+3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^5*d+2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c
^3*d^3-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c*d^5+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c
^6-2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^4*d^2-3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b
*c^2*d^4-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^6*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^4*d^3+3*(2*(c^2+d^2)^(1/2)+
2*c)^(1/2)*a*c^2*d^5-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d^7+6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^5*d^2+4*(2*(c^2+d
^2)^(1/2)+2*c)^(1/2)*b*c^3*d^4-2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c*d^6)*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*(-12*a*c^5*d^3-8*a*c^3*d^5+4*a*c*d^7-6*b*c^6*d^2+2*b*c^4*d
^4+6*b*c^2*d^6-2*b*d^8+1/2*(-(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^4+(c^2+d^2)^(3/2)*(2*(c^2+d^2)^
(1/2)+2*c)^(1/2)*b*d^4+3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^5*d+2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^
(1/2)+2*c)^(1/2)*a*c^3*d^3-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c*d^5+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^
(1/2)+2*c)^(1/2)*b*c^6-2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^4*d^2-3*(c^2+d^2)^(1/2)*(2*(c^2+d^2
)^(1/2)+2*c)^(1/2)*b*c^2*d^4-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^6*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^4*d^3+3
*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2*d^5-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d^7+6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b
*c^5*d^2+4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^3*d^4-2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c*d^6)*(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)))+1/4/d^2/(3*c^2-d^2)/(c^2+d^2)^(3/2)*(1/2*(-(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c
)^(1/2)*b*c^4+(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d^4+3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1
/2)*a*c^5*d+2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^3*d^3-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^
(1/2)*a*c*d^5+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^6-2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1
/2)*b*c^4*d^2-3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^2*d^4-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^6*
d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^4*d^3+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2*d^5-(2*(c^2+d^2)^(1/2)+2*c)^(1
/2)*a*d^7+6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^5*d^2+4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^3*d^4-2*(2*(c^2+d^2)^(
1/2)+2*c)^(1/2)*b*c*d^6)*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*(12*a*c^5*d^3+8*a*c^3*d^5-4*a*c*d^7+6*b*c^6*d^2-2*b*c^4*d^4-6*b*c^2*d^6+2*b*d^8-1/2*(-(c^2+d^2)^(3/2)*(2*
(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^4+(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d^4+3*(c^2+d^2)^(1/2)*(2*(c^2
+d^2)^(1/2)+2*c)^(1/2)*a*c^5*d+2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^3*d^3-(c^2+d^2)^(1/2)*(2*(c
^2+d^2)^(1/2)+2*c)^(1/2)*a*c*d^5+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^6-2*(c^2+d^2)^(1/2)*(2*(c^2
+d^2)^(1/2)+2*c)^(1/2)*b*c^4*d^2-3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^2*d^4-3*(2*(c^2+d^2)^(1/2
)+2*c)^(1/2)*a*c^6*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^4*d^3+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2*d^5-(2*(c^2
+d^2)^(1/2)+2*c)^(1/2)*a*d^7+6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^5*d^2+4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^3*d
^4-2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c*d^6)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arcta
n((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)))))

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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/(a+b*tan(f*x+e))/(c+d*tan(f*x+e))^(3/2),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. 31213 vs. \(2 (179) = 358\).
time = 208.29, size = 62414, normalized size = 295.80 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(f*x+e))/(c+d*tan(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

[1/4*(4*sqrt(2)*(((a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*c^11 - (a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*c^10*d +
3*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*c^9*d^2 - 3*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*c^8*d^3 + 2*(a^6*b +
3*a^4*b^3 + 3*a^2*b^5 + b^7)*c^7*d^4 - 2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*c^6*d^5 - 2*(a^6*b + 3*a^4*b^3
+ 3*a^2*b^5 + b^7)*c^5*d^6 + 2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*c^4*d^7 - 3*(a^6*b + 3*a^4*b^3 + 3*a^2*b
^5 + b^7)*c^3*d^8 + 3*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*c^2*d^9 - (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*c*
d^10 + (a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^11)*f^5*cos(f*x + e)^2 + 2*((a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^
7)*c^10*d - (a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*c^9*d^2 + 4*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*c^8*d^3 -
4*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*c^7*d^4 + 6*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*c^6*d^5 - 6*(a^7 + 3
*a^5*b^2 + 3*a^3*b^4 + a*b^6)*c^5*d^6 + 4*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*c^4*d^7 - 4*(a^7 + 3*a^5*b^2 +
3*a^3*b^4 + a*b^6)*c^3*d^8 + (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*c^2*d^9 - (a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a
*b^6)*c*d^10)*f^5*cos(f*x + e)*sin(f*x + e) + ((a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*c^9*d^2 - (a^7 + 3*a^5*b^
2 + 3*a^3*b^4 + a*b^6)*c^8*d^3 + 4*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*c^7*d^4 - 4*(a^7 + 3*a^5*b^2 + 3*a^3*
b^4 + a*b^6)*c^6*d^5 + 6*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*c^5*d^6 - 6*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^
6)*c^4*d^7 + 4*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*c^3*d^8 - 4*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*c^2*d^9
+ (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*c*d^10 - (a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^11)*f^5)*sqrt(((a^4
+ 2*a^2*b^2 + b^4)*c^6 + 3*(a^4 + 2*a^2*b^2 + b^4)*c^4*d^2 + 3*(a^4 + 2*a^2*b^2 + b^4)*c^2*d^4 + (a^4 + 2*a^2*
b^2 + b^4)*d^6 + ((a^6 + a^4*b^2 - a^2*b^4 - b^6)*c^9 - 6*(a^5*b + 2*a^3*b^3 + a*b^5)*c^8*d - 16*(a^5*b + 2*a^
3*b^3 + a*b^5)*c^6*d^3 - 6*(a^6 + a^4*b^2 - a^2*b^4 - b^6)*c^5*d^4 - 12*(a^5*b + 2*a^3*b^3 + a*b^5)*c^4*d^5 -
8*(a^6 + a^4*b^2 - a^2*b^4 - b^6)*c^3*d^6 - 3*(a^6 + a^4*b^2 - a^2*b^4 - b^6)*c*d^8 + 2*(a^5*b + 2*a^3*b^3 + a
*b^5)*d^9)*f^2*sqrt(1/(((a^4 + 2*a^2*b^2 + b^4)*c^6 + 3*(a^4 + 2*a^2*b^2 + b^4)*c^4*d^2 + 3*(a^4 + 2*a^2*b^2 +
b^4)*c^2*d^4 + (a^4 + 2*a^2*b^2 + b^4)*d^6)*f^4)))/(4*a^2*b^2*c^6 + 12*(a^3*b - a*b^3)*c^5*d + 3*(3*a^4 - 14*
a^2*b^2 + 3*b^4)*c^4*d^2 - 40*(a^3*b - a*b^3)*c^3*d^3 - 6*(a^4 - 8*a^2*b^2 + b^4)*c^2*d^4 + 12*(a^3*b - a*b^3)
*c*d^5 + (a^4 - 2*a^2*b^2 + b^4)*d^6))*sqrt((4*a^2*b^2*c^6 + 12*(a^3*b - a*b^3)*c^5*d + 3*(3*a^4 - 14*a^2*b^2
+ 3*b^4)*c^4*d^2 - 40*(a^3*b - a*b^3)*c^3*d^3 - 6*(a^4 - 8*a^2*b^2 + b^4)*c^2*d^4 + 12*(a^3*b - a*b^3)*c*d^5 +
(a^4 - 2*a^2*b^2 + b^4)*d^6)/(((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*c^12 + 6*(a^8 + 4*a^6*b^2 + 6*
a^4*b^4 + 4*a^2*b^6 + b^8)*c^10*d^2 + 15*(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*c^8*d^4 + 20*(a^8 + 4
*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*c^6*d^6 + 15*(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*c^4*d^8 +
6*(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*c^2*d^10 + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*
d^12)*f^4))*(1/(((a^4 + 2*a^2*b^2 + b^4)*c^6 + 3*(a^4 + 2*a^2*b^2 + b^4)*c^4*d^2 + 3*(a^4 + 2*a^2*b^2 + b^4)*c
^2*d^4 + (a^4 + 2*a^2*b^2 + b^4)*d^6)*f^4))^(3/4)*arctan(((2*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c^13 + 3*
(a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*c^12*d + 4*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c^11*d^2 + 14*(a^8 + 2*
a^6*b^2 - 2*a^2*b^6 - b^8)*c^10*d^3 - 10*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c^9*d^4 + 25*(a^8 + 2*a^6*b^2
- 2*a^2*b^6 - b^8)*c^8*d^5 - 40*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c^7*d^6 + 20*(a^8 + 2*a^6*b^2 - 2*a^2
*b^6 - b^8)*c^6*d^7 - 50*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c^5*d^8 + 5*(a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^
8)*c^4*d^9 - 28*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c^3*d^10 - 2*(a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*c^2*d
^11 - 6*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c*d^12 - (a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*d^13)*f^4*sqrt((4
*a^2*b^2*c^6 + 12*(a^3*b - a*b^3)*c^5*d + 3*(3*a^4 - 14*a^2*b^2 + 3*b^4)*c^4*d^2 - 40*(a^3*b - a*b^3)*c^3*d^3
- 6*(a^4 - 8*a^2*b^2 + b^4)*c^2*d^4 + 12*(a^3*b - a*b^3)*c*d^5 + (a^4 - 2*a^2*b^2 + b^4)*d^6)/(((a^8 + 4*a^6*b
^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*c^12 + 6*(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*c^10*d^2 + 15*(a^8
+ 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*c^8*d^4 + 20*(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*c^6*d^
6 + 15*(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*c^4*d^8 + 6*(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 +
b^8)*c^2*d^10 + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^12)*f^4))*sqrt(1/(((a^4 + 2*a^2*b^2 + b^4)*c
^6 + 3*(a^4 + 2*a^2*b^2 + b^4)*c^4*d^2 + 3*(a^4 + 2*a^2*b^2 + b^4)*c^2*d^4 + (a^4 + 2*a^2*b^2 + b^4)*d^6)*f^4)
) + (2*(a^5*b + 2*a^3*b^3 + a*b^5)*c^10 + 3*(a^6 + a^4*b^2 - a^2*b^4 - b^6)*c^9*d + 8*(a^6 + a^4*b^2 - a^2*b^4
- b^6)*c^7*d^3 - 12*(a^5*b + 2*a^3*b^3 + a*b^5)*c^6*d^4 + 6*(a^6 + a^4*b^2 - a^2*b^4 - b^6)*c^5*d^5 - 16*(a^5
*b + 2*a^3*b^3 + a*b^5)*c^4*d^6 - 6*(a^5*b + 2*...

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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 ) \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(f*x+e))/(c+d*tan(f*x+e))**(3/2),x)

[Out]

Integral(1/((a + b*tan(e + f*x))*(c + d*tan(e + f*x))**(3/2)), 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(1/(a+b*tan(f*x+e))/(c+d*tan(f*x+e))^(3/2),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 = 20.91, size = 2500, normalized size = 11.85 \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))*(c + d*tan(e + f*x))^(3/2)),x)

[Out]

(log(((((2*a^2*b^2*d^6*f^4 - b^4*d^6*f^4 - 4*a^2*b^2*c^6*f^4 - a^4*d^6*f^4 + 6*a^4*c^2*d^4*f^4 - 9*a^4*c^4*d^2
*f^4 + 6*b^4*c^2*d^4*f^4 - 9*b^4*c^4*d^2*f^4 - 48*a^2*b^2*c^2*d^4*f^4 + 42*a^2*b^2*c^4*d^2*f^4 + 12*a*b^3*c*d^
5*f^4 + 12*a*b^3*c^5*d*f^4 - 12*a^3*b*c*d^5*f^4 - 12*a^3*b*c^5*d*f^4 - 40*a*b^3*c^3*d^3*f^4 + 40*a^3*b*c^3*d^3
*f^4)^(1/2) - a^2*c^3*f^2 + b^2*c^3*f^2 + 3*a^2*c*d^2*f^2 - 3*b^2*c*d^2*f^2 - 2*a*b*d^3*f^2 + 6*a*b*c^2*d*f^2)
/(a^4*c^6*f^4 + a^4*d^6*f^4 + b^4*c^6*f^4 + b^4*d^6*f^4 + 2*a^2*b^2*c^6*f^4 + 2*a^2*b^2*d^6*f^4 + 3*a^4*c^2*d^
4*f^4 + 3*a^4*c^4*d^2*f^4 + 3*b^4*c^2*d^4*f^4 + 3*b^4*c^4*d^2*f^4 + 6*a^2*b^2*c^2*d^4*f^4 + 6*a^2*b^2*c^4*d^2*
f^4))^(1/2)*(((((2*a^2*b^2*d^6*f^4 - b^4*d^6*f^4 - 4*a^2*b^2*c^6*f^4 - a^4*d^6*f^4 + 6*a^4*c^2*d^4*f^4 - 9*a^4
*c^4*d^2*f^4 + 6*b^4*c^2*d^4*f^4 - 9*b^4*c^4*d^2*f^4 - 48*a^2*b^2*c^2*d^4*f^4 + 42*a^2*b^2*c^4*d^2*f^4 + 12*a*
b^3*c*d^5*f^4 + 12*a*b^3*c^5*d*f^4 - 12*a^3*b*c*d^5*f^4 - 12*a^3*b*c^5*d*f^4 - 40*a*b^3*c^3*d^3*f^4 + 40*a^3*b
*c^3*d^3*f^4)^(1/2) - a^2*c^3*f^2 + b^2*c^3*f^2 + 3*a^2*c*d^2*f^2 - 3*b^2*c*d^2*f^2 - 2*a*b*d^3*f^2 + 6*a*b*c^
2*d*f^2)/(a^4*c^6*f^4 + a^4*d^6*f^4 + b^4*c^6*f^4 + b^4*d^6*f^4 + 2*a^2*b^2*c^6*f^4 + 2*a^2*b^2*d^6*f^4 + 3*a^
4*c^2*d^4*f^4 + 3*a^4*c^4*d^2*f^4 + 3*b^4*c^2*d^4*f^4 + 3*b^4*c^4*d^2*f^4 + 6*a^2*b^2*c^2*d^4*f^4 + 6*a^2*b^2*
c^4*d^2*f^4))^(1/2)*(((((2*a^2*b^2*d^6*f^4 - b^4*d^6*f^4 - 4*a^2*b^2*c^6*f^4 - a^4*d^6*f^4 + 6*a^4*c^2*d^4*f^4
- 9*a^4*c^4*d^2*f^4 + 6*b^4*c^2*d^4*f^4 - 9*b^4*c^4*d^2*f^4 - 48*a^2*b^2*c^2*d^4*f^4 + 42*a^2*b^2*c^4*d^2*f^4
+ 12*a*b^3*c*d^5*f^4 + 12*a*b^3*c^5*d*f^4 - 12*a^3*b*c*d^5*f^4 - 12*a^3*b*c^5*d*f^4 - 40*a*b^3*c^3*d^3*f^4 +
40*a^3*b*c^3*d^3*f^4)^(1/2) - a^2*c^3*f^2 + b^2*c^3*f^2 + 3*a^2*c*d^2*f^2 - 3*b^2*c*d^2*f^2 - 2*a*b*d^3*f^2 +
6*a*b*c^2*d*f^2)/(a^4*c^6*f^4 + a^4*d^6*f^4 + b^4*c^6*f^4 + b^4*d^6*f^4 + 2*a^2*b^2*c^6*f^4 + 2*a^2*b^2*d^6*f^
4 + 3*a^4*c^2*d^4*f^4 + 3*a^4*c^4*d^2*f^4 + 3*b^4*c^2*d^4*f^4 + 3*b^4*c^4*d^2*f^4 + 6*a^2*b^2*c^2*d^4*f^4 + 6*
a^2*b^2*c^4*d^2*f^4))^(1/2)*(((((2*a^2*b^2*d^6*f^4 - b^4*d^6*f^4 - 4*a^2*b^2*c^6*f^4 - a^4*d^6*f^4 + 6*a^4*c^2
*d^4*f^4 - 9*a^4*c^4*d^2*f^4 + 6*b^4*c^2*d^4*f^4 - 9*b^4*c^4*d^2*f^4 - 48*a^2*b^2*c^2*d^4*f^4 + 42*a^2*b^2*c^4
*d^2*f^4 + 12*a*b^3*c*d^5*f^4 + 12*a*b^3*c^5*d*f^4 - 12*a^3*b*c*d^5*f^4 - 12*a^3*b*c^5*d*f^4 - 40*a*b^3*c^3*d^
3*f^4 + 40*a^3*b*c^3*d^3*f^4)^(1/2) - a^2*c^3*f^2 + b^2*c^3*f^2 + 3*a^2*c*d^2*f^2 - 3*b^2*c*d^2*f^2 - 2*a*b*d^
3*f^2 + 6*a*b*c^2*d*f^2)/(a^4*c^6*f^4 + a^4*d^6*f^4 + b^4*c^6*f^4 + b^4*d^6*f^4 + 2*a^2*b^2*c^6*f^4 + 2*a^2*b^
2*d^6*f^4 + 3*a^4*c^2*d^4*f^4 + 3*a^4*c^4*d^2*f^4 + 3*b^4*c^2*d^4*f^4 + 3*b^4*c^4*d^2*f^4 + 6*a^2*b^2*c^2*d^4*
f^4 + 6*a^2*b^2*c^4*d^2*f^4))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(512*a^9*b^9*d^37*f^9 + 512*a^11*b^7*d^37*f^9 -
512*a^13*b^5*d^37*f^9 - 512*a^15*b^3*d^37*f^9 - 512*b^18*c^9*d^28*f^9 - 5376*b^18*c^11*d^26*f^9 - 25344*b^18*
c^13*d^24*f^9 - 70656*b^18*c^15*d^22*f^9 - 129024*b^18*c^17*d^20*f^9 - 161280*b^18*c^19*d^18*f^9 - 139776*b^18
*c^21*d^16*f^9 - 82944*b^18*c^23*d^14*f^9 - 32256*b^18*c^25*d^12*f^9 - 7424*b^18*c^27*d^10*f^9 - 768*b^18*c^29
*d^8*f^9 - 18432*a^2*b^16*c^7*d^30*f^9 - 191744*a^2*b^16*c^9*d^28*f^9 - 897536*a^2*b^16*c^11*d^26*f^9 - 249062
4*a^2*b^16*c^13*d^24*f^9 - 4540416*a^2*b^16*c^15*d^22*f^9 - 5687808*a^2*b^16*c^17*d^20*f^9 - 4967424*a^2*b^16*
c^19*d^18*f^9 - 2996736*a^2*b^16*c^21*d^16*f^9 - 1204224*a^2*b^16*c^23*d^14*f^9 - 297216*a^2*b^16*c^25*d^12*f^
9 - 37376*a^2*b^16*c^27*d^10*f^9 - 1280*a^2*b^16*c^29*d^8*f^9 + 43008*a^3*b^15*c^6*d^31*f^9 + 446976*a^3*b^15*
c^8*d^29*f^9 + 2098176*a^3*b^15*c^10*d^27*f^9 + 5865984*a^3*b^15*c^12*d^25*f^9 + 10838016*a^3*b^15*c^14*d^23*f
^9 + 13870080*a^3*b^15*c^16*d^21*f^9 + 12515328*a^3*b^15*c^18*d^19*f^9 + 7934976*a^3*b^15*c^20*d^17*f^9 + 3446
784*a^3*b^15*c^22*d^15*f^9 + 969216*a^3*b^15*c^24*d^13*f^9 + 156672*a^3*b^15*c^26*d^11*f^9 + 10752*a^3*b^15*c^
28*d^9*f^9 - 64512*a^4*b^14*c^5*d^32*f^9 - 674304*a^4*b^14*c^7*d^30*f^9 - 3204352*a^4*b^14*c^9*d^28*f^9 - 9140
224*a^4*b^14*c^11*d^26*f^9 - 17392896*a^4*b^14*c^13*d^24*f^9 - 23190528*a^4*b^14*c^15*d^22*f^9 - 22116864*a^4*
b^14*c^17*d^20*f^9 - 15095808*a^4*b^14*c^19*d^18*f^9 - 7233024*a^4*b^14*c^21*d^16*f^9 - 2320896*a^4*b^14*c^23*
d^14*f^9 - 450816*a^4*b^14*c^25*d^12*f^9 - 40960*a^4*b^14*c^27*d^10*f^9 - 256*a^4*b^14*c^29*d^8*f^9 + 64512*a^
5*b^13*c^4*d^33*f^9 + 688128*a^5*b^13*c^6*d^31*f^9 + 3365376*a^5*b^13*c^8*d^29*f^9 + 9968640*a^5*b^13*c^10*d^2
7*f^9 + 19883520*a^5*b^13*c^12*d^25*f^9 + 28053504*a^5*b^13*c^14*d^23*f^9 + 28578816*a^5*b^13*c^16*d^21*f^9 +
21030912*a^5*b^13*c^18*d^19*f^9 + 10967040*a^5*b^13*c^20*d^17*f^9 + 3870720*a^5*b^13*c^22*d^15*f^9 + 840192*a^
5*b^13*c^24*d^13*f^9 + 89088*a^5*b^13*c^26*d^11*f^9 + 1536*a^5*b^13*c^28*d^9*f^9 - 43008*a^6*b^12*c^3*d^34*f^9
- 483840*a^6*b^12*c^5*d^32*f^9 - 2497536*a^6*b^12*c^7*d^30*f^9 - 7803136*a^6*b^12*c^9*d^28*f^9 - 16383488*a^6
*b^12*c^11*d^26*f^9 - 24254208*a^6*b^12*c^13*d^24*f^9 - 25817088*a^6*b^12*c^15*d^22*f^9 - 19751424*a^6*b^12*c^
17*d^20*f^9 - 10644480*a^6*b^12*c^19*d^18*f^9 -...

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