\(\int \frac {\csc ^2(x)}{a+b \cot (x)+c \csc (x)} \, dx\) [460]

Optimal result

Integrand size = 17, antiderivative size = 120 \[ \int \frac {\csc ^2(x)}{a+b \cot (x)+c \csc (x)} \, dx=-\frac {2 a c \text {arctanh}\left (\frac {a-(b-c) \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2-c^2}}\right )}{\left (b^2-c^2\right ) \sqrt {a^2+b^2-c^2}}+\frac {\log \left (\tan \left (\frac {x}{2}\right )\right )}{b+c}-\frac {b \log \left (b+c+2 a \tan \left (\frac {x}{2}\right )-(b-c) \tan ^2\left (\frac {x}{2}\right )\right )}{b^2-c^2} \]

[Out]

ln(tan(1/2*x))/(b+c)-b*ln(b+c+2*a*tan(1/2*x)-(b-c)*tan(1/2*x)^2)/(b^2-c^2)-2*a*c*arctanh((a-(b-c)*tan(1/2*x))/
(a^2+b^2-c^2)^(1/2))/(b^2-c^2)/(a^2+b^2-c^2)^(1/2)

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {4482, 12, 1642, 648, 632, 212, 642} \[ \int \frac {\csc ^2(x)}{a+b \cot (x)+c \csc (x)} \, dx=-\frac {2 a c \text {arctanh}\left (\frac {a-(b-c) \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2-c^2}}\right )}{\left (b^2-c^2\right ) \sqrt {a^2+b^2-c^2}}-\frac {b \log \left (2 a \tan \left (\frac {x}{2}\right )-\left ((b-c) \tan ^2\left (\frac {x}{2}\right )\right )+b+c\right )}{b^2-c^2}+\frac {\log \left (\tan \left (\frac {x}{2}\right )\right )}{b+c} \]

[In]

Int[Csc[x]^2/(a + b*Cot[x] + c*Csc[x]),x]

[Out]

(-2*a*c*ArcTanh[(a - (b - c)*Tan[x/2])/Sqrt[a^2 + b^2 - c^2]])/((b^2 - c^2)*Sqrt[a^2 + b^2 - c^2]) + Log[Tan[x
/2]]/(b + c) - (b*Log[b + c + 2*a*Tan[x/2] - (b - c)*Tan[x/2]^2])/(b^2 - c^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1642

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 4482

Int[u_, x_Symbol] :> Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\csc (x)}{c+b \cos (x)+a \sin (x)} \, dx \\ & = 2 \text {Subst}\left (\int \frac {1+x^2}{2 x \left (b+c+2 a x+(-b+c) x^2\right )} \, dx,x,\tan \left (\frac {x}{2}\right )\right ) \\ & = \text {Subst}\left (\int \frac {1+x^2}{x \left (b+c+2 a x+(-b+c) x^2\right )} \, dx,x,\tan \left (\frac {x}{2}\right )\right ) \\ & = \text {Subst}\left (\int \left (\frac {1}{(b+c) x}+\frac {2 (-a+b x)}{(b+c) \left (b+c+2 a x-(b-c) x^2\right )}\right ) \, dx,x,\tan \left (\frac {x}{2}\right )\right ) \\ & = \frac {\log \left (\tan \left (\frac {x}{2}\right )\right )}{b+c}+\frac {2 \text {Subst}\left (\int \frac {-a+b x}{b+c+2 a x+(-b+c) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{b+c} \\ & = \frac {\log \left (\tan \left (\frac {x}{2}\right )\right )}{b+c}-\frac {b \text {Subst}\left (\int \frac {2 a+2 (-b+c) x}{b+c+2 a x+(-b+c) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{b^2-c^2}+\frac {(2 a c) \text {Subst}\left (\int \frac {1}{b+c+2 a x+(-b+c) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{b^2-c^2} \\ & = \frac {\log \left (\tan \left (\frac {x}{2}\right )\right )}{b+c}-\frac {b \log \left (b+c+2 a \tan \left (\frac {x}{2}\right )-(b-c) \tan ^2\left (\frac {x}{2}\right )\right )}{b^2-c^2}-\frac {(4 a c) \text {Subst}\left (\int \frac {1}{4 \left (a^2+b^2-c^2\right )-x^2} \, dx,x,2 a+2 (-b+c) \tan \left (\frac {x}{2}\right )\right )}{b^2-c^2} \\ & = -\frac {2 a c \text {arctanh}\left (\frac {a-(b-c) \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2-c^2}}\right )}{\left (b^2-c^2\right ) \sqrt {a^2+b^2-c^2}}+\frac {\log \left (\tan \left (\frac {x}{2}\right )\right )}{b+c}-\frac {b \log \left (b+c+2 a \tan \left (\frac {x}{2}\right )-(b-c) \tan ^2\left (\frac {x}{2}\right )\right )}{b^2-c^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.98 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.87 \[ \int \frac {\csc ^2(x)}{a+b \cot (x)+c \csc (x)} \, dx=\frac {\frac {2 a c \text {arctanh}\left (\frac {a+(-b+c) \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2-c^2}}\right )}{\sqrt {a^2+b^2-c^2}}-(b+c) \log \left (\cos \left (\frac {x}{2}\right )\right )+(-b+c) \log \left (\sin \left (\frac {x}{2}\right )\right )+b \log (c+b \cos (x)+a \sin (x))}{(-b+c) (b+c)} \]

[In]

Integrate[Csc[x]^2/(a + b*Cot[x] + c*Csc[x]),x]

[Out]

((2*a*c*ArcTanh[(a + (-b + c)*Tan[x/2])/Sqrt[a^2 + b^2 - c^2]])/Sqrt[a^2 + b^2 - c^2] - (b + c)*Log[Cos[x/2]]
+ (-b + c)*Log[Sin[x/2]] + b*Log[c + b*Cos[x] + a*Sin[x]])/((-b + c)*(b + c))

Maple [A] (verified)

Time = 1.05 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.02

method	result	size
default	\(\frac {\frac {b \ln \left (-\tan \left (\frac {x}{2}\right )^{2} b +c \tan \left (\frac {x}{2}\right )^{2}+2 a \tan \left (\frac {x}{2}\right )+b +c \right )}{-b +c}+\frac {\left (-2 a -\frac {2 b a}{-b +c}\right ) \arctan \left (\frac {2 \left (-b +c \right ) \tan \left (\frac {x}{2}\right )+2 a}{2 \sqrt {-a^{2}-b^{2}+c^{2}}}\right )}{\sqrt {-a^{2}-b^{2}+c^{2}}}}{b +c}+\frac {\ln \left (\tan \left (\frac {x}{2}\right )\right )}{b +c}\)	\(123\)
risch	\(\text {Expression too large to display}\)	\(1342\)

[In]

int(csc(x)^2/(a+b*cot(x)+c*csc(x)),x,method=_RETURNVERBOSE)

[Out]

1/(b+c)*(b/(-b+c)*ln(-tan(1/2*x)^2*b+c*tan(1/2*x)^2+2*a*tan(1/2*x)+b+c)+(-2*a-2*b*a/(-b+c))/(-a^2-b^2+c^2)^(1/
2)*arctan(1/2*(2*(-b+c)*tan(1/2*x)+2*a)/(-a^2-b^2+c^2)^(1/2)))+ln(tan(1/2*x))/(b+c)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 260 vs. \(2 (110) = 220\).

Time = 1.67 (sec) , antiderivative size = 669, normalized size of antiderivative = 5.58 \[ \int \frac {\csc ^2(x)}{a+b \cot (x)+c \csc (x)} \, dx=\left [-\frac {\sqrt {a^{2} + b^{2} - c^{2}} a c \log \left (\frac {a^{4} + 3 \, a^{2} b^{2} + 2 \, b^{4} + {\left (a^{2} - b^{2}\right )} c^{2} + 2 \, {\left (a^{2} b + b^{3}\right )} c \cos \left (x\right ) + {\left (a^{4} - b^{4} - 2 \, {\left (a^{2} - b^{2}\right )} c^{2}\right )} \cos \left (x\right )^{2} + 2 \, {\left ({\left (a^{3} + a b^{2}\right )} c - {\left (a^{3} b + a b^{3} - 2 \, a b c^{2}\right )} \cos \left (x\right )\right )} \sin \left (x\right ) + 2 \, {\left (2 \, a b c \cos \left (x\right )^{2} - a b c + {\left (a^{3} + a b^{2}\right )} \cos \left (x\right ) - {\left (a^{2} b + b^{3} - {\left (a^{2} - b^{2}\right )} c \cos \left (x\right )\right )} \sin \left (x\right )\right )} \sqrt {a^{2} + b^{2} - c^{2}}}{2 \, b c \cos \left (x\right ) - {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + a^{2} + c^{2} + 2 \, {\left (a b \cos \left (x\right ) + a c\right )} \sin \left (x\right )}\right ) + {\left (a^{2} b + b^{3} - b c^{2}\right )} \log \left (2 \, b c \cos \left (x\right ) - {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + a^{2} + c^{2} + 2 \, {\left (a b \cos \left (x\right ) + a c\right )} \sin \left (x\right )\right ) - {\left (a^{2} b + b^{3} - b c^{2} - c^{3} + {\left (a^{2} + b^{2}\right )} c\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - {\left (a^{2} b + b^{3} - b c^{2} + c^{3} - {\left (a^{2} + b^{2}\right )} c\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right )}{2 \, {\left (a^{2} b^{2} + b^{4} + c^{4} - {\left (a^{2} + 2 \, b^{2}\right )} c^{2}\right )}}, \frac {2 \, \sqrt {-a^{2} - b^{2} + c^{2}} a c \arctan \left (\frac {{\left (b c \cos \left (x\right ) + a c \sin \left (x\right ) + a^{2} + b^{2}\right )} \sqrt {-a^{2} - b^{2} + c^{2}}}{{\left (a^{3} + a b^{2} - a c^{2}\right )} \cos \left (x\right ) - {\left (a^{2} b + b^{3} - b c^{2}\right )} \sin \left (x\right )}\right ) - {\left (a^{2} b + b^{3} - b c^{2}\right )} \log \left (2 \, b c \cos \left (x\right ) - {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + a^{2} + c^{2} + 2 \, {\left (a b \cos \left (x\right ) + a c\right )} \sin \left (x\right )\right ) + {\left (a^{2} b + b^{3} - b c^{2} - c^{3} + {\left (a^{2} + b^{2}\right )} c\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + {\left (a^{2} b + b^{3} - b c^{2} + c^{3} - {\left (a^{2} + b^{2}\right )} c\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right )}{2 \, {\left (a^{2} b^{2} + b^{4} + c^{4} - {\left (a^{2} + 2 \, b^{2}\right )} c^{2}\right )}}\right ] \]

[In]

integrate(csc(x)^2/(a+b*cot(x)+c*csc(x)),x, algorithm="fricas")

[Out]

[-1/2*(sqrt(a^2 + b^2 - c^2)*a*c*log((a^4 + 3*a^2*b^2 + 2*b^4 + (a^2 - b^2)*c^2 + 2*(a^2*b + b^3)*c*cos(x) + (
a^4 - b^4 - 2*(a^2 - b^2)*c^2)*cos(x)^2 + 2*((a^3 + a*b^2)*c - (a^3*b + a*b^3 - 2*a*b*c^2)*cos(x))*sin(x) + 2*
(2*a*b*c*cos(x)^2 - a*b*c + (a^3 + a*b^2)*cos(x) - (a^2*b + b^3 - (a^2 - b^2)*c*cos(x))*sin(x))*sqrt(a^2 + b^2
- c^2))/(2*b*c*cos(x) - (a^2 - b^2)*cos(x)^2 + a^2 + c^2 + 2*(a*b*cos(x) + a*c)*sin(x))) + (a^2*b + b^3 - b*c
^2)*log(2*b*c*cos(x) - (a^2 - b^2)*cos(x)^2 + a^2 + c^2 + 2*(a*b*cos(x) + a*c)*sin(x)) - (a^2*b + b^3 - b*c^2
- c^3 + (a^2 + b^2)*c)*log(1/2*cos(x) + 1/2) - (a^2*b + b^3 - b*c^2 + c^3 - (a^2 + b^2)*c)*log(-1/2*cos(x) + 1
/2))/(a^2*b^2 + b^4 + c^4 - (a^2 + 2*b^2)*c^2), 1/2*(2*sqrt(-a^2 - b^2 + c^2)*a*c*arctan((b*c*cos(x) + a*c*sin
(x) + a^2 + b^2)*sqrt(-a^2 - b^2 + c^2)/((a^3 + a*b^2 - a*c^2)*cos(x) - (a^2*b + b^3 - b*c^2)*sin(x))) - (a^2*
b + b^3 - b*c^2)*log(2*b*c*cos(x) - (a^2 - b^2)*cos(x)^2 + a^2 + c^2 + 2*(a*b*cos(x) + a*c)*sin(x)) + (a^2*b +
b^3 - b*c^2 - c^3 + (a^2 + b^2)*c)*log(1/2*cos(x) + 1/2) + (a^2*b + b^3 - b*c^2 + c^3 - (a^2 + b^2)*c)*log(-1
/2*cos(x) + 1/2))/(a^2*b^2 + b^4 + c^4 - (a^2 + 2*b^2)*c^2)]

Sympy [F]

\[ \int \frac {\csc ^2(x)}{a+b \cot (x)+c \csc (x)} \, dx=\int \frac {\csc ^{2}{\left (x \right )}}{a + b \cot {\left (x \right )} + c \csc {\left (x \right )}}\, dx \]

[In]

integrate(csc(x)**2/(a+b*cot(x)+c*csc(x)),x)

[Out]

Integral(csc(x)**2/(a + b*cot(x) + c*csc(x)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\csc ^2(x)}{a+b \cot (x)+c \csc (x)} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(csc(x)^2/(a+b*cot(x)+c*csc(x)),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(c^2-b^2-a^2>0)', see `assume?`
for more de

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.18 \[ \int \frac {\csc ^2(x)}{a+b \cot (x)+c \csc (x)} \, dx=\frac {2 \, {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, b + 2 \, c\right ) + \arctan \left (-\frac {b \tan \left (\frac {1}{2} \, x\right ) - c \tan \left (\frac {1}{2} \, x\right ) - a}{\sqrt {-a^{2} - b^{2} + c^{2}}}\right )\right )} a c}{\sqrt {-a^{2} - b^{2} + c^{2}} {\left (b^{2} - c^{2}\right )}} - \frac {b \log \left (-b \tan \left (\frac {1}{2} \, x\right )^{2} + c \tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, a \tan \left (\frac {1}{2} \, x\right ) + b + c\right )}{b^{2} - c^{2}} + \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{b + c} \]

[In]

integrate(csc(x)^2/(a+b*cot(x)+c*csc(x)),x, algorithm="giac")

[Out]

2*(pi*floor(1/2*x/pi + 1/2)*sgn(-2*b + 2*c) + arctan(-(b*tan(1/2*x) - c*tan(1/2*x) - a)/sqrt(-a^2 - b^2 + c^2)
))*a*c/(sqrt(-a^2 - b^2 + c^2)*(b^2 - c^2)) - b*log(-b*tan(1/2*x)^2 + c*tan(1/2*x)^2 + 2*a*tan(1/2*x) + b + c)
/(b^2 - c^2) + log(abs(tan(1/2*x)))/(b + c)

Mupad [B] (verification not implemented)

Time = 35.16 (sec) , antiderivative size = 531, normalized size of antiderivative = 4.42 \[ \int \frac {\csc ^2(x)}{a+b \cot (x)+c \csc (x)} \, dx=\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{b+c}-\frac {\ln \left (2\,a-2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )-\frac {\left (\mathrm {tan}\left (\frac {x}{2}\right )\,\left (-8\,a^2-8\,b^2+6\,b\,c+2\,c^2\right )-4\,a\,c+\frac {2\,\left (b-c\right )\,\left (a^2\,b-b\,c^2+b^3+a\,c\,\sqrt {a^2+b^2-c^2}\right )\,\left (4\,\mathrm {tan}\left (\frac {x}{2}\right )\,a^2+a\,b+a\,c+3\,\mathrm {tan}\left (\frac {x}{2}\right )\,b^2-3\,\mathrm {tan}\left (\frac {x}{2}\right )\,c^2\right )}{\left (b^2-c^2\right )\,\left (a^2+b^2-c^2\right )}\right )\,\left (a^2\,b-b\,c^2+b^3+a\,c\,\sqrt {a^2+b^2-c^2}\right )}{\left (b^2-c^2\right )\,\left (a^2+b^2-c^2\right )}\right )\,\left (b\,\left (a^2-c^2\right )+b^3+a\,c\,\sqrt {a^2+b^2-c^2}\right )}{\left (b^2-c^2\right )\,\left (a^2+b^2-c^2\right )}-\frac {\ln \left (2\,a-2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )-\frac {\left (\mathrm {tan}\left (\frac {x}{2}\right )\,\left (-8\,a^2-8\,b^2+6\,b\,c+2\,c^2\right )-4\,a\,c+\frac {2\,\left (b-c\right )\,\left (a^2\,b-b\,c^2+b^3-a\,c\,\sqrt {a^2+b^2-c^2}\right )\,\left (4\,\mathrm {tan}\left (\frac {x}{2}\right )\,a^2+a\,b+a\,c+3\,\mathrm {tan}\left (\frac {x}{2}\right )\,b^2-3\,\mathrm {tan}\left (\frac {x}{2}\right )\,c^2\right )}{\left (b^2-c^2\right )\,\left (a^2+b^2-c^2\right )}\right )\,\left (a^2\,b-b\,c^2+b^3-a\,c\,\sqrt {a^2+b^2-c^2}\right )}{\left (b^2-c^2\right )\,\left (a^2+b^2-c^2\right )}\right )\,\left (b\,\left (a^2-c^2\right )+b^3-a\,c\,\sqrt {a^2+b^2-c^2}\right )}{\left (b^2-c^2\right )\,\left (a^2+b^2-c^2\right )} \]

[In]

int(1/(sin(x)^2*(a + c/sin(x) + b*cot(x))),x)

[Out]

log(tan(x/2))/(b + c) - (log(2*a - 2*b*tan(x/2) - ((tan(x/2)*(6*b*c - 8*a^2 - 8*b^2 + 2*c^2) - 4*a*c + (2*(b -
c)*(a^2*b - b*c^2 + b^3 + a*c*(a^2 + b^2 - c^2)^(1/2))*(a*b + a*c + 4*a^2*tan(x/2) + 3*b^2*tan(x/2) - 3*c^2*t
an(x/2)))/((b^2 - c^2)*(a^2 + b^2 - c^2)))*(a^2*b - b*c^2 + b^3 + a*c*(a^2 + b^2 - c^2)^(1/2)))/((b^2 - c^2)*(
a^2 + b^2 - c^2)))*(b*(a^2 - c^2) + b^3 + a*c*(a^2 + b^2 - c^2)^(1/2)))/((b^2 - c^2)*(a^2 + b^2 - c^2)) - (log
(2*a - 2*b*tan(x/2) - ((tan(x/2)*(6*b*c - 8*a^2 - 8*b^2 + 2*c^2) - 4*a*c + (2*(b - c)*(a^2*b - b*c^2 + b^3 - a
*c*(a^2 + b^2 - c^2)^(1/2))*(a*b + a*c + 4*a^2*tan(x/2) + 3*b^2*tan(x/2) - 3*c^2*tan(x/2)))/((b^2 - c^2)*(a^2
+ b^2 - c^2)))*(a^2*b - b*c^2 + b^3 - a*c*(a^2 + b^2 - c^2)^(1/2)))/((b^2 - c^2)*(a^2 + b^2 - c^2)))*(b*(a^2 -
c^2) + b^3 - a*c*(a^2 + b^2 - c^2)^(1/2)))/((b^2 - c^2)*(a^2 + b^2 - c^2))