Integrand size = 19, antiderivative size = 102 \[ \int \csc (c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {(a+b)^3 \log (1-\cos (c+d x))}{2 d}-\frac {b \left (3 a^2+b^2\right ) \log (\cos (c+d x))}{d}-\frac {(a-b)^3 \log (1+\cos (c+d x))}{2 d}+\frac {3 a b^2 \sec (c+d x)}{d}+\frac {b^3 \sec ^2(c+d x)}{2 d} \] Output:
1/2*(a+b)^3*ln(1-cos(d*x+c))/d-b*(3*a^2+b^2)*ln(cos(d*x+c))/d-1/2*(a-b)^3* ln(1+cos(d*x+c))/d+3*a*b^2*sec(d*x+c)/d+1/2*b^3*sec(d*x+c)^2/d
Time = 0.72 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.87 \[ \int \csc (c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {-2 (a-b)^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-2 b \left (3 a^2+b^2\right ) \log (\cos (c+d x))+2 (a+b)^3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+6 a b^2 \sec (c+d x)+b^3 \sec ^2(c+d x)}{2 d} \] Input:
Integrate[Csc[c + d*x]*(a + b*Sec[c + d*x])^3,x]
Output:
(-2*(a - b)^3*Log[Cos[(c + d*x)/2]] - 2*b*(3*a^2 + b^2)*Log[Cos[c + d*x]] + 2*(a + b)^3*Log[Sin[(c + d*x)/2]] + 6*a*b^2*Sec[c + d*x] + b^3*Sec[c + d *x]^2)/(2*d)
Time = 0.55 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.16, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {3042, 4360, 25, 25, 3042, 25, 3316, 25, 27, 526, 2333, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \csc (c+d x) (a+b \sec (c+d x))^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a-b \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3}{\cos \left (c+d x-\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 4360 |
\(\displaystyle \int \csc (c+d x) \sec ^3(c+d x) \left (-(-a \cos (c+d x)-b)^3\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int -(b+a \cos (c+d x))^3 \csc (c+d x) \sec ^3(c+d x)dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \csc (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+b)^3dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\left (b-a \sin \left (c+d x-\frac {\pi }{2}\right )\right )^3}{\sin \left (c+d x-\frac {\pi }{2}\right )^3 \cos \left (c+d x-\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\left (b-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )\right )^3}{\cos \left (\frac {1}{2} (2 c-\pi )+d x\right ) \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )^3}dx\) |
\(\Big \downarrow \) 3316 |
\(\displaystyle \frac {a \int -\frac {(b+a \cos (c+d x))^3 \sec ^3(c+d x)}{a^2-a^2 \cos ^2(c+d x)}d(a \cos (c+d x))}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {a \int \frac {(b+a \cos (c+d x))^3 \sec ^3(c+d x)}{a^2-a^2 \cos ^2(c+d x)}d(a \cos (c+d x))}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {a^4 \int \frac {(b+a \cos (c+d x))^3 \sec ^3(c+d x)}{a^3 \left (a^2-a^2 \cos ^2(c+d x)\right )}d(a \cos (c+d x))}{d}\) |
\(\Big \downarrow \) 526 |
\(\displaystyle -\frac {a^4 \left (\frac {\int \frac {\left (\cos ^2(c+d x) a^4+3 b^2 a^2+b \left (3 a^2+b^2\right ) \cos (c+d x) a\right ) \sec ^2(c+d x)}{a^2 \left (a^2-a^2 \cos ^2(c+d x)\right )}d(a \cos (c+d x))}{a^2}-\frac {b^3 \sec ^2(c+d x)}{2 a^4}\right )}{d}\) |
\(\Big \downarrow \) 2333 |
\(\displaystyle -\frac {a^4 \left (\frac {\int \left (\frac {(a-b)^3}{2 a^2 (\cos (c+d x) a+a)}+\frac {3 b^2 \sec ^2(c+d x)}{a^2}+\frac {\left (b^3+3 a^2 b\right ) \sec (c+d x)}{a^3}+\frac {(a+b)^3}{2 a^2 (a-a \cos (c+d x))}\right )d(a \cos (c+d x))}{a^2}-\frac {b^3 \sec ^2(c+d x)}{2 a^4}\right )}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^4 \left (\frac {\frac {b \left (3 a^2+b^2\right ) \log (a \cos (c+d x))}{a^2}+\frac {(a-b)^3 \log (a \cos (c+d x)+a)}{2 a^2}-\frac {(a+b)^3 \log (a-a \cos (c+d x))}{2 a^2}-\frac {3 b^2 \sec (c+d x)}{a}}{a^2}-\frac {b^3 \sec ^2(c+d x)}{2 a^4}\right )}{d}\) |
Input:
Int[Csc[c + d*x]*(a + b*Sec[c + d*x])^3,x]
Output:
-((a^4*(-1/2*(b^3*Sec[c + d*x]^2)/a^4 + ((b*(3*a^2 + b^2)*Log[a*Cos[c + d* x]])/a^2 - ((a + b)^3*Log[a - a*Cos[c + d*x]])/(2*a^2) + ((a - b)^3*Log[a + a*Cos[c + d*x]])/(2*a^2) - (3*b^2*Sec[c + d*x])/a)/a^2))/d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((x_)^(m_)*((c_) + (d_.)*(x_))^(n_))/((a_) + (b_.)*(x_)^2), x_Symbol] : > With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemainder [(c + d*x)^n, x, x]}, Simp[R*(x^(m + 1)/(a*(m + 1))), x] + Simp[1/a Int[x ^(m + 1)*(ExpandToSum[a*Qx - b*R*x, x]/(a + b*x^2)), x], x]] /; FreeQ[{a, b , c, d}, x] && IGtQ[n, 1] && ILtQ[m, -1]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^m*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x, b* Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1) /2] && NeQ[a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Time = 0.39 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(\frac {a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )+3 a^{2} b \ln \left (\tan \left (d x +c \right )\right )+3 a \,b^{2} \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+b^{3} \left (\frac {1}{2 \cos \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(92\) |
default | \(\frac {a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )+3 a^{2} b \ln \left (\tan \left (d x +c \right )\right )+3 a \,b^{2} \left (\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )+b^{3} \left (\frac {1}{2 \cos \left (d x +c \right )^{2}}+\ln \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(92\) |
norman | \(\frac {\frac {6 a \,b^{2}}{d}-\frac {2 \left (3 a \,b^{2}-b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2}}+\frac {\left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {b \left (3 a^{2}+b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {b \left (3 a^{2}+b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(143\) |
parallelrisch | \(\frac {-3 b \left (a^{2}+\frac {b^{2}}{3}\right ) \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-3 b \left (a^{2}+\frac {b^{2}}{3}\right ) \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (a +b \right )^{3} \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 b^{2} \left (\frac {\left (a -\frac {b}{6}\right ) \cos \left (2 d x +2 c \right )}{2}+a \cos \left (d x +c \right )+\frac {a}{2}+\frac {b}{12}\right )}{d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(152\) |
risch | \(\frac {2 b^{2} \left (3 a \,{\mathrm e}^{3 i \left (d x +c \right )}+b \,{\mathrm e}^{2 i \left (d x +c \right )}+3 a \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) a^{2} b}{d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) a \,b^{2}}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{3}}{d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) a^{2} b}{d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) a \,b^{2}}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{3}}{d}-\frac {3 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) a^{2}}{d}-\frac {b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(261\) |
Input:
int(csc(d*x+c)*(a+b*sec(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(a^3*ln(csc(d*x+c)-cot(d*x+c))+3*a^2*b*ln(tan(d*x+c))+3*a*b^2*(1/cos(d *x+c)+ln(csc(d*x+c)-cot(d*x+c)))+b^3*(1/2/cos(d*x+c)^2+ln(tan(d*x+c))))
Time = 0.10 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.36 \[ \int \csc (c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {6 \, a b^{2} \cos \left (d x + c\right ) - 2 \, {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) - {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + b^{3}}{2 \, d \cos \left (d x + c\right )^{2}} \] Input:
integrate(csc(d*x+c)*(a+b*sec(d*x+c))^3,x, algorithm="fricas")
Output:
1/2*(6*a*b^2*cos(d*x + c) - 2*(3*a^2*b + b^3)*cos(d*x + c)^2*log(-cos(d*x + c)) - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(d*x + c)^2*log(1/2*cos(d*x + c ) + 1/2) + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c)^2*log(-1/2*cos(d*x + c) + 1/2) + b^3)/(d*cos(d*x + c)^2)
\[ \int \csc (c+d x) (a+b \sec (c+d x))^3 \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{3} \csc {\left (c + d x \right )}\, dx \] Input:
integrate(csc(d*x+c)*(a+b*sec(d*x+c))**3,x)
Output:
Integral((a + b*sec(c + d*x))**3*csc(c + d*x), x)
Time = 0.04 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.10 \[ \int \csc (c+d x) (a+b \sec (c+d x))^3 \, dx=-\frac {{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \log \left (\cos \left (d x + c\right ) + 1\right ) - {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \log \left (\cos \left (d x + c\right ) - 1\right ) + 2 \, {\left (3 \, a^{2} b + b^{3}\right )} \log \left (\cos \left (d x + c\right )\right ) - \frac {6 \, a b^{2} \cos \left (d x + c\right ) + b^{3}}{\cos \left (d x + c\right )^{2}}}{2 \, d} \] Input:
integrate(csc(d*x+c)*(a+b*sec(d*x+c))^3,x, algorithm="maxima")
Output:
-1/2*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*log(cos(d*x + c) + 1) - (a^3 + 3*a^2 *b + 3*a*b^2 + b^3)*log(cos(d*x + c) - 1) + 2*(3*a^2*b + b^3)*log(cos(d*x + c)) - (6*a*b^2*cos(d*x + c) + b^3)/cos(d*x + c)^2)/d
Time = 0.16 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.21 \[ \int \csc (c+d x) (a+b \sec (c+d x))^3 \, dx=-\frac {{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \log \left ({\left | \cos \left (d x + c\right ) + 1 \right |}\right )}{2 \, d} + \frac {{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \log \left ({\left | \cos \left (d x + c\right ) - 1 \right |}\right )}{2 \, d} - \frac {{\left (3 \, a^{2} b + b^{3}\right )} \log \left ({\left | \cos \left (d x + c\right ) \right |}\right )}{d} + \frac {6 \, a b^{2} \cos \left (d x + c\right ) + b^{3}}{2 \, d \cos \left (d x + c\right )^{2}} \] Input:
integrate(csc(d*x+c)*(a+b*sec(d*x+c))^3,x, algorithm="giac")
Output:
-1/2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*log(abs(cos(d*x + c) + 1))/d + 1/2*(a ^3 + 3*a^2*b + 3*a*b^2 + b^3)*log(abs(cos(d*x + c) - 1))/d - (3*a^2*b + b^ 3)*log(abs(cos(d*x + c)))/d + 1/2*(6*a*b^2*cos(d*x + c) + b^3)/(d*cos(d*x + c)^2)
Time = 0.14 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.83 \[ \int \csc (c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {\frac {\ln \left (\cos \left (c+d\,x\right )-1\right )\,{\left (a+b\right )}^3}{2}-\frac {\ln \left (\cos \left (c+d\,x\right )+1\right )\,{\left (a-b\right )}^3}{2}+\frac {\frac {b^3}{2}+3\,a\,\cos \left (c+d\,x\right )\,b^2}{{\cos \left (c+d\,x\right )}^2}-\ln \left (\cos \left (c+d\,x\right )\right )\,\left (3\,a^2\,b+b^3\right )}{d} \] Input:
int((a + b/cos(c + d*x))^3/sin(c + d*x),x)
Output:
((log(cos(c + d*x) - 1)*(a + b)^3)/2 - (log(cos(c + d*x) + 1)*(a - b)^3)/2 + (b^3/2 + 3*a*b^2*cos(c + d*x))/cos(c + d*x)^2 - log(cos(c + d*x))*(3*a^ 2*b + b^3))/d
Time = 0.18 (sec) , antiderivative size = 391, normalized size of antiderivative = 3.83 \[ \int \csc (c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {-6 \cos \left (d x +c \right ) a \,b^{2}-6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2} a^{2} b -2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2} b^{3}+6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a^{2} b +2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) b^{3}-6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2} a^{2} b -2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2} b^{3}+6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a^{2} b +2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) b^{3}+2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{2} a^{3}+6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{2} a^{2} b +6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{2} a \,b^{2}+2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{2} b^{3}-2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}-6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b -6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}-2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}-6 \sin \left (d x +c \right )^{2} a \,b^{2}-\sin \left (d x +c \right )^{2} b^{3}+6 a \,b^{2}}{2 d \left (\sin \left (d x +c \right )^{2}-1\right )} \] Input:
int(csc(d*x+c)*(a+b*sec(d*x+c))^3,x)
Output:
( - 6*cos(c + d*x)*a*b**2 - 6*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a* *2*b - 2*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*b**3 + 6*log(tan((c + d *x)/2) - 1)*a**2*b + 2*log(tan((c + d*x)/2) - 1)*b**3 - 6*log(tan((c + d*x )/2) + 1)*sin(c + d*x)**2*a**2*b - 2*log(tan((c + d*x)/2) + 1)*sin(c + d*x )**2*b**3 + 6*log(tan((c + d*x)/2) + 1)*a**2*b + 2*log(tan((c + d*x)/2) + 1)*b**3 + 2*log(tan((c + d*x)/2))*sin(c + d*x)**2*a**3 + 6*log(tan((c + d* x)/2))*sin(c + d*x)**2*a**2*b + 6*log(tan((c + d*x)/2))*sin(c + d*x)**2*a* b**2 + 2*log(tan((c + d*x)/2))*sin(c + d*x)**2*b**3 - 2*log(tan((c + d*x)/ 2))*a**3 - 6*log(tan((c + d*x)/2))*a**2*b - 6*log(tan((c + d*x)/2))*a*b**2 - 2*log(tan((c + d*x)/2))*b**3 - 6*sin(c + d*x)**2*a*b**2 - sin(c + d*x)* *2*b**3 + 6*a*b**2)/(2*d*(sin(c + d*x)**2 - 1))