\(\int \frac {\cot (d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx\) [19]

Optimal result

Integrand size = 33, antiderivative size = 79 \[ \int \frac {\cot (d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\frac {\text {arctanh}\left (\frac {2 a-b+(b-2 c) \cot ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 \sqrt {a-b+c} e} \]

[Out]

1/2*arctanh(1/2*(2*a-b+(b-2*c)*cot(e*x+d)^2)/(a-b+c)^(1/2)/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2))/e/(a-b+c)^
(1/2)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {3782, 1261, 738, 212} \[ \int \frac {\cot (d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\frac {\text {arctanh}\left (\frac {2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e \sqrt {a-b+c}} \]

[In]

Int[Cot[d + e*x]/Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]

[Out]

ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])
]/(2*Sqrt[a - b + c]*e)

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 738

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

Rule 1261

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 3782

Int[cot[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*(cot[(d_.) + (e_.)*(x_)]*(f_.))^(n_.) + (c_.)*(cot[(d_.) + (e
_.)*(x_)]*(f_.))^(n2_.))^(p_), x_Symbol] :> Dist[-f/e, Subst[Int[(x/f)^m*((a + b*x^n + c*x^(2*n))^p/(f^2 + x^2
)), x], x, f*Cot[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]

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

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.76 \[ \int \frac {\cot (d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\frac {\text {arctanh}\left (\frac {b-2 c+(2 a-b) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan ^2(d+e x)}{2 \sqrt {a-b+c} e \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}} \]

[In]

Integrate[Cot[d + e*x]/Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]

[Out]

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

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.29

method	result	size
derivativedivides	\(\frac {\ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+2 \sqrt {a -b +c}\, \sqrt {\left (\cot \left (e x +d \right )^{2}+1\right )^{2} c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}}{\cot \left (e x +d \right )^{2}+1}\right )}{2 e \sqrt {a -b +c}}\)	\(102\)
default	\(\frac {\ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+2 \sqrt {a -b +c}\, \sqrt {\left (\cot \left (e x +d \right )^{2}+1\right )^{2} c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}}{\cot \left (e x +d \right )^{2}+1}\right )}{2 e \sqrt {a -b +c}}\)	\(102\)

[In]

int(cot(e*x+d)/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2/e/(a-b+c)^(1/2)*ln((2*a-2*b+2*c+(b-2*c)*(cot(e*x+d)^2+1)+2*(a-b+c)^(1/2)*((cot(e*x+d)^2+1)^2*c+(b-2*c)*(co
t(e*x+d)^2+1)+a-b+c)^(1/2))/(cot(e*x+d)^2+1))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (69) = 138\).

Time = 0.46 (sec) , antiderivative size = 433, normalized size of antiderivative = 5.48 \[ \int \frac {\cot (d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\left [\frac {\log \left (2 \, {\left (a^{2} - 2 \, a b + b^{2} + 2 \, {\left (a - b\right )} c + c^{2}\right )} \cos \left (2 \, e x + 2 \, d\right )^{2} + 2 \, a^{2} - b^{2} + 2 \, c^{2} + 2 \, {\left ({\left (a - b + c\right )} \cos \left (2 \, e x + 2 \, d\right )^{2} - {\left (2 \, a - b\right )} \cos \left (2 \, e x + 2 \, d\right ) + a - c\right )} \sqrt {a - b + c} \sqrt {\frac {{\left (a - b + c\right )} \cos \left (2 \, e x + 2 \, d\right )^{2} - 2 \, {\left (a - c\right )} \cos \left (2 \, e x + 2 \, d\right ) + a + b + c}{\cos \left (2 \, e x + 2 \, d\right )^{2} - 2 \, \cos \left (2 \, e x + 2 \, d\right ) + 1}} - 4 \, {\left (a^{2} - a b + b c - c^{2}\right )} \cos \left (2 \, e x + 2 \, d\right )\right )}{4 \, \sqrt {a - b + c} e}, -\frac {\sqrt {-a + b - c} \arctan \left (\frac {{\left ({\left (a - b + c\right )} \cos \left (2 \, e x + 2 \, d\right )^{2} - {\left (2 \, a - b\right )} \cos \left (2 \, e x + 2 \, d\right ) + a - c\right )} \sqrt {-a + b - c} \sqrt {\frac {{\left (a - b + c\right )} \cos \left (2 \, e x + 2 \, d\right )^{2} - 2 \, {\left (a - c\right )} \cos \left (2 \, e x + 2 \, d\right ) + a + b + c}{\cos \left (2 \, e x + 2 \, d\right )^{2} - 2 \, \cos \left (2 \, e x + 2 \, d\right ) + 1}}}{{\left (a^{2} - 2 \, a b + b^{2} + 2 \, {\left (a - b\right )} c + c^{2}\right )} \cos \left (2 \, e x + 2 \, d\right )^{2} + a^{2} - b^{2} + 2 \, a c + c^{2} - 2 \, {\left (a^{2} - a b + b c - c^{2}\right )} \cos \left (2 \, e x + 2 \, d\right )}\right )}{2 \, {\left (a - b + c\right )} e}\right ] \]

[In]

integrate(cot(e*x+d)/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x, algorithm="fricas")

[Out]

[1/4*log(2*(a^2 - 2*a*b + b^2 + 2*(a - b)*c + c^2)*cos(2*e*x + 2*d)^2 + 2*a^2 - b^2 + 2*c^2 + 2*((a - b + c)*c
os(2*e*x + 2*d)^2 - (2*a - b)*cos(2*e*x + 2*d) + a - c)*sqrt(a - b + c)*sqrt(((a - b + c)*cos(2*e*x + 2*d)^2 -
2*(a - c)*cos(2*e*x + 2*d) + a + b + c)/(cos(2*e*x + 2*d)^2 - 2*cos(2*e*x + 2*d) + 1)) - 4*(a^2 - a*b + b*c -
c^2)*cos(2*e*x + 2*d))/(sqrt(a - b + c)*e), -1/2*sqrt(-a + b - c)*arctan(((a - b + c)*cos(2*e*x + 2*d)^2 - (2
*a - b)*cos(2*e*x + 2*d) + a - c)*sqrt(-a + b - c)*sqrt(((a - b + c)*cos(2*e*x + 2*d)^2 - 2*(a - c)*cos(2*e*x
+ 2*d) + a + b + c)/(cos(2*e*x + 2*d)^2 - 2*cos(2*e*x + 2*d) + 1))/((a^2 - 2*a*b + b^2 + 2*(a - b)*c + c^2)*co
s(2*e*x + 2*d)^2 + a^2 - b^2 + 2*a*c + c^2 - 2*(a^2 - a*b + b*c - c^2)*cos(2*e*x + 2*d)))/((a - b + c)*e)]

Sympy [F]

\[ \int \frac {\cot (d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\int \frac {\cot {\left (d + e x \right )}}{\sqrt {a + b \cot ^{2}{\left (d + e x \right )} + c \cot ^{4}{\left (d + e x \right )}}}\, dx \]

[In]

integrate(cot(e*x+d)/(a+b*cot(e*x+d)**2+c*cot(e*x+d)**4)**(1/2),x)

[Out]

Integral(cot(d + e*x)/sqrt(a + b*cot(d + e*x)**2 + c*cot(d + e*x)**4), x)

Maxima [F]

\[ \int \frac {\cot (d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\int { \frac {\cot \left (e x + d\right )}{\sqrt {c \cot \left (e x + d\right )^{4} + b \cot \left (e x + d\right )^{2} + a}} \,d x } \]

[In]

integrate(cot(e*x+d)/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x, algorithm="maxima")

[Out]

integrate(cot(e*x + d)/sqrt(c*cot(e*x + d)^4 + b*cot(e*x + d)^2 + a), x)

Giac [F(-1)]

Timed out. \[ \int \frac {\cot (d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\text {Timed out} \]

[In]

integrate(cot(e*x+d)/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot (d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\int \frac {\mathrm {cot}\left (d+e\,x\right )}{\sqrt {c\,{\mathrm {cot}\left (d+e\,x\right )}^4+b\,{\mathrm {cot}\left (d+e\,x\right )}^2+a}} \,d x \]

[In]

int(cot(d + e*x)/(a + b*cot(d + e*x)^2 + c*cot(d + e*x)^4)^(1/2),x)

[Out]

int(cot(d + e*x)/(a + b*cot(d + e*x)^2 + c*cot(d + e*x)^4)^(1/2), x)