\(\int \frac {\cot ^3(d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx\) [18]

Optimal result

Integrand size = 35, antiderivative size = 141 \[ \int \frac {\cot ^3(d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=-\frac {\text {arctanh}\left (\frac {2 a-b+(b-2 c) \cot ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 \sqrt {a-b+c} e}-\frac {\text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 \sqrt {c} e} \]

[Out]

-1/2*arctanh(1/2*(b+2*c*cot(e*x+d)^2)/c^(1/2)/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2))/e/c^(1/2)-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 = 0.35 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3782, 1265, 857, 635, 212, 738} \[ \int \frac {\cot ^3(d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=-\frac {\text {arctanh}\left (\frac {2 a+(b-2 c) \cot ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e \sqrt {a-b+c}}-\frac {\text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 \sqrt {c} e} \]

[In]

Int[Cot[d + e*x]^3/Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]

[Out]

-1/2*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])]/(Sqrt[a - b + c]*e) - ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d +
e*x]^4])]/(2*Sqrt[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 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 1265

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(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] &&
IntegerQ[(m - 1)/2]

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^3}{\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 {x}{(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}{\sqrt {a+b x+c x^2}} \, dx,x,\cot ^2(d+e x)\right )}{2 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 c-x^2} \, dx,x,\frac {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 {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}-\frac {\text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 \sqrt {c} e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.84 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.50 \[ \int \frac {\cot ^3(d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=-\frac {\left (\sqrt {c} \text {arctanh}\left (\frac {b-2 c+(2 a-b) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )+\sqrt {a-b+c} \text {arctanh}\left (\frac {2 c+b \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )\right ) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan ^2(d+e x)}{2 \sqrt {c} \sqrt {a-b+c} e \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}} \]

[In]

Integrate[Cot[d + e*x]^3/Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]

[Out]

-1/2*((Sqrt[c]*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*T
an[d + e*x]^4])] + Sqrt[a - b + c]*ArcTanh[(2*c + b*Tan[d + e*x]^2)/(2*Sqrt[c]*Sqrt[c + b*Tan[d + e*x]^2 + a*T
an[d + e*x]^4])])*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]*Tan[d + e*x]^2)/(Sqrt[c]*Sqrt[a - b + c]*e*Sqr
t[c + b*Tan[d + e*x]^2 + a*Tan[d + e*x]^4])

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(\frac {-\frac {\ln \left (\frac {\frac {b}{2}+c \cot \left (e x +d \right )^{2}}{\sqrt {c}}+\sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}\right )}{2 \sqrt {c}}-\frac {\ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+2 \sqrt {a -b +c}\, \sqrt {\left (\cot \left (e x +d \right )^{2}+1\right )^{2} c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}}{\cot \left (e x +d \right )^{2}+1}\right )}{2 \sqrt {a -b +c}}}{e}\)	\(153\)
default	\(\frac {-\frac {\ln \left (\frac {\frac {b}{2}+c \cot \left (e x +d \right )^{2}}{\sqrt {c}}+\sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}\right )}{2 \sqrt {c}}-\frac {\ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+2 \sqrt {a -b +c}\, \sqrt {\left (\cot \left (e x +d \right )^{2}+1\right )^{2} c +\left (b -2 c \right ) \left (\cot \left (e x +d \right )^{2}+1\right )+a -b +c}}{\cot \left (e x +d \right )^{2}+1}\right )}{2 \sqrt {a -b +c}}}{e}\)	\(153\)

[In]

int(cot(e*x+d)^3/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/e*(-1/2*ln((1/2*b+c*cot(e*x+d)^2)/c^(1/2)+(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2))/c^(1/2)-1/2/(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)*(cot(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. 406 vs. \(2 (121) = 242\).

Time = 1.03 (sec) , antiderivative size = 1695, normalized size of antiderivative = 12.02 \[ \int \frac {\cot ^3(d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\text {Too large to display} \]

[In]

integrate(cot(e*x+d)^3/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x, algorithm="fricas")

[Out]

[1/4*(sqrt(a - b + c)*c*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)*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)*c
os(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)) + (a - b + c)*sqrt(c)*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*cos(2*e*x
+ 2*d)^2 + b^2 + 4*(a + 2*b)*c + 8*c^2 - 4*((b - 2*c)*cos(2*e*x + 2*d)^2 - 2*b*cos(2*e*x + 2*d) + b + 2*c)*sqr
t(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*co
s(2*e*x + 2*d) + 1)) - 2*(b^2 + 4*a*c - 8*c^2)*cos(2*e*x + 2*d))/(cos(2*e*x + 2*d)^2 - 2*cos(2*e*x + 2*d) + 1)
))/(((a - b)*c + c^2)*e), -1/4*(2*(a - b + c)*sqrt(-c)*arctan(-1/2*((b - 2*c)*cos(2*e*x + 2*d)^2 - 2*b*cos(2*e
*x + 2*d) + b + 2*c)*sqrt(-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 - b)*c + c^2)*cos(2*e*x + 2*d)^2 + (a + b)*c + c^2 - 2*(a*c
- c^2)*cos(2*e*x + 2*d))) - sqrt(a - b + c)*c*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)*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)) - 4*(a^2 - a*b + b*c - c^2)*cos(2*e*x + 2*d)))/(((a - b)*c + c^2)*e), 1/4*(2*sqrt(-a + b -
c)*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)*cos(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)*sqrt(c)*log(((b^2 + 4*(a - 2*b)*c + 8*c^2)*cos(2*e*x + 2*d)^2 + b^2
+ 4*(a + 2*b)*c + 8*c^2 - 4*((b - 2*c)*cos(2*e*x + 2*d)^2 - 2*b*cos(2*e*x + 2*d) + b + 2*c)*sqrt(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)) - 2*(b^2 + 4*a*c - 8*c^2)*cos(2*e*x + 2*d))/(cos(2*e*x + 2*d)^2 - 2*cos(2*e*x + 2*d) + 1)))/(((a - b)*c
+ c^2)*e), 1/2*(sqrt(-a + b - c)*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)*cos(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)*sqrt(-c)*arctan(-1/2*((b - 2*c)*cos(2
*e*x + 2*d)^2 - 2*b*cos(2*e*x + 2*d) + b + 2*c)*sqrt(-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 - b)*c + c^2)*cos(2*e*x + 2*d)^2
+ (a + b)*c + c^2 - 2*(a*c - c^2)*cos(2*e*x + 2*d))))/(((a - b)*c + c^2)*e)]

Sympy [F]

\[ \int \frac {\cot ^3(d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\int \frac {\cot ^{3}{\left (d + e x \right )}}{\sqrt {a + b \cot ^{2}{\left (d + e x \right )} + c \cot ^{4}{\left (d + e x \right )}}}\, dx \]

[In]

integrate(cot(e*x+d)**3/(a+b*cot(e*x+d)**2+c*cot(e*x+d)**4)**(1/2),x)

[Out]

Integral(cot(d + e*x)**3/sqrt(a + b*cot(d + e*x)**2 + c*cot(d + e*x)**4), x)

Maxima [F]

\[ \int \frac {\cot ^3(d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\int { \frac {\cot \left (e x + d\right )^{3}}{\sqrt {c \cot \left (e x + d\right )^{4} + b \cot \left (e x + d\right )^{2} + a}} \,d x } \]

[In]

integrate(cot(e*x+d)^3/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x, algorithm="maxima")

[Out]

integrate(cot(e*x + d)^3/sqrt(c*cot(e*x + d)^4 + b*cot(e*x + d)^2 + a), x)

Giac [F(-1)]

Timed out. \[ \int \frac {\cot ^3(d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\text {Timed out} \]

[In]

integrate(cot(e*x+d)^3/(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. \[ \int \frac {\cot ^3(d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\int \frac {{\mathrm {cot}\left (d+e\,x\right )}^3}{\sqrt {c\,{\mathrm {cot}\left (d+e\,x\right )}^4+b\,{\mathrm {cot}\left (d+e\,x\right )}^2+a}} \,d x \]

[In]

int(cot(d + e*x)^3/(a + b*cot(d + e*x)^2 + c*cot(d + e*x)^4)^(1/2),x)

[Out]

int(cot(d + e*x)^3/(a + b*cot(d + e*x)^2 + c*cot(d + e*x)^4)^(1/2), x)