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

Optimal. Leaf size=79 \[ \frac {\tanh ^{-1}\left (\frac {2 a-b+(b-2 c) \cot ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 \sqrt {a-b+c} e} \]

[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)

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Rubi [A]
time = 0.08, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {3782, 1261, 738, 212} \begin {gather*} \frac {\tanh ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {\cot (d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx &=-\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 {\tanh ^{-1}\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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 74.79, size = 24736, normalized size = 313.11 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [A]
time = 0.36, size = 102, normalized size = 1.29

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

Verification of antiderivative is not currently implemented for this CAS.

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (71) = 142\).
time = 3.56, size = 447, normalized size = 5.66 \begin {gather*} \left [\frac {e^{\left (-1\right )} \log \left (2 \, {\left (a^{2} - 2 \, a b + b^{2} + 2 \, {\left (a - b\right )} c + c^{2}\right )} \cos \left (2 \, x e + 2 \, d\right )^{2} + 2 \, a^{2} - b^{2} + 2 \, c^{2} + 2 \, {\left ({\left (a - b + c\right )} \cos \left (2 \, x e + 2 \, d\right )^{2} - {\left (2 \, a - b\right )} \cos \left (2 \, x e + 2 \, d\right ) + a - c\right )} \sqrt {a - b + c} \sqrt {\frac {{\left (a - b + c\right )} \cos \left (2 \, x e + 2 \, d\right )^{2} - 2 \, {\left (a - c\right )} \cos \left (2 \, x e + 2 \, d\right ) + a + b + c}{\cos \left (2 \, x e + 2 \, d\right )^{2} - 2 \, \cos \left (2 \, x e + 2 \, d\right ) + 1}} - 4 \, {\left (a^{2} - a b + b c - c^{2}\right )} \cos \left (2 \, x e + 2 \, d\right )\right )}{4 \, \sqrt {a - b + c}}, -\frac {\sqrt {-a + b - c} \arctan \left (\frac {{\left ({\left (a - b + c\right )} \cos \left (2 \, x e + 2 \, d\right )^{2} - {\left (2 \, a - b\right )} \cos \left (2 \, x e + 2 \, d\right ) + a - c\right )} \sqrt {-a + b - c} \sqrt {\frac {{\left (a - b + c\right )} \cos \left (2 \, x e + 2 \, d\right )^{2} - 2 \, {\left (a - c\right )} \cos \left (2 \, x e + 2 \, d\right ) + a + b + c}{\cos \left (2 \, x e + 2 \, d\right )^{2} - 2 \, \cos \left (2 \, x e + 2 \, d\right ) + 1}}}{{\left (a^{2} - 2 \, a b + b^{2} + 2 \, {\left (a - b\right )} c + c^{2}\right )} \cos \left (2 \, x e + 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 \, x e + 2 \, d\right )}\right ) e^{\left (-1\right )}}{2 \, {\left (a - b + c\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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