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

Optimal. Leaf size=294 \[ \frac{\sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left (\frac{\sqrt{a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}-\frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left (\frac{-\sqrt{a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}} \]

[Out]

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

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Rubi [A]  time = 0.288611, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {3701, 1036, 1030, 208} \[ \frac{\sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left (\frac{\sqrt{a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}}-\frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left (\frac{-\sqrt{a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+a-c} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} e \sqrt{a^2-2 a c+b^2+c^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3701

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]

Rule 1036

Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q
= Rt[(c*d - a*f)^2 + a*c*e^2, 2]}, Dist[1/(2*q), Int[Simp[-(a*h*e) - g*(c*d - a*f - q) + (h*(c*d - a*f + q) -
g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - Dist[1/(2*q), Int[Simp[-(a*h*e) - g*(c*d - a*f + q
) + (h*(c*d - a*f - q) - g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f, g,
h}, x] && NeQ[e^2 - 4*d*f, 0] && NegQ[-(a*c)]

Rule 1030

Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[-2*
a*g*h, Subst[Int[1/Simp[2*a^2*g*h*c + a*e*x^2, x], x], x, Simp[a*h - g*c*x, x]/Sqrt[d + e*x + f*x^2]], x] /; F
reeQ[{a, c, d, e, f, g, h}, x] && EqQ[a*h^2*e + 2*g*h*(c*d - a*f) - g^2*c*e, 0]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{\cot (d+e x)}{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x}{\left (1+x^2\right ) \sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{e}\\ &=\frac{\operatorname{Subst}\left (\int \frac{-b+\left (a-c-\sqrt{a^2+b^2-2 a c+c^2}\right ) x}{\left (1+x^2\right ) \sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{2 \sqrt{a^2+b^2-2 a c+c^2} e}-\frac{\operatorname{Subst}\left (\int \frac{-b+\left (a-c+\sqrt{a^2+b^2-2 a c+c^2}\right ) x}{\left (1+x^2\right ) \sqrt{a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{2 \sqrt{a^2+b^2-2 a c+c^2} e}\\ &=\frac{\left (b \left (a-c-\sqrt{a^2+b^2-2 a c+c^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-2 b \left (a-c-\sqrt{a^2+b^2-2 a c+c^2}\right )+b x^2} \, dx,x,\frac{a-c-\sqrt{a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{a^2+b^2-2 a c+c^2} e}-\frac{\left (b \left (a-c+\sqrt{a^2+b^2-2 a c+c^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-2 b \left (a-c+\sqrt{a^2+b^2-2 a c+c^2}\right )+b x^2} \, dx,x,\frac{a-c+\sqrt{a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{a^2+b^2-2 a c+c^2} e}\\ &=-\frac{\sqrt{a-c-\sqrt{a^2+b^2-2 a c+c^2}} \tanh ^{-1}\left (\frac{a-c-\sqrt{a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt{2} \sqrt{a-c-\sqrt{a^2+b^2-2 a c+c^2}} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} \sqrt{a^2+b^2-2 a c+c^2} e}+\frac{\sqrt{a-c+\sqrt{a^2+b^2-2 a c+c^2}} \tanh ^{-1}\left (\frac{a-c+\sqrt{a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt{2} \sqrt{a-c+\sqrt{a^2+b^2-2 a c+c^2}} \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt{2} \sqrt{a^2+b^2-2 a c+c^2} e}\\ \end{align*}

Mathematica [C]  time = 11.0869, size = 253, normalized size = 0.86 \[ \frac{\tan (d+e x) \sqrt{a+b \cot (d+e x)+c \cot ^2(d+e x)} \left (\sqrt{a-i b-c} \tanh ^{-1}\left (\frac{(2 a+i b) \tan (d+e x)+b+2 i c}{2 \sqrt{a+i b-c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right )-i \sqrt{a+i b-c} \tan ^{-1}\left (\frac{(b+2 i a) \tan (d+e x)+i b+2 c}{2 \sqrt{a-i b-c} \sqrt{\tan (d+e x) (a \tan (d+e x)+b)+c}}\right )\right )}{2 e \sqrt{a-i b-c} \sqrt{a+i b-c} \sqrt{a \tan ^2(d+e x)+b \tan (d+e x)+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.047, size = 9339148, normalized size = 31765.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot{\left (d + e x \right )}}{\sqrt{a + b \cot{\left (d + e x \right )} + c \cot ^{2}{\left (d + e x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError