3.5 \(\int \frac{1}{a+b \cot ^2(c+d x)} \, dx\)

Optimal. Leaf size=49 \[ \frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \cot (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} d (a-b)}+\frac{x}{a-b} \]

[Out]

x/(a - b) + (Sqrt[b]*ArcTan[(Sqrt[b]*Cot[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)*d)

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Rubi [A]  time = 0.0726474, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3660, 3675, 205} \[ \frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \cot (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} d (a-b)}+\frac{x}{a-b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cot[c + d*x]^2)^(-1),x]

[Out]

x/(a - b) + (Sqrt[b]*ArcTan[(Sqrt[b]*Cot[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)*d)

Rule 3660

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> Simp[x/(a - b), x] - Dist[b/(a - b), Int[Sec[e
+ f*x]^2/(a + b*Tan[e + f*x]^2), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a, b]

Rule 3675

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/(c^(m - 1)*f), Subst[Int[(c^2 + ff^2*x^2)^(m/2 - 1)*(a + b*(ff*x)
^n)^p, x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2] && (IntegersQ[n, p
] || IGtQ[m, 0] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rule 205

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

Rubi steps

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

Mathematica [A]  time = 0.0530562, size = 49, normalized size = 1. \[ \frac{\tan ^{-1}(\tan (c+d x))-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{b}}\right )}{\sqrt{a}}}{a d-b d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cot[c + d*x]^2)^(-1),x]

[Out]

(ArcTan[Tan[c + d*x]] - (Sqrt[b]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[b]])/Sqrt[a])/(a*d - b*d)

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Maple [A]  time = 0.021, size = 64, normalized size = 1.3 \begin{align*} -{\frac{\pi }{2\,d \left ( a-b \right ) }}+{\frac{{\rm arccot} \left (\cot \left ( dx+c \right ) \right )}{d \left ( a-b \right ) }}+{\frac{b}{d \left ( a-b \right ) }\arctan \left ({b\cot \left ( dx+c \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/d/(a-b)*Pi+1/d/(a-b)*arccot(cot(d*x+c))+1/d*b/(a-b)/(a*b)^(1/2)*arctan(cot(d*x+c)*b/(a*b)^(1/2))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.77972, size = 574, normalized size = 11.71 \begin{align*} \left [\frac{4 \, d x - \sqrt{-\frac{b}{a}} \log \left (\frac{{\left (a^{2} + 6 \, a b + b^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )^{2} + 4 \,{\left (a^{2} - a b -{\left (a^{2} + a b\right )} \cos \left (2 \, d x + 2 \, c\right )\right )} \sqrt{-\frac{b}{a}} \sin \left (2 \, d x + 2 \, c\right ) + a^{2} - 6 \, a b + b^{2} - 2 \,{\left (a^{2} - b^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )^{2} + a^{2} + 2 \, a b + b^{2} - 2 \,{\left (a^{2} - b^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )}\right )}{4 \,{\left (a - b\right )} d}, \frac{2 \, d x + \sqrt{\frac{b}{a}} \arctan \left (\frac{{\left ({\left (a + b\right )} \cos \left (2 \, d x + 2 \, c\right ) - a + b\right )} \sqrt{\frac{b}{a}}}{2 \, b \sin \left (2 \, d x + 2 \, c\right )}\right )}{2 \,{\left (a - b\right )} d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(4*d*x - sqrt(-b/a)*log(((a^2 + 6*a*b + b^2)*cos(2*d*x + 2*c)^2 + 4*(a^2 - a*b - (a^2 + a*b)*cos(2*d*x +
2*c))*sqrt(-b/a)*sin(2*d*x + 2*c) + a^2 - 6*a*b + b^2 - 2*(a^2 - b^2)*cos(2*d*x + 2*c))/((a^2 - 2*a*b + b^2)*c
os(2*d*x + 2*c)^2 + a^2 + 2*a*b + b^2 - 2*(a^2 - b^2)*cos(2*d*x + 2*c))))/((a - b)*d), 1/2*(2*d*x + sqrt(b/a)*
arctan(1/2*((a + b)*cos(2*d*x + 2*c) - a + b)*sqrt(b/a)/(b*sin(2*d*x + 2*c))))/((a - b)*d)]

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Sympy [A]  time = 2.16574, size = 279, normalized size = 5.69 \begin{align*} \begin{cases} \frac{\tilde{\infty } x}{\cot ^{2}{\left (c \right )}} & \text{for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac{- x + \frac{1}{d \cot{\left (c + d x \right )}}}{b} & \text{for}\: a = 0 \\\frac{d x \cot ^{2}{\left (c + d x \right )}}{2 b d \cot ^{2}{\left (c + d x \right )} + 2 b d} + \frac{d x}{2 b d \cot ^{2}{\left (c + d x \right )} + 2 b d} - \frac{\cot{\left (c + d x \right )}}{2 b d \cot ^{2}{\left (c + d x \right )} + 2 b d} & \text{for}\: a = b \\\frac{x}{a + b \cot ^{2}{\left (c \right )}} & \text{for}\: d = 0 \\\frac{x}{a} & \text{for}\: b = 0 \\\frac{2 i \sqrt{a} d x \sqrt{\frac{1}{b}}}{2 i a^{\frac{3}{2}} d \sqrt{\frac{1}{b}} - 2 i \sqrt{a} b d \sqrt{\frac{1}{b}}} + \frac{\log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \cot{\left (c + d x \right )} \right )}}{2 i a^{\frac{3}{2}} d \sqrt{\frac{1}{b}} - 2 i \sqrt{a} b d \sqrt{\frac{1}{b}}} - \frac{\log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \cot{\left (c + d x \right )} \right )}}{2 i a^{\frac{3}{2}} d \sqrt{\frac{1}{b}} - 2 i \sqrt{a} b d \sqrt{\frac{1}{b}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x/cot(c)**2, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), ((-x + 1/(d*cot(c + d*x)))/b, Eq(a, 0)), (d*x*cot
(c + d*x)**2/(2*b*d*cot(c + d*x)**2 + 2*b*d) + d*x/(2*b*d*cot(c + d*x)**2 + 2*b*d) - cot(c + d*x)/(2*b*d*cot(c
+ d*x)**2 + 2*b*d), Eq(a, b)), (x/(a + b*cot(c)**2), Eq(d, 0)), (x/a, Eq(b, 0)), (2*I*sqrt(a)*d*x*sqrt(1/b)/(
2*I*a**(3/2)*d*sqrt(1/b) - 2*I*sqrt(a)*b*d*sqrt(1/b)) + log(-I*sqrt(a)*sqrt(1/b) + cot(c + d*x))/(2*I*a**(3/2)
*d*sqrt(1/b) - 2*I*sqrt(a)*b*d*sqrt(1/b)) - log(I*sqrt(a)*sqrt(1/b) + cot(c + d*x))/(2*I*a**(3/2)*d*sqrt(1/b)
- 2*I*sqrt(a)*b*d*sqrt(1/b)), True))

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Giac [B]  time = 1.21528, size = 178, normalized size = 3.63 \begin{align*} \frac{\frac{\pi \left \lfloor \frac{d x + c}{\pi } + \frac{1}{2} \right \rfloor + \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} \tan \left (d x + c\right )}{\sqrt{\frac{a + b + \sqrt{{\left (a + b\right )}^{2} - 4 \, a b}}{a}}}\right )}{{\left | a - b \right |}} - \frac{\sqrt{a b}{\left (\pi \left \lfloor \frac{d x + c}{\pi } + \frac{1}{2} \right \rfloor + \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} \tan \left (d x + c\right )}{\sqrt{\frac{a + b - \sqrt{{\left (a + b\right )}^{2} - 4 \, a b}}{a}}}\right )\right )}{\left | a \right |}}{a^{2}{\left | a - b \right |}}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((pi*floor((d*x + c)/pi + 1/2) + arctan(2*sqrt(1/2)*tan(d*x + c)/sqrt((a + b + sqrt((a + b)^2 - 4*a*b))/a)))/a
bs(a - b) - sqrt(a*b)*(pi*floor((d*x + c)/pi + 1/2) + arctan(2*sqrt(1/2)*tan(d*x + c)/sqrt((a + b - sqrt((a +
b)^2 - 4*a*b))/a)))*abs(a)/(a^2*abs(a - b)))/d