3.23 \(\int \sqrt{a+b \cot ^2(x)} \, dx\)

Optimal. Leaf size=65 \[ -\sqrt{a-b} \tan ^{-1}\left (\frac{\sqrt{a-b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right )-\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0452154, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3661, 402, 217, 206, 377, 203} \[ -\sqrt{a-b} \tan ^{-1}\left (\frac{\sqrt{a-b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right )-\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Cot[x]^2],x]

[Out]

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

Rule 3661

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

Rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Dist[b/d, Int[(a + b*x^2)^(p - 1), x], x]
- Dist[(b*c - a*d)/d, Int[(a + b*x^2)^(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d,
0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [C]  time = 0.391168, size = 167, normalized size = 2.57 \[ \frac{1}{2} i \left (\sqrt{a-b} \log \left (-\frac{4 i \left (\sqrt{a-b} \sqrt{a+b \cot ^2(x)}+a-i b \cot (x)\right )}{(a-b)^{3/2} (\cot (x)+i)}\right )-\sqrt{a-b} \log \left (\frac{4 i \left (\sqrt{a-b} \sqrt{a+b \cot ^2(x)}+a+i b \cot (x)\right )}{(a-b)^{3/2} (\cot (x)-i)}\right )+2 i \sqrt{b} \log \left (\sqrt{b} \sqrt{a+b \cot ^2(x)}+b \cot (x)\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*Cot[x]^2],x]

[Out]

(I/2)*(Sqrt[a - b]*Log[((-4*I)*(a - I*b*Cot[x] + Sqrt[a - b]*Sqrt[a + b*Cot[x]^2]))/((a - b)^(3/2)*(I + Cot[x]
))] - Sqrt[a - b]*Log[((4*I)*(a + I*b*Cot[x] + Sqrt[a - b]*Sqrt[a + b*Cot[x]^2]))/((a - b)^(3/2)*(-I + Cot[x])
)] + (2*I)*Sqrt[b]*Log[b*Cot[x] + Sqrt[b]*Sqrt[a + b*Cot[x]^2]])

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Maple [B]  time = 0.026, size = 137, normalized size = 2.1 \begin{align*} -\sqrt{b}\ln \left ( \cot \left ( x \right ) \sqrt{b}+\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}} \right ) +{\frac{1}{b \left ( a-b \right ) }\sqrt{{b}^{4} \left ( a-b \right ) }\arctan \left ({ \left ( a-b \right ){b}^{2}\cot \left ( x \right ){\frac{1}{\sqrt{{b}^{4} \left ( a-b \right ) }}}{\frac{1}{\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}}}}} \right ) }-{\frac{a}{ \left ( a-b \right ){b}^{2}}\sqrt{{b}^{4} \left ( a-b \right ) }\arctan \left ({ \left ( a-b \right ){b}^{2}\cot \left ( x \right ){\frac{1}{\sqrt{{b}^{4} \left ( a-b \right ) }}}{\frac{1}{\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b^(1/2)*ln(cot(x)*b^(1/2)+(a+b*cot(x)^2)^(1/2))+(b^4*(a-b))^(1/2)/b/(a-b)*arctan(b^2*(a-b)/(b^4*(a-b))^(1/2)/
(a+b*cot(x)^2)^(1/2)*cot(x))-a*(b^4*(a-b))^(1/2)/b^2/(a-b)*arctan(b^2*(a-b)/(b^4*(a-b))^(1/2)/(a+b*cot(x)^2)^(
1/2)*cot(x))

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.81478, size = 1285, normalized size = 19.77 \begin{align*} \left [\frac{1}{2} \, \sqrt{-a + b} \log \left (-{\left (a - b\right )} \cos \left (2 \, x\right ) + \sqrt{-a + b} \sqrt{\frac{{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + b\right ) + \frac{1}{2} \, \sqrt{b} \log \left (\frac{{\left (a - 2 \, b\right )} \cos \left (2 \, x\right ) + 2 \, \sqrt{b} \sqrt{\frac{{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) - a - 2 \, b}{\cos \left (2 \, x\right ) - 1}\right ), -\sqrt{a - b} \arctan \left (-\frac{\sqrt{a - b} \sqrt{\frac{{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right )}{{\left (a - b\right )} \cos \left (2 \, x\right ) + a - b}\right ) + \frac{1}{2} \, \sqrt{b} \log \left (\frac{{\left (a - 2 \, b\right )} \cos \left (2 \, x\right ) + 2 \, \sqrt{b} \sqrt{\frac{{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) - a - 2 \, b}{\cos \left (2 \, x\right ) - 1}\right ), \sqrt{-b} \arctan \left (\frac{\sqrt{-b} \sqrt{\frac{{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right )}{b \cos \left (2 \, x\right ) + b}\right ) + \frac{1}{2} \, \sqrt{-a + b} \log \left (-{\left (a - b\right )} \cos \left (2 \, x\right ) + \sqrt{-a + b} \sqrt{\frac{{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + b\right ), -\sqrt{a - b} \arctan \left (-\frac{\sqrt{a - b} \sqrt{\frac{{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right )}{{\left (a - b\right )} \cos \left (2 \, x\right ) + a - b}\right ) + \sqrt{-b} \arctan \left (\frac{\sqrt{-b} \sqrt{\frac{{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right )}{b \cos \left (2 \, x\right ) + b}\right )\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*sqrt(-a + b)*log(-(a - b)*cos(2*x) + sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x
) + b) + 1/2*sqrt(b)*log(((a - 2*b)*cos(2*x) + 2*sqrt(b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2
*x) - a - 2*b)/(cos(2*x) - 1)), -sqrt(a - b)*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1
))*sin(2*x)/((a - b)*cos(2*x) + a - b)) + 1/2*sqrt(b)*log(((a - 2*b)*cos(2*x) + 2*sqrt(b)*sqrt(((a - b)*cos(2*
x) - a - b)/(cos(2*x) - 1))*sin(2*x) - a - 2*b)/(cos(2*x) - 1)), sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(2*
x) - a - b)/(cos(2*x) - 1))*sin(2*x)/(b*cos(2*x) + b)) + 1/2*sqrt(-a + b)*log(-(a - b)*cos(2*x) + sqrt(-a + b)
*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) + b), -sqrt(a - b)*arctan(-sqrt(a - b)*sqrt(((a - b)
*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/((a - b)*cos(2*x) + a - b)) + sqrt(-b)*arctan(sqrt(-b)*sqrt(((a -
b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/(b*cos(2*x) + b))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + b*cot(x)**2), x)

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Giac [B]  time = 21.9382, size = 284, normalized size = 4.37 \begin{align*} -\frac{1}{2} \,{\left (\frac{2 \, \sqrt{-a + b} b \arctan \left (\frac{{\left (\sqrt{-a + b} \cos \left (x\right ) - \sqrt{-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a}\right )}^{2} + a - 2 \, b}{2 \, \sqrt{a b - b^{2}}}\right )}{\sqrt{a b - b^{2}}} + \sqrt{-a + b} \log \left ({\left (\sqrt{-a + b} \cos \left (x\right ) - \sqrt{-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a}\right )}^{2}\right )\right )} \mathrm{sgn}\left (\sin \left (x\right )\right ) + \frac{{\left (2 \, \sqrt{-a + b} b \arctan \left (-\frac{\sqrt{-a + b} \sqrt{b}}{\sqrt{a b - b^{2}}}\right ) + \sqrt{a b - b^{2}} \sqrt{-a + b} \log \left (-a - 2 \, \sqrt{-a + b} \sqrt{b} + 2 \, b\right )\right )} \mathrm{sgn}\left (\sin \left (x\right )\right )}{2 \, \sqrt{a b - b^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*sqrt(-a + b)*b*arctan(1/2*((sqrt(-a + b)*cos(x) - sqrt(-a*cos(x)^2 + b*cos(x)^2 + a))^2 + a - 2*b)/sqr
t(a*b - b^2))/sqrt(a*b - b^2) + sqrt(-a + b)*log((sqrt(-a + b)*cos(x) - sqrt(-a*cos(x)^2 + b*cos(x)^2 + a))^2)
)*sgn(sin(x)) + 1/2*(2*sqrt(-a + b)*b*arctan(-sqrt(-a + b)*sqrt(b)/sqrt(a*b - b^2)) + sqrt(a*b - b^2)*sqrt(-a
+ b)*log(-a - 2*sqrt(-a + b)*sqrt(b) + 2*b))*sgn(sin(x))/sqrt(a*b - b^2)