3.1.22 \(\int \cot ^2(x) \sqrt {a+b \cot ^2(x)} \, dx\) [22]

Optimal. Leaf size=89 \[ \sqrt {a-b} \text {ArcTan}\left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )-\frac {(a-2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{2 \sqrt {b}}-\frac {1}{2} \cot (x) \sqrt {a+b \cot ^2(x)} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {3751, 489, 537, 223, 212, 385, 209} \begin {gather*} \sqrt {a-b} \text {ArcTan}\left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )-\frac {1}{2} \cot (x) \sqrt {a+b \cot ^2(x)}-\frac {(a-2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{2 \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[x]^2*Sqrt[a + b*Cot[x]^2],x]

[Out]

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

Rule 209

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

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 223

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 385

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 489

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[e^(n - 1
)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(b*(m + n*(p + q) + 1))), x] - Dist[e^n/(b*(m + n*(p +
q) + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[a*c*(m - n + 1) + (a*d*(m - n + 1) - n*q*(b
*c - a*d))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 0] &&
GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 537

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 3751

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((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[(d*ff*(x/c))^m*((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, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(2105\) vs. \(2(89)=178\).
time = 22.86, size = 2105, normalized size = 23.65 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Cot[x]^2*Sqrt[a + b*Cot[x]^2],x]

[Out]

-1/2*(Sqrt[(-a - b + a*Cos[2*x] - b*Cos[2*x])/(-1 + Cos[2*x])]*Cot[x]) + ((-4*Sqrt[a - b]*Sqrt[b]*ArcTan[(Sqrt
[a - b]*(-1 + Tan[x/2]^2))/Sqrt[b*Cos[x]^2*Sec[x/2]^4 + 4*a*Tan[x/2]^2]] - (a - 2*b)*ArcTanh[(Sqrt[2]*(a + (-a
+ b)*Cos[x])*Sec[x/2]^2)/(Sqrt[b]*Sqrt[(a + b + (-a + b)*Cos[2*x])*Sec[x/2]^4])] + (a - 2*b)*ArcTanh[(2*a + b
*(-1 + Tan[x/2]^2))/(Sqrt[b]*Sqrt[b*Cos[x]^2*Sec[x/2]^4 + 4*a*Tan[x/2]^2])])*((b*Sqrt[-(a/(-1 + Cos[2*x])) - b
/(-1 + Cos[2*x]) + (a*Cos[2*x])/(-1 + Cos[2*x]) - (b*Cos[2*x])/(-1 + Cos[2*x])])/(-a - b + a*Cos[2*x] - b*Cos[
2*x]) - (a*Cos[2*x]*Sqrt[-(a/(-1 + Cos[2*x])) - b/(-1 + Cos[2*x]) + (a*Cos[2*x])/(-1 + Cos[2*x]) - (b*Cos[2*x]
)/(-1 + Cos[2*x])])/(-a - b + a*Cos[2*x] - b*Cos[2*x]) + (b*Cos[2*x]*Sqrt[-(a/(-1 + Cos[2*x])) - b/(-1 + Cos[2
*x]) + (a*Cos[2*x])/(-1 + Cos[2*x]) - (b*Cos[2*x])/(-1 + Cos[2*x])])/(-a - b + a*Cos[2*x] - b*Cos[2*x]))*Sqrt[
a + b*Cot[x]^2]*Tan[x/2])/(Sqrt[2]*Sqrt[b]*Sqrt[(a + b + (-a + b)*Cos[2*x])*Sec[x/2]^4]*(((-4*Sqrt[a - b]*Sqrt
[b]*ArcTan[(Sqrt[a - b]*(-1 + Tan[x/2]^2))/Sqrt[b*Cos[x]^2*Sec[x/2]^4 + 4*a*Tan[x/2]^2]] - (a - 2*b)*ArcTanh[(
Sqrt[2]*(a + (-a + b)*Cos[x])*Sec[x/2]^2)/(Sqrt[b]*Sqrt[(a + b + (-a + b)*Cos[2*x])*Sec[x/2]^4])] + (a - 2*b)*
ArcTanh[(2*a + b*(-1 + Tan[x/2]^2))/(Sqrt[b]*Sqrt[b*Cos[x]^2*Sec[x/2]^4 + 4*a*Tan[x/2]^2])])*Sqrt[a + b*Cot[x]
^2]*Sec[x/2]^2)/(2*Sqrt[2]*Sqrt[b]*Sqrt[(a + b + (-a + b)*Cos[2*x])*Sec[x/2]^4]) - (Sqrt[b]*(-4*Sqrt[a - b]*Sq
rt[b]*ArcTan[(Sqrt[a - b]*(-1 + Tan[x/2]^2))/Sqrt[b*Cos[x]^2*Sec[x/2]^4 + 4*a*Tan[x/2]^2]] - (a - 2*b)*ArcTanh
[(Sqrt[2]*(a + (-a + b)*Cos[x])*Sec[x/2]^2)/(Sqrt[b]*Sqrt[(a + b + (-a + b)*Cos[2*x])*Sec[x/2]^4])] + (a - 2*b
)*ArcTanh[(2*a + b*(-1 + Tan[x/2]^2))/(Sqrt[b]*Sqrt[b*Cos[x]^2*Sec[x/2]^4 + 4*a*Tan[x/2]^2])])*Cot[x]*Csc[x]^2
*Tan[x/2])/(Sqrt[2]*Sqrt[a + b*Cot[x]^2]*Sqrt[(a + b + (-a + b)*Cos[2*x])*Sec[x/2]^4]) - ((-4*Sqrt[a - b]*Sqrt
[b]*ArcTan[(Sqrt[a - b]*(-1 + Tan[x/2]^2))/Sqrt[b*Cos[x]^2*Sec[x/2]^4 + 4*a*Tan[x/2]^2]] - (a - 2*b)*ArcTanh[(
Sqrt[2]*(a + (-a + b)*Cos[x])*Sec[x/2]^2)/(Sqrt[b]*Sqrt[(a + b + (-a + b)*Cos[2*x])*Sec[x/2]^4])] + (a - 2*b)*
ArcTanh[(2*a + b*(-1 + Tan[x/2]^2))/(Sqrt[b]*Sqrt[b*Cos[x]^2*Sec[x/2]^4 + 4*a*Tan[x/2]^2])])*Sqrt[a + b*Cot[x]
^2]*Tan[x/2]*(-2*(-a + b)*Sec[x/2]^4*Sin[2*x] + 2*(a + b + (-a + b)*Cos[2*x])*Sec[x/2]^4*Tan[x/2]))/(2*Sqrt[2]
*Sqrt[b]*((a + b + (-a + b)*Cos[2*x])*Sec[x/2]^4)^(3/2)) + (Sqrt[a + b*Cot[x]^2]*Tan[x/2]*(-(((a - 2*b)*(-((Sq
rt[2]*(-a + b)*Sec[x/2]^2*Sin[x])/(Sqrt[b]*Sqrt[(a + b + (-a + b)*Cos[2*x])*Sec[x/2]^4])) + (Sqrt[2]*(a + (-a
+ b)*Cos[x])*Sec[x/2]^2*Tan[x/2])/(Sqrt[b]*Sqrt[(a + b + (-a + b)*Cos[2*x])*Sec[x/2]^4]) - ((a + (-a + b)*Cos[
x])*Sec[x/2]^2*(-2*(-a + b)*Sec[x/2]^4*Sin[2*x] + 2*(a + b + (-a + b)*Cos[2*x])*Sec[x/2]^4*Tan[x/2]))/(Sqrt[2]
*Sqrt[b]*((a + b + (-a + b)*Cos[2*x])*Sec[x/2]^4)^(3/2))))/(1 - (2*(a + (-a + b)*Cos[x])^2)/(b*(a + b + (-a +
b)*Cos[2*x])))) - (4*Sqrt[a - b]*Sqrt[b]*(-1/2*(Sqrt[a - b]*(-2*b*Cos[x]*Sec[x/2]^4*Sin[x] + 4*a*Sec[x/2]^2*Ta
n[x/2] + 2*b*Cos[x]^2*Sec[x/2]^4*Tan[x/2])*(-1 + Tan[x/2]^2))/(b*Cos[x]^2*Sec[x/2]^4 + 4*a*Tan[x/2]^2)^(3/2) +
(Sqrt[a - b]*Sec[x/2]^2*Tan[x/2])/Sqrt[b*Cos[x]^2*Sec[x/2]^4 + 4*a*Tan[x/2]^2]))/(1 + ((a - b)*(-1 + Tan[x/2]
^2)^2)/(b*Cos[x]^2*Sec[x/2]^4 + 4*a*Tan[x/2]^2)) + ((a - 2*b)*((Sqrt[b]*Sec[x/2]^2*Tan[x/2])/Sqrt[b*Cos[x]^2*S
ec[x/2]^4 + 4*a*Tan[x/2]^2] - ((-2*b*Cos[x]*Sec[x/2]^4*Sin[x] + 4*a*Sec[x/2]^2*Tan[x/2] + 2*b*Cos[x]^2*Sec[x/2
]^4*Tan[x/2])*(2*a + b*(-1 + Tan[x/2]^2)))/(2*Sqrt[b]*(b*Cos[x]^2*Sec[x/2]^4 + 4*a*Tan[x/2]^2)^(3/2))))/(1 - (
2*a + b*(-1 + Tan[x/2]^2))^2/(b*(b*Cos[x]^2*Sec[x/2]^4 + 4*a*Tan[x/2]^2)))))/(Sqrt[2]*Sqrt[b]*Sqrt[(a + b + (-
a + b)*Cos[2*x])*Sec[x/2]^4])))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(176\) vs. \(2(71)=142\).
time = 0.15, size = 177, normalized size = 1.99

method	result	size
derivativedivides	\(-\frac {\cot \left (x \right ) \sqrt {a +b \left (\cot ^{2}\left (x \right )\right )}}{2}-\frac {a \ln \left (\sqrt {b}\, \cot \left (x \right )+\sqrt {a +b \left (\cot ^{2}\left (x \right )\right )}\right )}{2 \sqrt {b}}+b \left (\frac {\ln \left (\sqrt {b}\, \cot \left (x \right )+\sqrt {a +b \left (\cot ^{2}\left (x \right )\right )}\right )}{\sqrt {b}}-\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (x \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \left (\cot ^{2}\left (x \right )\right )}}\right )}{b^{2} \left (a -b \right )}\right )+\frac {a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (x \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \left (\cot ^{2}\left (x \right )\right )}}\right )}{b^{2} \left (a -b \right )}\)	\(177\)
default	\(-\frac {\cot \left (x \right ) \sqrt {a +b \left (\cot ^{2}\left (x \right )\right )}}{2}-\frac {a \ln \left (\sqrt {b}\, \cot \left (x \right )+\sqrt {a +b \left (\cot ^{2}\left (x \right )\right )}\right )}{2 \sqrt {b}}+b \left (\frac {\ln \left (\sqrt {b}\, \cot \left (x \right )+\sqrt {a +b \left (\cot ^{2}\left (x \right )\right )}\right )}{\sqrt {b}}-\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (x \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \left (\cot ^{2}\left (x \right )\right )}}\right )}{b^{2} \left (a -b \right )}\right )+\frac {a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (x \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \left (\cot ^{2}\left (x \right )\right )}}\right )}{b^{2} \left (a -b \right )}\)	\(177\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(sqrt(b*cot(x)^2 + a)*cot(x)^2, x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (71) = 142\).
time = 2.49, size = 768, normalized size = 8.63 \begin {gather*} \left [\frac {2 \, \sqrt {-a + b} 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 ) \sin \left (2 \, x\right ) - {\left (a - 2 \, b\right )} \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 ) \sin \left (2 \, x\right ) - 2 \, {\left (b \cos \left (2 \, x\right ) + b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{4 \, b \sin \left (2 \, x\right )}, \frac {4 \, \sqrt {a - b} 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 ) \sin \left (2 \, x\right ) - {\left (a - 2 \, b\right )} \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 ) \sin \left (2 \, x\right ) - 2 \, {\left (b \cos \left (2 \, x\right ) + b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{4 \, b \sin \left (2 \, x\right )}, \frac {{\left (a - 2 \, 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 ) \sin \left (2 \, x\right ) + \sqrt {-a + b} 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 ) \sin \left (2 \, x\right ) - {\left (b \cos \left (2 \, x\right ) + b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{2 \, b \sin \left (2 \, x\right )}, \frac {2 \, \sqrt {a - b} 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 ) \sin \left (2 \, x\right ) + {\left (a - 2 \, 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 ) \sin \left (2 \, x\right ) - {\left (b \cos \left (2 \, x\right ) + b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{2 \, b \sin \left (2 \, x\right )}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(2*sqrt(-a + b)*b*log(-(a - b)*cos(2*x) - sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*si
n(2*x) + b)*sin(2*x) - (a - 2*b)*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))*sin(2*x) - 2*(b*cos(2*x) + b)*sqrt(((a - b)*cos(2*x) - a -
b)/(cos(2*x) - 1)))/(b*sin(2*x)), 1/4*(4*sqrt(a - b)*b*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(co
s(2*x) - 1))*sin(2*x)/((a - b)*cos(2*x) + a - b))*sin(2*x) - (a - 2*b)*sqrt(b)*log(((a - 2*b)*cos(2*x) - 2*sqr
t(b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) - a - 2*b)/(cos(2*x) - 1))*sin(2*x) - 2*(b*cos(2
*x) + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(b*sin(2*x)), 1/2*((a - 2*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))*sin(2*x) + sqrt(-a + b)*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)*sin(2*x) - (b*cos(
2*x) + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(b*sin(2*x)), 1/2*(2*sqrt(a - b)*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))*sin(2*x) + (a - 2*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))*sin(2*x) -
(b*cos(2*x) + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(b*sin(2*x))]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a + b \cot ^{2}{\left (x \right )}} \cot ^{2}{\left (x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + b*cot(x)**2)*cot(x)**2, x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(si

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {cot}\left (x\right )}^2\,\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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