3.22 \(\int \cot ^2(x) \sqrt{a+b \cot ^2(x)} \, dx\)
Optimal. Leaf size=89 \[ -\frac{1}{2} \cot (x) \sqrt{a+b \cot ^2(x)}+\sqrt{a-b} \tan ^{-1}\left (\frac{\sqrt{a-b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right )-\frac{(a-2 b) \tanh ^{-1}\left (\frac{\sqrt{b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right )}{2 \sqrt{b}} \]
[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
________________________________________________________________________________________
Rubi [A] time = 0.12499, antiderivative size = 89, normalized size of antiderivative = 1.,
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 =
{3670, 478, 523, 217, 206, 377, 203} \[ -\frac{1}{2} \cot (x) \sqrt{a+b \cot ^2(x)}+\sqrt{a-b} \tan ^{-1}\left (\frac{\sqrt{a-b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right )-\frac{(a-2 b) \tanh ^{-1}\left (\frac{\sqrt{b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right )}{2 \sqrt{b}} \]
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 3670
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]))
Rule 478
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 523
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 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 \cot ^2(x) \sqrt{a+b \cot ^2(x)} \, dx &=-\operatorname{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} \operatorname{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) \operatorname{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) \operatorname{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) \operatorname{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) \operatorname{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*}
Mathematica [B] time = 22.2456, size = 2105, normalized size = 23.65 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Cot[x]^2*Sqrt[a + b*Cot[x]^2],x]
[Out]
-(Sqrt[(-a - b + a*Cos[2*x] - b*Cos[2*x])/(-1 + Cos[2*x])]*Cot[x])/2 + ((-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[(Sq
rt[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)*Ar
cTanh[(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]*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])])*Cot[x]*Csc[x]^2*T
an[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[(Sq
rt[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)*Ar
cTanh[(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]*S
qrt[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)*(-((Sqrt
[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]*S
qrt[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]*(-(Sqrt[a - b]*(-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])*(-1 + Tan[x/2]^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*Sec
[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])))
________________________________________________________________________________________
Maple [B] time = 0.035, size = 174, normalized size = 2. \begin{align*} -{\frac{\cot \left ( x \right ) }{2}\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}}}-{\frac{a}{2}\ln \left ( \cot \left ( x \right ) \sqrt{b}+\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}} \right ){\frac{1}{\sqrt{b}}}}+\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(cot(x)^2*(a+b*cot(x)^2)^(1/2),x)
[Out]
-1/2*cot(x)*(a+b*cot(x)^2)^(1/2)-1/2*a/b^(1/2)*ln(cot(x)*b^(1/2)+(a+b*cot(x)^2)^(1/2))+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))
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \cot \left (x\right )^{2} + a} \cot \left (x\right )^{2}\,{d x} \end{align*}
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)
________________________________________________________________________________________
Fricas [B] time = 1.84215, size = 1901, normalized size = 21.36 \begin{align*} \text{result too large to display} \end{align*}
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))]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \cot ^{2}{\left (x \right )}} \cot ^{2}{\left (x \right )}\, dx \end{align*}
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)
________________________________________________________________________________________
Giac [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(x)^2*(a+b*cot(x)^2)^(1/2),x, algorithm="giac")
[Out]
Timed out