3.55 \(\int \frac{\cot ^2(x)}{(a+b \cot ^2(x))^{5/2}} \, dx\)

Optimal. Leaf size=94 \[ -\frac{(2 a+b) \cot (x)}{3 a (a-b)^2 \sqrt{a+b \cot ^2(x)}}-\frac{\cot (x)}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}+\frac{\tan ^{-1}\left (\frac{\sqrt{a-b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right )}{(a-b)^{5/2}} \]

[Out]

ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]]/(a - b)^(5/2) - Cot[x]/(3*(a - b)*(a + b*Cot[x]^2)^(3/2)) -
((2*a + b)*Cot[x])/(3*a*(a - b)^2*Sqrt[a + b*Cot[x]^2])

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Rubi [A]  time = 0.124137, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {3670, 471, 527, 12, 377, 203} \[ -\frac{(2 a+b) \cot (x)}{3 a (a-b)^2 \sqrt{a+b \cot ^2(x)}}-\frac{\cot (x)}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}+\frac{\tan ^{-1}\left (\frac{\sqrt{a-b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right )}{(a-b)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]^2/(a + b*Cot[x]^2)^(5/2),x]

[Out]

ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]]/(a - b)^(5/2) - Cot[x]/(3*(a - b)*(a + b*Cot[x]^2)^(3/2)) -
((2*a + b)*Cot[x])/(3*a*(a - b)^2*Sqrt[a + b*Cot[x]^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 471

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 + 1))/(n*(b*c - a*d)*(p + 1)), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 \frac{\cot ^2(x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx &=-\operatorname{Subst}\left (\int \frac{x^2}{\left (1+x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx,x,\cot (x)\right )\\ &=-\frac{\cot (x)}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{1-2 x^2}{\left (1+x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\cot (x)\right )}{3 (a-b)}\\ &=-\frac{\cot (x)}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}-\frac{(2 a+b) \cot (x)}{3 a (a-b)^2 \sqrt{a+b \cot ^2(x)}}+\frac{\operatorname{Subst}\left (\int \frac{3 a}{\left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\cot (x)\right )}{3 a (a-b)^2}\\ &=-\frac{\cot (x)}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}-\frac{(2 a+b) \cot (x)}{3 a (a-b)^2 \sqrt{a+b \cot ^2(x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \sqrt{a+b x^2}} \, dx,x,\cot (x)\right )}{(a-b)^2}\\ &=-\frac{\cot (x)}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}-\frac{(2 a+b) \cot (x)}{3 a (a-b)^2 \sqrt{a+b \cot ^2(x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-(-a+b) x^2} \, dx,x,\frac{\cot (x)}{\sqrt{a+b \cot ^2(x)}}\right )}{(a-b)^2}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{a-b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right )}{(a-b)^{5/2}}-\frac{\cot (x)}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}-\frac{(2 a+b) \cot (x)}{3 a (a-b)^2 \sqrt{a+b \cot ^2(x)}}\\ \end{align*}

Mathematica [C]  time = 6.63784, size = 200, normalized size = 2.13 \[ \frac{\tan (x) \left (-12 (a-b)^3 \cos ^4(x) \cot ^2(x) \left (a+b \cot ^2(x)\right ) \text{Hypergeometric2F1}\left (2,2,\frac{9}{2},\frac{(a-b) \cos ^2(x)}{a}\right )-\frac{35 a \sin ^2(x) \left (5 a+2 b \cot ^2(x)\right ) \left (a \csc ^2(x) \left ((a-4 b) \cot ^2(x)-3 a\right ) \sqrt{\frac{(a-b) \sin ^2(x) \cos ^2(x) \left (a+b \cot ^2(x)\right )}{a^2}}+3 \left (a+b \cot ^2(x)\right )^2 \sin ^{-1}\left (\sqrt{\frac{(a-b) \cos ^2(x)}{a}}\right )\right )}{\sqrt{\frac{(a-b) \sin ^2(x) \cos ^2(x) \left (a+b \cot ^2(x)\right )}{a^2}}}\right )}{315 a^3 (a-b)^2 \left (a+b \cot ^2(x)\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cot[x]^2/(a + b*Cot[x]^2)^(5/2),x]

[Out]

((-12*(a - b)^3*Cos[x]^4*Cot[x]^2*(a + b*Cot[x]^2)*Hypergeometric2F1[2, 2, 9/2, ((a - b)*Cos[x]^2)/a] - (35*a*
(5*a + 2*b*Cot[x]^2)*Sin[x]^2*(3*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*(a + b*Cot[x]^2)^2 + a*(-3*a + (a - 4*b)*C
ot[x]^2)*Csc[x]^2*Sqrt[((a - b)*Cos[x]^2*(a + b*Cot[x]^2)*Sin[x]^2)/a^2]))/Sqrt[((a - b)*Cos[x]^2*(a + b*Cot[x
]^2)*Sin[x]^2)/a^2])*Tan[x])/(315*a^3*(a - b)^2*(a + b*Cot[x]^2)^(3/2))

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Maple [B]  time = 0.022, size = 166, normalized size = 1.8 \begin{align*} -{\frac{\cot \left ( x \right ) }{3\,a} \left ( a+b \left ( \cot \left ( x \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,\cot \left ( x \right ) }{3\,{a}^{2}}{\frac{1}{\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}}}}}+{\frac{1}{ \left ( a-b \right ) ^{3}{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 ) }-{\frac{b\cot \left ( x \right ) }{ \left ( a-b \right ) ^{2}a}{\frac{1}{\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}}}}}-{\frac{b\cot \left ( x \right ) }{ \left ( 3\,a-3\,b \right ) a} \left ( a+b \left ( \cot \left ( x \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,b\cot \left ( x \right ) }{ \left ( 3\,a-3\,b \right ){a}^{2}}{\frac{1}{\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^2/(a+b*cot(x)^2)^(5/2),x)

[Out]

-1/3*cot(x)/a/(a+b*cot(x)^2)^(3/2)-2/3/a^2*cot(x)/(a+b*cot(x)^2)^(1/2)+1/(a-b)^3*(b^4*(a-b))^(1/2)/b^2*arctan(
b^2*(a-b)/(b^4*(a-b))^(1/2)/(a+b*cot(x)^2)^(1/2)*cot(x))-1/(a-b)^2*b*cot(x)/a/(a+b*cot(x)^2)^(1/2)-1/3/(a-b)*b
*cot(x)/a/(a+b*cot(x)^2)^(3/2)-2/3/(a-b)*b/a^2*cot(x)/(a+b*cot(x)^2)^(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(cot(x)^2/(a+b*cot(x)^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.07883, size = 1589, normalized size = 16.9 \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)^(5/2),x, algorithm="fricas")

[Out]

[-1/12*(3*(a^3 + 2*a^2*b + a*b^2 + (a^3 - 2*a^2*b + a*b^2)*cos(2*x)^2 - 2*(a^3 - a*b^2)*cos(2*x))*sqrt(-a + b)
*log(-2*(a^2 - 2*a*b + b^2)*cos(2*x)^2 + 2*((a - b)*cos(2*x) - b)*sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)
/(cos(2*x) - 1))*sin(2*x) + a^2 - 2*b^2 + 4*(a*b - b^2)*cos(2*x)) + 4*(3*a^3 - a^2*b - a*b^2 - b^3 - (3*a^3 -
5*a^2*b + a*b^2 + b^3)*cos(2*x))*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x))/(a^6 - a^5*b - 2*a^
4*b^2 + 2*a^3*b^3 + a^2*b^4 - a*b^5 + (a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*cos(2*x)^2
- 2*(a^6 - 3*a^5*b + 2*a^4*b^2 + 2*a^3*b^3 - 3*a^2*b^4 + a*b^5)*cos(2*x)), 1/6*(3*(a^3 + 2*a^2*b + a*b^2 + (a
^3 - 2*a^2*b + a*b^2)*cos(2*x)^2 - 2*(a^3 - a*b^2)*cos(2*x))*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) - b)) - 2*(3*a^3 - a^2*b - a*b^2 - b^3 - (3*a^3 - 5*
a^2*b + a*b^2 + b^3)*cos(2*x))*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x))/(a^6 - a^5*b - 2*a^4*
b^2 + 2*a^3*b^3 + a^2*b^4 - a*b^5 + (a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*cos(2*x)^2 -
2*(a^6 - 3*a^5*b + 2*a^4*b^2 + 2*a^3*b^3 - 3*a^2*b^4 + a*b^5)*cos(2*x))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot ^{2}{\left (x \right )}}{\left (a + b \cot ^{2}{\left (x \right )}\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)**2/(a+b*cot(x)**2)**(5/2),x)

[Out]

Integral(cot(x)**2/(a + b*cot(x)**2)**(5/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (x\right )^{2}}{{\left (b \cot \left (x\right )^{2} + a\right )}^{\frac{5}{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)^(5/2),x, algorithm="giac")

[Out]

integrate(cot(x)^2/(b*cot(x)^2 + a)^(5/2), x)