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

Optimal. Leaf size=78 \[ -\frac {1}{(a-b)^2 \sqrt {a+b \cot ^2(x)}}-\frac {1}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{5/2}} \]

[Out]

arctanh((a+b*cot(x)^2)^(1/2)/(a-b)^(1/2))/(a-b)^(5/2)-1/3/(a-b)/(a+b*cot(x)^2)^(3/2)-1/(a-b)^2/(a+b*cot(x)^2)^
(1/2)

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Rubi [A]  time = 0.09, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3670, 444, 51, 63, 208} \[ -\frac {1}{(a-b)^2 \sqrt {a+b \cot ^2(x)}}-\frac {1}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]/(a + b*Cot[x]^2)^(5/2),x]

[Out]

ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]/(a - b)^(5/2) - 1/(3*(a - b)*(a + b*Cot[x]^2)^(3/2)) - 1/((a - b)^2*
Sqrt[a + b*Cot[x]^2])

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rule 444

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

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]))

Rubi steps

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

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Mathematica [C]  time = 0.04, size = 47, normalized size = 0.60 \[ -\frac {\, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {b \cot ^2(x)+a}{a-b}\right )}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Cot[x]^2)/(a - b)]/((a - b)*(a + b*Cot[x]^2)^(3/2))

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fricas [B]  time = 0.83, size = 627, normalized size = 8.04 \[ \left [\frac {3 \, {\left ({\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} + a^{2} + 2 \, a b + b^{2} - 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, x\right )\right )} \sqrt {a - b} \log \left (-\sqrt {a - b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} {\left (\cos \left (2 \, x\right ) - 1\right )} - {\left (a - b\right )} \cos \left (2 \, x\right ) + a\right ) - 4 \, {\left (2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} + 2 \, a^{2} - a b - b^{2} - {\left (4 \, a^{2} - 5 \, a b + b^{2}\right )} \cos \left (2 \, x\right )\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{6 \, {\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5} + {\left (a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a^{5} - 3 \, a^{4} b + 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 3 \, a b^{4} + b^{5}\right )} \cos \left (2 \, x\right )\right )}}, \frac {3 \, {\left ({\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} + a^{2} + 2 \, a b + b^{2} - 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, x\right )\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}}}{a - b}\right ) - 2 \, {\left (2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} + 2 \, a^{2} - a b - b^{2} - {\left (4 \, a^{2} - 5 \, a b + b^{2}\right )} \cos \left (2 \, x\right )\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{3 \, {\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5} + {\left (a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a^{5} - 3 \, a^{4} b + 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 3 \, a b^{4} + b^{5}\right )} \cos \left (2 \, x\right )\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(a+b*cot(x)^2)^(5/2),x, algorithm="fricas")

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(a+b*cot(x)^2)^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:inde
x.cc index_m i_lex_is_greater Error: Bad Argument Value

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maple [A]  time = 0.14, size = 75, normalized size = 0.96 \[ -\frac {1}{\left (a -b \right )^{2} \sqrt {a +b \left (\cot ^{2}\relax (x )\right )}}-\frac {1}{3 \left (a -b \right ) \left (a +b \left (\cot ^{2}\relax (x )\right )\right )^{\frac {3}{2}}}-\frac {\arctan \left (\frac {\sqrt {a +b \left (\cot ^{2}\relax (x )\right )}}{\sqrt {-a +b}}\right )}{\left (a -b \right )^{2} \sqrt {-a +b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(a-b)^2/(a+b*cot(x)^2)^(1/2)-1/3/(a-b)/(a+b*cot(x)^2)^(3/2)-1/(a-b)^2/(-a+b)^(1/2)*arctan((a+b*cot(x)^2)^(1
/2)/(-a+b)^(1/2))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(a+b*cot(x)^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a-4*b>0)', see `assume?` for
more details)Is 4*a-4*b positive or negative?

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mupad [B]  time = 4.46, size = 82, normalized size = 1.05 \[ \frac {\mathrm {atanh}\left (\frac {\sqrt {b\,{\mathrm {cot}\relax (x)}^2+a}\,\left (2\,a^2-4\,a\,b+2\,b^2\right )}{2\,{\left (a-b\right )}^{5/2}}\right )}{{\left (a-b\right )}^{5/2}}-\frac {\frac {1}{3\,\left (a-b\right )}+\frac {b\,{\mathrm {cot}\relax (x)}^2+a}{{\left (a-b\right )}^2}}{{\left (b\,{\mathrm {cot}\relax (x)}^2+a\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)/(a + b*cot(x)^2)^(5/2),x)

[Out]

atanh(((a + b*cot(x)^2)^(1/2)*(2*a^2 - 4*a*b + 2*b^2))/(2*(a - b)^(5/2)))/(a - b)^(5/2) - (1/(3*(a - b)) + (a
+ b*cot(x)^2)/(a - b)^2)/(a + b*cot(x)^2)^(3/2)

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sympy [A]  time = 16.77, size = 70, normalized size = 0.90 \[ - \frac {1}{3 \left (a - b\right ) \left (a + b \cot ^{2}{\relax (x )}\right )^{\frac {3}{2}}} - \frac {1}{\left (a - b\right )^{2} \sqrt {a + b \cot ^{2}{\relax (x )}}} - \frac {\operatorname {atan}{\left (\frac {\sqrt {a + b \cot ^{2}{\relax (x )}}}{\sqrt {- a + b}} \right )}}{\sqrt {- a + b} \left (a - b\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(3*(a - b)*(a + b*cot(x)**2)**(3/2)) - 1/((a - b)**2*sqrt(a + b*cot(x)**2)) - atan(sqrt(a + b*cot(x)**2)/sq
rt(-a + b))/(sqrt(-a + b)*(a - b)**2)

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