3.51 \(\int \frac {\cot (x)}{(a+b \cot ^2(x))^{3/2}} \, dx\)

Optimal. Leaf size=55 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2}}-\frac {1}{(a-b) \sqrt {a+b \cot ^2(x)}} \]

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 5, 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} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2}}-\frac {1}{(a-b) \sqrt {a+b \cot ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]/(a + b*Cot[x]^2)^(3/2),x]

[Out]

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

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Mathematica [C]  time = 0.04, size = 44, normalized size = 0.80 \[ \frac {\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b \cot ^2(x)+a}{a-b}\right )}{(b-a) \sqrt {a+b \cot ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]/(a + b*Cot[x]^2)^(3/2),x]

[Out]

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

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fricas [B]  time = 0.47, size = 344, normalized size = 6.25 \[ \left [\frac {{\left ({\left (a - b\right )} \cos \left (2 \, x\right ) - a - b\right )} \sqrt {a - b} \log \left (\sqrt {a - b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} {\left (\cos \left (2 \, x\right ) - 1\right )} - {\left (a - b\right )} \cos \left (2 \, x\right ) + a\right ) + 2 \, {\left ({\left (a - b\right )} \cos \left (2 \, x\right ) - a + b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{2 \, {\left (a^{3} - a^{2} b - a b^{2} + b^{3} - {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cos \left (2 \, x\right )\right )}}, -\frac {{\left ({\left (a - b\right )} \cos \left (2 \, x\right ) - a - b\right )} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {-a + b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{a - b}\right ) - {\left ({\left (a - b\right )} \cos \left (2 \, x\right ) - a + b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{a^{3} - a^{2} b - a b^{2} + b^{3} - {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cos \left (2 \, x\right )}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(a+b*cot(x)^2)^(3/2),x, algorithm="fricas")

[Out]

[1/2*(((a - b)*cos(2*x) - a - b)*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) + 2*((a - b)*cos(2*x) - a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x)
- 1)))/(a^3 - a^2*b - a*b^2 + b^3 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(2*x)), -(((a - b)*cos(2*x) - a - b)*s
qrt(-a + b)*arctan(-sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))/(a - b)) - ((a - b)*cos(2*x)
- a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(a^3 - a^2*b - a*b^2 + b^3 - (a^3 - 3*a^2*b + 3*a*b^
2 - b^3)*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)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to divide, perhaps due to rounding error%%%{%%{[%%%{2,[1,2]%%%}+%%%{-2,[0,3]%%%},0]:[1,0,%%%{-1,[1,0]%%%}+%
%%{1,[0,1]%%%}]%%},[2]%%%}+%%%{%%%{4,[2,2]%%%}+%%%{-4,[1,3]%%%},[1]%%%}+%%%{%%{[%%%{2,[2,2]%%%},0]:[1,0,%%%{-1
,[1,0]%%%}+%%%{1,[0,1]%%%}]%%},[0]%%%} / %%%{%%%{1,[2,0]%%%}+%%%{-2,[1,1]%%%}+%%%{1,[0,2]%%%},[2]%%%}+%%%{%%{[
%%%{2,[2,0]%%%}+%%%{-2,[1,1]%%%},0]:[1,0,%%%{-1,[1,0]%%%}+%%%{1,[0,1]%%%}]%%},[1]%%%}+%%%{%%%{1,[3,0]%%%}+%%%{
-1,[2,1]%%%},[0]%%%} Error: Bad Argument Value

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maple [A]  time = 0.14, size = 56, normalized size = 1.02 \[ -\frac {1}{\left (a -b \right ) \sqrt {a +b \left (\cot ^{2}\relax (x )\right )}}-\frac {\arctan \left (\frac {\sqrt {a +b \left (\cot ^{2}\relax (x )\right )}}{\sqrt {-a +b}}\right )}{\left (a -b \right ) \sqrt {-a +b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)/(a+b*cot(x)^2)^(3/2),x)

[Out]

-1/(a-b)/(a+b*cot(x)^2)^(1/2)-1/(a-b)/(-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)^(3/2),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 = 1.78, size = 47, normalized size = 0.85 \[ \frac {\mathrm {atanh}\left (\frac {\sqrt {b\,{\mathrm {cot}\relax (x)}^2+a}}{\sqrt {a-b}}\right )}{{\left (a-b\right )}^{3/2}}-\frac {1}{\left (a-b\right )\,\sqrt {b\,{\mathrm {cot}\relax (x)}^2+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)/(a + b*cot(x)^2)^(3/2),x)

[Out]

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

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sympy [A]  time = 10.37, size = 48, normalized size = 0.87 \[ - \frac {1}{\left (a - b\right ) \sqrt {a + b \cot ^{2}{\relax (x )}}} - \frac {\operatorname {atan}{\left (\frac {\sqrt {a + b \cot ^{2}{\relax (x )}}}{\sqrt {- a + b}} \right )}}{\sqrt {- a + b} \left (a - b\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(a+b*cot(x)**2)**(3/2),x)

[Out]

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

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