3.19 \(\int \cot ^3(x) \sqrt{a+b \cot ^2(x)} \, dx\)

Optimal. Leaf size=66 \[ -\frac{\left (a+b \cot ^2(x)\right )^{3/2}}{3 b}+\sqrt{a+b \cot ^2(x)}-\sqrt{a-b} \tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.116081, antiderivative size = 66, 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, 446, 80, 50, 63, 208} \[ -\frac{\left (a+b \cot ^2(x)\right )^{3/2}}{3 b}+\sqrt{a+b \cot ^2(x)}-\sqrt{a-b} \tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]^3*Sqrt[a + b*Cot[x]^2],x]

[Out]

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

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 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(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] && IntegerQ[Simplify[(m + 1)/n]]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 50

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

Rubi steps

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

Mathematica [A]  time = 0.177717, size = 65, normalized size = 0.98 \[ -\frac{\sqrt{a+b \cot ^2(x)} \left (a+b \cot ^2(x)-3 b\right )}{3 b}-\sqrt{a-b} \tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]^3*Sqrt[a + b*Cot[x]^2],x]

[Out]

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

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Maple [A]  time = 0.055, size = 84, normalized size = 1.3 \begin{align*} -{\frac{1}{3\,b} \left ( a+b \left ( \cot \left ( x \right ) \right ) ^{2} \right ) ^{{\frac{3}{2}}}}+\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}}-{b\arctan \left ({\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}}{\frac{1}{\sqrt{-a+b}}}} \right ){\frac{1}{\sqrt{-a+b}}}}+{a\arctan \left ({\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}}{\frac{1}{\sqrt{-a+b}}}} \right ){\frac{1}{\sqrt{-a+b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^3*(a+b*cot(x)^2)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.27327, size = 782, normalized size = 11.85 \begin{align*} \left [\frac{3 \,{\left (b \cos \left (2 \, x\right ) - b\right )} \sqrt{a - b} \log \left (-2 \,{\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} - 2 \, a^{2} + b^{2} + 2 \,{\left ({\left (a - b\right )} \cos \left (2 \, x\right )^{2} -{\left (2 \, a - b\right )} \cos \left (2 \, x\right ) + a\right )} \sqrt{a - b} \sqrt{\frac{{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} + 4 \,{\left (a^{2} - a b\right )} \cos \left (2 \, x\right )\right ) - 4 \,{\left ({\left (a - 4 \, b\right )} \cos \left (2 \, x\right ) - a + 2 \, b\right )} \sqrt{\frac{{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{12 \,{\left (b \cos \left (2 \, x\right ) - b\right )}}, -\frac{3 \,{\left (b \cos \left (2 \, x\right ) - 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}}{\left (\cos \left (2 \, x\right ) - 1\right )}}{{\left (a - b\right )} \cos \left (2 \, x\right ) - a}\right ) + 2 \,{\left ({\left (a - 4 \, b\right )} \cos \left (2 \, x\right ) - a + 2 \, b\right )} \sqrt{\frac{{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{6 \,{\left (b \cos \left (2 \, x\right ) - b\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^3*(a+b*cot(x)^2)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \cot ^{2}{\left (x \right )}} \cot ^{3}{\left (x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)**3*(a+b*cot(x)**2)**(1/2),x)

[Out]

Integral(sqrt(a + b*cot(x)**2)*cot(x)**3, x)

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Giac [B]  time = 2.37974, size = 267, normalized size = 4.05 \begin{align*} \frac{1}{6} \,{\left (3 \, \sqrt{a - b} \log \left ({\left (\sqrt{a - b} \sin \left (x\right ) - \sqrt{a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2}\right ) + \frac{4 \,{\left (3 \,{\left (\sqrt{a - b} \sin \left (x\right ) - \sqrt{a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{4} \sqrt{a - b}{\left (a - 2 \, b\right )} + 6 \,{\left (\sqrt{a - b} \sin \left (x\right ) - \sqrt{a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2} \sqrt{a - b} b^{2} +{\left (a b^{2} - 4 \, b^{3}\right )} \sqrt{a - b}\right )}}{{\left ({\left (\sqrt{a - b} \sin \left (x\right ) - \sqrt{a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2} - b\right )}^{3}}\right )} \mathrm{sgn}\left (\sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^3*(a+b*cot(x)^2)^(1/2),x, algorithm="giac")

[Out]

1/6*(3*sqrt(a - b)*log((sqrt(a - b)*sin(x) - sqrt(a*sin(x)^2 - b*sin(x)^2 + b))^2) + 4*(3*(sqrt(a - b)*sin(x)
- sqrt(a*sin(x)^2 - b*sin(x)^2 + b))^4*sqrt(a - b)*(a - 2*b) + 6*(sqrt(a - b)*sin(x) - sqrt(a*sin(x)^2 - b*sin
(x)^2 + b))^2*sqrt(a - b)*b^2 + (a*b^2 - 4*b^3)*sqrt(a - b))/((sqrt(a - b)*sin(x) - sqrt(a*sin(x)^2 - b*sin(x)
^2 + b))^2 - b)^3)*sgn(sin(x))