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

Optimal. Leaf size=69 \[ -(a-b) \sqrt{a+b \cot ^2(x)}-\frac{1}{3} \left (a+b \cot ^2(x)\right )^{3/2}+(a-b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.091423, antiderivative size = 69, normalized size of antiderivative = 1., 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, 50, 63, 208} \[ -(a-b) \sqrt{a+b \cot ^2(x)}-\frac{1}{3} \left (a+b \cot ^2(x)\right )^{3/2}+(a-b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]*(a + b*Cot[x]^2)^(3/2),x]

[Out]

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

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

Mathematica [A]  time = 0.164916, size = 63, normalized size = 0.91 \[ (a-b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right )-\frac{1}{3} \sqrt{a+b \cot ^2(x)} \left (4 a+b \cot ^2(x)-3 b\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a - b)^(3/2)*ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]] - (Sqrt[a + b*Cot[x]^2]*(4*a - 3*b + b*Cot[x]^2))/3

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Maple [B]  time = 0.018, size = 136, normalized size = 2. \begin{align*} -{\frac{b \left ( \cot \left ( x \right ) \right ) ^{2}}{3}\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}}}-{\frac{4\,a}{3}\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}}}+b\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}}-{{b}^{2}\arctan \left ({\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}}{\frac{1}{\sqrt{-a+b}}}} \right ){\frac{1}{\sqrt{-a+b}}}}+2\,{\frac{ab}{\sqrt{-a+b}}\arctan \left ({\frac{\sqrt{a+b \left ( \cot \left ( x \right ) \right ) ^{2}}}{\sqrt{-a+b}}} \right ) }-{{a}^{2}\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)*(a+b*cot(x)^2)^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.28255, size = 803, normalized size = 11.64 \begin{align*} \left [-\frac{3 \,{\left ({\left (a - b\right )} \cos \left (2 \, x\right ) - a + 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 ) + 8 \,{\left (2 \,{\left (a - b\right )} \cos \left (2 \, x\right ) - 2 \, a + b\right )} \sqrt{\frac{{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{12 \,{\left (\cos \left (2 \, x\right ) - 1\right )}}, \frac{3 \,{\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}}{\left (\cos \left (2 \, x\right ) - 1\right )}}{{\left (a - b\right )} \cos \left (2 \, x\right ) - a}\right ) - 4 \,{\left (2 \,{\left (a - b\right )} \cos \left (2 \, x\right ) - 2 \, a + b\right )} \sqrt{\frac{{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{6 \,{\left (\cos \left (2 \, x\right ) - 1\right )}}\right ] \end{align*}

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]

[-1/12*(3*((a - b)*cos(2*x) - a + 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)) + 8*(2*(a - b)*cos(2*x) - 2*a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(cos(2*x)
- 1), 1/6*(3*((a - b)*cos(2*x) - a + b)*sqrt(-a + b)*arctan(-sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(co
s(2*x) - 1))*(cos(2*x) - 1)/((a - b)*cos(2*x) - a)) - 4*(2*(a - b)*cos(2*x) - 2*a + b)*sqrt(((a - b)*cos(2*x)
- a - b)/(cos(2*x) - 1)))/(cos(2*x) - 1)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 5.72451, size = 285, normalized size = 4.13 \begin{align*} -\frac{1}{6} \,{\left (3 \,{\left (a - b\right )}^{\frac{3}{2}} \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{8 \,{\left (3 \,{\left (a b - b^{2}\right )}{\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} - 3 \,{\left (a b^{2} - b^{3}\right )}{\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} + 2 \,{\left (a b^{3} - b^{4}\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)*(a+b*cot(x)^2)^(3/2),x, algorithm="giac")

[Out]

-1/6*(3*(a - b)^(3/2)*log((sqrt(a - b)*sin(x) - sqrt(a*sin(x)^2 - b*sin(x)^2 + b))^2) - 8*(3*(a*b - b^2)*(sqrt
(a - b)*sin(x) - sqrt(a*sin(x)^2 - b*sin(x)^2 + b))^4*sqrt(a - b) - 3*(a*b^2 - b^3)*(sqrt(a - b)*sin(x) - sqrt
(a*sin(x)^2 - b*sin(x)^2 + b))^2*sqrt(a - b) + 2*(a*b^3 - b^4)*sqrt(a - b))/((sqrt(a - b)*sin(x) - sqrt(a*sin(
x)^2 - b*sin(x)^2 + b))^2 - b)^3)*sgn(sin(x))