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

Optimal. Leaf size=45 \[ \frac{1}{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a \sec ^2(x)}}{\sqrt{a}}\right )-\frac{1}{2} \cot ^2(x) \sqrt{a \sec ^2(x)} \]

[Out]

(Sqrt[a]*ArcTanh[Sqrt[a*Sec[x]^2]/Sqrt[a]])/2 - (Cot[x]^2*Sqrt[a*Sec[x]^2])/2

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Rubi [A]  time = 0.089685, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {3657, 4124, 51, 63, 207} \[ \frac{1}{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a \sec ^2(x)}}{\sqrt{a}}\right )-\frac{1}{2} \cot ^2(x) \sqrt{a \sec ^2(x)} \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]^3*Sqrt[a + a*Tan[x]^2],x]

[Out]

(Sqrt[a]*ArcTanh[Sqrt[a*Sec[x]^2]/Sqrt[a]])/2 - (Cot[x]^2*Sqrt[a*Sec[x]^2])/2

Rule 3657

Int[(u_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*sec[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a, b]

Rule 4124

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> Dist[b/(2*f), Subst[In
t[(-1 + x)^((m - 1)/2)*(b*x)^(p - 1), x], x, Sec[e + f*x]^2], x] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p] &&
IntegerQ[(m - 1)/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 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \cot ^3(x) \sqrt{a+a \tan ^2(x)} \, dx &=\int \cot ^3(x) \sqrt{a \sec ^2(x)} \, dx\\ &=\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{(-1+x)^2 \sqrt{a x}} \, dx,x,\sec ^2(x)\right )\\ &=-\frac{1}{2} \cot ^2(x) \sqrt{a \sec ^2(x)}-\frac{1}{4} a \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{a x}} \, dx,x,\sec ^2(x)\right )\\ &=-\frac{1}{2} \cot ^2(x) \sqrt{a \sec ^2(x)}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-1+\frac{x^2}{a}} \, dx,x,\sqrt{a \sec ^2(x)}\right )\\ &=\frac{1}{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a \sec ^2(x)}}{\sqrt{a}}\right )-\frac{1}{2} \cot ^2(x) \sqrt{a \sec ^2(x)}\\ \end{align*}

Mathematica [A]  time = 0.0815088, size = 38, normalized size = 0.84 \[ -\frac{1}{2} \cos (x) \sqrt{a \sec ^2(x)} \left (\log \left (\sin \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )\right )+\cot (x) \csc (x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Cos[x]*(Cot[x]*Csc[x] - Log[Cos[x/2]] + Log[Sin[x/2]])*Sqrt[a*Sec[x]^2])/2

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Maple [A]  time = 0.089, size = 51, normalized size = 1.1 \begin{align*}{\frac{\cos \left ( x \right ) }{2\, \left ( \sin \left ( x \right ) \right ) ^{2}} \left ( \left ( \cos \left ( x \right ) \right ) ^{2}\ln \left ( -{\frac{\cos \left ( x \right ) -1}{\sin \left ( x \right ) }} \right ) -\cos \left ( x \right ) -\ln \left ( -{\frac{\cos \left ( x \right ) -1}{\sin \left ( x \right ) }} \right ) \right ) \sqrt{{\frac{a}{ \left ( \cos \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(cos(x)^2*ln(-(cos(x)-1)/sin(x))-cos(x)-ln(-(cos(x)-1)/sin(x)))*cos(x)*(a/cos(x)^2)^(1/2)/sin(x)^2

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Maxima [B]  time = 1.92552, size = 409, normalized size = 9.09 \begin{align*} -\frac{{\left (4 \,{\left (\cos \left (3 \, x\right ) + \cos \left (x\right )\right )} \cos \left (4 \, x\right ) - 4 \,{\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (3 \, x\right ) - 8 \, \cos \left (2 \, x\right ) \cos \left (x\right ) -{\left (2 \,{\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) +{\left (2 \,{\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) + 4 \,{\left (\sin \left (3 \, x\right ) + \sin \left (x\right )\right )} \sin \left (4 \, x\right ) - 8 \, \sin \left (3 \, x\right ) \sin \left (2 \, x\right ) - 8 \, \sin \left (2 \, x\right ) \sin \left (x\right ) + 4 \, \cos \left (x\right )\right )} \sqrt{a}}{4 \,{\left (2 \,{\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*(cos(3*x) + cos(x))*cos(4*x) - 4*(2*cos(2*x) - 1)*cos(3*x) - 8*cos(2*x)*cos(x) - (2*(2*cos(2*x) - 1)*c
os(4*x) - cos(4*x)^2 - 4*cos(2*x)^2 - sin(4*x)^2 + 4*sin(4*x)*sin(2*x) - 4*sin(2*x)^2 + 4*cos(2*x) - 1)*log(co
s(x)^2 + sin(x)^2 + 2*cos(x) + 1) + (2*(2*cos(2*x) - 1)*cos(4*x) - cos(4*x)^2 - 4*cos(2*x)^2 - sin(4*x)^2 + 4*
sin(4*x)*sin(2*x) - 4*sin(2*x)^2 + 4*cos(2*x) - 1)*log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1) + 4*(sin(3*x) + sin
(x))*sin(4*x) - 8*sin(3*x)*sin(2*x) - 8*sin(2*x)*sin(x) + 4*cos(x))*sqrt(a)/(2*(2*cos(2*x) - 1)*cos(4*x) - cos
(4*x)^2 - 4*cos(2*x)^2 - sin(4*x)^2 + 4*sin(4*x)*sin(2*x) - 4*sin(2*x)^2 + 4*cos(2*x) - 1)

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Fricas [A]  time = 1.41154, size = 167, normalized size = 3.71 \begin{align*} \frac{\sqrt{a} \log \left (\frac{a \tan \left (x\right )^{2} + 2 \, \sqrt{a \tan \left (x\right )^{2} + a} \sqrt{a} + 2 \, a}{\tan \left (x\right )^{2}}\right ) \tan \left (x\right )^{2} - 2 \, \sqrt{a \tan \left (x\right )^{2} + a}}{4 \, \tan \left (x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(sqrt(a)*log((a*tan(x)^2 + 2*sqrt(a*tan(x)^2 + a)*sqrt(a) + 2*a)/tan(x)^2)*tan(x)^2 - 2*sqrt(a*tan(x)^2 +
a))/tan(x)^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \left (\tan ^{2}{\left (x \right )} + 1\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+a*tan(x)**2)**(1/2),x)

[Out]

Integral(sqrt(a*(tan(x)**2 + 1))*cot(x)**3, x)

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Giac [A]  time = 1.10714, size = 68, normalized size = 1.51 \begin{align*} -\frac{1}{2} \, a^{2}{\left (\frac{\arctan \left (\frac{\sqrt{a \tan \left (x\right )^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{\sqrt{a \tan \left (x\right )^{2} + a}}{a^{2} \tan \left (x\right )^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a^2*(arctan(sqrt(a*tan(x)^2 + a)/sqrt(-a))/(sqrt(-a)*a) + sqrt(a*tan(x)^2 + a)/(a^2*tan(x)^2))