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

Optimal. Leaf size=37 \[ a \sqrt{a \sec ^2(x)}-a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a \sec ^2(x)}}{\sqrt{a}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0905474, antiderivative size = 37, normalized size of antiderivative = 1., 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 = {3657, 4124, 50, 63, 207} \[ a \sqrt{a \sec ^2(x)}-a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a \sec ^2(x)}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]*(a + a*Tan[x]^2)^(3/2),x]

[Out]

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

Mathematica [A]  time = 0.0380756, size = 34, normalized size = 0.92 \[ a \sqrt{a \sec ^2(x)} \left (\cos (x) \left (\log \left (\sin \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )\right )\right )+1\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*(1 + Cos[x]*(-Log[Cos[x/2]] + Log[Sin[x/2]]))*Sqrt[a*Sec[x]^2]

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Maple [A]  time = 0.069, size = 32, normalized size = 0.9 \begin{align*} \left ( \cos \left ( x \right ) \ln \left ( -{\frac{\cos \left ( x \right ) -1}{\sin \left ( x \right ) }} \right ) +\cos \left ( x \right ) +1 \right ) \left ( \cos \left ( x \right ) \right ) ^{2} \left ({\frac{a}{ \left ( \cos \left ( x \right ) \right ) ^{2}}} \right ) ^{{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(cos(x)*ln(-(cos(x)-1)/sin(x))+cos(x)+1)*cos(x)^2*(a/cos(x)^2)^(3/2)

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Maxima [B]  time = 1.90549, size = 181, normalized size = 4.89 \begin{align*} \frac{{\left (4 \, a \cos \left (2 \, x\right ) \cos \left (x\right ) + 4 \, a \sin \left (2 \, x\right ) \sin \left (x\right ) + 4 \, a \cos \left (x\right ) -{\left (a \cos \left (2 \, x\right )^{2} + a \sin \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) +{\left (a \cos \left (2 \, x\right )^{2} + a \sin \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right )\right )} \sqrt{a}}{2 \,{\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.43828, size = 140, normalized size = 3.78 \begin{align*} \frac{1}{2} \, a^{\frac{3}{2}} \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 ) + \sqrt{a \tan \left (x\right )^{2} + a} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^(3/2)*log((a*tan(x)^2 - 2*sqrt(a*tan(x)^2 + a)*sqrt(a) + 2*a)/tan(x)^2) + sqrt(a*tan(x)^2 + a)*a

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \left (\tan ^{2}{\left (x \right )} + 1\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+a*tan(x)**2)**(3/2),x)

[Out]

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

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Giac [A]  time = 1.08167, size = 50, normalized size = 1.35 \begin{align*}{\left (\frac{a \arctan \left (\frac{\sqrt{a \tan \left (x\right )^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \sqrt{a \tan \left (x\right )^{2} + a}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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