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((a*sec(x)^2)^(1/2)/a^(1/2))+a*(a*sec(x)^2)^(1/2)

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Rubi [A]  time = 0.09, antiderivative size = 37, 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 = {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 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])

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]

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*}

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Mathematica [A]  time = 0.04, 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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fricas [A]  time = 0.44, size = 49, normalized size = 1.32 \[ \frac {1}{2} \, a^{\frac {3}{2}} \log \left (\frac {a \tan \relax (x)^{2} - 2 \, \sqrt {a \tan \relax (x)^{2} + a} \sqrt {a} + 2 \, a}{\tan \relax (x)^{2}}\right ) + \sqrt {a \tan \relax (x)^{2} + a} a \]

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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giac [A]  time = 0.42, size = 42, normalized size = 1.14 \[ a^{2} {\left (\frac {\arctan \left (\frac {\sqrt {a \tan \relax (x)^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + \frac {\sqrt {a \tan \relax (x)^{2} + a}}{a}\right )} \]

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

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maple [A]  time = 0.42, size = 32, normalized size = 0.86 \[ \left (\cos \relax (x ) \ln \left (-\frac {-1+\cos \relax (x )}{\sin \relax (x )}\right )+\cos \relax (x )+1\right ) \left (\cos ^{2}\relax (x )\right ) \left (\frac {a}{\cos \relax (x )^{2}}\right )^{\frac {3}{2}} \]

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

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maxima [B]  time = 0.96, size = 134, normalized size = 3.62 \[ \frac {{\left (4 \, a \cos \left (2 \, x\right ) \cos \relax (x) + 4 \, a \sin \left (2 \, x\right ) \sin \relax (x) + 4 \, a \cos \relax (x) - {\left (a \cos \left (2 \, x\right )^{2} + a \sin \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a\right )} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) + {\left (a \cos \left (2 \, x\right )^{2} + a \sin \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a\right )} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right )\right )} \sqrt {a}}{2 \, {\left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right )}} \]

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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mupad [B]  time = 11.67, size = 33, normalized size = 0.89 \[ a\,\sqrt {a\,{\mathrm {tan}\relax (x)}^2+a}-a^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {a\,{\mathrm {tan}\relax (x)}^2+a}}{\sqrt {a}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)*(a + a*tan(x)^2)^(3/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \left (\tan ^{2}{\relax (x )} + 1\right )\right )^{\frac {3}{2}} \cot {\relax (x )}\, dx \]

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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