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]

1/2*arctanh((a*sec(x)^2)^(1/2)/a^(1/2))*a^(1/2)-1/2*cot(x)^2*(a*sec(x)^2)^(1/2)

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Rubi [A]  time = 0.09, antiderivative size = 45, normalized size of antiderivative = 1.00, 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 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])

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 ^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*}

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Mathematica [A]  time = 0.08, 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]

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

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fricas [A]  time = 0.43, size = 58, normalized size = 1.29 \[ \frac {\sqrt {a} \log \left (\frac {a \tan \relax (x)^{2} + 2 \, \sqrt {a \tan \relax (x)^{2} + a} \sqrt {a} + 2 \, a}{\tan \relax (x)^{2}}\right ) \tan \relax (x)^{2} - 2 \, \sqrt {a \tan \relax (x)^{2} + a}}{4 \, \tan \relax (x)^{2}} \]

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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giac [A]  time = 0.27, size = 42, normalized size = 0.93 \[ -\frac {a \arctan \left (\frac {\sqrt {a \tan \relax (x)^{2} + a}}{\sqrt {-a}}\right )}{2 \, \sqrt {-a}} - \frac {\sqrt {a \tan \relax (x)^{2} + a}}{2 \, \tan \relax (x)^{2}} \]

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

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maple [A]  time = 0.56, size = 51, normalized size = 1.13 \[ \frac {\left (\left (\cos ^{2}\relax (x )\right ) \ln \left (-\frac {-1+\cos \relax (x )}{\sin \relax (x )}\right )-\cos \relax (x )-\ln \left (-\frac {-1+\cos \relax (x )}{\sin \relax (x )}\right )\right ) \cos \relax (x ) \sqrt {\frac {a}{\cos \relax (x )^{2}}}}{2 \sin \relax (x )^{2}} \]

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

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maxima [B]  time = 0.86, size = 303, normalized size = 6.73 \[ -\frac {{\left (4 \, {\left (\cos \left (3 \, x\right ) + \cos \relax (x)\right )} \cos \left (4 \, x\right ) - 4 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (3 \, x\right ) - 8 \, \cos \left (2 \, x\right ) \cos \relax (x) - {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) + {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) + 4 \, {\left (\sin \left (3 \, x\right ) + \sin \relax (x)\right )} \sin \left (4 \, x\right ) - 8 \, \sin \left (3 \, x\right ) \sin \left (2 \, x\right ) - 8 \, \sin \left (2 \, x\right ) \sin \relax (x) + 4 \, \cos \relax (x)\right )} \sqrt {a}}{4 \, {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )}} \]

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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mupad [B]  time = 12.04, size = 37, normalized size = 0.82 \[ \frac {\sqrt {a}\,\mathrm {atanh}\left (\frac {\sqrt {a\,{\mathrm {tan}\relax (x)}^2+a}}{\sqrt {a}}\right )}{2}-\frac {\sqrt {a\,{\mathrm {tan}\relax (x)}^2+a}}{2\,{\mathrm {tan}\relax (x)}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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