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

Optimal. Leaf size=29 \[ \frac{\cot (x)}{\sqrt{a \csc ^2(x)}}+\frac{\csc (x) \sec (x)}{\sqrt{a \csc ^2(x)}} \]

[Out]

Cot[x]/Sqrt[a*Csc[x]^2] + (Csc[x]*Sec[x])/Sqrt[a*Csc[x]^2]

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Rubi [A]  time = 0.101283, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {3657, 4125, 2590, 14} \[ \frac{\cot (x)}{\sqrt{a \csc ^2(x)}}+\frac{\csc (x) \sec (x)}{\sqrt{a \csc ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Int[Tan[x]^2/Sqrt[a + a*Cot[x]^2],x]

[Out]

Cot[x]/Sqrt[a*Csc[x]^2] + (Csc[x]*Sec[x])/Sqrt[a*Csc[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 4125

Int[(u_.)*((b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sec[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Sec[e + f*x]^n)^FracPart[p])/(Sec[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Sec[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rule 2590

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[f^(-1), Subst[Int[(1 - x^2
)^((m + n - 1)/2)/x^n, x], x, Cos[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n - 1)/2]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.0302993, size = 19, normalized size = 0.66 \[ \frac{\cot (x)+\csc (x) \sec (x)}{\sqrt{a \csc ^2(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cot[x] + Csc[x]*Sec[x])/Sqrt[a*Csc[x]^2]

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Maple [A]  time = 0.069, size = 33, normalized size = 1.1 \begin{align*}{\frac{\sqrt{4} \left ( \sin \left ( x \right ) \right ) ^{3}}{2\,\cos \left ( x \right ) \left ( -1+\cos \left ( x \right ) \right ) ^{2}}{\frac{1}{\sqrt{-{\frac{a}{ \left ( \cos \left ( x \right ) \right ) ^{2}-1}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*4^(1/2)*sin(x)^3/(-a/(cos(x)^2-1))^(1/2)/cos(x)/(-1+cos(x))^2

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Maxima [A]  time = 1.49405, size = 24, normalized size = 0.83 \begin{align*} \frac{\tan \left (x\right )^{2} + 2}{\sqrt{\tan \left (x\right )^{2} + 1} \sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.5484, size = 97, normalized size = 3.34 \begin{align*} \frac{{\left (\tan \left (x\right )^{3} + 2 \, \tan \left (x\right )\right )} \sqrt{\frac{a \tan \left (x\right )^{2} + a}{\tan \left (x\right )^{2}}}}{a \tan \left (x\right )^{2} + a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.32829, size = 55, normalized size = 1.9 \begin{align*} -\frac{2 \, \mathrm{sgn}\left (\tan \left (x\right )\right )}{\sqrt{a}} + \frac{\sqrt{a \tan \left (x\right )^{2} + a} + \frac{a}{\sqrt{a \tan \left (x\right )^{2} + a}}}{a \mathrm{sgn}\left (\tan \left (x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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