\(\int \frac {\tan ^2(x)}{\sqrt {a+a \cot ^2(x)}} \, dx\) [18]

Optimal result

Integrand size = 17, antiderivative size = 29 \[ \int \frac {\tan ^2(x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\frac {\cot (x)}{\sqrt {a \csc ^2(x)}}+\frac {\csc (x) \sec (x)}{\sqrt {a \csc ^2(x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {3738, 4210, 2670, 14} \[ \int \frac {\tan ^2(x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\frac {\cot (x)}{\sqrt {a \csc ^2(x)}}+\frac {\csc (x) \sec (x)}{\sqrt {a \csc ^2(x)}} \]

[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 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]]

Rule 2670

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 3738

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 4210

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
]])

Rubi steps \begin{align*} \text {integral}& = \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) \text {Subst}\left (\int \frac {1-x^2}{x^2} \, dx,x,\cos (x)\right )}{\sqrt {a \csc ^2(x)}} \\ & = -\frac {\csc (x) \text {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] (verified)

Time = 0.14 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.66 \[ \int \frac {\tan ^2(x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\frac {\cot (x)+\csc (x) \sec (x)}{\sqrt {a \csc ^2(x)}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83

method	result	size
default	\(\frac {\sqrt {4}\, \left (\cos \left (x \right )+1\right )^{2} \sec \left (x \right ) \csc \left (x \right )}{2 \sqrt {a \csc \left (x \right )^{2}}}\)	\(24\)
risch	\(\frac {i \left ({\mathrm e}^{4 i x}+6 \,{\mathrm e}^{2 i x}+1\right )}{2 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right ) \left ({\mathrm e}^{2 i x}+1\right )}\)	\(55\)

[In]

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

[Out]

1/2*4^(1/2)*(cos(x)+1)^2/(a*csc(x)^2)^(1/2)*sec(x)*csc(x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21 \[ \int \frac {\tan ^2(x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\frac {{\left (\tan \left (x\right )^{3} + 2 \, \tan \left (x\right )\right )} \sqrt {\frac {a \tan \left (x\right )^{2} + a}{\tan \left (x\right )^{2}}}}{a \tan \left (x\right )^{2} + a} \]

[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)

Sympy [F]

\[ \int \frac {\tan ^2(x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\int \frac {\tan ^{2}{\left (x \right )}}{\sqrt {a \left (\cot ^{2}{\left (x \right )} + 1\right )}}\, dx \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.62 \[ \int \frac {\tan ^2(x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\frac {\tan \left (x\right )^{2} + 2}{\sqrt {\tan \left (x\right )^{2} + 1} \sqrt {a}} \]

[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))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {\tan ^2(x)}{\sqrt {a+a \cot ^2(x)}} \, dx=-\frac {2 \, \mathrm {sgn}\left (\sin \left (x\right )\right )}{\sqrt {a}} + \frac {\frac {1}{\cos \left (x\right )} + \cos \left (x\right )}{\sqrt {a} \mathrm {sgn}\left (\sin \left (x\right )\right )} \]

[In]

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

[Out]

-2*sgn(sin(x))/sqrt(a) + (1/cos(x) + cos(x))/(sqrt(a)*sgn(sin(x)))

Mupad [B] (verification not implemented)

Time = 13.35 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17 \[ \int \frac {\tan ^2(x)}{\sqrt {a+a \cot ^2(x)}} \, dx=\frac {{\mathrm {tan}\left (x\right )}^3\,\sqrt {\frac {1}{{\mathrm {tan}\left (x\right )}^2}}+2\,\mathrm {tan}\left (x\right )\,\sqrt {\frac {1}{{\mathrm {tan}\left (x\right )}^2}}}{\sqrt {a}\,\sqrt {{\mathrm {tan}\left (x\right )}^2+1}} \]

[In]

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

[Out]

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