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

Optimal. Leaf size=60 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right )}{\sqrt{a-b}} \]

[Out]

ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a]]/Sqrt[a] - ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]/Sqrt[a - b]

________________________________________________________________________________________

Rubi [A]  time = 0.0941914, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3670, 446, 86, 63, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right )}{\sqrt{a-b}} \]

Antiderivative was successfully verified.

[In]

Int[Tan[x]/Sqrt[a + b*Cot[x]^2],x]

[Out]

ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a]]/Sqrt[a] - ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]/Sqrt[a - b]

Rule 3670

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(c*ff)/f, Subst[Int[(((d*ff*x)/c)^m*(a + b*(ff*x)^n)^p)/(c^
2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 86

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), In
t[(e + f*x)^p/(a + b*x), x], x] - Dist[d/(b*c - a*d), Int[(e + f*x)^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d,
e, f, p}, x] &&  !IntegerQ[p]

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 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.0454356, size = 60, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right )}{\sqrt{a-b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a]]/Sqrt[a] - ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]/Sqrt[a - b]

________________________________________________________________________________________

Maple [C]  time = 0.14, size = 376, normalized size = 6.3 \begin{align*} -2\,{\frac{\sqrt{2}\sin \left ( x \right ) }{-1+\cos \left ( x \right ) }\sqrt{{\frac{\cos \left ( x \right ) \sqrt{a}\sqrt{a-b}-\sqrt{a}\sqrt{a-b}-a\cos \left ( x \right ) +b\cos \left ( x \right ) +a}{b \left ( \cos \left ( x \right ) +1 \right ) }}}\sqrt{-2\,{\frac{\cos \left ( x \right ) \sqrt{a}\sqrt{a-b}-\sqrt{a}\sqrt{a-b}+a\cos \left ( x \right ) -b\cos \left ( x \right ) -a}{b \left ( \cos \left ( x \right ) +1 \right ) }}} \left ({\it EllipticPi} \left ({\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }\sqrt{{\frac{2\,\sqrt{a}\sqrt{a-b}-2\,a+b}{b}}}},-{\frac{b}{2\,\sqrt{a}\sqrt{a-b}-2\,a+b}},{\sqrt{-{\frac{2\,\sqrt{a}\sqrt{a-b}+2\,a-b}{b}}}{\frac{1}{\sqrt{{\frac{2\,\sqrt{a}\sqrt{a-b}-2\,a+b}{b}}}}}} \right ) -{\it EllipticPi} \left ({\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }\sqrt{{\frac{2\,\sqrt{a}\sqrt{a-b}-2\,a+b}{b}}}},{\frac{b}{2\,\sqrt{a}\sqrt{a-b}-2\,a+b}},{\sqrt{-{\frac{2\,\sqrt{a}\sqrt{a-b}+2\,a-b}{b}}}{\frac{1}{\sqrt{{\frac{2\,\sqrt{a}\sqrt{a-b}-2\,a+b}{b}}}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{2\,\sqrt{a}\sqrt{a-b}-2\,a+b}{b}}}}}{\frac{1}{\sqrt{{\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}a-b \left ( \cos \left ( x \right ) \right ) ^{2}-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)/(a+b*cot(x)^2)^(1/2),x)

[Out]

-2/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)*2^(1/2)*(1/b*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(
x)+b*cos(x)+a)/(cos(x)+1))^(1/2)*(-2/b*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(c
os(x)+1))^(1/2)*(EllipticPi((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),-1/(2*a^(1/2)*(a-b)^(1/
2)-2*a+b)*b,(-(2*a^(1/2)*(a-b)^(1/2)+2*a-b)/b)^(1/2)/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2))-EllipticPi((-1+c
os(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),1/(2*a^(1/2)*(a-b)^(1/2)-2*a+b)*b,(-(2*a^(1/2)*(a-b)^(1/
2)+2*a-b)/b)^(1/2)/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)))*sin(x)/((cos(x)^2*a-b*cos(x)^2-a)/(cos(x)^2-1))^(
1/2)/(-1+cos(x))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(tan(x)/sqrt(b*cot(x)^2 + a), x)

________________________________________________________________________________________

Fricas [A]  time = 2.25094, size = 1079, normalized size = 17.98 \begin{align*} \left [\frac{{\left (a - b\right )} \sqrt{a} \log \left (2 \, a \tan \left (x\right )^{2} + 2 \, \sqrt{a} \sqrt{\frac{a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )^{2} + b\right ) + \sqrt{a - b} a \log \left (\frac{{\left (2 \, a - b\right )} \tan \left (x\right )^{2} - 2 \, \sqrt{a - b} \sqrt{\frac{a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2} + 1}\right )}{2 \,{\left (a^{2} - a b\right )}}, -\frac{2 \, a \sqrt{-a + b} \arctan \left (-\frac{\sqrt{-a + b} \sqrt{\frac{a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{a - b}\right ) -{\left (a - b\right )} \sqrt{a} \log \left (2 \, a \tan \left (x\right )^{2} + 2 \, \sqrt{a} \sqrt{\frac{a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )^{2} + b\right )}{2 \,{\left (a^{2} - a b\right )}}, -\frac{2 \, \sqrt{-a}{\left (a - b\right )} \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{a}\right ) - \sqrt{a - b} a \log \left (\frac{{\left (2 \, a - b\right )} \tan \left (x\right )^{2} - 2 \, \sqrt{a - b} \sqrt{\frac{a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2} + 1}\right )}{2 \,{\left (a^{2} - a b\right )}}, -\frac{\sqrt{-a}{\left (a - b\right )} \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{a}\right ) + a \sqrt{-a + b} \arctan \left (-\frac{\sqrt{-a + b} \sqrt{\frac{a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{a - b}\right )}{a^{2} - a b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*((a - b)*sqrt(a)*log(2*a*tan(x)^2 + 2*sqrt(a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b) + sqrt(a - b)
*a*log(((2*a - b)*tan(x)^2 - 2*sqrt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b)/(tan(x)^2 + 1)))/(a^2
- a*b), -1/2*(2*a*sqrt(-a + b)*arctan(-sqrt(-a + b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)/(a - b)) - (a - b)*sqrt(a
)*log(2*a*tan(x)^2 + 2*sqrt(a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b))/(a^2 - a*b), -1/2*(2*sqrt(-a)*(a
- b)*arctan(sqrt(-a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)/a) - sqrt(a - b)*a*log(((2*a - b)*tan(x)^2 - 2*sqrt(a -
b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b)/(tan(x)^2 + 1)))/(a^2 - a*b), -(sqrt(-a)*(a - b)*arctan(sqrt(
-a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)/a) + a*sqrt(-a + b)*arctan(-sqrt(-a + b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)/(
a - b)))/(a^2 - a*b)]

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.65363, size = 277, normalized size = 4.62 \begin{align*} -\frac{{\left (2 \, a \arctan \left (-\frac{a - b}{\sqrt{-a^{2} + a b}}\right ) - 2 \, b \arctan \left (-\frac{a - b}{\sqrt{-a^{2} + a b}}\right ) + \sqrt{-a^{2} + a b} \log \left (b\right )\right )} \mathrm{sgn}\left (\sin \left (x\right )\right )}{2 \, \sqrt{-a^{2} + a b} \sqrt{a - b}} + \frac{\sqrt{a - b} \arctan \left (\frac{{\left (\sqrt{a - b} \sin \left (x\right ) - \sqrt{a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2} - 2 \, a + b}{2 \, \sqrt{-a^{2} + a b}}\right )}{\sqrt{-a^{2} + a b} \mathrm{sgn}\left (\sin \left (x\right )\right )} + \frac{\log \left ({\left (\sqrt{a - b} \sin \left (x\right ) - \sqrt{a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2}\right )}{2 \, \sqrt{a - b} \mathrm{sgn}\left (\sin \left (x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*a*arctan(-(a - b)/sqrt(-a^2 + a*b)) - 2*b*arctan(-(a - b)/sqrt(-a^2 + a*b)) + sqrt(-a^2 + a*b)*log(b))
*sgn(sin(x))/(sqrt(-a^2 + a*b)*sqrt(a - b)) + sqrt(a - b)*arctan(1/2*((sqrt(a - b)*sin(x) - sqrt(a*sin(x)^2 -
b*sin(x)^2 + b))^2 - 2*a + b)/sqrt(-a^2 + a*b))/(sqrt(-a^2 + a*b)*sgn(sin(x))) + 1/2*log((sqrt(a - b)*sin(x) -
sqrt(a*sin(x)^2 - b*sin(x)^2 + b))^2)/(sqrt(a - b)*sgn(sin(x)))