\(\int \sqrt {a+b \cot ^2(x)} \tan ^2(x) \, dx\) [24]

Optimal result

Integrand size = 17, antiderivative size = 51 \[ \int \sqrt {a+b \cot ^2(x)} \tan ^2(x) \, dx=\sqrt {a-b} \arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )+\sqrt {a+b \cot ^2(x)} \tan (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3751, 486, 12, 385, 209} \[ \int \sqrt {a+b \cot ^2(x)} \tan ^2(x) \, dx=\sqrt {a-b} \arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )+\tan (x) \sqrt {a+b \cot ^2(x)} \]

[In]

Int[Sqrt[a + b*Cot[x]^2]*Tan[x]^2,x]

[Out]

Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]] + Sqrt[a + b*Cot[x]^2]*Tan[x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 385

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

Rule 486

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

Rule 3751

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]))

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.10 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.86 \[ \int \sqrt {a+b \cot ^2(x)} \tan ^2(x) \, dx=\sqrt {a+b \cot ^2(x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\frac {(a-b) \cot ^2(x)}{a+b \cot ^2(x)}\right ) \tan (x) \]

[In]

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

[Out]

Sqrt[a + b*Cot[x]^2]*Hypergeometric2F1[-1/2, 1, 1/2, -(((a - b)*Cot[x]^2)/(a + b*Cot[x]^2))]*Tan[x]

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(311\) vs. \(2(43)=86\).

Time = 0.87 (sec) , antiderivative size = 312, normalized size of antiderivative = 6.12

method	result	size
default	\(\frac {\sqrt {4}\, \sqrt {a +b \cot \left (x \right )^{2}}\, \left (\ln \left (4 \cos \left (x \right ) \sqrt {-a +b}\, \sqrt {-\frac {a \cos \left (x \right )^{2}-\cos \left (x \right )^{2} b -a}{\left (\cos \left (x \right )+1\right )^{2}}}-4 \cos \left (x \right ) a +4 b \cos \left (x \right )+4 \sqrt {-a +b}\, \sqrt {-\frac {a \cos \left (x \right )^{2}-\cos \left (x \right )^{2} b -a}{\left (\cos \left (x \right )+1\right )^{2}}}\right ) a \sin \left (x \right )-\ln \left (4 \cos \left (x \right ) \sqrt {-a +b}\, \sqrt {-\frac {a \cos \left (x \right )^{2}-\cos \left (x \right )^{2} b -a}{\left (\cos \left (x \right )+1\right )^{2}}}-4 \cos \left (x \right ) a +4 b \cos \left (x \right )+4 \sqrt {-a +b}\, \sqrt {-\frac {a \cos \left (x \right )^{2}-\cos \left (x \right )^{2} b -a}{\left (\cos \left (x \right )+1\right )^{2}}}\right ) b \sin \left (x \right )+\sqrt {-a +b}\, \sqrt {-\frac {a \cos \left (x \right )^{2}-\cos \left (x \right )^{2} b -a}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )+\sqrt {-a +b}\, \sqrt {-\frac {a \cos \left (x \right )^{2}-\cos \left (x \right )^{2} b -a}{\left (\cos \left (x \right )+1\right )^{2}}}\, \tan \left (x \right )\right )}{2 \sqrt {-a +b}\, \sqrt {-\frac {a \cos \left (x \right )^{2}-\cos \left (x \right )^{2} b -a}{\left (\cos \left (x \right )+1\right )^{2}}}\, \left (\cos \left (x \right )+1\right )}\)	\(312\)

[In]

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

[Out]

1/2*4^(1/2)/(-a+b)^(1/2)*(a+b*cot(x)^2)^(1/2)/(-(a*cos(x)^2-cos(x)^2*b-a)/(cos(x)+1)^2)^(1/2)/(cos(x)+1)*(ln(4
*cos(x)*(-a+b)^(1/2)*(-(a*cos(x)^2-cos(x)^2*b-a)/(cos(x)+1)^2)^(1/2)-4*cos(x)*a+4*b*cos(x)+4*(-a+b)^(1/2)*(-(a
*cos(x)^2-cos(x)^2*b-a)/(cos(x)+1)^2)^(1/2))*a*sin(x)-ln(4*cos(x)*(-a+b)^(1/2)*(-(a*cos(x)^2-cos(x)^2*b-a)/(co
s(x)+1)^2)^(1/2)-4*cos(x)*a+4*b*cos(x)+4*(-a+b)^(1/2)*(-(a*cos(x)^2-cos(x)^2*b-a)/(cos(x)+1)^2)^(1/2))*b*sin(x
)+(-a+b)^(1/2)*(-(a*cos(x)^2-cos(x)^2*b-a)/(cos(x)+1)^2)^(1/2)*sin(x)+(-a+b)^(1/2)*(-(a*cos(x)^2-cos(x)^2*b-a)
/(cos(x)+1)^2)^(1/2)*tan(x))

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 193, normalized size of antiderivative = 3.78 \[ \int \sqrt {a+b \cot ^2(x)} \tan ^2(x) \, dx=\left [\frac {1}{4} \, \sqrt {-a + b} \log \left (-\frac {a^{2} \tan \left (x\right )^{4} - 2 \, {\left (3 \, a^{2} - 4 \, a b\right )} \tan \left (x\right )^{2} + a^{2} - 8 \, a b + 8 \, b^{2} - 4 \, {\left (a \tan \left (x\right )^{3} - {\left (a - 2 \, b\right )} \tan \left (x\right )\right )} \sqrt {-a + b} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{\tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} + 1}\right ) + \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right ), \frac {1}{2} \, \sqrt {a - b} \arctan \left (\frac {2 \, \sqrt {a - b} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )}{a \tan \left (x\right )^{2} - a + 2 \, b}\right ) + \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )\right ] \]

[In]

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

[Out]

[1/4*sqrt(-a + b)*log(-(a^2*tan(x)^4 - 2*(3*a^2 - 4*a*b)*tan(x)^2 + a^2 - 8*a*b + 8*b^2 - 4*(a*tan(x)^3 - (a -
2*b)*tan(x))*sqrt(-a + b)*sqrt((a*tan(x)^2 + b)/tan(x)^2))/(tan(x)^4 + 2*tan(x)^2 + 1)) + sqrt((a*tan(x)^2 +
b)/tan(x)^2)*tan(x), 1/2*sqrt(a - b)*arctan(2*sqrt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)/(a*tan(x)^2 -
a + 2*b)) + sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (43) = 86\).

Time = 0.29 (sec) , antiderivative size = 239, normalized size of antiderivative = 4.69 \[ \int \sqrt {a+b \cot ^2(x)} \tan ^2(x) \, dx=\frac {1}{2} \, {\left (\sqrt {-a + b} \log \left ({\left (\sqrt {-a + b} \cos \left (x\right ) - \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a}\right )}^{2}\right ) - \frac {4 \, a \sqrt {-a + b}}{{\left (\sqrt {-a + b} \cos \left (x\right ) - \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a}\right )}^{2} - a}\right )} \mathrm {sgn}\left (\sin \left (x\right )\right ) - \frac {{\left (a \sqrt {-a + b} \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) - a \sqrt {b} \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) - \sqrt {-a + b} b \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) + b^{\frac {3}{2}} \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) + 2 \, a \sqrt {-a + b}\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{2 \, {\left (a + \sqrt {-a + b} \sqrt {b} - b\right )}} \]

[In]

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

[Out]

1/2*(sqrt(-a + b)*log((sqrt(-a + b)*cos(x) - sqrt(-a*cos(x)^2 + b*cos(x)^2 + a))^2) - 4*a*sqrt(-a + b)/((sqrt(
-a + b)*cos(x) - sqrt(-a*cos(x)^2 + b*cos(x)^2 + a))^2 - a))*sgn(sin(x)) - 1/2*(a*sqrt(-a + b)*log(-a - 2*sqrt
(-a + b)*sqrt(b) + 2*b) - a*sqrt(b)*log(-a - 2*sqrt(-a + b)*sqrt(b) + 2*b) - sqrt(-a + b)*b*log(-a - 2*sqrt(-a
+ b)*sqrt(b) + 2*b) + b^(3/2)*log(-a - 2*sqrt(-a + b)*sqrt(b) + 2*b) + 2*a*sqrt(-a + b))*sgn(sin(x))/(a + sqr
t(-a + b)*sqrt(b) - b)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+b \cot ^2(x)} \tan ^2(x) \, dx=\int {\mathrm {tan}\left (x\right )}^2\,\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a} \,d x \]

[In]

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

[Out]

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