Integrand size = 17, antiderivative size = 54 \[ \int \frac {\tan ^2(x)}{\sqrt {a+b \cot ^2(x)}} \, dx=\frac {\arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{\sqrt {a-b}}+\frac {\sqrt {a+b \cot ^2(x)} \tan (x)}{a} \]
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.85 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.48 \[ \int \frac {\tan ^2(x)}{\sqrt {a+b \cot ^2(x)}} \, dx=\frac {\left (1+\frac {b \cot ^2(x)}{a}\right ) \sin ^2(x) \left (\frac {4 (a-b) \cos ^2(x) \left (a+b \cot ^2(x)\right ) \operatorname {Hypergeometric2F1}\left (2,2,\frac {5}{2},\frac {(a-b) \cos ^2(x)}{a}\right )}{3 a^2}+\frac {\arcsin \left (\sqrt {\frac {(a-b) \cos ^2(x)}{a}}\right ) \left (a+2 b \cot ^2(x)\right )}{a \sqrt {\frac {(a-b) \cos ^2(x) \left (a+b \cot ^2(x)\right ) \sin ^2(x)}{a^2}}}\right ) \tan (x)}{\sqrt {a+b \cot ^2(x)}} \]
((1 + (b*Cot[x]^2)/a)*Sin[x]^2*((4*(a - b)*Cos[x]^2*(a + b*Cot[x]^2)*Hyper geometric2F1[2, 2, 5/2, ((a - b)*Cos[x]^2)/a])/(3*a^2) + (ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*(a + 2*b*Cot[x]^2))/(a*Sqrt[((a - b)*Cos[x]^2*(a + b*Co t[x]^2)*Sin[x]^2)/a^2]))*Tan[x])/Sqrt[a + b*Cot[x]^2]
Time = 0.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {3042, 4153, 382, 25, 27, 291, 216}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\tan ^2(x)}{\sqrt {a+b \cot ^2(x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan \left (x+\frac {\pi }{2}\right )^2 \sqrt {a+b \tan \left (x+\frac {\pi }{2}\right )^2}}dx\) |
\(\Big \downarrow \) 4153 |
\(\displaystyle -\int \frac {\tan ^2(x)}{\left (\cot ^2(x)+1\right ) \sqrt {b \cot ^2(x)+a}}d\cot (x)\) |
\(\Big \downarrow \) 382 |
\(\displaystyle \frac {\tan (x) \sqrt {a+b \cot ^2(x)}}{a}-\frac {\int -\frac {a}{\left (\cot ^2(x)+1\right ) \sqrt {b \cot ^2(x)+a}}d\cot (x)}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {a}{\left (\cot ^2(x)+1\right ) \sqrt {b \cot ^2(x)+a}}d\cot (x)}{a}+\frac {\tan (x) \sqrt {a+b \cot ^2(x)}}{a}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {1}{\left (\cot ^2(x)+1\right ) \sqrt {b \cot ^2(x)+a}}d\cot (x)+\frac {\tan (x) \sqrt {a+b \cot ^2(x)}}{a}\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \int \frac {1}{1-\frac {(b-a) \cot ^2(x)}{b \cot ^2(x)+a}}d\frac {\cot (x)}{\sqrt {b \cot ^2(x)+a}}+\frac {\tan (x) \sqrt {a+b \cot ^2(x)}}{a}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{\sqrt {a-b}}+\frac {\tan (x) \sqrt {a+b \cot ^2(x)}}{a}\) |
ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]]/Sqrt[a - b] + (Sqrt[a + b*Cot[x]^2]*Tan[x])/a
3.1.48.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/ (a*c*e*(m + 1))), x] - Simp[1/(a*c*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b* x^2)^p*(c + d*x^2)^q*Simp[(b*c + a*d)*(m + 3) + 2*(b*c*p + a*d*q) + b*d*(m + 2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[ b*c - a*d, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
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]}, Simp[c*(ff/f) Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2 + f f^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] && Ratio nalQ[n]))
Leaf count of result is larger than twice the leaf count of optimal. \(282\) vs. \(2(46)=92\).
Time = 0.91 (sec) , antiderivative size = 283, normalized size of antiderivative = 5.24
method | result | size |
default | \(\frac {\sqrt {4}\, \left (\sqrt {-\frac {a \cos \left (x \right )^{2}-\cos \left (x \right )^{2} b -a}{\left (\cos \left (x \right )+1\right )^{2}}}\, \ln \left (4 \cos \left (x \right ) \sqrt {-a +b}\, \sqrt {-\frac {a \cos \left (x \right )^{2}-\cos \left (x \right )^{2} b -a}{\left (\cos \left (x \right )+1\right )^{2}}}-4 \cos \left (x \right ) a +4 b \cos \left (x \right )+4 \sqrt {-a +b}\, \sqrt {-\frac {a \cos \left (x \right )^{2}-\cos \left (x \right )^{2} b -a}{\left (\cos \left (x \right )+1\right )^{2}}}\right ) a \cot \left (x \right )+\sqrt {-a +b}\, a \tan \left (x \right )+\sqrt {-a +b}\, b \cot \left (x \right )+\sqrt {-\frac {a \cos \left (x \right )^{2}-\cos \left (x \right )^{2} b -a}{\left (\cos \left (x \right )+1\right )^{2}}}\, \ln \left (4 \cos \left (x \right ) \sqrt {-a +b}\, \sqrt {-\frac {a \cos \left (x \right )^{2}-\cos \left (x \right )^{2} b -a}{\left (\cos \left (x \right )+1\right )^{2}}}-4 \cos \left (x \right ) a +4 b \cos \left (x \right )+4 \sqrt {-a +b}\, \sqrt {-\frac {a \cos \left (x \right )^{2}-\cos \left (x \right )^{2} b -a}{\left (\cos \left (x \right )+1\right )^{2}}}\right ) a \csc \left (x \right )\right )}{2 a \sqrt {-a +b}\, \sqrt {a +b \cot \left (x \right )^{2}}}\) | \(283\) |
1/2*4^(1/2)/a/(-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)*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-c os(x)^2*b-a)/(cos(x)+1)^2)^(1/2))*a*cot(x)+(-a+b)^(1/2)*a*tan(x)+(-a+b)^(1 /2)*b*cot(x)+(-(a*cos(x)^2-cos(x)^2*b-a)/(cos(x)+1)^2)^(1/2)*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* csc(x))
Time = 0.33 (sec) , antiderivative size = 229, normalized size of antiderivative = 4.24 \[ \int \frac {\tan ^2(x)}{\sqrt {a+b \cot ^2(x)}} \, dx=\left [-\frac {a \sqrt {-a + b} \log \left (-\frac {a^{2} \tan \left (x\right )^{4} - 2 \, {\left (3 \, a^{2} - 4 \, a b\right )} \tan \left (x\right )^{2} + a^{2} - 8 \, a b + 8 \, b^{2} + 4 \, {\left (a \tan \left (x\right )^{3} - {\left (a - 2 \, b\right )} \tan \left (x\right )\right )} \sqrt {-a + b} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{\tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} + 1}\right ) - 4 \, {\left (a - b\right )} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )}{4 \, {\left (a^{2} - a b\right )}}, \frac {\sqrt {a - b} a \arctan \left (\frac {2 \, \sqrt {a - b} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )}{a \tan \left (x\right )^{2} - a + 2 \, b}\right ) + 2 \, {\left (a - b\right )} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )}{2 \, {\left (a^{2} - a b\right )}}\right ] \]
[-1/4*(a*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)) - 4*(a - b)*sqrt((a *tan(x)^2 + b)/tan(x)^2)*tan(x))/(a^2 - a*b), 1/2*(sqrt(a - b)*a*arctan(2* sqrt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)/(a*tan(x)^2 - a + 2*b)) + 2*(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x))/(a^2 - a*b)]
\[ \int \frac {\tan ^2(x)}{\sqrt {a+b \cot ^2(x)}} \, dx=\int \frac {\tan ^{2}{\left (x \right )}}{\sqrt {a + b \cot ^{2}{\left (x \right )}}}\, dx \]
\[ \int \frac {\tan ^2(x)}{\sqrt {a+b \cot ^2(x)}} \, dx=\int { \frac {\tan \left (x\right )^{2}}{\sqrt {b \cot \left (x\right )^{2} + a}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (46) = 92\).
Time = 0.34 (sec) , antiderivative size = 216, normalized size of antiderivative = 4.00 \[ \int \frac {\tan ^2(x)}{\sqrt {a+b \cot ^2(x)}} \, dx=\frac {{\left (a \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) + \sqrt {-a + b} \sqrt {b} \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) - b \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) + 2 \, a - 2 \, b\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{2 \, {\left (a \sqrt {-a + b} - a \sqrt {b} - \sqrt {-a + b} b + b^{\frac {3}{2}}\right )}} - \frac {\frac {\log \left ({\left (\sqrt {-a + b} \cos \left (x\right ) - \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a}\right )}^{2}\right )}{\sqrt {-a + b}} + \frac {4 \, \sqrt {-a + b}}{{\left (\sqrt {-a + b} \cos \left (x\right ) - \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a}\right )}^{2} - a}}{2 \, \mathrm {sgn}\left (\sin \left (x\right )\right )} \]
1/2*(a*log(-a - 2*sqrt(-a + b)*sqrt(b) + 2*b) + sqrt(-a + b)*sqrt(b)*log(- a - 2*sqrt(-a + b)*sqrt(b) + 2*b) - b*log(-a - 2*sqrt(-a + b)*sqrt(b) + 2* b) + 2*a - 2*b)*sgn(sin(x))/(a*sqrt(-a + b) - a*sqrt(b) - sqrt(-a + b)*b + b^(3/2)) - 1/2*(log((sqrt(-a + b)*cos(x) - sqrt(-a*cos(x)^2 + b*cos(x)^2 + a))^2)/sqrt(-a + b) + 4*sqrt(-a + b)/((sqrt(-a + b)*cos(x) - sqrt(-a*cos (x)^2 + b*cos(x)^2 + a))^2 - a))/sgn(sin(x))
Timed out. \[ \int \frac {\tan ^2(x)}{\sqrt {a+b \cot ^2(x)}} \, dx=\int \frac {{\mathrm {tan}\left (x\right )}^2}{\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a}} \,d x \]