Integrand size = 15, antiderivative size = 90 \[ \int \cot (x) \sqrt {a+b \cot ^4(x)} \, dx=\frac {1}{2} \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \cot ^2(x)}{\sqrt {a+b \cot ^4(x)}}\right )+\frac {1}{2} \sqrt {a+b} \text {arctanh}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )-\frac {1}{2} \sqrt {a+b \cot ^4(x)} \]
1/2*arctanh(cot(x)^2*b^(1/2)/(a+b*cot(x)^4)^(1/2))*b^(1/2)+1/2*arctanh((a- b*cot(x)^2)/(a+b)^(1/2)/(a+b*cot(x)^4)^(1/2))*(a+b)^(1/2)-1/2*(a+b*cot(x)^ 4)^(1/2)
Time = 0.19 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.96 \[ \int \cot (x) \sqrt {a+b \cot ^4(x)} \, dx=\frac {1}{2} \left (\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \cot ^2(x)}{\sqrt {a+b \cot ^4(x)}}\right )+\sqrt {a+b} \text {arctanh}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )-\sqrt {a+b \cot ^4(x)}\right ) \]
(Sqrt[b]*ArcTanh[(Sqrt[b]*Cot[x]^2)/Sqrt[a + b*Cot[x]^4]] + Sqrt[a + b]*Ar cTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])] - Sqrt[a + b*Co t[x]^4])/2
Time = 0.32 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.96, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.733, Rules used = {3042, 25, 4153, 25, 1577, 493, 719, 224, 219, 488, 219}
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.
\(\displaystyle \int \cot (x) \sqrt {a+b \cot ^4(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\tan \left (x+\frac {\pi }{2}\right ) \sqrt {a+b \tan \left (x+\frac {\pi }{2}\right )^4}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \tan \left (x+\frac {\pi }{2}\right ) \sqrt {b \tan \left (x+\frac {\pi }{2}\right )^4+a}dx\) |
\(\Big \downarrow \) 4153 |
\(\displaystyle \int -\frac {\cot (x) \sqrt {a+b \cot ^4(x)}}{\cot ^2(x)+1}d\cot (x)\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\cot (x) \sqrt {b \cot ^4(x)+a}}{\cot ^2(x)+1}d\cot (x)\) |
\(\Big \downarrow \) 1577 |
\(\displaystyle -\frac {1}{2} \int \frac {\sqrt {b \cot ^4(x)+a}}{\cot ^2(x)+1}d\cot ^2(x)\) |
\(\Big \downarrow \) 493 |
\(\displaystyle \frac {1}{2} \left (-\int \frac {a-b \cot ^2(x)}{\left (\cot ^2(x)+1\right ) \sqrt {b \cot ^4(x)+a}}d\cot ^2(x)-\sqrt {a+b \cot ^4(x)}\right )\) |
\(\Big \downarrow \) 719 |
\(\displaystyle \frac {1}{2} \left (b \int \frac {1}{\sqrt {b \cot ^4(x)+a}}d\cot ^2(x)-(a+b) \int \frac {1}{\left (\cot ^2(x)+1\right ) \sqrt {b \cot ^4(x)+a}}d\cot ^2(x)-\sqrt {a+b \cot ^4(x)}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {1}{2} \left (b \int \frac {1}{1-b \cot ^4(x)}d\frac {\cot ^2(x)}{\sqrt {b \cot ^4(x)+a}}-(a+b) \int \frac {1}{\left (\cot ^2(x)+1\right ) \sqrt {b \cot ^4(x)+a}}d\cot ^2(x)-\sqrt {a+b \cot ^4(x)}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left (-(a+b) \int \frac {1}{\left (\cot ^2(x)+1\right ) \sqrt {b \cot ^4(x)+a}}d\cot ^2(x)+\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \cot ^2(x)}{\sqrt {a+b \cot ^4(x)}}\right )-\sqrt {a+b \cot ^4(x)}\right )\) |
\(\Big \downarrow \) 488 |
\(\displaystyle \frac {1}{2} \left ((a+b) \int \frac {1}{-\cot ^4(x)+a+b}d\frac {a-b \cot ^2(x)}{\sqrt {b \cot ^4(x)+a}}+\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \cot ^2(x)}{\sqrt {a+b \cot ^4(x)}}\right )-\sqrt {a+b \cot ^4(x)}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left (\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \cot ^2(x)}{\sqrt {a+b \cot ^4(x)}}\right )+\sqrt {a+b} \text {arctanh}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )-\sqrt {a+b \cot ^4(x)}\right )\) |
(Sqrt[b]*ArcTanh[(Sqrt[b]*Cot[x]^2)/Sqrt[a + b*Cot[x]^4]] + Sqrt[a + b]*Ar cTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])] - Sqrt[a + b*Co t[x]^4])/2
3.1.60.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (c + d*x)^(n + 1)*((a + b*x^2)^p/(d*(n + 2*p + 1))), x] + Simp[2*(p/(d*(n + 2*p + 1))) Int[(c + d*x)^n*(a + b*x^2)^(p - 1)*(a*d - b*c*x), x], x] /; FreeQ[{a, b, c, d, n}, x] && GtQ[p, 0] && NeQ[n + 2*p + 1, 0] && ( !Rationa lQ[n] || LtQ[n, 1]) && !ILtQ[n + 2*p, 0] && IntQuadraticQ[a, 0, b, c, d, n , p, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; Free Q[{a, c, d, e, 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]))
Time = 0.24 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.54
method | result | size |
derivativedivides | \(-\frac {\sqrt {b \left (\cot \left (x \right )^{2}+1\right )^{2}-2 b \left (\cot \left (x \right )^{2}+1\right )+a +b}}{2}+\frac {\sqrt {b}\, \ln \left (\frac {b \left (\cot \left (x \right )^{2}+1\right )-b}{\sqrt {b}}+\sqrt {b \left (\cot \left (x \right )^{2}+1\right )^{2}-2 b \left (\cot \left (x \right )^{2}+1\right )+a +b}\right )}{2}+\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 b \left (\cot \left (x \right )^{2}+1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\cot \left (x \right )^{2}+1\right )^{2}-2 b \left (\cot \left (x \right )^{2}+1\right )+a +b}}{\cot \left (x \right )^{2}+1}\right )}{2}\) | \(139\) |
default | \(-\frac {\sqrt {b \left (\cot \left (x \right )^{2}+1\right )^{2}-2 b \left (\cot \left (x \right )^{2}+1\right )+a +b}}{2}+\frac {\sqrt {b}\, \ln \left (\frac {b \left (\cot \left (x \right )^{2}+1\right )-b}{\sqrt {b}}+\sqrt {b \left (\cot \left (x \right )^{2}+1\right )^{2}-2 b \left (\cot \left (x \right )^{2}+1\right )+a +b}\right )}{2}+\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 b \left (\cot \left (x \right )^{2}+1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\cot \left (x \right )^{2}+1\right )^{2}-2 b \left (\cot \left (x \right )^{2}+1\right )+a +b}}{\cot \left (x \right )^{2}+1}\right )}{2}\) | \(139\) |
-1/2*(b*(cot(x)^2+1)^2-2*b*(cot(x)^2+1)+a+b)^(1/2)+1/2*b^(1/2)*ln((b*(cot( x)^2+1)-b)/b^(1/2)+(b*(cot(x)^2+1)^2-2*b*(cot(x)^2+1)+a+b)^(1/2))+1/2*(a+b )^(1/2)*ln((2*a+2*b-2*b*(cot(x)^2+1)+2*(a+b)^(1/2)*(b*(cot(x)^2+1)^2-2*b*( cot(x)^2+1)+a+b)^(1/2))/(cot(x)^2+1))
Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (72) = 144\).
Time = 0.45 (sec) , antiderivative size = 1063, normalized size of antiderivative = 11.81 \[ \int \cot (x) \sqrt {a+b \cot ^4(x)} \, dx=\text {Too large to display} \]
[1/4*sqrt(a + b)*log(1/2*(a^2 + 2*a*b + b^2)*cos(2*x)^2 + 1/2*a^2 + 1/2*b^ 2 + 1/2*((a + b)*cos(2*x)^2 - 2*a*cos(2*x) + a - b)*sqrt(a + b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1) ) - (a^2 - b^2)*cos(2*x)) + 1/4*sqrt(b)*log(-((a + 2*b)*cos(2*x)^2 - 2*(co s(2*x)^2 - 1)*sqrt(b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)) - 2*(a - 2*b)*cos(2*x) + a + 2*b)/(cos(2 *x)^2 - 2*cos(2*x) + 1)) - 1/2*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2* x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)), 1/2*sqrt(-b)*arctan(sqrt(-b)*s qrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos( 2*x) + 1))*(cos(2*x) - 1)/(b*cos(2*x) + b)) + 1/4*sqrt(a + b)*log(1/2*(a^2 + 2*a*b + b^2)*cos(2*x)^2 + 1/2*a^2 + 1/2*b^2 + 1/2*((a + b)*cos(2*x)^2 - 2*a*cos(2*x) + a - b)*sqrt(a + b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*co s(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)) - (a^2 - b^2)*cos(2*x)) - 1 /2*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2* cos(2*x) + 1)), -1/2*sqrt(-a - b)*arctan(((a + b)*cos(2*x)^2 - 2*a*cos(2*x ) + a - b)*sqrt(-a - b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1))/((a^2 + 2*a*b + b^2)*cos(2*x)^2 + a^2 + 2*a*b + b^2 - 2*(a^2 - b^2)*cos(2*x))) + 1/4*sqrt(b)*log(-((a + 2*b)*cos (2*x)^2 - 2*(cos(2*x)^2 - 1)*sqrt(b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)* cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)) - 2*(a - 2*b)*cos(2*x)...
\[ \int \cot (x) \sqrt {a+b \cot ^4(x)} \, dx=\int \sqrt {a + b \cot ^{4}{\left (x \right )}} \cot {\left (x \right )}\, dx \]
\[ \int \cot (x) \sqrt {a+b \cot ^4(x)} \, dx=\int { \sqrt {b \cot \left (x\right )^{4} + a} \cot \left (x\right ) \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (72) = 144\).
Time = 0.30 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.27 \[ \int \cot (x) \sqrt {a+b \cot ^4(x)} \, dx=-\frac {b \arctan \left (-\frac {\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} - \frac {1}{2} \, \sqrt {a + b} \log \left ({\left | -{\left (\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}\right )} {\left (a + b\right )} + \sqrt {a + b} b \right |}\right ) - \frac {{\left (\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}\right )} b - \sqrt {a + b} b}{{\left (\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}\right )}^{2} - b} \]
-b*arctan(-(sqrt(a + b)*sin(x)^2 - sqrt(a*sin(x)^4 + b*sin(x)^4 - 2*b*sin( x)^2 + b))/sqrt(-b))/sqrt(-b) - 1/2*sqrt(a + b)*log(abs(-(sqrt(a + b)*sin( x)^2 - sqrt(a*sin(x)^4 + b*sin(x)^4 - 2*b*sin(x)^2 + b))*(a + b) + sqrt(a + b)*b)) - ((sqrt(a + b)*sin(x)^2 - sqrt(a*sin(x)^4 + b*sin(x)^4 - 2*b*sin (x)^2 + b))*b - sqrt(a + b)*b)/((sqrt(a + b)*sin(x)^2 - sqrt(a*sin(x)^4 + b*sin(x)^4 - 2*b*sin(x)^2 + b))^2 - b)
Timed out. \[ \int \cot (x) \sqrt {a+b \cot ^4(x)} \, dx=\int \mathrm {cot}\left (x\right )\,\sqrt {b\,{\mathrm {cot}\left (x\right )}^4+a} \,d x \]