Integrand size = 17, antiderivative size = 127 \[ \int \cot ^2(x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=(a-b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )-\frac {\left (3 a^2-12 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{8 \sqrt {b}}-\frac {1}{8} (5 a-4 b) \cot (x) \sqrt {a+b \cot ^2(x)}-\frac {1}{4} b \cot ^3(x) \sqrt {a+b \cot ^2(x)} \]
(a-b)^(3/2)*arctan(cot(x)*(a-b)^(1/2)/(a+b*cot(x)^2)^(1/2))-1/8*(3*a^2-12* a*b+8*b^2)*arctanh(cot(x)*b^(1/2)/(a+b*cot(x)^2)^(1/2))/b^(1/2)-1/8*(5*a-4 *b)*cot(x)*(a+b*cot(x)^2)^(1/2)-1/4*b*cot(x)^3*(a+b*cot(x)^2)^(1/2)
Time = 1.32 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.99 \[ \int \cot ^2(x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=\frac {\sqrt {-a-b+(a-b) \cos (2 x)} \csc (x) \left (8 \sqrt {2} (a-b)^2 \sqrt {-b} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a-b} \cos (x)}{\sqrt {-a-b+(a-b) \cos (2 x)}}\right )+\sqrt {a-b} \left (-\sqrt {2} \left (3 a^2-12 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {-b} \cos (x)}{\sqrt {-a-b+(a-b) \cos (2 x)}}\right )+\sqrt {-b} \sqrt {-a-b+(a-b) \cos (2 x)} \cot (x) \csc (x) \left (5 a-6 b+2 b \csc ^2(x)\right )\right )\right )}{8 \sqrt {2} \sqrt {a-b} \sqrt {-b} \sqrt {-\left ((-a-b+(a-b) \cos (2 x)) \csc ^2(x)\right )}} \]
(Sqrt[-a - b + (a - b)*Cos[2*x]]*Csc[x]*(8*Sqrt[2]*(a - b)^2*Sqrt[-b]*ArcT anh[(Sqrt[2]*Sqrt[a - b]*Cos[x])/Sqrt[-a - b + (a - b)*Cos[2*x]]] + Sqrt[a - b]*(-(Sqrt[2]*(3*a^2 - 12*a*b + 8*b^2)*ArcTanh[(Sqrt[2]*Sqrt[-b]*Cos[x] )/Sqrt[-a - b + (a - b)*Cos[2*x]]]) + Sqrt[-b]*Sqrt[-a - b + (a - b)*Cos[2 *x]]*Cot[x]*Csc[x]*(5*a - 6*b + 2*b*Csc[x]^2))))/(8*Sqrt[2]*Sqrt[a - b]*Sq rt[-b]*Sqrt[-((-a - b + (a - b)*Cos[2*x])*Csc[x]^2)])
Time = 0.40 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.588, Rules used = {3042, 4153, 379, 444, 27, 398, 224, 219, 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.
| \(\displaystyle \int \cot ^2(x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan \left (x+\frac {\pi }{2}\right )^2 \left (a+b \tan \left (x+\frac {\pi }{2}\right )^2\right )^{3/2}dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle -\int \frac {\cot ^2(x) \left (b \cot ^2(x)+a\right )^{3/2}}{\cot ^2(x)+1}d\cot (x)\) |
| \(\Big \downarrow \) 379 |
| \(\displaystyle -\frac {1}{4} \int \frac {\cot ^2(x) \left ((5 a-4 b) b \cot ^2(x)+a (4 a-3 b)\right )}{\left (\cot ^2(x)+1\right ) \sqrt {b \cot ^2(x)+a}}d\cot (x)-\frac {1}{4} b \cot ^3(x) \sqrt {a+b \cot ^2(x)}\) |
| \(\Big \downarrow \) 444 |
| \(\displaystyle \frac {1}{4} \left (\frac {\int \frac {b \left (a (5 a-4 b)-\left (3 a^2-12 b a+8 b^2\right ) \cot ^2(x)\right )}{\left (\cot ^2(x)+1\right ) \sqrt {b \cot ^2(x)+a}}d\cot (x)}{2 b}-\frac {1}{2} (5 a-4 b) \cot (x) \sqrt {a+b \cot ^2(x)}\right )-\frac {1}{4} b \cot ^3(x) \sqrt {a+b \cot ^2(x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \int \frac {a (5 a-4 b)-\left (3 a^2-12 b a+8 b^2\right ) \cot ^2(x)}{\left (\cot ^2(x)+1\right ) \sqrt {b \cot ^2(x)+a}}d\cot (x)-\frac {1}{2} (5 a-4 b) \cot (x) \sqrt {a+b \cot ^2(x)}\right )-\frac {1}{4} b \cot ^3(x) \sqrt {a+b \cot ^2(x)}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (8 (a-b)^2 \int \frac {1}{\left (\cot ^2(x)+1\right ) \sqrt {b \cot ^2(x)+a}}d\cot (x)-\left (3 a^2-12 a b+8 b^2\right ) \int \frac {1}{\sqrt {b \cot ^2(x)+a}}d\cot (x)\right )-\frac {1}{2} (5 a-4 b) \cot (x) \sqrt {a+b \cot ^2(x)}\right )-\frac {1}{4} b \cot ^3(x) \sqrt {a+b \cot ^2(x)}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (8 (a-b)^2 \int \frac {1}{\left (\cot ^2(x)+1\right ) \sqrt {b \cot ^2(x)+a}}d\cot (x)-\left (3 a^2-12 a b+8 b^2\right ) \int \frac {1}{1-\frac {b \cot ^2(x)}{b \cot ^2(x)+a}}d\frac {\cot (x)}{\sqrt {b \cot ^2(x)+a}}\right )-\frac {1}{2} (5 a-4 b) \cot (x) \sqrt {a+b \cot ^2(x)}\right )-\frac {1}{4} b \cot ^3(x) \sqrt {a+b \cot ^2(x)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (8 (a-b)^2 \int \frac {1}{\left (\cot ^2(x)+1\right ) \sqrt {b \cot ^2(x)+a}}d\cot (x)-\frac {\left (3 a^2-12 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{\sqrt {b}}\right )-\frac {1}{2} (5 a-4 b) \cot (x) \sqrt {a+b \cot ^2(x)}\right )-\frac {1}{4} b \cot ^3(x) \sqrt {a+b \cot ^2(x)}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (8 (a-b)^2 \int \frac {1}{1-\frac {(b-a) \cot ^2(x)}{b \cot ^2(x)+a}}d\frac {\cot (x)}{\sqrt {b \cot ^2(x)+a}}-\frac {\left (3 a^2-12 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{\sqrt {b}}\right )-\frac {1}{2} (5 a-4 b) \cot (x) \sqrt {a+b \cot ^2(x)}\right )-\frac {1}{4} b \cot ^3(x) \sqrt {a+b \cot ^2(x)}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (8 (a-b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )-\frac {\left (3 a^2-12 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )}{\sqrt {b}}\right )-\frac {1}{2} (5 a-4 b) \cot (x) \sqrt {a+b \cot ^2(x)}\right )-\frac {1}{4} b \cot ^3(x) \sqrt {a+b \cot ^2(x)}\) |
-1/4*(b*Cot[x]^3*Sqrt[a + b*Cot[x]^2]) + ((8*(a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]] - ((3*a^2 - 12*a*b + 8*b^2)*ArcTanh[(Sq rt[b]*Cot[x])/Sqrt[a + b*Cot[x]^2]])/Sqrt[b])/2 - ((5*a - 4*b)*Cot[x]*Sqrt [a + b*Cot[x]^2])/2)/4
3.1.27.3.1 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[((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/(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[d*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*e*(m + 2*(p + q) + 1))), x] + Simp[1/(b*(m + 2*(p + q) + 1)) Int[(e *x)^m*(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*((b*c - a*d)*(m + 1) + b*c*2 *(p + q)) + (d*(b*c - a*d)*(m + 1) + d*2*(q - 1)*(b*c - a*d) + b*c*d*2*(p + q))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d, 0 ] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) , x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} , x]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[f*g*(g*x)^(m - 1)*(a + b*x^2)^ (p + 1)*((c + d*x^2)^(q + 1)/(b*d*(m + 2*(p + q + 1) + 1))), x] - Simp[g^2/ (b*d*(m + 2*(p + q + 1) + 1)) Int[(g*x)^(m - 2)*(a + b*x^2)^p*(c + d*x^2) ^q*Simp[a*f*c*(m - 1) + (a*f*d*(m + 2*q + 1) + b*(f*c*(m + 2*p + 1) - e*d*( m + 2*(p + q + 1) + 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && GtQ[m, 1]
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. \(285\) vs. \(2(105)=210\).
Time = 0.03 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.25
| method | result | size |
| derivativedivides | \(-\frac {\cot \left (x \right ) \left (a +b \cot \left (x \right )^{2}\right )^{\frac {3}{2}}}{4}-\frac {3 a \cot \left (x \right ) \sqrt {a +b \cot \left (x \right )^{2}}}{8}-\frac {3 a^{2} \ln \left (\sqrt {b}\, \cot \left (x \right )+\sqrt {a +b \cot \left (x \right )^{2}}\right )}{8 \sqrt {b}}-b^{\frac {3}{2}} \ln \left (\sqrt {b}\, \cot \left (x \right )+\sqrt {a +b \cot \left (x \right )^{2}}\right )+\frac {b \cot \left (x \right ) \sqrt {a +b \cot \left (x \right )^{2}}}{2}+\frac {3 \sqrt {b}\, a \ln \left (\sqrt {b}\, \cot \left (x \right )+\sqrt {a +b \cot \left (x \right )^{2}}\right )}{2}+\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (x \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (x \right )^{2}}}\right )}{a -b}-\frac {2 a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (x \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (x \right )^{2}}}\right )}{b \left (a -b \right )}+\frac {a^{2} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (x \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (x \right )^{2}}}\right )}{b^{2} \left (a -b \right )}\) | \(286\) |
| default | \(-\frac {\cot \left (x \right ) \left (a +b \cot \left (x \right )^{2}\right )^{\frac {3}{2}}}{4}-\frac {3 a \cot \left (x \right ) \sqrt {a +b \cot \left (x \right )^{2}}}{8}-\frac {3 a^{2} \ln \left (\sqrt {b}\, \cot \left (x \right )+\sqrt {a +b \cot \left (x \right )^{2}}\right )}{8 \sqrt {b}}-b^{\frac {3}{2}} \ln \left (\sqrt {b}\, \cot \left (x \right )+\sqrt {a +b \cot \left (x \right )^{2}}\right )+\frac {b \cot \left (x \right ) \sqrt {a +b \cot \left (x \right )^{2}}}{2}+\frac {3 \sqrt {b}\, a \ln \left (\sqrt {b}\, \cot \left (x \right )+\sqrt {a +b \cot \left (x \right )^{2}}\right )}{2}+\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (x \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (x \right )^{2}}}\right )}{a -b}-\frac {2 a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (x \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (x \right )^{2}}}\right )}{b \left (a -b \right )}+\frac {a^{2} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (x \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (x \right )^{2}}}\right )}{b^{2} \left (a -b \right )}\) | \(286\) |
-1/4*cot(x)*(a+b*cot(x)^2)^(3/2)-3/8*a*cot(x)*(a+b*cot(x)^2)^(1/2)-3/8*a^2 /b^(1/2)*ln(b^(1/2)*cot(x)+(a+b*cot(x)^2)^(1/2))-b^(3/2)*ln(b^(1/2)*cot(x) +(a+b*cot(x)^2)^(1/2))+1/2*b*cot(x)*(a+b*cot(x)^2)^(1/2)+3/2*b^(1/2)*a*ln( b^(1/2)*cot(x)+(a+b*cot(x)^2)^(1/2))+(b^4*(a-b))^(1/2)/(a-b)*arctan(b^2*(a -b)/(b^4*(a-b))^(1/2)/(a+b*cot(x)^2)^(1/2)*cot(x))-2*a/b*(b^4*(a-b))^(1/2) /(a-b)*arctan(b^2*(a-b)/(b^4*(a-b))^(1/2)/(a+b*cot(x)^2)^(1/2)*cot(x))+a^2 *(b^4*(a-b))^(1/2)/b^2/(a-b)*arctan(b^2*(a-b)/(b^4*(a-b))^(1/2)/(a+b*cot(x )^2)^(1/2)*cot(x))
Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (105) = 210\).
Time = 0.33 (sec) , antiderivative size = 1134, normalized size of antiderivative = 8.93 \[ \int \cot ^2(x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=\text {Too large to display} \]
[1/16*(8*(a*b - b^2 - (a*b - b^2)*cos(2*x))*sqrt(-a + b)*log(-(a - b)*cos( 2*x) + sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2* x) + b)*sin(2*x) - (3*a^2 - 12*a*b + 8*b^2 - (3*a^2 - 12*a*b + 8*b^2)*cos( 2*x))*sqrt(b)*log(((a - 2*b)*cos(2*x) + 2*sqrt(b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) - a - 2*b)/(cos(2*x) - 1))*sin(2*x) + 2*( 4*b^2*cos(2*x) - (5*a*b - 6*b^2)*cos(2*x)^2 + 5*a*b - 2*b^2)*sqrt(((a - b) *cos(2*x) - a - b)/(cos(2*x) - 1)))/((b*cos(2*x) - b)*sin(2*x)), -1/8*((3* a^2 - 12*a*b + 8*b^2 - (3*a^2 - 12*a*b + 8*b^2)*cos(2*x))*sqrt(-b)*arctan( sqrt(-b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/(b*cos(2 *x) + b))*sin(2*x) - 4*(a*b - b^2 - (a*b - b^2)*cos(2*x))*sqrt(-a + b)*log (-(a - b)*cos(2*x) + sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x ) - 1))*sin(2*x) + b)*sin(2*x) - (4*b^2*cos(2*x) - (5*a*b - 6*b^2)*cos(2*x )^2 + 5*a*b - 2*b^2)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/((b* cos(2*x) - b)*sin(2*x)), -1/16*(16*(a*b - b^2 - (a*b - b^2)*cos(2*x))*sqrt (a - b)*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1) )*sin(2*x)/((a - b)*cos(2*x) + a - b))*sin(2*x) + (3*a^2 - 12*a*b + 8*b^2 - (3*a^2 - 12*a*b + 8*b^2)*cos(2*x))*sqrt(b)*log(((a - 2*b)*cos(2*x) + 2*s qrt(b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) - a - 2*b) /(cos(2*x) - 1))*sin(2*x) - 2*(4*b^2*cos(2*x) - (5*a*b - 6*b^2)*cos(2*x)^2 + 5*a*b - 2*b^2)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/((b*...
\[ \int \cot ^2(x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=\int \left (a + b \cot ^{2}{\left (x \right )}\right )^{\frac {3}{2}} \cot ^{2}{\left (x \right )}\, dx \]
\[ \int \cot ^2(x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=\int { {\left (b \cot \left (x\right )^{2} + a\right )}^{\frac {3}{2}} \cot \left (x\right )^{2} \,d x } \]
Exception generated. \[ \int \cot ^2(x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \cot ^2(x) \left (a+b \cot ^2(x)\right )^{3/2} \, dx=\int {\mathrm {cot}\left (x\right )}^2\,{\left (b\,{\mathrm {cot}\left (x\right )}^2+a\right )}^{3/2} \,d x \]