Integrand size = 25, antiderivative size = 287 \[ \int \frac {\cot ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {\cot (e+f x)}{3 a f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {(3 a+4 b) \cot (e+f x)}{3 a^2 (a+b) f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(7 a+8 b) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 (a+b) f}-\frac {(7 a+8 b) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 (a+b) f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {4 \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 a^2 f \sqrt {a+b \sin ^2(e+f x)}} \]
1/3*cot(f*x+e)/a/f/(a+b*sin(f*x+e)^2)^(3/2)+1/3*(3*a+4*b)*cot(f*x+e)/a^2/( a+b)/f/(a+b*sin(f*x+e)^2)^(1/2)-1/3*(7*a+8*b)*cot(f*x+e)*(a+b*sin(f*x+e)^2 )^(1/2)/a^3/(a+b)/f-1/3*(7*a+8*b)*EllipticE(sin(f*x+e),(-b/a)^(1/2))*sec(f *x+e)*(cos(f*x+e)^2)^(1/2)*(a+b*sin(f*x+e)^2)^(1/2)/a^3/(a+b)/f/(1+b*sin(f *x+e)^2/a)^(1/2)+4/3*EllipticF(sin(f*x+e),(-b/a)^(1/2))*sec(f*x+e)*(cos(f* x+e)^2)^(1/2)*(1+b*sin(f*x+e)^2/a)^(1/2)/a^2/f/(a+b*sin(f*x+e)^2)^(1/2)
Time = 3.24 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.73 \[ \int \frac {\cot ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {-\frac {\left (24 a^3+68 a^2 b+69 a b^2+24 b^3-4 b \left (11 a^2+19 a b+8 b^2\right ) \cos (2 (e+f x))+b^2 (7 a+8 b) \cos (4 (e+f x))\right ) \cot (e+f x)}{\sqrt {2}}-2 a^2 (7 a+8 b) \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} E\left (e+f x\left |-\frac {b}{a}\right .\right )+8 a^2 (a+b) \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )}{6 a^3 (a+b) f (2 a+b-b \cos (2 (e+f x)))^{3/2}} \]
(-(((24*a^3 + 68*a^2*b + 69*a*b^2 + 24*b^3 - 4*b*(11*a^2 + 19*a*b + 8*b^2) *Cos[2*(e + f*x)] + b^2*(7*a + 8*b)*Cos[4*(e + f*x)])*Cot[e + f*x])/Sqrt[2 ]) - 2*a^2*(7*a + 8*b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticE[ e + f*x, -(b/a)] + 8*a^2*(a + b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)* EllipticF[e + f*x, -(b/a)])/(6*a^3*(a + b)*f*(2*a + b - b*Cos[2*(e + f*x)] )^(3/2))
Time = 0.56 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.07, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 3675, 371, 25, 441, 25, 445, 399, 323, 321, 330, 327}
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.
| \(\displaystyle \int \frac {\cot ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (e+f x)^2 \left (a+b \sin (e+f x)^2\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3675 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \int \frac {\csc ^2(e+f x) \sqrt {1-\sin ^2(e+f x)}}{\left (b \sin ^2(e+f x)+a\right )^{5/2}}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 371 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{3 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\int -\frac {\csc ^2(e+f x) \left (4-3 \sin ^2(e+f x)\right )}{\sqrt {1-\sin ^2(e+f x)} \left (b \sin ^2(e+f x)+a\right )^{3/2}}d\sin (e+f x)}{3 a}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\int \frac {\csc ^2(e+f x) \left (4-3 \sin ^2(e+f x)\right )}{\sqrt {1-\sin ^2(e+f x)} \left (b \sin ^2(e+f x)+a\right )^{3/2}}d\sin (e+f x)}{3 a}+\frac {\sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{3 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 441 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {(3 a+4 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}-\frac {\int -\frac {\csc ^2(e+f x) \left (-\left ((3 a+4 b) \sin ^2(e+f x)\right )+7 a+8 b\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a (a+b)}}{3 a}+\frac {\sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{3 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {\int \frac {\csc ^2(e+f x) \left (-\left ((3 a+4 b) \sin ^2(e+f x)\right )+7 a+8 b\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a (a+b)}+\frac {(3 a+4 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a}+\frac {\sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{3 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {-\frac {\int \frac {b (7 a+8 b) \sin ^2(e+f x)+a (3 a+4 b)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a}-\frac {(7 a+8 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}}{a (a+b)}+\frac {(3 a+4 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a}+\frac {\sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{3 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {-\frac {(7 a+8 b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)-4 a (a+b) \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a}-\frac {(7 a+8 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}}{a (a+b)}+\frac {(3 a+4 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a}+\frac {\sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{3 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {-\frac {(7 a+8 b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)-\frac {4 a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}d\sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}}{a}-\frac {(7 a+8 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}}{a (a+b)}+\frac {(3 a+4 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a}+\frac {\sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{3 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {-\frac {(7 a+8 b) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)-\frac {4 a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(e+f x)}}}{a}-\frac {(7 a+8 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}}{a (a+b)}+\frac {(3 a+4 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a}+\frac {\sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{3 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {-\frac {\frac {(7 a+8 b) \sqrt {a+b \sin ^2(e+f x)} \int \frac {\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {4 a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(e+f x)}}}{a}-\frac {(7 a+8 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}}{a (a+b)}+\frac {(3 a+4 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a}+\frac {\sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{3 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {-\frac {\frac {(7 a+8 b) \sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {4 a (a+b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(e+f x)}}}{a}-\frac {(7 a+8 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}}{a (a+b)}+\frac {(3 a+4 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a}+\frac {\sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{3 a \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
(Sqrt[Cos[e + f*x]^2]*Sec[e + f*x]*((Csc[e + f*x]*Sqrt[1 - Sin[e + f*x]^2] )/(3*a*(a + b*Sin[e + f*x]^2)^(3/2)) + (((3*a + 4*b)*Csc[e + f*x]*Sqrt[1 - Sin[e + f*x]^2])/(a*(a + b)*Sqrt[a + b*Sin[e + f*x]^2]) + (-(((7*a + 8*b) *Csc[e + f*x]*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[a + b*Sin[e + f*x]^2])/a) - (( (7*a + 8*b)*EllipticE[ArcSin[Sin[e + f*x]], -(b/a)]*Sqrt[a + b*Sin[e + f*x ]^2])/Sqrt[1 + (b*Sin[e + f*x]^2)/a] - (4*a*(a + b)*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/Sqrt[a + b*Sin[e + f*x]^ 2])/a)/(a*(a + b)))/(3*a)))/f
3.6.41.3.1 Defintions of rubi rules used
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && !GtQ[c, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2] Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && !GtQ[a, 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/(a *e*2*(p + 1))), x] + Simp[1/(a*2*(p + 1)) Int[(e*x)^m*(a + b*x^2)^(p + 1) *(c + d*x^2)^(q - 1)*Simp[c*(m + 2*(p + 1) + 1) + d*(m + 2*(p + q + 1) + 1) *x^2, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && Lt Q[p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ )*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*g*2*(b*c - a*d)*(p + 1))), x] + Si mp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(g*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2 )^q*Simp[c*(b*e - a*f)*(m + 1) + e*2*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, q}, x] && LtQ[p, -1]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^ (m_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff^(m + 1 )*(Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x])) Subst[Int[x^m*((a + b*ff^2*x^2) ^p/(1 - ff^2*x^2)^((m + 1)/2)), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b , e, f, p}, x] && IntegerQ[m/2] && !IntegerQ[p]
Time = 5.29 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.43
| method | result | size |
| default | \(\frac {\left (-7 a \,b^{2}-8 b^{3}\right ) \left (\cos ^{6}\left (f x +e \right )\right )+\left (11 a^{2} b +26 a \,b^{2}+16 b^{3}\right ) \left (\cos ^{4}\left (f x +e \right )\right )-\sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, a b \left (4 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a +4 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b -7 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a -8 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b \right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (-3 a^{3}-14 a^{2} b -19 a \,b^{2}-8 b^{3}\right ) \left (\cos ^{2}\left (f x +e \right )\right )+\sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, a \left (4 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}+8 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b +4 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{2}-7 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}-15 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b -8 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{2}\right ) \sin \left (f x +e \right )}{3 \sin \left (f x +e \right ) a^{3} \left (a +b \right ) {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}} \cos \left (f x +e \right ) f}\) | \(411\) |
1/3*((-7*a*b^2-8*b^3)*cos(f*x+e)^6+(11*a^2*b+26*a*b^2+16*b^3)*cos(f*x+e)^4 -(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*(cos(f*x+e)^2)^(1/2)*a*b*(4*EllipticF(s in(f*x+e),(-1/a*b)^(1/2))*a+4*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*b-7*Ell ipticE(sin(f*x+e),(-1/a*b)^(1/2))*a-8*EllipticE(sin(f*x+e),(-1/a*b)^(1/2)) *b)*cos(f*x+e)^2*sin(f*x+e)+(-3*a^3-14*a^2*b-19*a*b^2-8*b^3)*cos(f*x+e)^2+ (-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*(cos(f*x+e)^2)^(1/2)*a*(4*EllipticF(sin( f*x+e),(-1/a*b)^(1/2))*a^2+8*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*a*b+4*El lipticF(sin(f*x+e),(-1/a*b)^(1/2))*b^2-7*EllipticE(sin(f*x+e),(-1/a*b)^(1/ 2))*a^2-15*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*a*b-8*EllipticE(sin(f*x+e) ,(-1/a*b)^(1/2))*b^2)*sin(f*x+e))/sin(f*x+e)/a^3/(a+b)/(a+b*sin(f*x+e)^2)^ (3/2)/cos(f*x+e)/f
Result contains complex when optimal does not.
Time = 0.22 (sec) , antiderivative size = 1595, normalized size of antiderivative = 5.56 \[ \int \frac {\cot ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \]
1/6*((2*(-7*I*a^3*b^2 - 22*I*a^2*b^3 - 23*I*a*b^4 - 8*I*b^5 + (-7*I*a*b^4 - 8*I*b^5)*cos(f*x + e)^4 - 2*(-7*I*a^2*b^3 - 15*I*a*b^4 - 8*I*b^5)*cos(f* x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2)*sin(f*x + e) - (14*I*a^4*b + 51*I *a^3*b^2 + 68*I*a^2*b^3 + 39*I*a*b^4 + 8*I*b^5 + (14*I*a^2*b^3 + 23*I*a*b^ 4 + 8*I*b^5)*cos(f*x + e)^4 + 2*(-14*I*a^3*b^2 - 37*I*a^2*b^3 - 31*I*a*b^4 - 8*I*b^5)*cos(f*x + e)^2)*sqrt(-b)*sin(f*x + e))*sqrt((2*b*sqrt((a^2 + a *b)/b^2) + 2*a + b)/b)*elliptic_e(arcsin(sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*(cos(f*x + e) + I*sin(f*x + e))), (8*a^2 + 8*a*b + b^2 - 4*(2 *a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) + (2*(7*I*a^3*b^2 + 22*I*a^2*b^3 + 23*I*a*b^4 + 8*I*b^5 + (7*I*a*b^4 + 8*I*b^5)*cos(f*x + e)^4 - 2*(7*I*a^2* b^3 + 15*I*a*b^4 + 8*I*b^5)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2) *sin(f*x + e) - (-14*I*a^4*b - 51*I*a^3*b^2 - 68*I*a^2*b^3 - 39*I*a*b^4 - 8*I*b^5 + (-14*I*a^2*b^3 - 23*I*a*b^4 - 8*I*b^5)*cos(f*x + e)^4 + 2*(14*I* a^3*b^2 + 37*I*a^2*b^3 + 31*I*a*b^4 + 8*I*b^5)*cos(f*x + e)^2)*sqrt(-b)*si n(f*x + e))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_e(arcsi n(sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*(cos(f*x + e) - I*sin(f*x + e))), (8*a^2 + 8*a*b + b^2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) - 2*(2*(-3*I*a^4*b - 17*I*a^3*b^2 - 33*I*a^2*b^3 - 27*I*a*b^4 - 8*I*b^5 + (-3*I*a^2*b^3 - 11*I*a*b^4 - 8*I*b^5)*cos(f*x + e)^4 + 2*(3*I*a^3*b^2 + 1 4*I*a^2*b^3 + 19*I*a*b^4 + 8*I*b^5)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 ...
\[ \int \frac {\cot ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\cot ^{2}{\left (e + f x \right )}}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\cot ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\cot \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\cot ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\cot \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\cot ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {{\mathrm {cot}\left (e+f\,x\right )}^2}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]