Integrand size = 25, antiderivative size = 322 \[ \int \frac {\csc ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {b \cot (e+f x)}{3 a (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {2 b (3 a+2 b) \cot (e+f x)}{3 a^2 (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\left (3 a^2+13 a b+8 b^2\right ) \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 (a+b)^2 f}-\frac {\left (3 a^2+13 a b+8 b^2\right ) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 a^3 (a+b)^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {(3 a+4 b) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 a^2 (a+b) f \sqrt {a+b \sin ^2(e+f x)}} \]
1/3*b*cot(f*x+e)/a/(a+b)/f/(a+b*sin(f*x+e)^2)^(3/2)+2/3*b*(3*a+2*b)*cot(f* x+e)/a^2/(a+b)^2/f/(a+b*sin(f*x+e)^2)^(1/2)-1/3*(3*a^2+13*a*b+8*b^2)*cot(f *x+e)*(a+b*sin(f*x+e)^2)^(1/2)/a^3/(a+b)^2/f-1/3*(3*a^2+13*a*b+8*b^2)*Elli pticE(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)^2/f/(1+b*sin(f*x+e)^2/a)^(1/2)+1/3*(3*a+4*b)*Ellip ticF(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/(a+b)/f/(a+b*sin(f*x+e)^2)^(1/2)
Time = 1.77 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.66 \[ \int \frac {\csc ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {4 a^2 \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} \left (-\left (\left (3 a^2+13 a b+8 b^2\right ) E\left (e+f x\left |-\frac {b}{a}\right .\right )\right )+\left (3 a^2+7 a b+4 b^2\right ) \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )\right )-2 \sqrt {2} \left (3 (a+b)^2 (2 a+b-b \cos (2 (e+f x)))^2 \cot (e+f x)+2 a b^2 (a+b) \sin (2 (e+f x))+b^2 (7 a+5 b) (2 a+b-b \cos (2 (e+f x))) \sin (2 (e+f x))\right )}{12 a^3 (a+b)^2 f (2 a+b-b \cos (2 (e+f x)))^{3/2}} \]
(4*a^2*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*(-((3*a^2 + 13*a*b + 8*b^2 )*EllipticE[e + f*x, -(b/a)]) + (3*a^2 + 7*a*b + 4*b^2)*EllipticF[e + f*x, -(b/a)]) - 2*Sqrt[2]*(3*(a + b)^2*(2*a + b - b*Cos[2*(e + f*x)])^2*Cot[e + f*x] + 2*a*b^2*(a + b)*Sin[2*(e + f*x)] + b^2*(7*a + 5*b)*(2*a + b - b*C os[2*(e + f*x)])*Sin[2*(e + f*x)]))/(12*a^3*(a + b)^2*f*(2*a + b - b*Cos[2 *(e + f*x)])^(3/2))
Time = 0.62 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.09, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3042, 3667, 374, 25, 441, 25, 445, 27, 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.
| \(\displaystyle \int \frac {\csc ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (e+f x)^2 \left (a+b \sin (e+f x)^2\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3667 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \int \frac {\csc ^2(e+f x)}{\sqrt {1-\sin ^2(e+f x)} \left (b \sin ^2(e+f x)+a\right )^{5/2}}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 374 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {b \sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{3 a (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\int -\frac {\csc ^2(e+f x) \left (-3 b \sin ^2(e+f x)+3 a+4 b\right )}{\sqrt {1-\sin ^2(e+f x)} \left (b \sin ^2(e+f x)+a\right )^{3/2}}d\sin (e+f x)}{3 a (a+b)}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\int \frac {\csc ^2(e+f x) \left (-3 b \sin ^2(e+f x)+3 a+4 b\right )}{\sqrt {1-\sin ^2(e+f x)} \left (b \sin ^2(e+f x)+a\right )^{3/2}}d\sin (e+f x)}{3 a (a+b)}+\frac {b \sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{3 a (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 441 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {2 b (3 a+2 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}-\frac {\int -\frac {\csc ^2(e+f x) \left (3 a^2+13 b a+8 b^2-2 b (3 a+2 b) \sin ^2(e+f x)\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a (a+b)}}{3 a (a+b)}+\frac {b \sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{3 a (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {\int \frac {\csc ^2(e+f x) \left (3 a^2+13 b a+8 b^2-2 b (3 a+2 b) \sin ^2(e+f x)\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a (a+b)}+\frac {2 b (3 a+2 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a (a+b)}+\frac {b \sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{3 a (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {-\frac {\int \frac {b \left (\left (3 a^2+13 b a+8 b^2\right ) \sin ^2(e+f x)+2 a (3 a+2 b)\right )}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a}-\frac {\left (3 a^2+13 a b+8 b^2\right ) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}}{a (a+b)}+\frac {2 b (3 a+2 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a (a+b)}+\frac {b \sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{3 a (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {-\frac {b \int \frac {\left (3 a^2+13 b a+8 b^2\right ) \sin ^2(e+f x)+2 a (3 a+2 b)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a}-\frac {\left (3 a^2+13 a b+8 b^2\right ) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}}{a (a+b)}+\frac {2 b (3 a+2 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a (a+b)}+\frac {b \sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{3 a (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {-\frac {b \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{b}-\frac {a (a+b) (3 a+4 b) \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{b}\right )}{a}-\frac {\left (3 a^2+13 a b+8 b^2\right ) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}}{a (a+b)}+\frac {2 b (3 a+2 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a (a+b)}+\frac {b \sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{3 a (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {-\frac {b \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{b}-\frac {a (a+b) (3 a+4 b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}d\sin (e+f x)}{b \sqrt {a+b \sin ^2(e+f x)}}\right )}{a}-\frac {\left (3 a^2+13 a b+8 b^2\right ) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}}{a (a+b)}+\frac {2 b (3 a+2 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a (a+b)}+\frac {b \sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{3 a (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {-\frac {b \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{b}-\frac {a (a+b) (3 a+4 b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{b \sqrt {a+b \sin ^2(e+f x)}}\right )}{a}-\frac {\left (3 a^2+13 a b+8 b^2\right ) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}}{a (a+b)}+\frac {2 b (3 a+2 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a (a+b)}+\frac {b \sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{3 a (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {-\frac {b \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \sqrt {a+b \sin ^2(e+f x)} \int \frac {\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{b \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {a (a+b) (3 a+4 b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{b \sqrt {a+b \sin ^2(e+f x)}}\right )}{a}-\frac {\left (3 a^2+13 a b+8 b^2\right ) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}}{a (a+b)}+\frac {2 b (3 a+2 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a (a+b)}+\frac {b \sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{3 a (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {-\frac {b \left (\frac {\left (3 a^2+13 a b+8 b^2\right ) \sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{b \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {a (a+b) (3 a+4 b) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{b \sqrt {a+b \sin ^2(e+f x)}}\right )}{a}-\frac {\left (3 a^2+13 a b+8 b^2\right ) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a}}{a (a+b)}+\frac {2 b (3 a+2 b) \sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{a (a+b) \sqrt {a+b \sin ^2(e+f x)}}}{3 a (a+b)}+\frac {b \sqrt {1-\sin ^2(e+f x)} \csc (e+f x)}{3 a (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
(Sqrt[Cos[e + f*x]^2]*Sec[e + f*x]*((b*Csc[e + f*x]*Sqrt[1 - Sin[e + f*x]^ 2])/(3*a*(a + b)*(a + b*Sin[e + f*x]^2)^(3/2)) + ((2*b*(3*a + 2*b)*Csc[e + f*x]*Sqrt[1 - Sin[e + f*x]^2])/(a*(a + b)*Sqrt[a + b*Sin[e + f*x]^2]) + ( -(((3*a^2 + 13*a*b + 8*b^2)*Csc[e + f*x]*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[a + b*Sin[e + f*x]^2])/a) - (b*(((3*a^2 + 13*a*b + 8*b^2)*EllipticE[ArcSin[Si n[e + f*x]], -(b/a)]*Sqrt[a + b*Sin[e + f*x]^2])/(b*Sqrt[1 + (b*Sin[e + f* x]^2)/a]) - (a*(a + b)*(3*a + 4*b)*EllipticF[ArcSin[Sin[e + f*x]], -(b/a)] *Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(b*Sqrt[a + b*Sin[e + f*x]^2])))/a)/(a*(a + b)))/(3*a*(a + b))))/f
3.2.70.3.1 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[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[(-b)*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*e*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(e*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[b*c*(m + 1) + 2*(b*c - a*d)*(p + 1) + d*b*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -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[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^( p_.), 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/Sqrt[1 - ff^2*x^2]), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && !IntegerQ[p]
Time = 3.70 (sec) , antiderivative size = 527, normalized size of antiderivative = 1.64
| method | result | size |
| default | \(\frac {\left (-3 a^{2} b^{2}-13 a \,b^{3}-8 b^{4}\right ) \left (\cos ^{6}\left (f x +e \right )\right )+\left (6 a^{3} b +26 a^{2} b^{2}+38 a \,b^{3}+16 b^{4}\right ) \left (\cos ^{4}\left (f x +e \right )\right )-\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, a b \left (3 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}+7 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}-3 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}-13 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 ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (-3 a^{4}-12 a^{3} b -26 a^{2} b^{2}-25 a \,b^{3}-8 b^{4}\right ) \left (\cos ^{2}\left (f x +e \right )\right )+\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, a \left (3 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}+10 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b +11 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2}+4 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{3}-3 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}-16 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b -21 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2}-8 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{3}\right ) \sin \left (f x +e \right )}{3 {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}} \left (a +b \right )^{2} \sin \left (f x +e \right ) a^{3} \cos \left (f x +e \right ) f}\) | \(527\) |
1/3*((-3*a^2*b^2-13*a*b^3-8*b^4)*cos(f*x+e)^6+(6*a^3*b+26*a^2*b^2+38*a*b^3 +16*b^4)*cos(f*x+e)^4-(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/ 2)*a*b*(3*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*a^2+7*EllipticF(sin(f*x+e), (-1/a*b)^(1/2))*a*b+4*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*b^2-3*EllipticE (sin(f*x+e),(-1/a*b)^(1/2))*a^2-13*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)*cos(f*x+e)^2*sin(f*x+e)+(-3* a^4-12*a^3*b-26*a^2*b^2-25*a*b^3-8*b^4)*cos(f*x+e)^2+(cos(f*x+e)^2)^(1/2)* (-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*a*(3*EllipticF(sin(f*x+e),(-1/a*b)^(1/2) )*a^3+10*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*a^2*b+11*EllipticF(sin(f*x+e ),(-1/a*b)^(1/2))*a*b^2+4*EllipticF(sin(f*x+e),(-1/a*b)^(1/2))*b^3-3*Ellip ticE(sin(f*x+e),(-1/a*b)^(1/2))*a^3-16*EllipticE(sin(f*x+e),(-1/a*b)^(1/2) )*a^2*b-21*EllipticE(sin(f*x+e),(-1/a*b)^(1/2))*a*b^2-8*EllipticE(sin(f*x+ e),(-1/a*b)^(1/2))*b^3)*sin(f*x+e))/(a+b*sin(f*x+e)^2)^(3/2)/(a+b)^2/sin(f *x+e)/a^3/cos(f*x+e)/f
Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 1719, normalized size of antiderivative = 5.34 \[ \int \frac {\csc ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \]
1/6*((2*(-3*I*a^4*b - 19*I*a^3*b^2 - 37*I*a^2*b^3 - 29*I*a*b^4 - 8*I*b^5 + (-3*I*a^2*b^3 - 13*I*a*b^4 - 8*I*b^5)*cos(f*x + e)^4 - 2*(-3*I*a^3*b^2 - 16*I*a^2*b^3 - 21*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) - (6*I*a^5 + 41*I*a^4*b + 93*I*a^3*b^2 + 95*I*a^2*b ^3 + 45*I*a*b^4 + 8*I*b^5 + (6*I*a^3*b^2 + 29*I*a^2*b^3 + 29*I*a*b^4 + 8*I *b^5)*cos(f*x + e)^4 + 2*(-6*I*a^4*b - 35*I*a^3*b^2 - 58*I*a^2*b^3 - 37*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*(3*I*a^4*b + 19*I*a^3*b^ 2 + 37*I*a^2*b^3 + 29*I*a*b^4 + 8*I*b^5 + (3*I*a^2*b^3 + 13*I*a*b^4 + 8*I* b^5)*cos(f*x + e)^4 - 2*(3*I*a^3*b^2 + 16*I*a^2*b^3 + 21*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) - (-6*I*a^5 - 41*I*a^4*b - 93*I*a^3*b^2 - 95*I*a^2*b^3 - 45*I*a*b^4 - 8*I*b^5 + (-6*I* a^3*b^2 - 29*I*a^2*b^3 - 29*I*a*b^4 - 8*I*b^5)*cos(f*x + e)^4 + 2*(6*I*a^4 *b + 35*I*a^3*b^2 + 58*I*a^2*b^3 + 37*I*a*b^4 + 8*I*b^5)*cos(f*x + e)^2)*s qrt(-b)*sin(f*x + e))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*ellipt ic_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) - 4*((-9*I*a^4*b - 35*I*a^3*b^2 - 51*I*a^2*b^3 - 33*I*a*b^4 ...
\[ \int \frac {\csc ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\csc ^{2}{\left (e + f x \right )}}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\csc ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\csc \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\csc ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\csc \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\csc ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {1}{{\sin \left (e+f\,x\right )}^2\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]